Petru

Cosmin

Eduardo, Houda, Jérémy, Johann, Khaled, Marwa Rana David

Victor Salwa

Xian

Zeina Xin

Ainsi Qu'au "marnais

Marwa B Rhana

Harry Merci Spécialement À Mes Cobureaux Laurent

Keywords: Ginzburg-Landau model, Superconductivity, Prescribed degrees, Free boundary problems, Loss of compactness, Bubbling, Harmonic maps, Hopf differential, Minimal surfaces, Vorticity measure

Pendant ces trois années de thèse de nombreuses personnes m'ont aidé ou accompagné de diverses manières. Je souhaite les remercier dans ces pages.

Je tiens tout d'abord à remercier profondément mes

"Rien ne vaut la recherche lorsqu'on veut trouver quelque chose" -J.R.R Tolkien, Bilbo le Hobbit

Résumé

Cette thèse est dédiée à l'analyse de quelques problèmes variationnels motivés par le modèle de Ginzburg-Landau en supraconductivité. Dans la première partie on étudie l'existence de solutions pour les équations de Ginzburg-Landau sans champ magnétique et avec données au bord de type semi-rigides. Ces données consistent à prescrire le module de la fonction sur le bord du domaine ainsi que son degré topologique. Ici la méthode directe du calcul des variations ne peut pas s'appliquer car le degré n'est pas continu pour la convergence faible dans l'espace de Sobolev adapté. On dit que c'est un problème sans compacité : un phénomène de "bubbling" apparaît.

Dans le Chapitre 1 on étudie des conditions sous lesquelles la différence entre deux niveaux d'énergie est strictement optimale. Pour cela on adapte une technique due à Brezis-Coron. Ceci nous permet de redémontrer un résultat (précédemment obtenu par Berlaynd Rybalko et Dos Santos) d'existence de solutions stables pour les équations de Ginzburg-Landau dans des domaines multiplement connexes.

Dans le Chapitre 2 on considère les applications harmoniques à valeurs dans R 2 avec des conditions au bord de type degrés prescrits sur un anneau. On fait un lien avec la théorie des surfaces minimales dans R 3 grâce à la différentielle de Hopf. Ceci nous conduit à l'étude des surfaces minimales bordées par deux cercles dans des plans parallèles. On prouve l'existence de telles surfaces qui ne sont pas des caténoïdes grâce à un résultat de bifurcation.

On utilise alors les résultats obtenus pour déduire des théorèmes d'existence et de non existence de minimiseurs de l'énergie de Ginzburg-Landau à degrés prescrits dans un anneau. Dans ce troisième Chapitre on obtient des conclusions pour ε grand.

Le Chapitre 4 est dédié aux problèmes à degrés prescrits en dimension n ≥ 3. On y montre la non existence des minimiseurs de la n-énergie de Ginzburg-Landau dans un domaine difféomorphe à une boule. On étudie ensuite des points critiques de type min-max pour une énergie perturbée.

La deuxième partie est consacrée à l'analyse asymptotique des solutions des équations de Ginzburg-Landau lorsque ε tend vers zéro. Sandier et Serfaty ont étudié le comportement asymptotique des mesures de vorticité associées aux équations. Ils ont notamment trouvé des conditions critiques sur les mesures limites dans le cas des équations avec et sans champ magnétique. Nous nous intéressons alors à ces conditions dans le cas sans champ magnétique. Le problème de la régularité des mesures limites se ramène ainsi à l'étude de la régularité des fonctions stationnaires harmoniques dont le Laplacien est une mesure. Nous montrons que localement de telles mesures sont supportées par une union de lignes appartenant à l'ensemble des zéros d'une fonction harmonique.

Mots clés : Modèle de Ginzburg-Landau, Supraconductivité, Degrés prescrits, Perte de compacité, Bubbling, Applications harmoniques, Différentielle de Hopf, Surfaces minimales, Mesure de vorticité.

Table des matières

Remerciements i

Résumé iv

Introduction Générale

Cette thèse est composée de deux parties indépendantes. Toutes deux sont motivées par l'étude du modèle de Ginzburg-Landau lié à la théorie de la supraconductivité. Dans la première partie on étudie l'existence et la non-existence de solutions des équations de Ginzburg-Landau sans champ magnétique avec données au bord semi-rigides (ou à bord libre). Ceci nous conduit à étudier aussi l'équation des applications harmoniques avec le même type de données au bord et à faire un lien avec les surfaces minimales. La deuxième partie est consacrée à l'étude de la régularité des limites de mesures de vorticité associées aux équations de Ginzburg-Landau en l'absence de champ magnétique. Dans les deux parties on voit apparaître l'importance de l'étude des variations internes pour certaines énergies. Les méthodes utilisées sont celles du calcul des variations et plus particulièrement celles liées aux problèmes variationnels sans compacité. Dans la première partie, on est amené à considérer des problèmes de minimisation. Leur étude fait appel à l'établissement d'inégalités strictes ou larges entre certains niveaux d'énergie. On utilise également des méthodes de bifurcation, la théorie des surfaces minimales dans l'espace euclidien ou la mise en place d'un procédé de min-max. Dans la seconde partie on utilise des techniques d'analyse complexe et la théorie des ensembles de périmètre fini pour traiter un problème de régularité de mesures.

1.1 La supraconductivité et la théorie de Ginzburg-Landau 1.1.1 La supraconductivité Lorsque l'on fait passer un courant électrique dans un matériau conducteur on observe une dissipation d'énergie sous forme de chaleur due à la résistance électrique du matériau : c'est ce qu'on appelle l'effet Joule. On sait depuis longtemps que la résistance d'un conducteur diminue lorsque la température diminue. Toutefois en 1911 le physicien hollandais Kammerlingh Onnes fit une découverte surprenante : en dessous d'une certaine température, dite critique, certains matériaux n'opposent plus aucune résistance au passage du courant. C'est par exemple le cas du mercure refroidi à -263 degrés Celsius. La résistance électrique devient strictement nulle et le matériau passe dans un état dit supraconducteur. Ainsi, un courant électrique peut continuer à circuler indéfiniment dans un anneau supraconducteur même si on ne l'alimente pas en énergie.

Une seconde propriété, distincte de la précédente et peut-être plus importante encore, caractérise les matériaux supraconducteurs. Ils ont la capacité d'expulser un champ magnétique auquel ils sont soumis. Ceci est connu sous le nom d'effet Meissner et a été découvert par W. Meissner et R.Ochsenfeld en 1933. Lorsque l'on soumet un supraconducteur à un champ magnétique, par exemple en approchant un aimant, des courants supraconducteurs se créent à la surface du matériau. Ces courants induisent un nouveau champ magnétique qui compensent exactement le champ magnétique initial à l'intérieur du matériau. De plus le nouveau champ magnétique exerce une force sur l'aimant et le repousse. L'aimant lévite au-dessus du supraconducteur. En utilisant cette propriété de lévitation afin de réduire les frottements, un train japonais le JR Maglev a pu atteindre la vitesse record de 609 km/h pendant 11 secondes le 21 avril 2015. Toutefois lorsque l'on applique un champ magnétique trop élevé, celui-ci finit par pénétrer dans l'échantillon et la supraconductivité à l'intérieur est détruite. En fait, on classe les supraconducteurs en deux catégories selon leur réponse à une augmentation de l'intensité d'un champ magnétique. Les supraconducteurs de type I (mercure, plomb) réagissent de la manière suivante : il existe une intensité critique du champ magnétique extérieur H c 1 telle qu'en dessous de cette valeur le champ magnétique ne pénètre pas dans l'échantillon et le matériau est dans un état supraconducteur tandis qu'au-dessus de cette valeur le champ magnétique pénètre totalement dans l'échantillon et la supraconductivité est détruite. Les supraconducteurs de type II (les cuprates) eux montrent une réponse progressive à l'augmentation de l'intensité du champ magnétique. Il existe trois valeurs critiques de l'intensité du champ magnétique H c 1 < H c 2 < H c 3 telles que : en-dessous de H c 1 le matériau est dans un état supraconducteur parfait et repousse totalement le champ magnétique. Entre H c 1 et H c 2 le champ magnétique pénètre dans l'échantillon le long de petites zones en forme de filaments appelées "vortex". À proximité de ces vortex se mettent en place des courants supraconducteurs qui tournent autour des filaments d'où la dénomination de "tourbillons". Plus le champ magnétique est intense, plus les vortex à apparaître dans l'échantillon sont nombreux. Au-dessus de l'intensité H c 2 le champ magnétique pénètre totalement à l'intérieur du matériau mais la supraconductivité subsiste à la surface du 1.1 La supraconductivité et la théorie de Ginzburg-Landau matériau. Enfin au-dessus de H c 3 toute supraconductivité est détruite pour le matériau et il se retrouve dans un état normal. Les questions liées à l'apparition des vortex, à leur localisation dans l'échantillon ou à leur évolution dans le temps a suscité et suscite toujours beaucoup d'intérêt. Nous renvoyons à [START_REF] Tinkham | Introduction to superconductivity[END_REF] et au site internet [eS11] pour plus d'informations sur la supraconductivité et pour des illustrations. Afin de décrire mathématiquement le phénomène de supraconductivité dans un matériau Ω ⊂ R 3 , V.Ginzburg et L.Landau ont proposé un modèle dans les années 50. Dans ce modèle, on caractérise l'état d'un matériau supraconducteur par deux quantités : une fonction d'onde (ou paramètre d'ordre) u : Ω → C et la donnée du potentiel vecteur du champ magnétique A : Ω → R 3 . À un certain état (u, A) du matériau on associe une quantité, appelée énergie de Ginzburg-Landau qui peut s'écrire (après adimensionalisation et certaines réductions) :

G ε (u, A) = 1 2 Ω |∇u -iAu| 2 + | curl A -h ext | 2 + (1 -|u| 2 ) 2 2ε 2 . (1.1)
Dans cette expression, h = curl A (curl est une notation pour le rotationel) est le champ magnétique à l'intérieur de l'échantillon, h ext désigne lui le champ magnétique à l'extérieur du matériau et ε est un paramètre qui dépend des propriétés du supraconducteur et de la température ambiante 1 . La fonction d'onde u représente l'état local du matériau : |u|2 est la densité des paires d'électrons supraconducteurs (dites paires de Cooper). Avec la normalisation utilisée on a |u| ≤ 1 et lorsque |u| est proche de 1, le matériau est dans un état supraconducteur alors que lorsque |u| est proche de 0 le matériau est dans un état normal : les deux états, ou phases, pouvant coexister. Ainsi les vortex correspondent à des petites régions où |u| 0. La théorie prédit que les états du matériau que l'on observe sont des minimiseurs de l'énergie G ε . On dispose donc d'un outil mathématique pour étudier l'apparition des vortex, leurs emplacements, l'existence et la valeur des différents champs critiques (voir [START_REF] Sandier | Progress in Nonlinear Partial Differential Equations Birkhauser[END_REF] pour plus d'informations sur le modèle de Ginzburg-Landau).

1.1.3 L'énergie de Ginzburg-Landau sans champ magnétique avec donnée au bord de Dirichlet

Des résultats rigoureux sur l'énergie de Ginzburg-Landau ont commencé à être obtenus dans les années 80. La très grande majorité des études portent sur le modèle de Ginzburg-Landau en deux dimensions. On considère que les grandeurs physiques sont invariantes par translation verticale et on s'intéresse à des sections horizontales du matériau. Toutefois même en deux dimensions, l'analyse du modèle complet de Ginzburg-Landau reste difficile. Dans un travail pionnier [START_REF] Bethuel | Progress in Nonlinear Differential Equations and their Applications[END_REF], Béthuel-Brézis-Hélein ont considéré un modèle simplifié, sans champ magnétique. Ils ont étudié l'énergie suivante :

E ε (u) = 1 2 Ω |∇u| 2 + (1 -|u| 2 ) 2 2ε 2 . (1.2)
On a vu que la cause de l'apparition des vortex dans un supraconducteur de type II est la présence du champ magnétique. Cependant, ils ont observé que l'on pouvait mimer l'effet du champ magnétique en imposant une donnée au bord de type Dirichlet g : ∂Ω → S 1 avec degré non nul (pour la définition du degré nous renvoyons à (1.4)). En effet les points critiques de l'énergie (1.2) présentent des vortex qui ont les propriétés suivantes : l'énergie est concentrée autour des vortex et est quantifiée, la circulation des courants autour des vortex est elle aussi quantifiée. De plus les vortex se repoussent et sont également repoussés par le bord. Ces propriétés correspondent aux observations physiques. Cependant Béthuel-Brézis-Hélein observent qu'imposer une donnée au bord u = g sur ∂Ω ne correspond pas à une réalité physique car seul |u| 2 représente une quantité physique 2 . Dans [START_REF] Bethuel | Vortices for a variational problem related to superconductivity[END_REF] Béthuel et Rivière ont étudié la minimisation de l'énergie de Ginzburg-Landau avec champ magnétique et avec des données au bord de type Dirichlet, mais invariantes par changement de jauge donc pouvant représenter des quantités physiques. Plus tard, grâce à des outils mathématiques développés par Jerrard et Sandier dans [START_REF] Jerrard | Lower bounds for generalized Ginzburg-Landau functionals[END_REF] et [START_REF]Lower bounds for the energy of unit vector fields and applications[END_REF], l'étude du modèle complet de Ginzburg-Landau est devenue accessible et de nombreux résultats ont été obtenus (cf. [START_REF] Sandier | Progress in Nonlinear Partial Differential Equations Birkhauser[END_REF]). Notons que le modèle de Ginzburg-Landau, (ou des modèles voisins) permet d'étudier d'autres phénomènes de transition de phase comme la superfluidité ou les condensats de Bose-Einstein.

1.2 L'énergie de Ginzburg-Landau avec données au bords semirigides Les équations (1.3) sont obtenues en faisant des variations de la forme u t = u + tψ pour ψ ∈ C ∞ c (Ω, R 2 ) et u t = ue itψ pour ψ ∈ C ∞ (Ω, R). Il se trouve que les solutions de (1.3) sont régulières (C ∞ ) jusqu'au bord et que si l'on écrit u = |u|e iϕ près du bord du domaine, la deuxième condition au bord devient ∂ ν ϕ = 0 sur ∂Ω. On obtient donc une condition de Dirichlet pour le module et une condition de Neuman homogène pour la phase. C'est un problème mixte, d'où la dénomination semi-rigide. En effet dans la littérature concernant Ginzburg-Landau, le problème de Ginzburg-Landau avec donnée au bord de Dirichlet et appelé problème "soft", tandis que le problème avec donnée au bord de Neumann est dénommé "stiff". Comme les constantes dans S 1 sont solutions de (1.3), afin de trouver des solutions non triviales, il est naturel d'imposer le degré de la solution sur le bord. C'est pourquoi on appelle aussi ce problème : problème à degrés prescrits. Il n'est pas tout à fait évident que le degré soit bien défini pour des fonctions dans l'espace I, car a priori, le degré classique est défini seulement pour des fonctions continues. Cependant Boutet-de-Monvel et Gabber ont remarqué dans l'appendice de [START_REF] Boutet De Monvel-Berthier | A boundary value problem related to the Ginzburg-Landau model[END_REF] que l'on pouvait étendre la définition du degré à des applications dans l'espace H 1/2 (∂Ω, S 1 ) et donc à l'espace I. À la suite de ces travaux Brézis et Nirenberg ont donné un cadre général pour la définition du degré pour des applications non régulières (cf. [START_REF] Brezis | Degree theory and BMO. I. Compact manifolds without boundaries[END_REF], [START_REF]Degree theory and BMO. II. Compact manifolds with boundaries[END_REF], [START_REF]Degree theory : old and new, Topological nonlinear analysis[END_REF]). Ce cadre et celui de l'espace fonctionnel VMO. Remarquons qu'il peut aussi être intéressant d'étudier l'énergie de Ginzburg-Landau avec champ magnétique sous contrainte de degrés prescrits au bord comme dans [START_REF]Minimizers of the magnetic Ginzburg-Landau functional in simply connected domain with prescribed degree on the boundary[END_REF], [START_REF] Berlyand | Near boundary vortices in a magnetic Ginzburg-Landau model : their locations via tight energy bounds[END_REF] et [START_REF] Rybalko | Local minimizers of the magnetic Ginzburg-Landau functional with S 1 -valued order parameter on the boundary[END_REF].

     -∆u = 1 ε 2 u(1 -|u| 2 ) dans Ω |u| = 1 sur ∂Ω u ∧ ∂ ν u = 0 sur ∂Ω.

Un problème à bord libre

On peut également noter que le modèle précédent rentre dans la catégorie des problèmes dits à bord libre (ou frontière libre). On n'impose pas la donnée au bord, mais on demande seulement que la solution du problème prenne, sur le bord, ses valeurs dans la sphère S 1 . Ainsi, la valeur de la fonction au bord est une inconnue du problème. Ce type de conditions a été étudié auparavant pour un autre problème variationnel : les surfaces minimales. Ce sont des surfaces dont la courbure moyenne est nulle. Cette condition apparaît lorsqu'on cherche à minimiser l'aire parmi une famille de surfaces avec une contrainte fixée. On peut par exemple chercher à minimiser l'aire parmi toutes les surfaces s'appuyant sur un contour donné, c'est le problème dit de Plateau. Physiquement les surfaces minimales correspondent à des bulles (ou plutôt à des films) de savon. Le problème de Plateau a été résolu par Douglas et Rado et a valu la médaille fields à Douglas en 1936. Pour des références sur les surfaces minimales et le problème de Plateau nous renvoyons à [DHS10], [START_REF] Colding | A course in minimal surfaces[END_REF] et aux références citées dans ces ouvrages. À la suite de cela les mathématiciens se sont intéressés à une généralisation de ce problème, on n'impose plus le contour entier mais on demande seulement que le contour reste sur une certaine surface prescrite : c'est le problème des surfaces minimales à bord libre, (voir par exemple [START_REF] Dierkes | Regularity of minimal surfaces[END_REF], [START_REF] Nitsche | Stationary partitioning of convex bodies[END_REF]). Mathématiquement, on obtient une condition sur la surface minimale : elle rencontre le bord de la surface prescrite orthogonalement. À la lumière de cette observation on peut interpréter la condition u ∧ ∂ ν u = 0 sur ∂Ω en disant que la solution u rencontre la sphère S 1 orthogonalement.

Un problème sans compacité

La difficulté principale du modèle avec données au bord semi-rigides est que le degré n'est pas continu pour la convergence faible dans l'espace de Sobolev H 1 (Ω) (où Ω est un domaine de R 2 ). Ceci peut se voir grâce à l'exemple suivant :

1.2 Les données au bord semi-rigides Exemple 1.1. Soit M n : D → D défini par M n (z) = z-(1-1/n) 1-(1-1/n)z , alors M n -1 faiblement dans H 1 , deg(M n (z), S 1 ) = 1 pour tout n ∈ N mais deg(-1, S 1 ) = 0.

Ainsi on ne peut pas utiliser la méthode directe du calcul des variations pour trouver des minimiseurs de l'énergie E ε dans des sous-ensembles à degrés prescrits. On dit que le problème présente un manque de compacité. Plus généralement on appelle problème sans compacité un problème variationnel où l'énergie F est définie sur un espace non compact et ne satisfait pas (a priori) la condition, dite de Palais-Smale, suivante :

Pour toute suite (u n ) telle que F (u n ) reste bornée et F (u n ) → 0 il existe une sous-suite qui converge fortement dans l'espace fonctionnel adéquat. (PS) De nombreux problèmes provenant de la physique ou de la géométrie sont non compacts. On peut citer par exemple le célèbre problème de Yamabe ou de la courbure prescrite, les problèmes d'existence de surfaces minimales ou d'applications harmoniques dans des classes d'homotopie, le problème d'existence de surfaces à courbure moyenne constante et le problème de Willmore3 . Pour des références précises sur les problèmes sans compacité on renvoie à [START_REF] Brezis | Points critiques dans les problèmes variationnels sans compacité[END_REF]. Dans les problèmes cités l'absence de compacité est due à l'invariance de l'énergie par un groupe noncompact (groupe des transformations conformes par exemple). On se trouve également dans un cas limite des injections de Sobolev, pour cela on parle de problèmes critiques. Les problèmes sans compacité ont été étudiés depuis plus de 30 ans. Dans des travaux novateurs Sacks-Uhlenbeck pour les applications harmoniques (cf. [START_REF] Sacks | The existence of minimal immersions of 2-spheres[END_REF]) et Wente pour les surfaces à courbure moyenne constante (cf. [START_REF] Wente | Large solutions to the volume constrained Plateau problem[END_REF]) ont mis en évidence indépendamment le phénomène de "bubbling" ou "séparation des sphères". Ceci correspond à peu près au fait qu'un suite de Palais-Smale peut se décomposer en une limite faible (solution du problème) plus une somme de bulles, où les bulles sont des solutions d'une équation "limite". De plus, l'énergie des bulles étant quantifiée, on peut par des arguments d'énergie prouver que les bulles n'apparaissent pas et qu'en fait la compacité a bien lieu. C'est également ce qu'exprime le principe de concentration-compacité dû à P.L Lions (voir [START_REF]The concentration-compactness principle in the calculus of variations. The limit case. I[END_REF], [START_REF]The concentration-compactness principle in the calculus of variations. The limit case. II[END_REF], [START_REF] Lions | La méthode de concentration-compacité en calcul des variations[END_REF]). Pour résumer ce principe affirme que si une suite minimisante pour un problème non compact ne converge pas fortement alors une concentration d'énergie apparaît. Dans certains cas des inégalités strictes entre certaines énergies permettent de conclure à la compacité du problème. Outre des arguments de comparaison d'énergies, d'autres méthodes on été établies comme par exemple des méthodes topologiques ( [START_REF] Bahri | On a nonlinear elliptic equation involving the critical Sobolev exponent : the effect of the topology of the domain[END_REF], [START_REF] Bahri | Critical points at infinity in some variational problems[END_REF], [START_REF] Malchiodi | Topological methods for an elliptic equation with exponential nonlinearities[END_REF]). Signalons enfin que les problèmes sans compacité qui ne se trouvent pas dans des cas limites d'injections de Sobolev, les problèmes dits sur-critiques sont encore très mal compris.

Pour les équations de Ginzburg-Landau à degrés prescrits, le problème a ceci d'original que le bubbling a lieu sur le bord : les bulles sont de plus en plus concentrées près du bord et finissent par "s'échapper" du bord comme dans l'exemple 1.1. Il existe à présent une assez grande littérature concernant ce problème : [START_REF] Berlyand | Symmetry breaking in annular domains for a ginzburg-landau superconductivity model[END_REF], [START_REF] Golovaty | On uniqueness of vector-valued minimizers of the Ginzburg-Landau functional in annular domains[END_REF], [BM], [START_REF] Berlyand | Nonexistence of Ginzburg-Landau minimizers with prescribed degree on the boundary of a doubly connected domain[END_REF], [START_REF]Ginzburg-Landau minimizers with prescribed degrees. Capacity of the domain and emergence of vortices[END_REF], [START_REF] Berlyand | Solutions with vortices of a semi-stiff boundary value problem for the Ginzburg-Landau equation[END_REF], [START_REF] Santos | Local minimizers of the Ginzburg-Landau functional with prescribed degrees[END_REF], [START_REF] Farina | Uniqueness of vortexless Ginzburg-Landau type minimizers in two dimensions[END_REF], [START_REF] Berlyand | Minimax critical points in Ginzburg-Landau problems with semi-stiff boundary conditions : existence and bubbling[END_REF], [START_REF] Lamy | Existence of critical points with semi-stiff boundary conditions for singular perturbation problems in simply connected planar domains[END_REF], voir en particulier le survey [START_REF]Size of planar domains and existence of minimizers of the ginzburg-landau energy with semistiff boundary conditions[END_REF]. Les auteurs ont montré l'existence ou la non existence de solutions dans des domaines simplement connexes ou multiplement connexes en étudiant la perte de compacité et en utilisant des arguments d'énergie. En particulier dans [START_REF] Berlyand | Minimax critical points in Ginzburg-Landau problems with semi-stiff boundary conditions : existence and bubbling[END_REF] un théorème de décomposition de l'énergie des suites de Palais-Smale est établi. Mentionnons que dans [START_REF] Lamy | Existence of critical points with semi-stiff boundary conditions for singular perturbation problems in simply connected planar domains[END_REF], des solutions sont construites par des méthodes nonvariationnelles. Toutefois le problème n'est pas totalement compris et dans la première partie de cette thèse, nous nous sommes intéressés à apporter quelques réponses supplémentaires à ce problème.

Contributions aux problèmes à degrés prescrits

Dans cette partie, on considère un domaine doublement connexe A = Ω \ ω (où ω ⊂ Ω sont deux ouverts bornés et réguliers de R 2 ). On s'intéresse à l'existence de minimiseurs locaux de l'énergie de Ginzburg-Landau simplifiée avec I p,q = {v ∈ I; deg(v, ∂Ω) = p et deg(v, ∂ω) = q}. Boutet-de-Monvel et Gabber ont montré dans [START_REF] Boutet De Monvel-Berthier | A boundary value problem related to the Ginzburg-Landau model[END_REF] que le degré est bien défini pour des applications de I, et que les ensembles I p,q constituent les composantes connexes de I. De plus, elles sont ouvertes et fermées pour la topologie induite par la norme H 1 sur I. Ainsi, pour produire des minimiseurs locaux de E ε on peut chercher à minimiser cette énergie dans les classes I p,q pour p, q dans Z. Soit 0 < ε < +∞, on pose m ε (p, q) = inf{E ε (v); v ∈ I p,q }.

E ε (u) = 1 2 A |∇u| 2 + (1 -|u| 2 ) 2 2ε
Comme on a vu que le degré n'est pas continu pour la convergence faible (voir Exemple 1.1), l'existence de minimiseur dans ces classes n'est pas assurée. Les premiers résultats de minimisation obtenus concernent les configurations de degrés suivantes : q ≤ 0 < p ou q > 0 ≥ p. On a alors :

Proposition 1.2 ( [BM]). Soit (p, q) ∈ Z 2 tels que q < 0 ≤ p (ou q > 0 ≥ p), alors pour 0 < ε < +∞ m ε (p, q) = π(|p| + |q|) et n'est pas atteint.

Le cas des degrés p = q = 1 a ensuite été considéré. À la suite des travaux [BM], [START_REF]Ginzburg-Landau minimizers with prescribed degrees. Capacity of the domain and emergence of vortices[END_REF], [START_REF] Golovaty | On uniqueness of vector-valued minimizers of the Ginzburg-Landau functional in annular domains[END_REF], [START_REF] Berlyand | Nonexistence of Ginzburg-Landau minimizers with prescribed degree on the boundary of a doubly connected domain[END_REF] la question de l'existence de minimiseurs dans I 1,1 a été complètement résolue. Elle fait apparaître le rôle de la capacité du domaine A dont nous rappelons la définition.

Définition 1.3. Soit A = Ω \ ω comme avant. Soit V la solution du problème suivant

    
∆V = 0 dans A V = 1 sur ∂Ω V = 0 sur ∂ω.

(1.5) La capacité de A est définie par cap(A) :

= 1 2 A |∇V | 2 .
La capacité d'un domaine mesure son "épaisseur" et plus la capacité est grande plus l'anneau est fin. Par exemple, pour un domaine circulaire A = A ρ = {z ∈ C; ρ < |z| < 1} on a cap(A) = -1 2π ln(ρ). On dira qu'un anneau est fin si sa capacité est grande et épais dans le cas contraire. On a alors Théorème 1.4 (Berlyand-Mironescu, Berlyand-Rybalko-Golovaty). Soit A un domaine annulaire. 1) Si cap(A) ≥ π alors pour tout 0 < ε ≤ +∞, on a : m ε (1, 1, ) < 2π et est atteint.

2) Si cap(A) < π alors il existe 0 < ε 0 < +∞ tel que : * si ε ≥ ε 0 alors m ε (1, 1) ≤ 2π est atteint, * si ε ≤ ε 0 alors m ε (1, 1, ) = 2π et n'est pas atteint.

1.3.1 Insertion de bulles au bord pour l'énergie de Ginzburg-Landau (en collaboration avec E.Sandier, publié : [START_REF] Rodiac | Insertion of bubbles at the boundary for the Ginzburg-Landau functional[END_REF])

Berlyand et Rybalko ont alors eu l'idée, dans [START_REF] Berlyand | Solutions with vortices of a semi-stiff boundary value problem for the Ginzburg-Landau equation[END_REF], de chercher des minimiseurs locaux de E ε dans des sous ensembles des classes I p,q . Ils ont obtenus le résultat suivant :

Théorème 1.5 (Théorème 1 dans [START_REF] Berlyand | Solutions with vortices of a semi-stiff boundary value problem for the Ginzburg-Landau equation[END_REF]). Pour tout (p, q) ∈ Z 2 et M ∈ N * , il existe ε 1 = ε 1 (A, p, q, M ) tel que si 0 < ε < ε 1 alors E ε possède au moins M minimiseurs locaux dans I p,q .

Ce théorème a été généralisé au cas des domaines multiplement connexes (avec plusieurs trous) par Dos Santos dans [START_REF] Santos | Local minimizers of the Ginzburg-Landau functional with prescribed degrees[END_REF]. L'idée de la preuve de ce théorème consiste à remarquer que si on considère des fonctions v ∈ I qui vérifient E ε (v) ≤ Λ où Λ ∈ R + est indépendant de ε, alors pour ε assez petit ces fonctions sont "presque" à valeurs dans S 1 . Ceci permet de définir un degré approché moyen, qui peut être vu comme le degré sur une courbe à l'intérieur de l'anneau. Ce degré est noté abdeg(u) et est continu pour la convergence faible. On cherche alors à minimiser E ε dans les classes

I d p,q = {v ∈ I p,q ; d -1/2 < abdeg(v) < d + 1/2}, et on pose m ε (p, q, d) = inf{E ε (v); v ∈ I d p,q }.
E ε (v) < E ε (u ε ) + π. (1.6)
La même conclusion reste vraie pour un ṽ ∈ I d p,q+1 .

Ce lemme, essentiel dans la démonstration du Théorème 1.5, a été redémontré par Dos Santos dans [START_REF] Santos | Local minimizers of the Ginzburg-Landau functional with prescribed degrees[END_REF]. Il a donné une construction locale, ce qui lui a permis d'adapter le résultat à des domaines multiplement connexes. Dans ce premier chapitre, nous donnons une troisième démonstration de ce résultat. Pour cela, en collaboration avec Etienne Sandier, nous adaptons la technique, dite d'insertion de bulle, développée par Brézis et Coron dans [START_REF] Brezis | Large solutions for harmonic maps in two dimensions[END_REF] dans le contexte des applications harmoniques à valeurs dans la sphère S 2 . Cette technique a été raffinée par Soyeur dans [START_REF] Soyeur | The Dirichlet problem for harmonic maps from the disc into the 2-sphere[END_REF] et Kuwert dans [START_REF] Kuwert | Minimizing the energy of maps from a surface into a 2-sphere with prescribed degree and boundary values[END_REF]. L'idée est de modifier une fonction de départ u (pas nécessairement une solution de l'équation de Ginzburg-Landau) dans un petit voisinage du bord en la remplaçant par une fonction holomorphe dont l'image recouvre presque tout le disque. De cette manière on augmente le degré d'une unité et l'énergie supplémentaire (égale à l'aire du recouvrement de la fonction modifiée) est strictement plus petite que π car la fonction holomorphe recouvre "un peu moins" que le disque. La nouveauté par rapport aux constructions de [START_REF] Brezis | Large solutions for harmonic maps in two dimensions[END_REF] et [START_REF] Kuwert | Minimizing the energy of maps from a surface into a 2-sphere with prescribed degree and boundary values[END_REF] est que dans notre cas la construction est faite sur le bord du domaine. Ceci impose certaines conditions sur la fonction de départ u sur le bord. Notons que des inégalités strictes du type (1.6) sont très importantes dans les problèmes variationnels sans compacité. En effet de telles inégalités permettent de conclure à la compacité de suites minimisantes ou de suites de Palais-Smale.

Dans [START_REF] Berlyand | Solutions with vortices of a semi-stiff boundary value problem for the Ginzburg-Landau equation[END_REF], les auteurs démontrent l'existence de minimiseurs locaux, pour ε petit, dans les classes I d p,q telles que d > 0 et d ≥ max{p, q}. Il a été conjecturé que cette condition (ou son analogue d < 0 et < d ≤ min{p, q}) est nécessaire est suffisante pour que m ε (p, q, d) soit atteint. Cette conjecture a été partiellement résolue dans [START_REF] Misiats | The necessary conditions for the existence of local Ginzburg-Landau minimizers with prescribed degrees on the boundary[END_REF]. On pouvait espérer qu'une meilleure compréhension du Lemme 1.6 permette de résoudre cette conjecture. En fait il apparaît que c'est une version opposée de ce lemme qui permettrait de répondre à cette question. Il s'agirait de montrer que pour

u ∈ I d d,d un minimiseur de m ε (d, d, d) on a E ε (v) > E ε (u) + π(|p -d| + |q -d|) pour v ∈ I d p,q avec p ≥ d ou q ≥ d.
1.3.2 Applications harmoniques à degrés prescrits et surfaces minimales (en collaboration avec L.Hauswirth, soumis [HR] 2015 )

Jusqu'ici on sait qu'il existe des minimiseurs locaux de E ε dans tous les I p,q pour ε suffisamment petit (et donc des solutions stables de (1.3)) mais la question de l'existence des minimiseurs globaux dans I p,q n'est pas entièrement résolue. Afin d'apporter des réponses complémentaires à ce problème nous avons décidé de regarder un problème voisin : celui de la minimisation de l'énergie de Dirichlet

E(u) = 1 2 A |∇u| 2
dans les espaces I p,q . Formellement on peut voir cette énergie comme une énergie de Ginzburg-Landau avec ε = +∞. On note

m(p, q) = inf{E(v); v ∈ I p,q }.
Remarquons que dans [BM], Berlyand et Mironescu, avaient déjà déduit des résultats sur l'énergie de Ginzburg-Landau en s'appuyant sur des résultats obtenus pour l'énergie de Dirichlet. C'est comme cela qu'ils ont montré que pour ε grand, m ε (1, 1) est toujours atteint (quelque soit la capacité de l'anneau). De plus, dans [START_REF] Berlyand | Minimax critical points in Ginzburg-Landau problems with semi-stiff boundary conditions : existence and bubbling[END_REF], où les auteurs s'intéressent aux points critiques de type min-max de l'énergie E ε avec données au bord semi-rigides, la classification des solutions de de l'équation de Laplace avec données au bord semi-rigides joue un rôle important. C'est pourquoi dans le chapitre 2 de cette thèse, écrit en collaboration avec Laurent Hauswirth, on s'intéresse aux solutions de

     ∆u = 0 dans A |u| = 1 sur ∂A u ∧ ∂ ν u = 0 sur ∂A,
(1.7) avec une attention particulière pour les solutions qui minimisent l'énergie dans les classes I p,q . L'énergie de Dirichlet présente un avantage par rapport à l'énergie de Ginzburg-Landau : elle est invariante par transformation conforme. Ainsi on peut supposer dans ce chapitre que A est un anneau circulaire, i.e.,

A = A ρ = {z ∈ C; ρ < |z| < 1}.
Avant de rappeler les résultats connus sur ce problème et de décrire ceux que l'on a obtenus, remarquons que l'on peut faire un lien étroit avec un autre problème celui des applications 1/2-harmoniques à valeurs dans la sphère. Cette notion a été définie par Da Lio et Rivière dans [START_REF]Three-term commutator estimates and the regularity of 1 2 -harmonic maps into spheres[END_REF] de la manière suivante. On note R 2 + = {(x, y) ∈ R 2 ; y > 0}, et on peut voir R comme le bord de ce domaine. On considère des applications u : R → S 1 et on leur associe une énergie

L(u) = inf{E(v); tr |R v = u}.
Cette énergie est laissée invariante par les transformations de Möbius qui envoient le demi-plan supérieur sur le demi-plan supérieur tout en envoyant son bord sur lui même.

Définition 1.7. On appelle application 1/2-harmonique à valeurs dans S 1 un point critique de la fonctionnelle L dans l'espace H 1 2 (R; S 1 ).

La dénomination 1/2-harmonique vient du fait que l'on peut ré-écrire

L(u) = R |∆ 1/4 u(x)| 2 dx = u Ḣ1/2 (R)
(1.8) avec ∆ 1/4 défini comme un multiplicateur de Fourier (cf. [START_REF]Three-term commutator estimates and the regularity of 1 2 -harmonic maps into spheres[END_REF]) et • Ḣ1/2 qui représente la semi-norme homogène de l'espace de Sobolev H 1 2 . En remarquant que R 2 + est conformément équivalent au disque 4 on obtient la définition d'une application 1/2-harmonique dans un domaine. Plus généralement on peut définir les applications 1/2-harmoniques comme suit. Soit Ω ⊂ R 2 un domaine borné et M ⊂ R n une sous-variété lisse qui a un bord ∂M = N . On dit que g : ∂Ω → ∂M est une application 1/2-harmonique si il existe u : Ω → M harmonique telle que tr |∂Ω u = g et l'énergie de u est stationnaire par rapport aux variations de la forme u t qui vérifient que tr |∂Ω (u t ) ⊂ ∂M . En faisant de telles variations on obtient alors que la dérivée normale ∂ ν u(x) est parallèle à la normale de ∂M au point u(x). Ainsi, on peut voir que le problème (1.7) est équivalent à trouver des applications 1/2-harmoniques g : ∂A → S 1 . Les applications 1/2-harmoniques apparaissent également comme limites d'une équation de Ginzburg-Landau fractionnaire (voir [START_REF] Millot | On a fractional Ginzburg-Landau equation and 1/2-harmonic maps into spheres[END_REF]). Pour en savoir plus sur ces applications nous renvoyons à [DLR11b], [START_REF] Lio | Compactness and bubble analysis for 1/2-harmonic maps[END_REF], [START_REF] Da Lio | Sub-criticality of non-local Schrödinger systems with antisymmetric potentials and applications to half-harmonic maps[END_REF], [START_REF] Moser | Intrinsic semiharmonic maps[END_REF]. Remarquons que les applications 1/2-harmoniques jouent un rôle important dans les surfaces minimales à bord libre comme expliqué dans [START_REF] Fraser | The first Steklov eigenvalue, conformal geometry, and minimal surfaces[END_REF] (voir aussi [START_REF] Schoen | Existence and geometric structure of metrics on surfaces which extremize eigenvalues[END_REF], [START_REF]Minimal surfaces and eigenvalue problems, Geometric analysis, mathematical relativity, and nonlinear partial differential equations[END_REF]).

Le premier résultat concernant le problème (1.7) a été établi par Berlyand et Mironescu dans [BM]. Ils ont obtenu Théorème 1.8. L'infimum de l'énergie E dans la classe I 1,1 , noté m(1, 1), est toujours atteint (i.e. pour toute valeur de la capacité de A). De plus, les minimiseurs sont à symétrie radiale, ils sont de la forme

u(z) = α 1 1 + ρ (r + r ρ )e iθ où (r, θ) sont les coordonnées polaires dans A ρ et α ∈ S 1 .
Leur approche pour montrer ce théorème consiste à écrire les données au bord en série de Fourier et à minimiser directement l'énergie d'une extension harmonique de ces données au bord en utilisant la formule du degré en fonction des coefficients de Fourier 5 (voir [START_REF]Degree theory : old and new, Topological nonlinear analysis[END_REF] ou [START_REF]New questions related to the topological degree[END_REF] pour une description de cette formule). Cette approche ne fonctionne pas si on s'intéresse à la minimisation de E dans I p,p pour p > 2. On note

u p (z) = 1 1 + ρ p (r p + r p ρ p )e ipθ
(1.9)

4. et donc à tout domaine borné simplement connexe. 5. Cette formule peut s'écrire deg(u, S 1 ) = n∈Z n|an| 2 où u = n∈Z ane inθ sur S 1 .

On peut vérifier que ces fonctions sont solutions de (1.7) et E(u p ) = 1-ρ p 1+ρ p . En utilisant des argument de comparaisons d'énergies ainsi qu'un résultat établi par Golovaty et Berlyand dans [START_REF] Golovaty | On uniqueness of vector-valued minimizers of the Ginzburg-Landau functional in annular domains[END_REF] on a obtenu : Théorème 1.9 (Théorème 3.6, Chapitre 3). Pour tout p ∈ N * , il existe des valeurs critiques de ρ, notées ρ p et ρ p telles que ρ p ≤ ρ p et : 1) si ρ ≥ ρ p alors m(p, p) est atteint et le minimiseur est unique (modulo les rotations). De plus le minimiseur est à symétrie radiale. 2) Si ρ < ρ p alors la solution à symétrie radiale u p (z) = 1 1+ρ p (r p + ρ p r p )e ipθ ne minimise pas E dans I p,p .

Ce résultat jouera un rôle dans le Chapitre 4 sur la minimisation de l'énergie de Ginzburg-Landau à degrés prescrits.

Un deuxième résultat relatif aux applications harmoniques a été démontré dans [START_REF] Berlyand | Minimax critical points in Ginzburg-Landau problems with semi-stiff boundary conditions : existence and bubbling[END_REF] et concerne les domaines simplement connexes. Par invariance conforme de l'énergie de Dirichlet, on peut alors supposer que le domaine est le disque unité D. On appelle produit de Blaschke une application de la forme Pour démontrer ce résultat, les auteurs ont utilisé un outil de la théorie des applications harmoniques : la différentielle de Hopf. Pour u ∈ H 1 (A, R 2 ), on définit sa différentielle de Hopf par

B α,z 1 ,...,z d = α d j=1 z -z j 1 -z j z , z ∈ D, α ∈ S 1 , z j ∈ D pour j = 1, ...d. ( 1
H u (z) = |∂ x u| 2 -|∂ y u| 2 -2i ∂ x u, ∂ y u . (1.11)
Il se trouve que si u est harmonique, sa différentielle de Hopf est holomorphe et que si la différentielle de Hopf est nulle alors u est conforme (i.e. holomorphe ou antiholomorphe). En utilisant la première propriété et les conditions au bord semi-rigides on peut montrer que dans un domaine circulaire (un disque ou un anneau) les solutions u de (1.7) ont une différentielle de Hopf qui vérifie z 2 H u (z) = c avec c une constante réelle. Dans un disque on voit que l'on a nécessairement H u = 0 et donc toute solution est conforme6 . On peut alors montrer que les solutions sont les produits de Blaschke. Dans un anneau, rien ne garantit que c = 0, et il faut étudier les cas c = 0.

Il existe des liens étroits entre les applications harmoniques et les surfaces minimales. Un de ces liens peut être fait grâce à la différentielle de Hopf. Étant donnée une application harmonique u : A → R 2 , on peut localement la "relever" en surface minimale. Plus précisément on peut trouver une fonction hauteur h, qui est définie comme la partie réelle d'une primitive de la racine carrée de la différentielle de Hopf, telle que (u, h) : A → R 3 soit la paramétrisation conforme d'une surface minimale. Pour plus d'informations sur ce procédé nous renvoyant au Chapitre 3, ainsi qu'à [START_REF] Hauswirth | Associate and conjugate minimal immersions in M × R[END_REF], [START_REF] Iwaniec | Doubly connected minimal surfaces and extremal harmonic mappings[END_REF]. Si la différentielle de Hopf s'écrit H u (z) = c z 2 avec c = 0, on doit distinguer deux cas. Si c > 0 le relèvement en surface minimale n'est pas globalement défini dans l'anneau et la fonction hauteur est h = θ, où (r, θ) sont les coordonnées polaires dans A. Mais si c < 0 alors le relèvement en surface minimale est bien défini sur tout l'anneau A et la fonction hauteur est h = a ln(r) + b (avec r = |z|). Ainsi chercher des solutions de (1.7) qui vérifient H u (z) = c z 2 , avec c < 0 conduit à chercher des surfaces minimales dans R 3 bordées par deux cercles concentriques dans des plans parallèles. On peut même montrer que les deux problèmes sont équivalent. On remarque que les degrés de u sur les bords de A correspondent aux nombres de fois que les cercles sont parcourus. Intuitivement, les degrés sur les bords doivent être les mêmes, et on peut formaliser cet argument pour obtenir la non-existence de solutions de (1.7) avec des degrés pq > 0 et p = q (cf. Proposition 3.36 au Chapitre 3).

Quand on relève les solutions à symétrie radiale u p (cf. (1.9) en surfaces minimales on obtient des caténoïdes parcourues p-fois. L'étude des surfaces minimales dans R 3 bordées par des cercles a été conduite par M.Schiffman dans [START_REF] Shiffman | On surfaces of stationary area bounded by two circles, or convex curves, in parallel planes[END_REF]. Il démontre notamment que si une surface minimale est bordée par des cercles dans des plans parallèles alors les intersections de cette surface avec des plans parallèles aux cercles du bord sont encore des cercles. Ceci implique que si les cercles sont concentriques, la surface minimale est nécessairement une portion de caténoïde. Mais dans l'article [START_REF] Shiffman | On surfaces of stationary area bounded by two circles, or convex curves, in parallel planes[END_REF] les cercles ne sont parcourus qu'une seule fois. Ainsi son résultat ne s'applique pas à notre situation si on veut trouver des solutions dans I p,p avec p ≥ 2. L'idée principale de ce chapitre est alors de chercher des surfaces minimales bordées par des cercles à partir de bifurcations d'une famille de caténoïdes parcourues p-fois, p ≥ 2. Le paramètre de bifurcation peut être vu comme le type conforme des caténoïdes, c'est-à-dire que l'on fait varier 0 < ρ < 1. On considère une famille de solutions u p (ρ, z) = 1 1+ρ p (r p + r p ρ p )e ipθ et on cherche une bifurcation. On peut alors vérifier que le théorème de bifurcation à partir d'une valeur propre simple, dit de Crandall-Rabinowitz, peut être appliqué et on obtient :

Théorème 1.11 (Théorème 3.53, Chapitre 3). Il existe des surfaces minimales dans R 3 bordées par deux cercles concentriques dans des plans parallèles qui ne sont pas des caténoïdes. Ces surfaces sont seulement immergées.

Outre l'intérêt géométrique de ce théorème, il permet aussi de trouver des solutions non radiales de l'équation (1.7) (dans I p,p , p ≥ 2). En utilisant la différentielle de Hopf et le lien avec les surfaces minimales on peut donc énoncer : Théorème 1.12 (Théorèmes 3.5 et 3.8, Chapitre 3). Soit A = A ρ = {z ∈ C; ρ < |z| < 1}. Soient (p, q) ∈ Z 2 . 1) Si pq > 0 et p = q il n'existe pas de solutions de (1.7) dans I p,q . 2) Si p = q ≥ 2, pour certaines valeurs de ρ il existe des solutions non radiales de l'équation

(1.7) dans I p,p .

Mentionnons que dans une série de travaux

0 = Id on a d dt |t=0 E(u•φ t ) = 0. Ici E désigne l'énergie de Dirichlet.
De manière équivalente u est stationnaire harmonique si div(T u ) = 0, où

T u = |∂ y u| 2 -|∂ x u| 2 -2 ∂ x u, ∂ y u -2 ∂ x u, ∂ y u |∂ x u| 2 -|∂ y u| 2
et la divergence d'une matrice est la divergence des lignes. On appelle T u le tenseur énergieimpulsion associé à l'énergie E. On peut généraliser la notion de stationnarité à d'autres fonctionnelles F . Si F est de classe C 2 et si u est également C 2 alors si u est point critique de F pour les variations standards c'est également un point stationnaire. Cependant il y a des cas où les deux notions ne coïncident pas (cf. [START_REF] Rivière | Everywhere discontinuous harmonic maps into spheres[END_REF]). Dans le cas de l'énergie de Dirichlet, la stationnarité est équivalente au fait que la différentielle de Hopf soit holomorphe. De plus dans le problème considéré dans [START_REF]Mappings of least Dirichlet energy and their Hopf differentials[END_REF], [START_REF] Iwaniec | n-harmonic mappings between annuli : the art of integrating free Lagrangians[END_REF] comme dans le notre, on peut faire des variations internes qui "glissent le long des bords 7 ". Plus précisément les variations de la forme u t (x) = u(x + tϕ(x)), pour t petit et avec ϕ ∈ C 1 (Ω) telle que ϕ(x) est tangent à ∂Ω pour x ∈ ∂Ω, sont autorisées. Elles impliquent alors que la différentielle de Hopf s'écrit H u (z) = c z 2 dans un domaine annulaire. Le point commun entre le problème de [START_REF]Mappings of least Dirichlet energy and their Hopf differentials[END_REF] et le notre est que l'on est intéressé par des minimiseurs de l'énergie de Dirichlet parmi des applications avec une classe d'homotopie fixée au bord. La principale différence est que dans [START_REF]Mappings of least Dirichlet energy and their Hopf differentials[END_REF] les auteurs considèrent des limites d'homéomorphismes pour la topologie forte de l'espace de Sobolev H 1 et donc des applications de degré 1.

Terminologie empruntée à Iwaniec et Onninen dans [IO12]

1.3.3 Minimiseurs de l'énergie de Ginzburg-Landau à degrés prescrits dans un anneau (en collaboration avec M.Dos Santos : soumis [DSR] 2015

)
Dans ce chapitre, écrit en collaboration avec Mickaël Dos Santos, on utilise les résultats obtenus au chapitre précédent pour donner des conclusions sur les minimiseurs (globaux) de l'énergie de Ginzburg-Landau simplifiée dans les classes I p,q . Nous obtenons un résultat d'existence dans la configuration p = q ≥ 2 et un résultat de non-existence pour pq > 0 et p = q. Nous renvoyons au début de la section pour les définitions et les notations. Concernant l'existence de minimiseurs, outre le cas p = q = 1 déjà mentionné, le théorème suivant dû à Golovaty et Berlyand traite le cas des degrés p = q ≥ 1 dans un anneau circulaire.

Théorème 1.14 (Théorème 2.13 dans [START_REF] Golovaty | On uniqueness of vector-valued minimizers of the Ginzburg-Landau functional in annular domains[END_REF]).

Soit A ρ = {z ∈ C; ρ < |z| < 1}, soit p ∈ N * .
Il existe un rayon intérieur critique 0 < ρ p < 1 tel que pour ρ p < ρ < 1, m ε (p, p) est atteint par un unique (à multiplication par une constante dans S 1 près) minimiseur à symétrie radiale pour tout 0 < ε < +∞.

Nous étendons ce résultat à des domaines doublement connexes non nécessairement circulaires lorsque le paramètre ε est grand.

Théorème 1.15 (Théorème 4.2 et Proposition 4.18, Chapitre 4). Soit A un domaine doublement connexe. Si cap(A) > -2π ln(ρp) , avec ρ p défini dans le Théorème 1.14, alors il existe ε p > 0 tel que m ε (p, p) est atteint pour tout ε p < ε ≤ +∞.

Le premier argument de la preuve de ce résultat est un lemme qui affirme que des suites minimisantes pour E ou pour E ε ne peuvent que "perdre" des bulles lors du passage à la limite faible (le degré doit diminuer en valeur absolue). Ce lemme est inspiré par la preuve du résultat de [START_REF] Kuwert | Minimizing the energy of maps from a surface into a 2-sphere with prescribed degree and boundary values[END_REF]. Il a été également utilisé dans le chapitre précédent.

Lemme 1.16 (Lemme 3.20 et Lemme 4.15). Soit (u n ) n une suite minimisante pour E dans I p,q . Supposons que p ≥ 0 et q ≥ 0. Alors il existe u ∈ H 1 tel que, quitte à extraire,

u n u dans H 1 . De plus u ∈ I p ,q avec 0 ≤ p ≤ p et 0 ≤ q ≤ q et E(u) = lim inf n→+∞ E(u n ) -π (p -p ) + (q -q ) .
On peut adapter la conclusion pour des configurations différentes de degrés p, q ∈ Z. Une fois ce lemme établi on observe qu'une certaine inégalité stricte empêche la formation de bulles et restaure la compacité des suites minimisantes de E ε dans I p,p pour p ≥ 2 et ε grand.

Proposition 1.17 (Proposition 4.16). Soit A un domaine annulaire et p ∈ N * . Supposons que m(p, p) < m(p -1, p -1) + 2π.

(1.12)

Alors pour ε suffisamment grand, les suites minimisantes pour m ε (p, p) sont relativement compactes dans H 1 .

On remarque que l'inégalité (1.12) fait intervenir le problème de minimisation pour l'énergie de Dirichlet qui est invariante par transformations conformes. Ainsi pour établir l'inégalité (1.12) on peut travailler dans un anneau circulaire et utiliser le Théorème 1.14. Grâce à ce théorème on déduit que pour un anneau fin les solutions radiales u p et u p-1 (cf. (1.9)) sont minimisantes pour l'énergie de Dirichlet dans I p,p et I p-1,p-1 . On vérifie alors que l'inégalité (1.12) a lieue pour ρ p < ρ < 1. Signalons que le résultat 1.14 avait été précédemment généralisé à des anneaux non nécessairement circulaires mais très fins dans [START_REF] Farina | Uniqueness of vortexless Ginzburg-Landau type minimizers in two dimensions[END_REF]. Les techniques que nous utilisons sont différentes de celles utilisées dans [START_REF] Farina | Uniqueness of vortexless Ginzburg-Landau type minimizers in two dimensions[END_REF].

Le deuxième résultat de ce chapitre a pour objet la non existence de minimiseurs de E ε dans I p,q avec p, q différents mais de même signe. Ce résultat est un résultat asymptotique et est obtenu pour ε grand. Si on exclut le cas facile pq ≤ 0 avec (p, q) = (0, 0), le premier résultat de non-existence concernant les minimiseurs de E ε à degrés prescrits se trouve dans [START_REF] Berlyand | Nonexistence of Ginzburg-Landau minimizers with prescribed degree on the boundary of a doubly connected domain[END_REF]. Dans cet article les auteurs ont mis en place une technique pour prouver que m ε (1, 1) n'est pas atteint si ε est assez petit et si l'anneau est assez large. Cette technique a ensuite été utilisée et perfectionnée par Misiats dans [START_REF] Misiats | The necessary conditions for the existence of local Ginzburg-Landau minimizers with prescribed degrees on the boundary[END_REF] pour montrer la non existence de minimiseurs locaux dans les classes I d p,q , pour ε petit avec des configurations du type d > 0, d ≤ min{p, q}. Il a répondu ainsi par l'affirmative à une partie de la conjecture faite par Berlyand et Rybalko dans [START_REF] Berlyand | Solutions with vortices of a semi-stiff boundary value problem for the Ginzburg-Landau equation[END_REF] et mentionnée dans la première sous partie. Enfin cette technique a été également employée par Mironescu dans [START_REF]Size of planar domains and existence of minimizers of the ginzburg-landau energy with semistiff boundary conditions[END_REF] pour montrer que m ε (p, q) n'est pas atteint si pq > 0, si l'anneau est assez épais et si ε est assez petit.

Théorème 1.18 (Théorème 4.16 dans [START_REF]Size of planar domains and existence of minimizers of the ginzburg-landau energy with semistiff boundary conditions[END_REF]). Soit p, q ∈ N * , pq > 0, alors ils existe une valeur critique de la capacité C min{p,q} tel que si cap(A) < C min{p,q} alors m ε (p, q) n'est pas atteint pour ε suffisamment petit.

Nous étendons la conclusion de ce résultat aux situations où ε est grand, p = q et pq > 0 et la capacité de l'anneau est grande. Toutefois notre résultat est limité au cas des anneaux circulaires.

Théorème 1.19 (Théorème 4.5, Chapitre 4). Soit A = A ρ un anneau circulaire, soit (p, q) ∈ N * tels que p = q et pq > 0. Alors il existe ρ min{p,q} > 0 tel que si ρ min{p,q} < ρ < 1 alors m ε (p, q) n'est pas atteint si ε est suffisament grand.

La preuve est également basée sur une adaptation de la technique citée plus haut. Nous esquissons maintenant les grandes lignes de cette technique dans notre cas. On raisonne par l'absurde et on suppose que m ε (p, q) est atteint pour ε grand et pour p > q > 0 (par exemple). On étudie alors le comportement asymptotique des minimiseurs u ε lorsque ε → +∞. On peut montrer que ces minimiseurs convergent faiblement vers u ∞ ∈ I q,q si l'anneau est assez fin. On utilise ensuite un lemme de décomposition pour écrire

E ε (u ε ) = E ε (u q ε ) + G ε,u q ε (v)
avec u q ε un minimiseur de E ε dans I q,q (qui existe si on suppose l'anneau assez fin), G ε,u q ε une certaine fonctionnelle et u ε = u q ε v avec v ∈ I p-q,0 . On majore alors G ε par une fonctionnelle quadratique, puis on utilise les séries de Fourier pour montrer que G ε (v) > π(p -q) ce qui fournit une contradiction. Le nouvel ingrédient qui permet de faire cette démonstration est en fait le résultat de non existence de points critiques de l'énergie de Dirichlet dans I p,q avec pq > 0 obtenu au chapitre précédent. C'est ce résultat qui permet d'étudier le comportement asymptotique d'éventuels minimiseurs de E ε dans I p,q pour ε → +∞, ce qui est crucial dans la preuve. De manière un peu paradoxale la technique utilisée se révèle plus difficile dans le cas ε grand que dans le cas ε petit. Ceci est dû au fait que la limite faible d'hypothétiques minimiseurs ne prend pas ses valeurs dans la sphère S 1 lorsque ε → +∞. C'est notamment à cause de cette dernière remarque que l'on ne parvient pas à étendre le résultat pour des anneaux de formes quelconques, alors que les résultats obtenus par Misiats et Mironescu étaient valables pour des anneaux généraux.

1.3.4 Points critiques de l'énergie de Ginzburg-Landau à degrés prescrits en dimension n ≥ 3

Dans ce chapitre on s'intéresse à une généralisation aux dimensions n ≥ 3 du problème décrit précédemment. Plus précisément on considère la n-Ginzburg-Landau énergie (sans champ magnétique)

E ε,n (u) = 1 n Ω |du| n + 1 4ε n Ω (1 -|u| 2 ) 2 , où du est la différentielle d'une application u : Ω ⊂ R n → R n et |du| 2 = tr(du t du) = n
i=1 |∂ x i u| 2 est la norme de Hilbert-Schmidt de la matrice du. Une motivation possible pour s'intéresser à cette énergie est la suivante. En dimension n = 2, Béthuel-Brézis-Hélein [START_REF] Bethuel | Progress in Nonlinear Differential Equations and their Applications[END_REF] ont montré que pour ε petit les solutions des équations de Ginzburg-Landau sans champ magnétique convergent (en un sens à préciser) vers des applications harmoniques à valeurs dans la sphère S 1 . En dimension n ≥ 3, la généralisation des applications harmoniques est la notion d'applications n-harmoniques. Ce sont les points critiques de la n-énergie :

E n (u) = 1 n Ω |du| n .
Une des propriétés importantes de la n-énergie pour n ≥ 2 est son invariance par les transformations conformes. Dans la littérature, certains travaux [START_REF] Strzelecki | Asymptotics for the minimization of a Ginzburg-Landau energy in n dimensions[END_REF], [START_REF] Hong | Asymptotic behavior for minimizers of a Ginzburg-Landau-type functional in higher dimensions associated with n-harmonic maps[END_REF], [START_REF] Han | Degenerate elliptic systems and applications to Ginzburg-Landau type equations. I[END_REF], [GSZ] ont étudié la convergence des solutions de la n-Ginzburg-Landau énergie vers des applications harmoniques. Ici on considère plutôt l'analogue n-dimensionnel (n ≥ 3) du problème des équations de Ginzburg-Landau à degrés prescrits. Soit Ω ⊂ R n un ouvert borné lisse, simplement connexe et difféomorphe à la boule unité B n . On pose

I = {v ∈ W 1,n (Ω, R n ); | tr |∂Ω v| = 1}.
Comme pour v ∈ I on a tr |∂Ω v ∈ W 1-1 n ,n (∂Ω, S n-1 ) → VMO(∂Ω, S n-1 ), la théorie du degré pour des applications de VMO développée par Brézis et Nirenberg (voir [START_REF] Brezis | Degree theory and BMO. I. Compact manifolds without boundaries[END_REF]) permet d'affirmer que le degré sur ∂Ω des applications de I est bien défini. On peut alors chercher des minimiseurs de E ε dans les espaces

I d = {v ∈ I; deg(v, ∂Ω) = d}. On pose m ε (d, Ω) = inf{E ε (v); v ∈ I d }.
On rencontre le même problème de compacité qu'aux paragraphes précédents. Pour Ω difféomorphe à une boule, on obtient le même résultat qu'en dimension 2 (cf. Lemme 3.4 dans [START_REF] Berlyand | Minimax critical points in Ginzburg-Landau problems with semi-stiff boundary conditions : existence and bubbling[END_REF]) : Devant ces résultats de non-existence on peut alors essayer de chercher des points critiques de E ε dans I 1 par exemple. Par points critiques 8 on entend des solutions de

       -∆ n u = 1 ε 2 (1 -|u| 2 )u dans Ω |u| = 1 sur ∂Ω ∂ ν u |u| = 0 sur ∂Ω.
(1.13) avec ∆ n u = div(|du| n-2 du). Les travaux [START_REF] Berlyand | Minimax critical points in Ginzburg-Landau problems with semi-stiff boundary conditions : existence and bubbling[END_REF] et [START_REF] Lamy | Bifurcation Analysis in a Frustrated Nematic Cell[END_REF] sont consacrés à ce problème en dimension 2. On cherche à généraliser [START_REF] Berlyand | Minimax critical points in Ginzburg-Landau problems with semi-stiff boundary conditions : existence and bubbling[END_REF] et à obtenir des points critiques de type min-max grâce à l'application du théorème du col. Une première grande difficulté et qu'en dimension n ≥ 3 on ne peut pas supposer que Ω est la boule unité quitte à rajouter un poids β(x) dans le terme potentiel dans l'énergie 9 . En effet les difféomorphes à une boule ne sont pas conformément équivalents en dimension n ≥ 3. On remarque que dans la boule il existe effectivement une solution de (1.13), qui est une solution à symétrie radiale dans I 1 . Ainsi on s'intéresse à des domaines qui ne sont pas des boules. Dans ce chapitre nous n'obtenons pas le résultat pour l'énergie de Ginzburg-Landau. On considère une énergie perturbée

E α ε (u) = 1 n + α Ω |du| n+α + 1 4ε n Ω (1 -|u| 2 ) 2 dans l'espace I α = I ∩ W 1,n+α (Ω, R n ), pour α petit. Grâce à l'injection de Sobolev W 1,n+α (Ω, R n ) → C 0 (Ω, R n )
on peut montrer que la fonctionnelle E α ε vérifie la condition de Palais-Smale. On met alors en place un schéma de min-max pour cette fonctionnelle et on obtient des points critiques de 8. Pour trouver les équations d'Euler-Lagrange on utilise ici la structure de variété de I ∩ C 0 (Ω, R n ) cf. Définition 5.14 9. c'est ce qui avait été fait dans [START_REF] Berlyand | Minimax critical points in Ginzburg-Landau problems with semi-stiff boundary conditions : existence and bubbling[END_REF] E α ε dans I 1 ∩ I α . Il serait alors naturel de faire tendre α vers 0 et d'étudier la convergence de ces points critiques vers une éventuelle solution non-triviale de (1.13). Nous ne réalisons pas cette étude dans ce chapitre. Elle nécessiterait une analyse de "blow-up" et l'établissement d'un théorème de décomposition d'énergie (ou théorème de bubbling) similaire au Théorème 8.13 dans [START_REF] Berlyand | Minimax critical points in Ginzburg-Landau problems with semi-stiff boundary conditions : existence and bubbling[END_REF].

L'idée d'étudier des problèmes sans compacité en utilisant une fonctionnelle perturbée qui vérifie la condition de Palais-Smale remonte aux travaux de Sacks-Uhlenbeck [START_REF] Sacks | The existence of minimal immersions of 2-spheres[END_REF] pour les applications harmonqies dans des classes d'homotopie. Cette idée a ensuite été utilisée pour la recherche de surfaces minimales à bord libre dans [START_REF] Struwe | On a free boundary problem for minimal surfaces[END_REF] et [START_REF] Fraser | On the free boundary variational problem for minimal disks[END_REF]. Ce contexte est assez proche de notre problème.

1.4 Limites de mesures de vorticité associées aux équations de Ginzburg-Landau (soumis [Rod] 2015

)
Dans cette seconde partie de la thèse on ne s'intéresse plus à l'existence de solutions des équations de Ginzburg-Landau mais plutôt à leur comportement asymptotique dans la "limite de London" ε → 0. Ce régime asymptotique correspond physiquement à des supraconducteurs dits de type II-extrème. On a déjà mentionné dans la description du modèle de Ginzburg-Landau qu'un des phénomènes importants pour les supraconducteurs de type II est l'apparition de vortex. Ceux-ci correspondent à des zones extrêmement étroites où l'échantillon se trouve dans un état normal, et autour desquelles circulent des courants supraconducteurs. Dans la limite ε → 0, les vortex peuvent être décrits comme des zéros isolés de u autour desquels u a un degré topologique non nul. Dans la littérature on s'intéresse particulièrement au nombre de ces vortex (en fonction du champ magnétique appliqué par exemple) et à leurs positions. Dans le Chapitre 13 de [START_REF] Sandier | Progress in Nonlinear Partial Differential Equations Birkhauser[END_REF] (voir aussi [START_REF] Sandier | Limiting vorticities for the Ginzburg-Landau equations[END_REF]) Sandier et Serfaty ont étudié l'asymptotique de solutions générales des équations de Ginzburg-Landau, avec et sans champ magnétique. En particulier ils ont obtenu des conditions nécessaires sur les vorticités limites de ces équations. Dans ce cinquième chapitre on considère des solutions des équations de Ginzburg-Landau sans champ magnétique. On rappelle en premier lieu la notion de vorticité limite ainsi que les résultats concernant ce problème obtenus dans [START_REF] Sandier | Progress in Nonlinear Partial Differential Equations Birkhauser[END_REF] au Chapitre 13.

Soit Ω ⊂ R 2 un ouvert lisse de du plan que l'on suppose simplement connexe. Soient {u ε } ε>0 des solutions de

-∆u ε = u ε ε 2 (1 -|u ε | 2 ) dans Ω (1.14) on suppose de plus que |u ε | ≤ 1 dans Ω et E ε (u ε ) < C 0 ε α-1 , α > 2 3 (1.15)
pour tout ε > 0 (avec E ε définie par (1.2)). On pose

j ε = iu ε , ∇u ε , µ ε = curl j ε . (1.16)
a ε i ∈ Ω et d ε i ∈ Z pour tout i, telle que en posant n ε = i |d ε i |, n ε ≤ C E ε (u ε , B ε ) | ln ε| , (1.18)
où B ε est une union de boules de rayons plus petit que Cε 3/2 et telle que

µ ε -ν ε W -1,p (Ω µ ε -ν ε C 0 (Ω) * → 0, ( 1
ε L ∞ (Ω) ≤ C ε . Alors on a lorsque ε → 0 : 1) si n ε = o E ε (u ε )
, alors quitte à extraire :

h ε E ε (u ε ) h faiblement dans H 1 (Ω) et fortement dans W 1,p loc (Ω), p < 2 (1.20) avec ∆h = 0. 2) Si E ε (u ε ) = o(n ε ) alors νε nε
0 pour la topologie faible sur les mesures.

10. Attention ici hε ne désigne pas le champ magnétique. 11. Afin de ne pas confondre µε et νε, on appelle parfois µε le déterminant Jacobien de uε.

3) Si

n ε ∼ C E ε (u ε ) (C > 0) alors, quitte à extraite, h ε n ε h faiblement dans H 1 (Ω) et fortment dans W 1,p loc (Ω), p < 2, (1.21) ν ε n ε µ = ∆h ∈ M(Ω) ∩ H -1 (Ω) faiblement pour la topologie des mesures (1.22)
de plus la différentielle de Hopf de h définit par

ω h = (∂ x h) 2 -(∂ y h) 2 -2i∂ x h∂ y h (1.23)
est holomorphe dans Ω. De plus si µ ∈ L p (Ω ) pour p > 1 et pour un sous domaine Ω ⊆ Ω alors µ = 0 dans Ω .

On a noté M(Ω) l'espace des mesures de Radon sur Ω. Dans ce chapitre on entreprend de décrire la régularité locale de mesures µ singulières qui vérifient de telles conditions critiques. Ceci correspond au problème ouvert nR14 de [START_REF] Sandier | Progress in Nonlinear Partial Differential Equations Birkhauser[END_REF] dans le cas des equations de Ginzburg-Landau sans champ magnétique. De manière plus précise soit (µ,

h µ ) ∈ M(Ω) × H 1 (Ω) telles que ∆h µ = µ (1.24) et ω hµ = (∂ x h µ ) 2 -(∂ y h µ ) 2 -2i∂ x h µ ∂ y h µ est holomorphe dans Ω, (1.25)
que peut-on dire sur µ et sur son support ? Remarquons que, comme |∇h µ | 2 = |ω hµ | 2 , on a |∇h µ | ∈ C 0 (Ω) et les points critiques de h µ sont bien définis. On peut alors résumer les résultats obtenus dans ce chapitre de la manière suivante :

Théorème 1.24 (Théorèmes 6.3, 6.4, 6.6). Soit (µ, h µ ) ∈ M(Ω) × H 1 (Ω) qui vérifient (1.24) et (1.25). 1) Les points z où ∇h µ (z) = 0 sont isolés dans Ω, si O est compact il y a un nombre fini de tels points dans Ω. 2) Soit z 0 ∈ supp µ, qui vérifie ∇h µ (z 0 ) = 0. On suppose h µ (z 0 ) = 0. Dans un voisinage de ce point noté V on peut trouver une fonction harmonique H telle que

h µ = |H| ou h µ = -|H| dans V . De plus supp µ V = {z ∈ V ; H(z) = 0}. 3) Soit z 0 ∈ supp µ tel que ∇h µ (z 0 ) = 0. On suppose toujours h µ (z 0 ) = 0. Alors il existe un voisinage de z 0 noté V et une fonction harmonique dans V notée H tels que supp µ V ⊆ {z ∈ V ; H(z) = 0}.
On savait déjà qu'une mesure de vorticité limite pour les équations de Ginzburg-Landau sans champ magnétique ne pouvait pas être supportée par un ouvert de R 2 (cf. 3) du Théorème 1.23).

Le résultat obtenu affine cela en montrant qu'une telle mesure ne peut être supportée que par des lignes ou des points (mais dans ce dernier cas la mesure n'appartient pas à H -1 (Ω)). La preuve de ce résultat repose sur l'observation suivante : on peut écrire

ω hµ = (∂ x h µ -i∂ y h µ ) 2 .
Ainsi près des points z où ω hµ (z) = 0 (ou de manière équivalente ∇h µ (z) = 0), formellement on obtient 1.5 Perspectives 1.5.1 Perspectives de la première partie

(∂ x h µ -i∂ y h µ )(z) = θ(z) √ ω hµ avec θ(z) ∈ {±1}.
Concernant le premier chapitre, après les travaux de Misiats [START_REF] Misiats | The necessary conditions for the existence of local Ginzburg-Landau minimizers with prescribed degrees on the boundary[END_REF] une partie de la conjecture de Berlyand et Rybalko reste ouverte. Cette conjecture a pour objet la non existence de minimiseurs locaux dans les classes I d p,q définies plus haut. Ainsi on peut essayer de prouver que si d > 0, p > d > q (ou q > d > p) alors m ε (p, q, d) = m ε (p, d, d) + π(q -d) et m ε (p, q, d) n'est pas atteint (pour la définition de m ε (p, q, d) nous renvoyons à la section 1.3.1).

Dans le deuxième chapitre en construisant de nouvelles surfaces minimales dans R 3 bordées par deux cercles dans des plans parallèles on obtient des solutions de (1.7) non-radiales. Une question intéressante est alors de savoir si ces solutions minimisent l'énergie de Dirichlet dans les classes I p,p , p ≥ 2. Plus généralement, comme l'on sait que lorsque l'anneau est trop épais la solution radiale n'est plus minimisante dans I p,p on peut se demander si l'infimum m(p, p) est atteint dans ce cas. Par exemple pour p = 2, à la lumière de l'analyse effectuée dans ce chapitre on sait que m(2, 2) est atteint si et seulement si il existe v ∈ I 2,2 telle que

E(v) ≤ E(u 1 ) + 2π où u 1 = 1 1+ρ (r + ρ r
)e iθ est le minimiseur de m(1, 1) (toujours atteint). Dans cette partie on a également vu qu'il existait des solutions radiales avec une différentielle de Hopf qui vérifie H u (z) = c z 2 où c > 0. Ces solutions correspondent à des hélicoïdes lorsqu'elles sont relevées en surfaces minimales par le procédé expliqué dans ce chapitre. On peut alors se demander si il existe d'autres solutions non-radiales dans les classes I p,p avec une différentielle de Hopf H u (z) = c z 2 , c > 0. Ces solutions produiraient des surfaces minimales dans R 3 bordées par deux hélices.

Bien que le troisième chapitre apporte des réponses supplémentaires au problème de minimisation à degrés prescrits de l'énergie de Ginzburg-Landau sur des domaines doublement connexes, ce problème dans toute sa généralité reste ouvert. On peut notamment chercher à prouver que pour tout domaine annulaire, et pour tout ε > 0, m ε (p, q) n'est pas atteint lorsque pq > 0, p = q. C'est ce résultat qui semble naturel de conjecturer à la lumière des Théorèmes 1.18 et 1.19. D'un autre côté pour le cas p = q ≥ 2, existe-t-il des minimiseurs de E ε si on ne suppose plus l'anneau fin ? Cette question est très liée à la même question pour l'énergie de Dirichlet. La question de l'existence de points critiques non minimisants de E ε , avec ε grand, dans les classes I p,q , pq > 0, p = q est également intéressante. A ce sujet on peut remarquer que les auteurs de [START_REF] Berlyand | Minimax critical points in Ginzburg-Landau problems with semi-stiff boundary conditions : existence and bubbling[END_REF] ont prouvé l'existence de tels points critiques dans la classe I 1,0 .

En dimension n ≥ 3 beaucoup de questions sont ouvertes pour les problèmes à degrés prescrits. On peut chercher à déterminer toutes les solutions de (1.13) pour ε = +∞ et Ω = B n . En dimension 2 on sait que ces solutions sont des produits de transformations de Möbius (cf. Théorème 1.10). Est-ce qu'en dimension n ≥ 3 les solutions de ces équations sont aussi nécessairement conformes ? On remarque qu'on peut ainsi définir une généralisation des applications 1/2-harmoniques grâce à ce problème. Ces solutions jouent un rôle particulier puisque ce sont les "bulles" pour le problème (1.13). C'est-à-dire que lorsqu'on fait une analyse de "blowup" pour déterminer le défaut de compacité de l'équation on tombe sur ces solutions. C'est pour cela qu'une meilleure connaissance de ces "bulles" pourraient servir à prouver l'existence de points critiques de type min-max de la n-énergie de Ginzburg-Landau pour ε > 0 grand. On peut également penser à prouver l'existence de points-critiques non triviaux de la n-énergie de Dirichlet pour des domaines Ω non difféomorphes à la boule unité B n .

Dans [START_REF] Bahri | On a nonlinear elliptic equation involving the critical Sobolev exponent : the effect of the topology of the domain[END_REF], Bahri et Coron ont développé une méthode, faisant appel à la topologie algébrique pour trouver des solutions de l'équation de Yamabe (voir aussi [START_REF] Bahri | Critical points at infinity in some variational problems[END_REF], [START_REF] Bahri | Vers une théorie des points critiques à l'infini, Bony-Sjöstrand-Meyer seminar[END_REF]). Puisque la perte de compacité de l'équation de Yamabe et similaire à celle rencontrée pour les problèmes de Ginzburg-Landau à degrés prescrits, dans le cas n = 2, on peut essayer d'appliquer la technique de Bahri-Coron pour trouver des points critiques (non minimisants) de Ginzburg-Landau avec données semi-rigides dans Ω simplement connexe. Cette méthode, dite des points critiques à l'infini, nécessite une analyse des "interactions" entre les bulles. On peut également essayer de construire des solutions en utilisant des méthodes non variationnelles.

Perspectives de la deuxième partie

Après l'étude de la régularité des limites des mesures de vorticité associées au système de Ginzburg-Landau sans champ magnétique une question, naturelle est : qu'en est-il du cas avec champ magnétique ? Là encore Sandier et Serfaty ont obtenus des conditions critiques pour de telles mesures au Chapitre 5 de [START_REF] Sandier | Progress in Nonlinear Partial Differential Equations Birkhauser[END_REF]. Avec les notations du Chapitre 5, l'analogue du problème rencontré dans ce chapitre lorsque l'on prend en compte le champ magnétique peut s'énoncer comme suit. Soit (µ, h) ∈ M(Ω) ∩ H 1 (Ω) telle que

-∆h + h = µ dans Ω (1.27) div(T µ ) = 0 (1.28) avec T µ = 1 2 (∂ y h) 2 -(∂ x h) 2 + h 2 -∂ x h∂ y h -∂ x h∂ y h 1 2 (∂ x h) 2 -(∂ y h) 2 + h 2 .
Que peut-on dire sur le support de µ ? En particulier est-ce que ce support est nécessairement constitué de lignes ou d'ouverts de R 2 ? Notons que si µ est concentrée en des points alors h / ∈ H 1 (Ω). On remarque aussi que contrairement au cas sans champ magnétique ici µ peut être supportée par un ouvert.

Introduction

In their paper [START_REF] Berlyand | Solutions with vortices of a semi-stiff boundary value problem for the Ginzburg-Landau equation[END_REF] Berlyand and Rybalko studied solutions of the Ginzburg-Landau (G.L in short) equation with "semi-stiff" boundary conditions

     -∆u + 1 ε 2 u(|u| 2 -1) = 0 in A, |u| = 1 on ∂A u ∧ ∂ ν u = 0 on ∂A. (2.1)
where ε is a positive parameter (the inverse of the GL parameter), u is a complex-valued map, ν is the outer normal to the boundary, ∂ ν u is the partial derivative of u in the direction ν and where, if a, b are complex numbers, we let a ∧ b = i 2 (ab -ab). Here A is a smooth, bounded domain in R 2 .

Insertion of bubbles

Solutions of problem (2.1) are critical points of the energy functional

E ε (u) = 1 2 A |∇u| 2 dx + 1 4ε 2 A (|u| 2 -1) 2 dx (2.2)
with respect to variations which leave the modulus of u on the boundary unchanged or, in other words, of variations which stay on the space An important feature of the model is the existence of vortices, which we can define either as a certain radially symmetric entire solution to the equations or as very small (as ε → 0) regions in the domain where an arbitrary solution u behaves like the radially symmetric vortex, and in particular vanishes at some point in the region, with a certain degree. The driving force for the appearing of such vortices is the magnetic field in the full Ginzburg-Landau model. However in their pioneering work [START_REF] Bethuel | Progress in Nonlinear Differential Equations and their Applications[END_REF] Bethuel, Brézis and Hélein showed that critical points of the G.L energy (2.2) subject to a Dirichlet condition g ∈ C 1 (∂A, S 1 ) with non zero topological degree exhibit similar quantized vortices as caused by a magnetic field in type II superconductors or by rotation in superfluids. The relaxation of the Dirichlet boundary condition to semi-stiff boundary conditions is natural. It corresponds to prescribing the physical quantity |u| on the boundary, together with the winding number of u, which both have physical significance.

I = {u ∈ H 1 (A; R 2 ); |u| =
Minimizing the energy E ε over I produces only constant solutions. On the other hand, Boutet de Monvel and Gabber [START_REF] Boutet De Monvel-Berthier | A boundary value problem related to the Ginzburg-Landau model[END_REF] proved that the connected components of I are described using the degree, where the degree of a map u ∈ H 1 2 (γ, S 1 ) on γ (where γ is a simple, smooth, closed curve) is the integer

deg(u, γ) = 1 2π γ u ∧ ∂u ∂τ dτ, (2.4)
and where the integral is understood via H 1/2 -H -1/2 duality, and ∂ ∂τ is the tangential derivative with respect to the counterclockwise orientation on γ.

From now on we consider A to be a doubly connected domain and write A = Ω \ ω where Ω and ω are simply connected smooth domains and ω ⊂ Ω ⊂ R 2 . Throughout the paper we denote by ν the outward pointing normal to ∂A and by τ the tangent unit vector to ∂A pointing in the counter-clockwise direction, so that (ν, τ ) is a direct frame on ∂Ω and an indirect frame on ∂ω. For more explanation about the definition of the degree we refer to [START_REF]Degree theory : old and new, Topological nonlinear analysis[END_REF].

The connected components of I are labeled by a pair of integers (p, q) :

I p,q = {u ∈ I; deg(u, ∂Ω) = p, deg(u, ∂ω) = q}. (2.5)
They are open and closed for the strong H 1 topology, and we may try to find solutions of (2.1) by minimizing E ε in I p,q . This is a nontrivial problem because the degree is not continuous with respect to weak convergence in H 1 . Hence I p,q is not weakly closed, and, as it turns out, the minimum of E ε in I p,q is not achieved when p > 0 ≥ q. The typical behaviour of a minimizing sequence jumping from one class to the other would be, in the case of the simply connected disk, the sequence of Möebius transformations u n (z) = (z -a n )/(1 -ān z) for a sequence {a n } n in the disk converging to 1. Then for each n the map u n has degree 1, and the weak limit is the constant -1 which has degree 0.

Berlyand and Rybalko devised a tool, the approximate bulk degree, allowing them to find local minimizers of E ε in I. The idea is the following : if E ε (u) ≤ Λ for some finite Λ, and if ε is small enough, then u is "almost" S 1 -valued (its values are everywhere close to 1 except for some small regions where they are close to 0). Hence the degree of u over a simple closed curve lying in A is well defined, and we can take the average of all these degrees. An other point of view about the approximate bulk degree is the following. Let us recall a lemma of Mironescu [?]

Lemma 2.1. Let Λ > 0 and let u be a solution of the G.L equation such that

|u| ≤ 1, E ε (u) ≤ Λ,
where Λ is indepent of ε. Then

1 -|u(x)| 2 ≤ Cε 2 dist 2 (x, ∂A) , (2.6) and |D k u(x)| ≤ C k dist 2 (x, ∂A) (2.7)
where C, C k are independent of ε.

This lemma says that the zeros of a G.L equation are near to the boundary when ε is small. Thus the degree of u over a curve lying in A is well defined if the curve is far away enough from the boundary. The approximate degree of u is the degree of u over any such curve. The important feature is that the approximate degree is continuous for the weak H 1 convergence, which will result from the following definition. (2.8)

Then the approximate bulk degree of u in A is the following scalar quantity :

Insertion of bubbles abdeg(u) = 1 2π A u ∧ (∂ x V ∂ y u -∂ y V ∂ x u)dx.
(2.9)

The properties of the approximate bulk degree can be found in [START_REF] Berlyand | Solutions with vortices of a semi-stiff boundary value problem for the Ginzburg-Landau equation[END_REF], we list here those useful to us :

1. If A is an annulus A := {x; R 1 < |x| < R 2 } then abdeg(u) = 1 log(R 2 /R 1 ) R 2 R 1 1 2π |x|=ξ u ∧ ∂u ∂τ ds dξ ξ .
This formula clarifies that it is an average of the degree over the curves lying in A. 2. The key property of abdeg(u) is that it is preserved in weak H 1 limits, i.e., abdeg

(u n ) → abdeg(u) if u n u in H 1 (A). 3. If u ∈ H 1 (A, S 1 ), then abdeg(u) = deg(u, ∂Ω) = deg(u, ∂ω). 4. If u ε ∈ E Λ ε := {u ∈ I; E ε (u)
< Λ}, then, up to choosing a subsequence, u ε u in H 1 (A) and abdeg(u ε ) → deg(u, L), ε → 0, where L is any simple closed smooth curve, which lies in A and encloses ∂ω. In particular, abdeg(u ε ) is close to an integer for small ε. The method developped by Berlyand and Rybalko (see [START_REF] Berlyand | Solutions with vortices of a semi-stiff boundary value problem for the Ginzburg-Landau equation[END_REF] and also [START_REF] Santos | Local minimizers of the Ginzburg-Landau functional with prescribed degrees[END_REF]) in order to find solutions of (2.1) is to study the minimization problem

m ε (p, q, d) := inf{E ε (u); u ∈ I d p,q } (2.10)
where

I d p,q = {u ∈ I p,q ; d -1/2 ≤ abdeg(u) ≤ d + 1/2} (2.11)
They obtained the following result Theorem 2.3. ([BR10]) For any integers (p, q) ∈ Z 2 and d > 0 with d ≥ max{p, q} there exists ε 1 = ε 1 (p, q, d) > 0 such that the infimum in (2.10) is always attained when ε < ε 1 . Additionally, each minimizer lies in I d p,q with its open neighborhood, therefore, the minimizers in I d p,q are disctinct local minimizers in I.

The essential ingredient in the proof of the theorem is the following lemma, which is reminiscent of Lemma 2 in [START_REF] Brezis | Large solutions for harmonic maps in two dimensions[END_REF].

Lemma 2.4 (Fundamental lemma). Let u ∈ I d p,q be any function which satisfies

|u| ≤ 1 in A and ∂ ν |u| > 0, u ∧ ∂ ν u = 0 on ∂Ω. (2.12) i) Assume that there is x 0 ∈ ∂Ω such that u ∧ ∂ τ u(x 0 ) > -u • ∂ ν u(x 0 ) (2.13) then there exists v ∈ I d p-k,q such that E ε (v) < E ε (u) + kπ.
ii) Assume that there is x 0 ∈ ∂Ω such that

u ∧ ∂ τ u(x 0 ) < u • ∂ ν u(x 0 ) (2.14)
then there exists v ∈ I d p+k,q such that

E ε (v) < E ε (u) + kπ.
Remark : An analogous lemma can be stated considering the other boundary ∂ω.

We will explain why this lemma is of fundamental importance in the next section. It was proved by Berland and Rybalko in [START_REF] Berlyand | Solutions with vortices of a semi-stiff boundary value problem for the Ginzburg-Landau equation[END_REF] under the stronger hypothesis -which replaces (2.13)that u is a local minimizer of the Ginzburg-Landau energy and ε is small. These two facts allow them to find a point x 0 ∈ ∂Ω such that u ∧ ∂ τ u(x 0 ) > 0 ( see Lemma (2.10) below or Lemma 25 in [START_REF] Berlyand | Solutions with vortices of a semi-stiff boundary value problem for the Ginzburg-Landau equation[END_REF]) and they can use this to construct a test function which realizes the strict inequality in the lemma. Dos Santos in [START_REF] Santos | Local minimizers of the Ginzburg-Landau functional with prescribed degrees[END_REF] generalized their result to the case of multiply connected domains.

Our aim here is to prove this lemma with a method inspired by that of Soyeur in [START_REF] Soyeur | The Dirichlet problem for harmonic maps from the disc into the 2-sphere[END_REF], which requires the weaker assumptions (2.13) or (2.14). Assumption (2.13) is indeed weaker since if

u ∧ ∂ τ u(x 0 ) > 0 then automatically u ∧ ∂ τ u(x 0 ) > -u • ∂ ν u(x 0 )
. This is because of assumption (2.12) and because u ∧ ∂ τ u(x 0 ) = ∂θ ∂τ where θ is the phase of u (which is only locally defined but its gradient is globally defined) and u•∂ ν u(x 0 ) = ∂ρ ∂ν where ρ := |u|. Note that assumption (2.12) is automatically satisfied by a solution of the G.L equation with semi-stiff boundary conditions thanks to the maximum principle and the Hopf lemma.

The paper is organized as follows : in section 2 we give the proof of the fundamental lemma. In section 3 we gather some lemmas and preliminary results for the existence result of Berlyand and Rybalko, whose proof we reproduce in Section 4.

Proof of the fundamental lemma

Idea of the proof. Let u ∈ I d p,q and x 0 ∈ ∂Ω, we can write

u(z) = u(x 0 ) + a(z -x 0 ) + b(z -x 0 ) + O(|z -x 0 | 2 ) for z ∈ V ∩ A where V is a neigbourhood of x 0 , because u ∈ C ∞ (A, R 2
). The idea is to replace u with a holomorphic function in a small neighborhood of x 0 in order to cover the disc one more time, and so increase the degree by one unit. This idea was already used in [START_REF] Soyeur | The Dirichlet problem for harmonic maps from the disc into the 2-sphere[END_REF] (see also [START_REF] Kuwert | Minimizing the energy of maps from a surface into a 2-sphere with prescribed degree and boundary values[END_REF]) and is in fact a refinement of the constructions presented by Brézis and Coron [START_REF] Brezis | Large solutions for harmonic maps in two dimensions[END_REF] and Jost [START_REF] Jost | The Dirichlet problem for harmonic maps from a surface with boundary onto a 2-sphere with nonconstant boundary values[END_REF]. All of these constructions were done for x 0 in the interior of the domain, though. In our case the construction takes place on the boundary and thus we need to preserve the modulus boundary condition. This is one of the reasons why conditions (2.13) and (2.14) come up.

We will prove case ii) of the theorem, since case i) may be deduced from it by applying case ii) to the complex conjugate ū.

Assume then that x 0 ∈ ∂Ω is such that u ∧ ∂ τ u(x 0 ) = 0. We assume, without loss of generality, that x 0 = 0 and that u(0) = 1 (otherwise we change variables and multiply u by a suitable phase). We also assume, rotating the domain if necessary that τ (0) = (0, 1) is vertical. Then ν(0) = (1, 0), and Ω lies to the left of the boundary, which is vertical at the point 0.

We start by proving the result in the case where ∂Ω is flat near in a neighbourhood of 0. This hypothesis will be removed in the end by using suitable diffeomorphism.

Step1 : Image of a little neighborhood of 0 under du(0). In a neighborhood of 0 we can write :

u(z) = 1 + az + bz + O(|z| 2 ),
where

a = ∂ z u(0) = a 1 +ia 2 and b = ∂ z u(0) = b 1 +ib 2 .
Then the matrix of du(0) in the canonical basis of R 2 , which is also (ν(0), τ (0)), is

du(0) = a 1 + b 1 b 2 -a 2 a 2 + b 2 a 1 -b 1 = ∂ ν u 1 ∂ τ u 1 ∂ ν u 2 ∂ τ u 2
where u 1 , u 2 stand for the coordinates of function u on the basis (ν(0), τ (0)). Hypothesis (2.12) and (2.14) give us some information about a 1 , a 2 , b 1 , b 2 : i) a 2 = b 2 .This follows from the fact that

∂ τ |u| 2 = u • ∂ τ u = 0 on ∂A, since |u| = 1 on ∂A. ii) a 1 + b 1 > 0. Indeed, from (2.12) we have |u| ≤ 1 in A and ∂ ν |u| 2 = u • ∂ ν u > 0 on ∂A.
iii) a 2 = 0. This follows from the fact that u(0

) ∧ ∂ ν u(0) = 0 hence ∂ ν u 2 (0) = 0, which means that a 2 + b 2 =0 but a 2 = b 2 hence a 2 = 0 = b 2 . iv) b 1 > 0. This follows from (2.14) since u(0) • ∂ ν u(0) = a 1 + b 1 and u(0) ∧ ∂ τ u(0) = a 1 -b 1 .
We let E -:= du(0)(D -), where

D -:= D(0, 1) ∩ C -and C -:= {z ∈ C; (z) < 0}.
E -is a half-ellipse centered at 0, with horizontal (resp., vertical) semi-axis a 1 + b 1 (resp.,

|a 1 -b 1 |).
Step 2 : Construction of the bubble. Consider the conformal map

w(z) = a 1 z + b 1 z .
For z ∈ ∂D ∩ C -we have z = 1/z thus w(z) = du(0)(z). This is what will allow us to glue a bubble to u.

We can describe the image of D -under w using Cartesian coordinates or we can also write w in polar coordinates. Let z = re iφ , then

w(z) = a 1 x + b 1 x x 2 + y 2 + i a 1 y - b 1 y x 2 + y 2 O |z| = µ |z| = 1 z → w(z) E - [-2 √ a 1 b 1 , 0]
w(re iφ ) = (a 1 r + b 1 r ) cos(φ) + i(a 1 r - b 1 r ) sin(φ).
From this expression we see that circles centered at the origin of D(0, 1) are mapped to ellipses except possibly for the circle r

= b 1 a 1 which is mapped to a segment [-2 √ a 1 b 1 , 2 √ a 1 b 1 ] if a 1 > 0 and b 1 a 1 < 1.
Studying the variations of the functions

f (r) = a 1 r + b 1 r and g(r) = a 1 r - b 1 r
one can see that two cases can occur (the notation v : A B means that v is a bijection from A to B see Figure 2.1 and 2.2) :

i) if a 1 ≥ b 1 > 0 then, letting µ = b 1 /a 1 , w : {re iφ ; 0 < r < µ, π 2 < φ < 3π 2 } C -\ [-2 (a 1 b 1 ), 0] and w : {re iφ ; µ < r < 1, π 2 < φ < 3π 2 } E -\ [-2 (a 1 b ), 0]. ii) if b 1 > |a 1 | ≥ 0 then w : D -C -\ E -. |z| = 1 |z| = µ O z → w(z) C - E - Figure 2
.2: Image of {re iφ ; 0 < r < µ} by w in the case i).

Now we compose w with a suitable holomorphic function so that the result takes values in D(0, 1). Let

ϕ(z) = 2 + z 2 -z .
We can check that ϕ is a holomorphic diffeomorphism between C -and D(0, 1) such that

ϕ(0) = ϕ (0) = 1 and ϕ({ (z) = 0}) = S 1 .
We let, for any σ ∈ (0, 1),

ψ σ (z) = ϕ (σw(z/σ)) , so that ψ σ (z) = a 1 z 2 + b 1 σ 2 + 2z -a 1 z 2 -b 1 σ 2 + 2z . (2.15)
Note that, since ϕ(0) = ϕ (0) = 1 and since for any θ ∈ R we have w(e iθ ) = du(0)(e iθ ), we have,

uniformly in z ∈ ∂D σ ∩ C -, ψ σ (z) = ϕ(0) + σw(z/σ) + O(σ 2 ) = 1 + du(0)(z) + O(σ 2 ) = u(z) + O(σ 2 ).
(2.16)

The map ψ σ is the bubble, which we now glue with u, in a slightly different way from [START_REF] Kuwert | Minimizing the energy of maps from a surface into a 2-sphere with prescribed degree and boundary values[END_REF], [START_REF] Soyeur | The Dirichlet problem for harmonic maps from the disc into the 2-sphere[END_REF]. Choosing τ ∈ (σ, 1) (to be chosen later) we let, for z ∈ A, and assuming τ is small

enough so that D - τ ⊂ A, v σ =      ψ σ (z) if |z| < σ, u(z) if |z| > τ, I(z) if σ ≤ |z| ≤ τ,
(2.17

)
where I is defined by interpolating the modulus and argument of u and ψ σ . More precisely z) , where

I(z) = f (z)e ig(
f (re iθ ) = r -σ τ -σ |u(τ e iθ )| + τ -r τ -σ |ψ σ (σe iθ )|, g(re iθ ) = r -σ τ -σ arg(u(τ e iθ )) + τ -r τ -σ arg(ψ σ (σe iθ )).
Note that the argument of u(τ e iθ ) and ψ σ (σe iθ ) is well defined since both complex numbers are close to 1, if τ is small enough. Note also that this method of interpolating guarantees that |v σ | = 1 on ∂A, which would have been lost by linear interpolation.

Step 3 : Computation of the energy of v σ . We have

E ε (v σ ) = E ε (u) -E ε (u, D - τ ) + E ε (ψ σ , D - σ ) + E ε (I, D - τ \ D - σ ),
where

D - r := D(0, r) ∩ C -. i) Computation of E ε (ψ σ , D - σ ).
We have that the Dirichlet energy of ψ σ , denoted by E, is equal to the area of the image of ψ σ (counted with multiplicity) because it is conformal. Using the fact that ϕ (0) = 1 we find that

|ϕ(σE -)| = σ 2 |E -| + O(σ 3 ) therefore in case i) we have E(ψ σ , D - σ ) = |ϕ(C -)| + |ϕ(σE -)| = π + σ 2 |E -| + O(σ 3 ) = π + π 2 σ 2 (a 1 2 -b 1 2 ) + O(σ 3 ).
Similarly, in case ii) we find

E(ψ σ , D - σ ) = |ϕ(C -\ σE -)| = π -σ 2 |E -| + O(σ 3 ) = π - π 2 σ 2 (a 1 2 -b 1 2 ) + O(σ 3 ). (2.18) For the potential term D - σ (|ψ σ | 2 -1)
2 ), we write :

D - σ (|ψ σ | 2 -1) 2 ) = D - σ 3/2 (|ψ σ | 2 -1) 2 ) I + D - σ \D - σ 3/2 (|ψ σ | 2 -1) 2 ) II . For I we have (|ψ σ | 2 -1) 2 ≤ 1 hence I ≤ π 2 σ 3 .
For the term II we check that, from (2.15), for any z such that |z| ≥ σ 3/2 we have (1

-|ψ σ | 2 ) = O(σ 1/2 ) hence II = O(σ 3 ),
and finally

E ε (ψ σ , D - σ ) ≤ π + π 2 (a 2 1 -b 2 1 )σ 2 + O(σ 3 ). (2.19) ii) Computation of E ε (u, D τ ).
We also have

E ε (u, D - τ ) ≥ 1 2 D - τ |∇u| 2 = π 2 (a 1 + b 1 ) 2 + (a 1 -b 1 ) 2 2 τ 2 + O(τ 3 ), hence E ε (u, D - τ ) ≥ π 2 (a 1 2 + b 1 2 )τ 2 + O(τ 3 ). (2.20) iii) Computation of E ε (I, D - τ \ D - σ ). Recall that I(z) = f (z)e ig(z) with |f | ≤ 1, hence |∇I(z)| 2 ≤ 2|∇f (z)| 2 + 2|∇g(z)| 2 and 1 2 D - τ \D - σ |∇I(z)| 2 ≤ D - τ \D - σ |∇f (z)| 2 + D - τ \D - σ |∇g(z)| 2 .
We are going to estimate the right-hand side. For any function h :

D - τ \ D - σ → R defined by h(re iθ ) = r -σ τ -σ h 1 (τ e iθ ) + τ -r τ -σ h 2 (σe iθ ), we have 1 2 D - τ \D - σ |∇h| 2 = 1 2 D - τ \D - σ |∂ r h| 2 rdrdθ E rad + 1 2 D - τ \D - σ |∂ θ h| 2 1 r drdθ Eang .
A straightforward computation yields

E rad (h) = 1 4 τ + σ τ -σ 3π/2 π/2 |h 1 (τ e iθ ) -h 2 (σe iθ )| 2 dθ, E ang (h) = 1 2(τ -σ) 2 × τ 2 -σ 2 2 3π/2 π/2 |∂ θ h 1 (τ e iθ ) -∂ θ h 2 (σe iθ )| 2 dθ +2(τ -σ) 3π/2 π/2 ∂ θ h 1 (τ e iθ ) -∂ θ h 2 (σe iθ ) τ ∂ θ h 2 (σe iθ ) -σ∂ θ h 1 (τ e iθ ) dθ + ln( τ σ ) 3π/2 π/2 τ h 2 (σe iθ ) -σ∂ θ h 1 (τ e iθ ) 2 dθ .
We are going to use these formulas with

f (re iθ ) = r -σ τ -σ |u(τ e iθ )| f 1 + τ -r τ -σ |ψ σ (σe iθ | f 2 g(re iθ ) = r -σ τ -σ arg(u(τ e iθ )) g 1 + τ -r τ -σ arg(ψ σ (σe iθ ) g 2
).

The resulting terms are estimated by using Taylor expansions for u and ψ σ . Without going through the details we have

|u(τ e iθ )| = 1 -τ (a 1 + b 1 ) cos θ + O(τ 2 ), |ψ σ (σe iθ )| = 1 -σ(a 1 + b 1 ) cos θ + O(σ 2 ), this leads to E rad (f ) = π 8 (a 1 + b 1 ) 2 (τ 2 -σ 2 ) + O(τ 3 ). (2.21)
For the angular energy we have, writing f 1 for f 1 (τ e iθ ) and f 2 for f 2 (σe iθ ),

∂ θ f 1 = τ (a 1 + b 1 ) sin θ + O(τ 2 ), ∂ θ f 2 = σ(a 1 + b 1 ) sin θ + O(σ 2 ), therefore (∂ θ f 1 -∂ θ f 2 ) = (τ -σ)(a 1 + b 1 ) cos θ + O(τ 2 ), (τ ∂ θ f 2 -σ∂ θ f 1 ) = O(τ 3 ).
It follows that

E ang (f ) = π 8 (a 1 + b 1 ) 2 (τ 2 -σ 2 ) + O(τ 3 ) + O τ 5 /(τ -σ) . (2.22)
By a similar computation for g, we obtain

g 1 (τ e iθ ) = τ ∇θ(1).((a 1 + b 1 ) cos θ, (a 1 -b 1 ) sin θ) + O(τ 2 ) = τ (a 1 -b 1 ) sin θ + O(τ 2 ), g 2 (σe iθ ) = σ(a 1 -b 1 ) sin θ + O(σ 2 ). Thus |g 1 -g 2 | 2 = (τ -σ) 2 (a 1 -b 1 ) 2 sin 2 θ + O(τ 2 ), hence E rad (g) = π 8 (a 1 -b 1 ) 2 (τ 2 -σ 2 ) + O(τ 3 ). (2.23)
Concerning the angular energy we have

∂ θ g 1 = (a 1 -b 1 )τ sin θ + O(τ 2 ) ∂ θ g 2 = (a 1 -b 1 )σ sin θ + O(σ 2 )
Inserting these expressions in the formula for E ang (g) we find

E ang (g) = π 8 (a 1 -b 1 ) 2 (τ 2 -σ 2 ) + O(τ 3 ) + O τ 5 /(τ -σ) .
(2.24)

The potential term is easy to control because |I(z

)| = |f (z)| ≤ 1 because |u| ≤ 1 and |ψ σ | ≤ 1 thus D - τ \D - σ (|I(z)| 2 -1) 2 = O(τ 2 -σ 2 ). (2.25) Now we choose τ = σ + σ 3/2 ,
E ε (v σ ) = E ε (u) + π - π 2 b 1 2 σ 2 + O(σ 5/2 ),
where the main contributions come from (2.19), (2.20). Thus for σ small enough we have

E ε (v σ ) < E(u) + π. (2.26)
It remains to check that v σ belongs to I d p+k,q . We already know that |v σ | = 1 on ∂A, thus it remains to compute the degrees of v σ restricted to each component of ∂A, and abdeg(v σ ).

Step 4 : Computation of the approximate bulk degree of v σ . Since v σ tends to u almost everywhere when σ goes to 0 and since v σ is uniformly bounded in H 1 , up to a subsequence we have v σ converges weakly in H 1 to u. This implies, using the properties of the approximate bulk degree, that for σ small enough abdeg(v σ ) ∈ (d -1 2 , d + 1 2 ).

Step 5 : Computation of the degree of v σ on ∂Ω. First note that the degree on ∂ω does not change because u = v σ on ∂ω. Now we set p := deg(v σ , ∂Ω), and letting J(u) denote the jacobian determinant of u we note that

p -q = 1 π A J(v σ ) = 1 π D - σ J(ψ σ ) A + 1 π D - τ \D - σ J(I) B + 1 π A\D - τ J(u). C We have, as σ → 0, C 1 π A J(u) = p -q.
Also, we defined ψ σ as a holomorphic function whose restriction to D - σ covers the unit disk once, except for region of area O(σ 2 ). It follows that A 1.

Finally, we can estimate

|B| ≤ 1 π Dτ \Dσ |∇I| 2 = O(σ 5/2 ).
Hence, for σ small enough, since the degree is an integer we have

deg(v σ , ∂Ω) = p + 1,
which concludes the proof of the lemma under the hypothesis that ∂Ω is flat near x 0 .

Step 6 : The case of a nonflat boundary. We still assume that x 0 = 0 ∈ ∂Ω, and that τ (0) = (0, 1).

Since we assumed ∂Ω is smooth, there exist σ 0 > 0 small enough and a diffeomorphism φ from D τ - 0 , with τ 0 = σ 0 + σ 3 0 , to a neighborhood of 0, such that

φ(0) = 0, dφ(0) = Id, φ(D - τ 0 ) = A ∩ φ(D τ 0 ).
Then, for any σ < σ 0 /2, if we let

τ = σ + σ 3 we have φ(D - τ ) = A ∩ φ(D τ ) =: V τ and there exists C > 0 such that if τ < τ 0 /2 then φ -Id C 1 (Dτ ) ≤ Cτ, φ -1 -Id C 1 (Dτ ) ≤ Cτ. Now let w : D - τ → C and w = w • φ -1 : V τ → C, we have : E(w, V τ ) = 1 2 Vτ |∇( w • φ -1 )| 2 dx = 1 2 Vτ |∇φ -1 | 2 |∇ w(φ -1 (x))| 2 dx = 1 2 D - τ |∇φ -1 (φ(y))| 2 |∇ w(y)| 2 | det J φ -1 (y)|dy besides |∇φ -1 φ(y)| 2 = 1 + O(τ ), ∀ y ∈ D - τ , | det J φ -1 (y)| = 1 + O(τ ), ∀ y ∈ D - τ , thus E(w, V τ ) = E( w, D - τ ) + O(τ )E( w, D - τ ) = E( w, D - τ ) + O(τ 3 ) because lim τ →0 E( w, D - τ ) = π 2 (|∂ x w(0)| 2 + |∂ y w(0)| 2 )τ 2 . We also have Vτ (|w(x)| 2 -1) 2 dx = D - τ (| w| 2 (y) -1) 2 |detJ φ(y) -1 |dy = D - τ (| w| 2 -1) 2 + O(τ ) D - τ (| w| 2 -1)dy = D - τ (| w| 2 -1) 2 + O(τ 3 )
and then

E ε (w, V τ ) = E ε ( w, D + τ ) + O(τ 3 ).
Hence if we do not assume anymore that ∂A is flat near 0, for τ small enough, one can find a neighborhood V τ and a diffeomorphism φ like before, and we keep the same notation. If u satisfies the "semi-stiff" boundary conditions, so does ũ because dφ(-1) = Id. We have

E ε (u) = E ε (u, V τ ) + E ε (u, A \ V τ ) = E ε (ũ, D - τ ) + E ε (u, A \ V τ ) + O(τ 3 ).
But the first part of the proof allow us to construct a function χτ :

D - τ → C such that E ε ( χτ , D - τ ) = E ε (ũ, D - τ ) + π - πb 2 1 σ 2 2 + O(σ 3/2 ).
( χτ contains the bubble ψ σ and the interpolation part

I) Let χ τ = χτ • φ -1 and v σ = χ τ if z ∈ V τ , u(z) if z ∈ A \ V τ .
(2.27)

Then E ε (v σ ) = E ε (u) + π - πb 2 1 σ 2 2 + O(σ 3/2 ),
hence for σ small enough we have

E ε (v σ ) < E ε (u) + π.
It remains to check that, for σ small enough, the approximate bulk degree of v σ is the same as u and that deg(v σ , ∂Ω) = deg((u, ∂Ω) + 1. To prove these two facts we can argue as in Steps 4 and 5. Indeed v σ tends to u almost everywhere so Step 4 shows that for σ small enough abdeg(v σ ) = abdeg(u). Besides to compute the degree on ∂Ω we need to compute the area of the image of v σ but this area is unchanged by reparametrisation by diffeomorphism thus Step 5 proves that deg(v σ , ∂Ω) = deg(u, ∂Ω) + 1.

Remarks : Many remarks are in order concerning this proof. 1) We did the proof only to increase the degree by one unit, but we can repeat the construction in different points of the boundary to increase the degree by several units. 2) We did the construction to increase the degree, the construction to decrease the degree is similar, just use w -(z) = a 1 z + b 1 z instead of w and adapt the proof.

3) The construction is local, thus the lemma can be applied to the case of several holes in the domain. 4) We have named the construction of test function in the proof of the lemma "insertion of bubbles" in reference to the paper of Kuwert [START_REF] Kuwert | Minimizing the energy of maps from a surface into a 2-sphere with prescribed degree and boundary values[END_REF]. In fact in our proof we insert "disc", i.e, we modify the original function in a little neighborhood in order to make it cover the disc one more time. In [START_REF] Kuwert | Minimizing the energy of maps from a surface into a 2-sphere with prescribed degree and boundary values[END_REF] the author did the same except that he wants to cover the sphere S 2 which is a common representation of a bubble.

Proof of Theorem 2.3

First we collect some useful lemmas. For the proofs we refer to [START_REF]Ginzburg-Landau minimizers with prescribed degrees. Capacity of the domain and emergence of vortices[END_REF] and [START_REF] Santos | Local minimizers of the Ginzburg-Landau functional with prescribed degrees[END_REF].

Lemma 2.5. (Price Lemma) Let {u (n) } ⊂ I d p,q be a sequence which converges to u weakly in

H 1 (A, R 2 ) with u ∈ I d r,s . Then 1 2 A |∇u| 2 dx ≤ lim inf n→+∞ A |∇u (n) | 2 -π(|p -r| + |q -s|)
or equivalently (by Sobolev embeddings)

E ε (u) ≤ lim inf n→+∞ E ε (u (n) ) -π(|p -r| + |q -s|).
(2.28)

Lemma 2.6. For u ∈ I d p,q , (r, s) ∈ Z × Z and δ > 0 there exists v ∈ I d r,s such that

E ε (v) ≤ E ε (u) + π(|p -r| + |q -s|) + δ.
In particular

m ε (r, s, d) ≤ m ε (p, q, d) + π(|p -r| + |q -s|)
Using these results together, one obtains :

Lemma 2.7. Assume that a minimizing sequence {u (n) } ⊂ I d p,q for m ε (p, q, d) converges weakly to some u ∈ I d r,s . Then

E ε (u) = lim inf n→+∞ E ε (u (n) ) -π(|p -r| + |q -s|),
and u minimizes the energy in I d r,s that is

E ε (u) = m ε (r, s, d).
Proof. Thanks to Lemma (2.5) we have

E ε (u) ≤ lim inf n→+∞ E ε (u (n) ) -π(|p -r| + |q -s|).
But thanks to lemma (2.6)

E ε (u) ≥ m ε (r, s, d) ≥ m ε (p, q, d) -π(|p -r| + |q -s|) thus E ε (u) = m ε (p, q, d) -π(|p -r| + |q -s|)
and we can reapply Lemma (2.6) again to obtain

E ε (u) ≤ m ε (r, s, d) E ε (u) = m ε (r, s, d).
Remark : Lemma (2.7) can be found in a slightly different form in [START_REF] Santos | Local minimizers of the Ginzburg-Landau functional with prescribed degrees[END_REF] Proposition 4. It says that a weak limit of a minimizing sequence in I d p,q may fall in I d r,s but in this case the weak limit is a minimizer in I d r,s and we know exactly what the jump in energy is, namely π(|p-r|+|q -s|). The fundamental lemma is important because it says that under certain assumptions we can jump back to the original class I d p,q by adding strictly less energy, and then contradict the fact that we had a minimizing sequence, to conclude that there was no jump after all.

The next lemma is useful to compare the G.L problem and the problem of harmonic maps with prescribed degrees on the boundary. In [START_REF] Bethuel | Progress in Nonlinear Differential Equations and their Applications[END_REF] the authors consider the following problem

I 0 (d, A) = inf{ 1 2 A |∇u| 2 dx; u ∈ H 1 (A, S 1 ), deg(u, ∂Ω) = d}.
(2.29)

They proved the following result Theorem 2.8.

([BBH94]

) There exists a unique (up to multiplication by constants with unit modulus) solution u of the minimization problem (2.29), and u is a regular harmonic map in A i.e -∆u = u|∇u| 2 in A, u ∈ H 1 (A, S 1 ), u ∧ ∂ ν u = 0 on ∂A.

It will be useful to note that u

∈ I d d,d because u ∈ H 1 (A, S 1 ) thus m ε (d, d, d) ≤ I 0 (d, A).
Lemma 2.9.

([DS09],[BR10]) Let (p, q, d) ∈ Z 3 .
Then there exists a function o ε (1) of ε and tending to 0 as ε → 0 such for any u ∈ I d p,q we have

E ε (u) ≥ I 0 (d, A) + π(|p -d| + |q -d|) + o ε (1).
(2.30)

The next lemma will be useful to satisfy the hypothesis of the fundamental lemma. It was proved in [START_REF] Berlyand | Solutions with vortices of a semi-stiff boundary value problem for the Ginzburg-Landau equation[END_REF] (Lemma 25) and in [START_REF] Santos | Local minimizers of the Ginzburg-Landau functional with prescribed degrees[END_REF] (Lemma 8).

Lemma 2.10. Let (p, q, d) ∈ Z 3 and Λ > 0. There exists ε 2 (p, q, d, Λ) > 0 such that for 0 < ε < ε 2 and u ∈ I d p,q a solution of (2.1)

with E ε (u) ≤ Λ, if d > 0, then there is x 0 ∈ ∂Ω (or y 0 ∈ ∂ω) such that u ∧ ∂ τ u(x 0 ) > 0 (respectively u ∧ ∂ τ u(y 0 ) > 0).
We may now prove Theorem 2.3 : Let d in N * 1) The case where p = q = d. We first prove that m

ε (d, d, d) is attained. For ε > 0 let (u ε n ) n be a minimizing sequence of E ε in I d d,d
, then for all n ∈ N, E ε (u n ) < Λ for some finite Λ thus there is a subsequence (still denoted u ε n ) which satisfies u ε n converges weakly to some u ε in H 1 . Thanks to Lemma (2.7) u ε minimizes E ε in some I d r,s with r, s possibly different from d. Applying Lemma (2.5) we have A). However using Lemma (2.30) we can say that there exists

E ε (u ε ) ≤ m ε (d, d, d) -π(|r -d| + |s -d|) ≤ I 0 (d, A) -π(|r -d| + |s -d|) (2.31) because m ε (d, d, d) ≤ I 0 (d,
ε 2 > 0 such that E ε (u ε ) ≥ I 0 (d, A) - π 2 + π(|r -d| + |s -d|), (2.32) hence |r -d| + |s -d| ≤ 1 4 and finally r = s = d. That is : u ε minimizes E ε in I d d,d .
2) The case where p ≤ d, q ≤ d Let (u ε n ) n be a minimizing sequence for E ε in I d p,q , the sequence is bounded in H 1 thus, up to a subsequence, we can find u ε ∈ I d r,s with (r, s) possibly different from (p, q) such that u ε n converges weakly to u ε in H 1 and the Lemma (2.7) says that u ε minimizes E ε in I d r,s . We have thanks to Lemma (2.6) that

m ε (p, q, d) = lim inf n→+∞ E ε (u ε n ) ≤ I 0 (d, A) + π(|p -d| + |q -d|) (2.33)
using the Price Lemma (2.5) we find that

lim inf n→+∞ E ε (u ε n ) ≥ E ε (u ε ) + π(|p -r| + |q -r|).
(2.34)

For ε small enough the Lemma (2.30) can be used to say that

E ε (u ε ) ≥ I 0 (d, A) - π 2 + π(|r -d| + |s -d|).
(2.35)

We now use the last three inequalities (2.33), (2.34), (2.35) to obtain

I 0 (d, A) + π(|p -d| + |q -d|) ≥ I 0 (d, A) - π 2 + π(|r -d| + |s -d|), |p -r| + |q -s| + |r -d| + |s -d| ≤ |p -d| + |q -d| + 1 2 .
This implies, because p, q, r, s, d are integers, that

p ≤ r ≤ d, q ≤ s ≤ d.
Now suppose that there exists ε 3 > 0 small enough such that u ε does not belong to I d p,q . Because d > 0, Lemma (2.10) allows us to use the fundamental Lemma. Hence we can find a v ∈ I d p,q such that

m ε (p, q, d) ≤ E ε (v) < E ε (u ε ) + π(|r -p| + |s -q|) ≤ lim inf n→+∞ E ε (u ε n ) = m ε (p, q, d) (2.36)
thus there is a contradiction and u ε is a minimizer of E ε in I p,q for ε small enough.

Conclusion

Finally, we remark that it would also be important to have a converse of the fundamental lemma, i.e, to know when, given u ∈ I d p,q a minimizer of the G.L energy we have for (r, s)

∈ Z 2 given, that E(v) > E(u) + π(|p -r| + |q -s|)
for all v ∈ I d r,s . Such result could be useful to prove the following conjecture of Berlyand and Rybalko which we state more precisely :

Conjecture 2.1. Let d > 0, i) If p ≥ d and q ≥ d with p = q then for ε small enough we have m ε (p, q, d) = m ε (d, d, d) + π[(p -d) + (q -d)] and m ε (p, q, d) is not attained. ii) If p > d > q then for ε small enough m ε (p, q, d) = m ε (d, q, d) + π(p -d) and is not attained. iii) If q > d > p then m ε (p, q, d) = m ε (p, d, d) + π(q -d)
and it is not attained.

The part i) of the conjecture was proved by Misiats in [START_REF] Misiats | The necessary conditions for the existence of local Ginzburg-Landau minimizers with prescribed degrees on the boundary[END_REF]. He proved that for all v ∈ I d p,q for p > d and q ≥ d, for ε small enough we have

E ε (v) > m ε (d, d, d) + π[(p -d) + (q -d)].
He also solved the case where d = 0. The two last items of the conjecture remain open. As an application of the fundamental lemma and of the result of Misiats [START_REF] Misiats | The necessary conditions for the existence of local Ginzburg-Landau minimizers with prescribed degrees on the boundary[END_REF] we obtain :

Proposition 2.11. Let u ε be a minimizer of the G.L energy in

I d d,d then u ε ∧ ∂ τ u ε (x) ≥ u ε • ∂ ν u ε (x) > 0 for all x in ∂A.
This means that the phase of u ε on each connected component of ∂A is strictly monotonic.

Proof. Let p > d and q ≥ d, the result of Misiats (part i) of the conjecture) states that there is no minimizer in I d p,q and

m ε (p, q, d) = m ε (d, d, d) + π[(p -d) + (q -d)].
(2.37)

Let u ε be a minimizer of the G.L energy in I d d,d . Using the fundamental lemma we obtain that

u ε ∧ ∂ τ u ε (x) ≥ u ε • ∂ ν u ε (x) ∀x ∈ ∂Ω. (2.38)
Indeed by contradiction if there exists x 0 in ∂Ω such that

u ∧ ∂ τ u(x 0 ) < u • ∂ ν u(x 0 ),
then the fundamental lemma (associated with Lemma (2.6)) gives the existence of a function v in

I d p,q such that E ε (v) < E ε (u ε ) + π[p -d) + (q -d)]
which contradicts (2.37). But we also have that

u ε • ∂ ν u ε (x) = ∂ ν |u ε | > 0 thanks to the Hopf's lemma. We can conclude that u ε ∧ ∂ τ u ε (x) > 0, ∀ x ∈ ∂Ω.
The same argument gives the result on the other boundary ∂ω.

Chapitre 3

Harmonic maps with prescribed degrees on the boundary and bifurcation of catenoids

Abstract : Let A ⊂ R 2 be a smooth bounded doubly connected domain. We consider the Dirichlet energy E(u) = A |∇u| 2 , where u : A → C, and look for critical points of this energy with prescribed modulus |u| = 1 on ∂A and with prescribed degrees on the two connected components of ∂A. This variational problem is a problem with lack of compactness. Hence we can not use the direct methods of calculus of variations. Our analysis relies on the so-called Hopf quadratic differential and on a strong link between this problem and the problem of finding all minimal surfaces bounded by two p-coverings of circles in parallel planes. We then construct new immersed minimal surfaces in R 3 with this property. These surfaces are obtained by bifurcation from a family of p-coverings of catenoids.

Introduction and statement of the results

Let A ⊂ R 2 C be a smooth bounded doubly connected domain of the form A = Ω \ ω, where Ω and ω are smooth bounded simply connected domains and ω ⊂ Ω ⊂ R 2 . We are interested here in solutions u : A → C of the following equations

     ∆u = 0 in A, |u| = 1 on ∂A, u ∧ ∂ ν u = 0 on ∂A, (3.1)
where ν stands for the outer unit normal to ∂A and a ∧ b stands for the determinant of two vectors a, b in R 2 . Solutions of (3.1) are critical points of the Dirichlet energy

E(u) = 1 2 A |∇u| 2 dx, in the space I = {u ∈ H 1 (A, C); |u| = 1 a.e. on ∂A}. (3.2)
By critical point we mean critical with respect to variations of the form

u t = u + tϕ for all ϕ in C ∞ c (A, C) and u t = ue itψ for all ψ in C ∞ (A, R).
It is classic that the first type of variations gives that ∆u = 0 in A and we can see that the second type of variations implies that u ∧ ∂ ν u = 0, a.e. on ∂A.

(3.3)

Let us explain in more details what the condition (3.3) means. We can see that a solution u of (3.1) is sufficiently regular (cf. Proposition 3.14). Then we can write, locally near the boundary, u = |u|e iϕ . The boundary conditions give us that Dirichlet boundary conditions are prescribed for the modulus, |u| = 1 on ∂A and homogeneous Neumann conditions are prescribed for the phase, ∂ϕ ∂ν = 0 on ∂A. In the literature these boundary conditions are called "semi-stiff".

Solutions of (3.1) are linked to 1 2 -harmonic maps defined by Da Lio and Rivière in [START_REF]Three-term commutator estimates and the regularity of 1 2 -harmonic maps into spheres[END_REF].

Definition 3.1. Let U be a smooth bounded open set in R 2 , g : ∂U → S 1 is a 1 2 -harmonic map if it is a critical point, for variations of the form g t = ge itψ with ψ ∈ H 1/2 (∂U, R), of the functional F (g) = g 2 Ḣ1/2 (∂U ) = 1 2 U |∇ũ(g)| 2 ,
where g Ḣ1/2 (∂U ) is the homogeneous H 1/2 Sobolev norm and ũ(g) denotes the harmonic extension of g in U .

One can then see that g : ∂A → C is a 1 2 -harmonic map if and only if its harmonic extension ũ(g) is a solution of (3.1). The notion of 1 2 -harmonic maps is linked to free boundary minimal surfaces in the euclidean ball B n for further references on this topic see [START_REF] Fraser | The first Steklov eigenvalue, conformal geometry, and minimal surfaces[END_REF], [START_REF]Minimal surfaces and eigenvalue problems, Geometric analysis, mathematical relativity, and nonlinear partial differential equations[END_REF], [START_REF] Schoen | Existence and geometric structure of metrics on surfaces which extremize eigenvalues[END_REF].

The physical motivation for studying (3.1) comes from the following fact. While they were examining the Ginzburg-Landau model (G.L in short), the authors of [START_REF] Bethuel | Progress in Nonlinear Differential Equations and their Applications[END_REF] suggested to consider a simplified Ginzburg-Landau functional

E ε (u) = 1 2 G |∇u| 2 + 1 4ε 2 (1 -|u| 2 ) 2 dx (3.4)
and to prescribe a Dirichlet boundary condition u = g on the boundary ∂G (in their work G was a simply connected domain). They observed that a boundary data such that |u| = 1 on ∂G and deg(g, ∂G) = d creates "[...]the same "quantized vortices" as a magnetic field in type-II superconductors or as angular rotation in superfluids." However they mentioned that "in physical situations the Dirichlet condition is not realistic" because in the G.L theory only |u| 2 has a physical meaning (it is the density of Cooper pairs of electrons see e.g. [START_REF] Sandier | Progress in Nonlinear Partial Differential Equations Birkhauser[END_REF]). That is why, some years later in the work [START_REF] Berlyand | Symmetry breaking in annular domains for a ginzburg-landau superconductivity model[END_REF] the authors tried to relax this condition by imposing only the condition |u| = 1 on the boundary and by prescribing winding numbers on the connected components of the boundary. The Dirichlet energy can be viewed as a limit, when ε goes to infinity of the Ginzburg-landau energy and it was studied by Berlyand, Rybalko, Mironescu, Sandier in the work [START_REF] Berlyand | Minimax critical points in Ginzburg-Landau problems with semi-stiff boundary conditions : existence and bubbling[END_REF] in the case of a simply connected domain.

Functions of the space I are classified by their degrees on the two boundaries of the domain. If g ∈ C 1 (Γ, S 1 ), where Γ is a smooth, simple, closed curve then by definition

deg(g, Γ) = Γ g ∧ ∂ τ gdτ. (3.5)
We can still define the degree for maps in H 1 2 (Γ, S 1 ). We refer to section 2 for more information and references on the degree. We set

I p,q = {u ∈ I; deg(u, ∂Ω) = p and deg(u, ∂ω) = q}, and m(p, q) = inf{E(v); v ∈ I p,q }. (3.6)
The main results of this paper describe the existence and non-existence of solutions of (3.1) in each I p,q , (p, q) ∈ Z 2 with a special emphasis on minimizing solutions (i.e. minimizers of the Dirichlet energy E in spaces I p,q ). Let us point out that the degree is not continuous under the weak convergence in H 1 2 as shown by the following example.

Example 3.1. Let M n : D → D defined by M n (z) = z-(1-1/n) 1-(1-1/n)z , then M n -1 weakly in H 1 , deg(M n (z), S 1 ) = 1 for all n ∈ N but deg(-1, S 1 ) = 0.
Hence we can not use the direct methods of calculus of variations. We are in presence of a problem of lack of compactness. The same phenomenon occurs for the study of the G.L equations with semi-stiff boundary conditions which were studied in [START_REF] Berlyand | Symmetry breaking in annular domains for a ginzburg-landau superconductivity model[END_REF], [START_REF]Ginzburg-Landau minimizers with prescribed degrees. Capacity of the domain and emergence of vortices[END_REF], [START_REF] Berlyand | Nonexistence of Ginzburg-Landau minimizers with prescribed degree on the boundary of a doubly connected domain[END_REF], [START_REF] Golovaty | On uniqueness of vector-valued minimizers of the Ginzburg-Landau functional in annular domains[END_REF], [BM], [START_REF] Berlyand | Solutions with vortices of a semi-stiff boundary value problem for the Ginzburg-Landau equation[END_REF], [START_REF] Santos | Local minimizers of the Ginzburg-Landau functional with prescribed degrees[END_REF], [START_REF] Berlyand | Minimax critical points in Ginzburg-Landau problems with semi-stiff boundary conditions : existence and bubbling[END_REF], [START_REF] Lamy | Existence of critical points with semi-stiff boundary conditions for singular perturbation problems in simply connected planar domains[END_REF] and [START_REF]Size of planar domains and existence of minimizers of the ginzburg-landau energy with semistiff boundary conditions[END_REF].

In the case of a simply connected domain Ω, in [START_REF] Berlyand | Minimax critical points in Ginzburg-Landau problems with semi-stiff boundary conditions : existence and bubbling[END_REF] the authors obtained all the critical points of the Dirichlet energy with prescribed degrees. Using the conformal invariance of the Dirichlet energy and the Riemann's theorem they assumed that Ω = D and ∂Ω = S 1 . Recall that a Blaschke product is a map of the form

B α,z 1 ,...,z d = α d j=1 z -z j 1 -z j z , z ∈ D, α ∈ S 1 , z j ∈ D, for j = 1, ...d.
Then using a lemma similar to Lemma 3.16 and a tool called the Hopf quadratic differential (cf. Section 3) they proved :

Theorem 3.2 ([BMRS14]). The critical points of E in I d = {u ∈ I; deg(u, S 1 ) = d} are precisely a) the d-Blaschke products if d > 0, b) the conjugates of (-d)-Blaschke products if d < 0, c) constant of modulus 1 if d = 0. d) All these solutions are minimizing in I d .
In the case of a doubly connected domain A, the conformal invariance of the energy E allows us to assume that A = {z ∈ C; < |z| < 1}, where 2π ln(1/ ) = cap(A) and cap(A) is the capacity of the domain (cf. Section 2 for a precise definition of the capacity). In the case p = q = 1, Berlyand and Mironescu in [BM] have showed that Proposition 3.3 ( [BM]). The only minimizers of E in I 1,1 are of the form

u 1 (z) = α 1 1 + (r + r )e iθ ,
where α is a constant of modulus one. Moreover m(1, 1) = 2π 1- 1+ < 2π.

In order to prove this result they used the fact that m(p, q) = inf{E(ũ(g)), g : ∂A → S 1 }, where g satisfies deg(g, ∂Ω) = p, deg(g, ∂ω) = q and ũ(g) is the harmonic extension of g in A. This is the 1 2 -harmonic maps point of view. Their approach consisted in writing E(ũ(g)) for any g boundary data with degree one on the two boundaries in function of the Fourier coefficient of g. Then they minimized that quantity under the constraint deg(g, ∂Ω) = deg(g, ∂ω) = 1 directly, using the expression of the degree in terms of the Fourier coefficients of the boundary data (see [START_REF]Degree theory : old and new, Topological nonlinear analysis[END_REF] for a formula of the degree involving Fourier coefficient). We can not use the same method when p > 1 or p = q.

Thanks to the properties of the degree (cf. Lemma 3.11), one can see that if u is a critical point of of E in I p,q then ū is a critical point of E in I -p,-q , and u( √ z ) is a critical point of E in I q,p (this latter fact is true because of the conformal invariance of the Dirichlet Energy). That is why we can restrict ourselves to three different cases : p ≥ 0 ≥ q, p = q > 0, p > q > 0. The main results of the paper are the followings : Theorem 3.4. Let p ≥ 0 ≥ q then m(p, q) = π(p + |q|) and 1) If p = q = 0 then the only minimizing solutions of (3.1) are constants in S 1 . 2) If p > 0 and q = 0 then there is no solution of (3.1) in I p,0 , in particular there is no minimizer. 3) If p > 0 > q then there exist an infinite number of solutions of (3.1), all solutions are holomorphic and energy minimizing, i.e., m(p,q) is attained.

Theorem 3.5. Let p > q > 0 then there is no solution of (3.1) in I p,q . In particular there is no minimizer of E in I p,q and we have m(p, q) = m(q, q) + π(p -q).

Theorem 3.6. Let p = q ≥ 2 then there exist critical values of called p and p such that p ≤ p and 1) If > p then m(p, p) is attained by a unique (modulo rotations) radially symmetric minimizer. Hence m(p, p)

= 2πp 1-p 1+ p . 2) If < p then the radially symmetric solution u p (z) = 1 1+ p (r p + p r p )e ipθ is not minimizing. Furthermore it holds that √ 2 -1 = 2 < 3 < ... < p < ... < 1. 3) In the case p = q = 2, we have that if 2 ≤ ≤ 2 then m(2, 2) is attained.
Remark 3.7. In point 3) of the previous theorem we do not know if m(2, 2) is attained by the radially symmetric solution u 2 , neither if minimizers are unique (up to rotations).

Since radial solutions are not always minimizing for p = q ≥ 2 one can wonder if other non-radial solutions of (3.1) exist in I p,p . We obtained non radially symmetric solutions of (3.1) in I p,p which could be minimizer of E in I p,p .

Theorem 3.8. There exist non radial solutions of (3.1) in I p,p if p ≥ 2. However we do not know if these solutions are indeed minimizing or even if a minimizer of E in I p,p exists when the radial solution u p is not minimizing.

The essential tool we used to obtain these results are the so-called Hopf quadratic differential (cf. Definition 3.21). Using the three conditions of equation (3.1) we can prove that the Hopf differential has the following form

Q(u) = H u (z)(dz) 2 = c z 2 (dz) 2 , in A (3.7)
with c a constant real number. We also use a deep relation between harmonic maps and minimal surfaces. Roughly speaking given a harmonic map u such that ∆u = 0, locally we can find a harmonic function h such that X = (u, h) is a conformal parametrization of a minimal surface. The function h is given in terms of the Hopf differential (see Lemma 3.27 for a precise statement). It turns out that finding solutions of (3.1) with H u (z) = c z 2 (dz) 2 where c < 0 is equivalent to finding minimal surfaces bounded by two concentric p-coverings of circles in parallel planes. We call p-covering of a circle a parametrization of a circle of degree p.

In 1956 in a beautiful paper [START_REF] Shiffman | On surfaces of stationary area bounded by two circles, or convex curves, in parallel planes[END_REF], M.Shiffman proved that if S is a minimal surface bounded by two concentric circles in parallel planes then S is (part of) a catenoid. However in this theorem we assume that the circles are described only once. In terms of solutions of (3.1) it is equivalent to ask that deg(u, S 1 ) = deg(u, C ) = 1. In order to find non-radially symmetric solutions of (3.1), with deg(u, S 1 ) = deg(u, C ) = p ≥ 2 one can look for immersed minimal surfaces bounded by two p-coverings of circles in parallel planes that are not rotationally symmetric. We obtained such surfaces by bifurcation of a p-covering of catenoids. Then thanks to the link between equation (3.1) and the minimal surface problem we deduce Theorem 3.8 from Theorem 3.9. There exist non rotationally symmetric, immersed minimal surfaces in R 3 bounded by two concentric p-coverings of circles in parallel planes if p ≥ 2.

There are some results in the literature concerning bifurcation of constant mean curvature (CMC) submanifolds. In particular in [START_REF] Koiso | Bifurcation and symmetry breaking of nodoids with fixed boundary[END_REF] the authors studied bifurcation of (compact portions of) CMC nodoids in R 3 whose boundary consists of two fixed coaxial circles of the same radius lying in parallel planes. This situation presents some similarity with our problem. However we used different techniques to obtain bifurcation of catenoids. The novelty in our approach is to consider bifurcation of p-coverings of catenoids. If p ≥ 2 we can apply the theorem of Crandall-Rabinowitz (see [START_REF] Crandall | Bifurcation from simple eigenvalues[END_REF]) to prove that bifurcation occurs and produces non rotationally symmetric minimal surfaces.

Let us mention that in a series of papers (see [START_REF] Iwaniec | Doubly connected minimal surfaces and extremal harmonic mappings[END_REF], [START_REF] Iwaniec | n-harmonic mappings between annuli : the art of integrating free Lagrangians[END_REF], [START_REF]Mappings of least Dirichlet energy and their Hopf differentials[END_REF] and the references therein) T. Iwaniec and J.Onninen studied harmonic mappings with the same form of Hopf differential i.e. H u (z) = c z 2 with c real. One of their purpose was to minimize the Dirichlet Energy on an annulus among some class of homeomorphisms. The common feature of their problem and ours is that we are both interested in critical points of the Dirichlet energy among maps with a given homotopy class at the boundary. The main difference is that they considered one-to-one mapping while we allow maps which are not one to one. They also made a link between such mappings and minimal surfaces (see [START_REF] Iwaniec | Doubly connected minimal surfaces and extremal harmonic mappings[END_REF]).

The paper is organized as follows : part 2 is devoted to known analytic results and to the analysis of lack of compactness for minimizing sequences of the energy, in Section 3 we present the properties of the Hopf quadratic differential of solutions of our problem and make a link with minimal surfaces theory. In Section 4 we study holomorphic solutions of the problem. Section 5 is devoted to a discussion on radial solutions of (3.1). In Section 6 we prove Theorem 3.9.

Preliminaries

Notations and definitions

Throughout the paper we use the following notations : * The vectors a = (a 1 , a 2 ) are identified with complex numbers a = a 1 + ia 2 . * a ∧ b stands for the vector product a * The orientation of simple curves (in particular ∂ω and ∂Ω) in R 2 is assumed to be counterclockwise. We denote by τ the unit tangent vector pointing counter-clockwise and ν the outer unit normal vector, hence (ν, τ ) is direct on ∂Ω and (ν, τ ) is indirect on ∂ω.

∧ b = a 1 b 2 -a 2 b 1 = i 2 (a b -āb) = det(
Let us recall the definition and some properties of the degree, we refer to [START_REF]Degree theory : old and new, Topological nonlinear analysis[END_REF], [START_REF]New questions related to the topological degree[END_REF] and the references therein for more on this subject. Let Γ be a smooth, simple, closed curve.

Definition 3.10. Let g : Γ → S 1 , the degree of g is the following quantity :

deg(g, Γ) = 1 2π Γ g ∧ ∂ τ gdτ.
The degree of a function is an integer and it measures the algebraic change of phase of g. Boutet de Monvel and Gabber have noticed that we can still define a degree for maps g ∈ H 1 2 (Γ, S 1 ) (see [START_REF] Boutet De Monvel-Berthier | A boundary value problem related to the Ginzburg-Landau model[END_REF] and [START_REF]Degree theory : old and new, Topological nonlinear analysis[END_REF]). This degree is defined by approximation, indeed C ∞ (Γ, S 1 ) is dense in H 1 2 (Γ, S 1 ) and we can see that the degree is continuous with respect to the strong H 1 2 convergence. Here are well-known properties of the degree (cf. [START_REF]Degree theory : old and new, Topological nonlinear analysis[END_REF] or [START_REF]New questions related to the topological degree[END_REF]) : Lemma 3.11. Let g, h ∈ H 1/2 (Γ, S 1 ). Then the following hold. 1) If g is continuous, then the degree of g in the sense H 1/2 maps is the same as the degree of g in the sense of continuous maps.

2) deg(gh) = deg(g) + deg(h). 3) deg(g) = -deg(g). 4) deg(g/h) = deg(g) -deg(h).
The degree can be used to characterize the connected components of I. These are exactly the sets I p,q = {u ∈ I; deg(u, ∂Ω) = p and deg(u, ∂ω) = q}, defined in introduction : Proposition 3.12 ([BdMBGP91]). The I p,q are the connected components of I. They are open and closed in I for the topology induced by the H 1 (A) norm.

I = p,q∈Z 2 I p,q .
Since the I p,q 's are open if a minimizer for the energy E in the class I p,q exists then it is a local minimizer of the energy in I and hence a solution of (3.1). However the degree is not continuous under the weak H 1 convergence. That is why finding solutions of (3.1) is a non trivial problem.

We now recall the notion of capacity of a doubly connected domain.

Lemma 3.17 ([BM], [START_REF]Ginzburg-Landau minimizers with prescribed degrees. Capacity of the domain and emergence of vortices[END_REF]). Let {u (n) } ⊂ I p,q be a sequence that converges to u weakly in

H 1 (A, R 2 ) with u ∈ I r,s . Then E(u) ≤ lim inf n→+∞ E(u (n) ) -π(|p -r| + |q -s|).
The combination of these lemmas allow us to make a first description of what can happen to a minimizing sequence in I p,q for E. The proof of the following lemma is inspired by [START_REF] Kuwert | Minimizing the energy of maps from a surface into a 2-sphere with prescribed degree and boundary values[END_REF].

Lemma 3.18. Assume that a minimizing sequence {u (n) } ⊂ I p,q for m(p, q) converges weakly to some u ∈ I r,s . Then

E(u) = lim inf n→+∞ E(u (n) ) -π(|p -r| + |q -s|),
and u minimizes energy in I r,s that is E(u) = m(r, s).

Proof. Using Lemma 3.17 we have

E(u) ≤ lim inf n→+∞ E(u (n) ) -π(|p -r| + |q -s|). But thanks to Lemma 3.16 E(u) ≥ m(r, s) ≥ m(p, q) -π(|p -r| + |q -s|) thus E(u) = m(p, q) -π(|p -r| + |q -s|)
and we can apply Lemma 3.16 again to obtain m(r, s)

≤ E(u) ≤ m(r, s) hence E(u) = m(r, s).
Lemma 3.18 means that if the infimum m(p, q) is not attained then the weak limit of a minimizing sequence falls into an another class where the infimum is attained. We can give a better description of the behavior of the minimizing sequences thanks to the following lemma :

Lemma 3.19. ([RS14]) Let u ∈ I p,q be a solution of the Laplace Equation with semi-stiff boundary conditions and k ∈ N * . i) Assume that there is

x 0 ∈ ∂Ω such that u ∧ ∂ τ u(x 0 ) > -u, ∂ ν u (x 0 ), then there exists v ∈ I p-k,q such that E(v) < E(u) + kπ.
ii) Assume that there is

x 0 ∈ ∂Ω such that u ∧ ∂ τ u(x 0 ) < u, ∂ ν u (x 0 ), then there exists v ∈ I d p+k,q such that E(v) < E(u) + kπ.
Note that if one writes locally near the boundary u = e iϕ then u ∧ ∂ τ u(x 0 ) = ∂ τ ϕ(x 0 ), and

u, ∂ ν u (x 0 ) = ∂ ν (x 0 ).
Lemma 3.20. Let (u n ) ⊂ I p,q be a minimizing sequence for E, (lim n→∞ E(u n ) = m(p, q)), up to extraction we have u n converges weakly to some u in H 1 and u ∈ I r,s for some (r, s) ∈ N 2 . 1) If p > 0 then r ≤ p. If q > 0 then s ≤ q.

2) If p < 0 then r ≥ p. If q < 0 then s ≥ q.

Proof. Let us suppose that p > 0. By contradiction if r > p writing locally u = e iϕ we have

∂ϕ ∂τ (x) ≤ - ∂ ∂ν (x) ∀x ∈ S 1 .
(3.10) Indeed assume that (3.10) is not true. Then thanks to Lemma 3.19 you can find v ∈ I p,q such that E(v) < E(u) + π(|p -r| + |q -s|).

However using Lemma 3.18 we have

E(u) = m(p, q) -π(|p -r| + |q -s|)
and then E(v) < m(p, q). This is a contradiction since v ∈ I p,q .

Furthermore the Hopf maximum principle tells us that ∂ ∂ν ≥ 0 on ∂A thus ∂ϕ ∂τ ≤ 0 on ∂Ω = S 1 and by integrating over ∂Ω we find that 2πr ≤ 0.

The proof is the same for the other cases.

Hopf differentials of solutions of (3.1)

In this section we present the main tool used to prove the results of this paper : the Hopf quadratic differential. We refer to [START_REF] Hélein | Constant mean curvature surfaces, harmonic maps and integrable systems[END_REF] for properties of the Hopf differential.

Definition 3.21. Let u : A → C, the Hopf quadratic differential of u is Q(u) = H u (z)(dz) 2 = 1 4 |∂ x u| 2 -|∂ y u| 2 -2i ∂ x u, ∂ y u (dz) 2 = (∂ z u)(∂ z u)(dz) 2 (3.11) Proposition 3.22. Let u : A → C. 1) If u is harmonic (∆u = 0), then H u is holomorphic.
2) H u = 0 is equivalent to u conformal (i.e., u holomorphic or u anti-holomorphic).

Proof. 1) Assume that ∆u = 0. Recall that ∆v = 4∂ z ∂ z v and let us compute

∂ z H u (z) = 4(∂ z ∂ z )u∂ z ū + 4∂ z u(∂ z ∂ z )ū = ∆u∂ z ū + ∂ z u∆ū = 0. Hence ∂ z H u (z) = 0 that is H u is holomorphic.
2) H u = 0 is equivalent to ∂ x u, ∂ y u = 0 and |∂ x u| = |∂ y u|. This means precisely that the differential of u (which is a 2 × 2 matrix) is a similitude, and that is the definition of a conformal application.

The conformal invariance of the Dirichlet energy allows us to work in an annulus which is a rotational symmetric domain, then we work in polar coordinates z = re iθ , with < r < 1 and 0 ≤ θ < 2π. A simple computation shows that we can write

4z 2 H u (z) = r 2 |∂ r u| 2 -|∂ θ u| 2 -2ir ∂ r u, ∂ θ u .
(3.12)

Lemma 3.23. Let u be a solution of (3.1), then there exists c ∈ R such that z 2 H u (z) = c in A.

Proof. We have that u is smooth up to the boundary and that z 2 H u (z) is holomoprhic in A thanks to Proposition 3.22. On ∂Ω = S 1 we have

∂ ν u = ∂ r u and ∂ τ u = ∂ θ u. Hence ∂ r u, ∂ θ u = 0, on S 1 .
Indeed on S 1 we have u ∧ ∂ r u = 0 according to (3.1) and u, ∂ θ u = 0 because |u| 2 = 1. The same method with

∂ ν u = -∂ r u and ∂ τ u = 1 ∂ θ u on C leads to ∂ r u, ∂ θ u = 0, on C .
Thus z → z 2 H u (z) is holomorphic in A and takes real values on ∂A. We can conclude that this function is real valued in all A. Indeed the imaginary part of z 2 H u (z) is harmonic in A and null on ∂A thus it is identically null on A. The function z 2 H u (z) being holomorphic and real-valued we deduce that it is constant in A.

Thanks to this lemma we can say that if u is a solution of (3.1) then

r 2 |∂ r u| 2 -|∂ θ u| 2 = c, and ∂ r u, ∂ θ u = 0 in A. (3.13)
There is a strong link between C-valued harmonic function, their Hopf differential and minimal surfaces. In order to explain this link let us recall few facts about minimal surfaces theory. We refer to [START_REF] Hauswirth | Associate and conjugate minimal immersions in M × R[END_REF] for more explanations on that link.

Definition 3.24. Let V ⊂ R 2 C be a domain. X : V → R 3 is a conformal (or isothermal) parametrization of a surface if X is an immersion (i.e. |∂ x X ∧ ∂ y X| is never zero in V ) and ∂ x X, ∂ y X = 0, and |∂ x X| 2 = |∂ y X| 2 = 0 in V. (3.14)
It is well-known (see [START_REF] Dierkes | Grundlehren der Mathematischen Wissenschaften[END_REF] p.77) that we can represent every regular surface of class C 2 by conformal parameters. Now we take advantage of the complex variable. If we set X = (u, h) = (u 1 , u 2 , h) then u : V → C and the conformality relations (3.14) reduce to one complex equation

∂ z u 1 2 + ∂ z u 2 2 + ∂ z h 2 = 0. (3.15)
A direct computation shows that

∂ z u 2 1 + ∂ z u 2 2 = (∂ z u 1 + i∂ z u 2 )(∂ z u 1 -i∂ z u 2 ) = ∂ z u∂ z u = H u . (3.16)
Hence the conformality relations mean that Proposition 3.27. Let u : V → C be a complex harmonic function and H u (z) = ∂ z u∂ z u its Hopf quadratic differential. Then locally outside the zeros of odd order of H u if we set

H u + ∂ z h 2 = 0 (3.17
h(z) = Re z z 0 -2i H u (z)dz (3.18)
where the line integral runs along any smooth curve γ ⊂ V beginning at a given point z 0 ∈ V and terminating at z, then X = (u, h), defined locally, satisfies ∆X = 0 and (3.14) holds. If X is an immersion then it is a conformal parametrization of a minimal surface.

Proof. Locally, near a point z 0 such that H u (z 0 ) = 0 or near every zero of even order of H u , in simply connected sub-domains, we can always define h by the relation (3.18). This function h is harmonic because it is the imaginary part of an holomorphic function. We show that X = (u, h) satisfies (3.14), equivalently we show

H u + ∂ z h 2 = 0.
This fact comes from the definition of h and from the fact that if U is a holomorphic function then

U (z) = ∂ z U = 2 ∂Re(U )
∂z thus with the definition (3.18) we find that

∂ z h = i H u
and then H u + ∂ z h 2 = 0. Now if X is an immersion thanks to Proposition 3.26, then X defines a minimal surface.

Remark 3.28. The surface is planar (h ≡ 0) if and only if u is conformal. The global lifting exists provided the imaginary part of the integral in (3.18) is single valued. Since we are interested in H u (z) = c z 2 it will be the case if c < 0, because in this case the imaginary part of this term will be ln |z|, but not in the case where c > 0 (the imaginary part of the term will be arg(z)).

The case c = 0 : holomorphic solutions

In this section we study, existence, non-existence and minimizing properties of solutions of (3.1) which satisfy H u (z) = 0. Such solutions are conformal by Proposition 3.22. We will restrict ourselves to holomorphic solutions, the antiholomorphic case being obtained by conjugation. We first recall a formula which is very useful in this section (see e.g. [START_REF]Degree theory : old and new, Topological nonlinear analysis[END_REF]). This formula is obtained via the divergence formula.

Proposition 3.29. Let u ∈ I p,q then A ∂ x u ∧ ∂ y u = π(p -q).
Lemma 3.30. Let u be holomorphic and u ∈ I p,q then p ≥ q and u possesses exactly p -q zeros in A counted with their multiplicities.

Proof. If u is holomorphic in A we have ∂ x u ∧ ∂ y u ≥ 0 because the differential of u is a direct similitude. Hence we obtain thanks to the previous lemma that A ∂ x u ∧ ∂ y u = π(p -q) ≥ 0. Now since u is holomorphic its zeros are isolated in A, and hence there is a finite number of zeros of u in A because A is compact. Let us denote by N the number of zeros of u in A (counted with multiplicity) and by m the number of distinct zeros. Let z 1 , ..., z m be the zeros of u in A. Let r > 0 small enough for D(z i , r) to contain only z i as a zero of u for all 1 ≤ i ≤ m. We set

B := A \ m i=1 D(z i , r
). The function u does not vanish in B. Thus we can set T := u |u| in B. We then have

B ∂ x T ∧ ∂ y T = deg(T, ∂Ω) -deg(T, ∂ω) - m i=1 deg(T, ∂D(z i , r)) (3.19)
thanks to the divergence formula (this is a formula analog to 3.29). However because T is S 1 valued we have B ∂ x T ∧ ∂ y T = 0 and hence we find that

deg(T, ∂Ω) -deg(T, ∂ω) = m i=1 deg(T, ∂D(z i , r)) deg(u, ∂Ω) -deg(u, ∂ω) = m i=1 deg(T, ∂D(z i , r)) p -q = m i=1 deg(T, ∂D(z i , r)). (3.20)
We now claim that deg(T, ∂D(z i , r)) is the multiplicity of z i as a zero of u. Indeed deg(T, ∂D(z i , r)) measures the algebraic change of phase of T (and hence of u because T = u |u| ) on ∂D(z i , r). The variation of the argument (or the phase) of u is also given by

1 2πi ∂D(z i ,r) u (z) u(z) dz.
But the argument principle for holomorphic functions (see e.g. [START_REF] Stein | Complex analysis[END_REF] th.4.1 p.90) gives us that

1 2πi ∂D(z i ,r) u (z)
u(z) dz is the number of zeros of u inside D(z i , r) counted with multiplicity thus it is equal to the multiplicity of z i as a zero of u. And thus (3.20) implies that

N = p -q.
Lemma 3.31. Let u be holomorphic, u ∈ I p,q then p > 0 and q < 0.

Proof. If u is holomorphic in A and u ∈ I p,q thanks to the previous lemma p ≥ q and u possesses exactly p -q zeros in A. Let < µ < 1 with µ close enough to 1 so that u has no zero in A µ := {z ∈ C; µ < |z| < 1}. We set G = ln |u|, the logarithm of the modulus of an holomorphic function which does not vanish is an harmonic function thus we have

     ∆G = 0 in A µ , G = 0 on S 1 , G = ln |u| on C µ .
(3.21) Now the maximum principle for holomorphic functions tells us that |u| < 1 inside the ring A, hence G = ln |u| < 0, on C µ . Applying the Hopf's lemma (see e.g. [START_REF]Partial differential equations[END_REF] p.330) we find that

∂G ∂ν (x) > 0, ∀x ∈ S 1 . But ∂G ∂ν (x) = ∂(ln |u|) ∂ν = 1 |u| ∂|u| ∂ν = ∂|u| ∂ν (x), ∀x ∈ S 1 (recall that |u| = 1 on S 1
). We also have

∂|u| ∂ν = 1 2|u| ∂|u| 2 ∂ν = 1 |u| u, ∂u ∂ν = 1 |u| u ∧ ∂u ∂τ
the last equality being true because (ν, τ ) is direct on S 1 then ∂u ∂ν and ∂u ∂τ are directly orthogonal since u holomorphic. Hence

2πp = S 1 u ∧ ∂ τ u = S 1 ∂G ∂ν > 0
and p > 0. We can apply a similar argument on the other boundary C but this time (ν, τ ) is indirect thus u, ∂u ∂ν = -u ∧ ∂u ∂τ and q < 0.

Lemma 3.32. Let p ≥ 0 ≥ q then m(p, q) = π(p + |q|) and u is a minimizer of E in I p,q if and only if u is holomorphic.

Proof. If p = q = 0 then constant solutions in S 1 are minimizers. During the proof we always assume that p > 0 ≥ q. We have the following point-wise equalities

1 2 |∇u| 2 = ∂ x u ∧ ∂ y u + 2|∂ z u| 2 , 1 2 |∇u| 2 = -∂ x u ∧ ∂ y u + 2|∂ z u| 2 .
Hence we have

E(u) = 1 2 A |∇u| 2 ≥ | A ∂ x u ∧ ∂ y u|
and an integration by parts (formula 3.29) gives E(u) ≥ π|p -q|. Then if p > 0 and q ≤ 0

E(u) ≥ π(p + |q|)
with equality if and only if u is holomorphic. But Lemma 3.16 tells us that m(p, q) ≤ π(p + |q|). Thus we can conclude that m(p, q) = π(p + |q|) and that if a minimizer exists it must be holomorphic. Conversely if u is holomorphic thanks to the previous computation we find that u minimizes the energy in I p,q . This lemma allows us to prove the following part of point 2) of Theorem 3.4. Proposition 3.33. There is no minimizer of E in I p,0 with p > 0.

Proof. Indeed there is no minimizer in I p,0 with p > 0 because if this minimizer exists it should be holomorphic. But Lemma 3.31 says that there is no holomorphic function in I p,0 .

A similar result was obtained with a different proof in [START_REF] Berlyand | Minimax critical points in Ginzburg-Landau problems with semi-stiff boundary conditions : existence and bubbling[END_REF] (see Lemma 9.9 p.1001) but only in the case I 1,0 . The authors proved the non existence of holomorphic functions in I 1,0 . Proposition 3.34. If p > 0 > q then there is an infinite number of critical points of E in I p,q and these are holomorphic solutions which can be written explicitly. Furthermore every solution is energy minimizing.

For the proof of this proposition we give an explicit formula for a solution and we check that it satisfies all the desired properties. We explain in the remark below the heuristic of the derivation of this formula. Let p > 0 > q be two integers. We choose p -q points x 1 , x 2 , ..., x p-q in A which satisfy :

- p-q i=1 ln |x i | ln = q. (3.22)
Note that it is possible to realize these conditions since 0 < p-q i=1

ln |x i | ln < p -q because < |x i | < 1 for all 1 ≤ i ≤ p -q and p -q > |q|. We then set u(z) := z q p-q i=1 |x i | 1 -z x i 1 -zx i +∞ k=1 (1 - 2k z x i )(1 - 2k x i z ) (1 - 2k zx i )(1 -2k zx i )
.

(3.23)

We want to show that 1) u is holomorphic in A, 2) u ∈ I, i.e., |u| = 1 on S 1 and on C , 3) deg(u, C ) = q and deg(u, S 1 ) = p.

To this end we set

f x i (z) := 1 -z x i 1 -zx i +∞ k=1 (1 - 2k z x i )(1 - 2k x i z ) (1 - 2k zx i )(1 -2k zx i ) (3.24)
we can thus rewrite

u(z) = z q p-q i=1 x i f x i (z). Lemma 3.35. For all 1 ≤ i ≤ p -q i) f x i is a meromorphic function on C.
ii) f x i has simple zeros at the points x i 2k for all k ∈ N and 2k x i for all k ∈ N * . iii) f x i has simple poles at the points 2k

x i for all k ∈ N and 1

2k x i for all k ∈ N * . iv) f x i (z) = f x i (z) for all z ∈ C. v) For all z ∈ C it holds f x i (z)f x i ( 1 z ) = 1 |x i | 2 and f x i (z)f x i ( 2 z ) = 1.
Proof. i) To prove the first point it suffices to prove that each infinite product converges and is a holomorphic function in C. For example for +∞ k=1 (1

- 2k z x i ) it holds that | 2k z x i | < 2k-1 |z| because |x i | > . Besides since < 1 the sum k≥1
2k-1 |z| is finite for all z ∈ C. This implies (see e.g. Proposition 3.2 p.141 in [START_REF] Stein | Complex analysis[END_REF]) that +∞ k=1 (1 -2k z

x i ) < +∞ and this product is holomorphic in C. The proof goes the same way for the three other infinite products in f x i .

Points ii),iii) and iv) are immediate from the definition of f x i . The point v) is obtained via the following computations :

f x i (z)f x i ( 1 z ) = (1 -z x i )(1 -1 x i z ) (1 -zx i )(1 -x i z ) +∞ k=1 (1 - 2k z x i )(1 - 2k x i z )(1 - 2k x i z )(1 -2k x i z) (1 - 2k x i z )(1 -2k x i z)(1 - 2k z x i )(1 - 2k x i z ) = (1 -z x i )(1 -1 x i z ) (1 -zx i )(1 -x i z ) = 1 |x i | 2 and f x i (z)f x i ( 2 z ) = (1 -z x i )(1 - 2 x i z ) (1 -zx i )(1 - 2 x i z ) +∞ k=1 (1 - 2k z x i )(1 - 2k x i z )(1 - 2k+2 x i z )(1 -2k-2 x i z) (1 - 2k x i z )(1 -2k x i z)(1 - 2k-2 z x i )(1 - 2k+2 x i z ) = 1 -z x i 1 -zx i × 1 -zx i 1 -z x i +∞ k=1 (1 - 2k z x i )(1 - 2k x i z )(1 - 2k x i z )(1 -2k x i z) (1 - 2k x i z )(1 -2k x i z)(1 - 2k z x i )(1 - 2k x i z ) = 1.
Proof of Proposition 3.34. To prove that u is holomorphic in A it suffices to apply the previous Lemma 3. 35 and to observe that u has no pole in A (the points 2k

x i for k ∈ N and 1

2k x i for k ∈ N * are not in A).
We now prove point 2) that is u ∈ I. For all z ∈ S 1 we have z = 1 z . Thus for all z ∈ S 1 :

|u(z)| 2 = u(z)u(z) = p-q i=1 |x i | 2 f x i (z)f x i (z).
we use points iv) and v) of Lemma 3.35 to obtain that for all z ∈ S 1

|u(z)| 2 = p-q i=1 |x i | 2 f x i (z)f x i (z) = p-q i=1 |x i | 2 f x i (z)f x i ( 1 z ) = p-q i=1 |x i | 2 1 |x i | 2 = 1. (3.25) Hence |u| = 1 on S 1 . Likewise for all z ∈ C |u(z)| 2 = u(z)u(z) = |z| 2q p-q i=1 |x i | 2 f x i (z)f x i (z) = 2q p-q i=1 |x i | 2 f x i (z)f x i (z) = 2q p-q i=1 |x i | 2 f x i (z)f x i ( 2 z ) = 2q p-q i=1 |x i | 2 = 1 (3.26)
the last equality being true because of the choice (3.22) that is q = -p-q i=1 ln |x i | ln . We then have u ∈ I.

To conclude the proof we only need to show that deg(u, C ) = q. Indeed since u has p -q zeros in A and since deg(u, S 1 ) -deg(u, C ) = number of zeros of u in A (cf. Proposition 3.30) we will deduce that deg(u, S 1 ) = p. For all N ∈ N * we set

u N (z) = z q p-q i=1 x i 1 -z x i 1 -zx i N k=1 (1 - 2k z x i )(1 - 2k x i z ) (1 - 2k zx i )(1 -2k zx i ) .
The functions u N are meromorphic on C. We count the number of zeros and poles of u N inside the disk D . The zeros of u N inside this disk are at the points 2k x i for 1 ≤ k ≤ N . We can see that 0 is a pole of order |q| (let us recall that q < 0) of u N and its other poles inside D are simple and are at 2k

x i for 1 ≤ k ≤ N . We then use the variation of argument principle (see for example [START_REF] Stein | Complex analysis[END_REF] p.90) to obtain that

C u N (z) u N (z) dz = -2iπ|q|. (3.27)
But we have that u N converges uniformly to u near C . Thus u N converges uniformly to u near C . We can pass to the limit and obtain

1 2iπ C u (z) u(z) dz = 1 2π C u ∧ ∂ τ u = deg(u, C ) = q.
This concludes the proof of the proposition because we showed that every holomorphic functions in I is minimizing in its class I p,q for p > 0 > q. (cf. Lemma 3.32).

We want to indicate how we derived the formula (3.23) for holomorphic solutions in I. If we assume that such a function u ∈ I p,q exists. Then u has p -q zeros counted with their multiplicities (cf. Lemma 3.30). We denote them by x 1 , x 2 , ..., x p-q . We then set G = ln |u|,

since u is holomorphic in A, the function G satisfies ∆G = 2π p-q i=1 δ x i in A, G = 0 on ∂A (3.28)
where δ x i denotes the Dirac measure at x i . Since the problem (3.28) is linear, to solve it, we only need to find a solution of

∆G = 2πδ x 0 in A, G = 0 on ∂A (3.29)
where x 0 is a point in A. By definition a solution of (3.29) is called a Green's function. Hence we must find a Green function in the ring A. This Green's function is unique for each x 0 ∈ A.

We can find the construction of such a Green's function in the book of Courant and Hilbert [CH89] p.388-389. If x 0 ∈ C then thanks to similar computations done in Proposition 3.34 it can be shown that the solution of (3.29) is

G x 0 = ln |F x 0 |, where F x 0 = |x 0 |z -ln |x 0 | ln 1 -z x 0 1 -zx 0 +∞ k=1 (1 - 2k z x 0 )(1 - 2k x 0 z ) (1 - 2k zx 0 )(1 -2k zx 0 ) . Note that F x 0 is always a multi-valued function because of the term z -ln |x 0 | ln , ( -1 < -ln |x 0 | ln < 0). A solution of (3.28) is given by G = ln |F x 1 F x 2 ...F x p-q |
and we can show that F x 1 F x 2 ...F x p-q is an holomorphic (single-valued) function if and only if p-q i=1 ln |x i | ln is an integer. This is exactly the condition (3.22). Thus this geometric condition on the positions of the x i is sufficient and necessary for an holomorphic function in I with prescribed zeros at the x i 's to exist.

3.5

The case c = 0 : properties of radial solutions 3.5.1 Non existence of solutions with c = 0 in I p,q if p = q Proposition 3.36. Let u be a solution of (3.1) in I p,q with c = 0, where c is the constant in the Hopf differential (3.11) then p = q.

Proof. Let u be a solution of (3.1) with c = 0, we have thanks to Lemma 3.23

r 2 |∂ r u| 2 -|∂ θ u| 2 = c.
* If c > 0 then we can see that ∂ r u does not vanish in A. We can then set T := ∂ru |∂ru| : A → S 1 . The map T takes all its values in S 1 then it has same degree on the two boundaries of A, i.e. on S 1 and C . This comes from the fact that ∂ x T ∧ ∂ y T = 0 because T is S 1 -valued and from the following formula which is true for all T in I (cf. Proposition 3.29) We can write λ 1 = ∂ru u because |u| = 1 on S 1 , thus λ 1 is continuous and does not vanish thus λ 1 has constant sign. Then T = u on S 1 or T = -u on S 1 and hence deg(T, S 1 ) = deg(u, S 1 ). On the other hand for the same reason we also have on C the existence of |∂ θ u| . We can conclude by the same argument except that this time there exists a function λ defined on ∂A such that ∂ θ u = iλu on ∂A. We have λ = ∂ θ u u on S 1 then λ is continuous and does not vanish on S 1 . This implies that R = iu or R = -iu on S 1 and we can deduce that deg(R, S 1 ) = deg(u, S 1 ). The same argument on C implies that deg(R, C ) = deg(u, C ) and then deg(u, S 1 ) = deg(u, C ).

A ∂ x T ∧ ∂ y T = deg(T, S 1 ) -deg(T, C ). ( 3
λ 2 such that ∂ r u( e iθ ) = λ 2 (θ)u( e iθ
Corollary 3.37. The following hold : 1) For all p ∈ N * there is no solution of (3.1) in I p,0 . 2) Let p > 0 > q two integers, there is no non-holomorphic solution of (3.1) in I p,q and hence no non-minimizing solution of (3.1) in I p,q .

Proof. Proposition 3.36 implies that there is no solution in I p,0 nor in I p,q for p > 0 > q integers with c = 0 (where c is the constant in the Hopf differential H u (z) = c z 2 ). Hence 1) if there exists a solution of (3.1) in I p,0 for p > 0 then its Hopf differential satisfies c = 0 and H u (z) = 0. Thus it must be holomorphic or antiholomorphic. However Lemma 3.31 (and its straightforward adaptation to the antiholomorphic case) implies that there is no holomorphic functions in I p,0 . 2) Again if there exists a solution of (3.1) in I p,q with p > 0 > q then thanks that what precedes it must be holomorphic or antiholomorphic. But we have seen in Lemma 3.32 that holomorphic solutions (or antiholomorphic solutions) minimize the Dirichlet energy in I p,q for p > 0 > q.

Corollary 3.37 and the results obtained in the previous section imply Theorem 3.4.

Corollary 3.38. There is no solution of (3.1) in I p,q with p > q > 0, in particular there is no minimizer of E in I p,q with p > q > 0.

Proof. This corollary is obtained by combining the preceding Proposition 3.36 and Lemma 3.31. Indeed we have seen that there is no solution of (3.1) with c = 0 in I p,q if p > q > 0. But if there is a solution with c = 0 then it must be conformal and because of Lemma 3.31 we must have p and q of opposite signs.

In order to complete the proof of Theorem 3.5 it remains to show Proposition 3.39. Let p > q > 0 we have m(p, q) = m(q, q) + π(p -q).

Proof. Thanks to the previous Corollary 3.38 we know that m(p, q) is not attained. Let (u n ) be a minimizing sequence for m(p, q) then, up to a subsequence, u n converges weakly in H 1 to some u ∈ I r,s . Thanks to Lemma 3.20 we have r ≤ p and s ≤ q. But applying Lemma 3.18 we also have that u minimizes the Dirichlet energy in its class, thus u is a solution of (3.1) and

E(u) = m(r, s) = m(p, q) -π[(p -r) + (q -s)].
We prove that u belongs to some I d,d for some d ≥ 1. Indeed u can not belong to some I p ,q for p = q and p , q > 0 because of Corollary 3.38. Furthermore because u is a solution of (3.1), we have several possibilities : either u ∈ I d,d for some d ∈ Z or u ∈ I r,s with r > 0 > s and u is holomorphic or u ∈ I r,s with s > 0 > r and u is antiholomorphic. We claim that the two last cases do not occur. Indeed if u ∈ I r,s with r > 0 > s and u is holomorphic for example. We then have

E(u) = π(r + |s|) = lim n→+∞ E(u n ) -π(p -r + q + |s|) = m(p, q) -π(p + q) + π(r -|s|). (3.31)
The last equalities are obtained by using Lemma 3.18 and Lemma 3.32. However because we always have m(1, 1) = 2π 1- 1+ < 2π (cf. Theorem 3.3) we obtain that m(p, q) < 2π+π(p-1+q-1) for p > q > 0 by Lemma 3.16. Then m(p, q) < π(p + q) which implies, using (3.31) that π(r + |s|) < π(r -|s|). This is a contradiction. The reasoning for u antiholomorphic is the same. Thus we obtain that u ∈ I d,d for some d ∈ Z and

E(u) = m(d, d) = lim n→+∞ E(u n ) -π(p -d + q -d) = m(p, q) -π(p -d + q -d).
(3.32)

Hence we deduce that d ≥ 1. Indeed because m(p, q) < π(p + q) if d ≤ 0 then E(u) < 0 which is a contradiction. Thus thanks to what precedes we have that

m(p, q) = m(d, d) + π(p -d + q -d). (3.33)
for some 1 ≤ d ≤ q. If d = q the proposition is proved, but it can occur that d < q, in this case we claim that m(q, q) = m(d, d) + π(q -d).

Indeed because of Lemma 3.16 we have m(q, q) ≤ m(d, d) + 2π(q -d). But if m(q, q) < m(d, d) + 2π(q -d) applying Lemma 3.16 again we obtain m(p, q) ≤ m(q, q) + π(p -q) < m(d, d)

+ 2π(q - d) + π(p -q) < m(d, d) + π(p -d + q -d)
which is a contradiction with (3.33). This proves that m(p, q) = m(q, q) + π(p -q).

Corollary 3.38 and Proposition 3.39 proves Theorem 3.5.

The case c < 0 : geometry of the catenoid

From now on, because of the previous Corollary 3.38 we are looking for critical points of E in I p,p for p > 0 (the case p < 0 being obtained by complex conjugation). In this subsection we are interested in solutions of (3.1) with c < 0 (where c is the constant in the Hopf differential). This case is of particular importance because if u is a minimizer of E in I p,p then its Hopf differential satisfies c < 0 as shown in the following proposition.

Proposition 3.40. Let u be a minimizer of E in I p,p then ∂u ∂r = 0 on C √ and c < 0.

Proof. If ∂u ∂r = 0 on C √ we have on C √ : c = 0 -|∂ θ u| 2 where c is the constant in the Hopf differential (3.23), hence c ≤ 0 in A. However if c = 0 then u is holomorphic or antiholomorphic and then u ∈ I r,s with r > 0 > s or r < 0 < s thanks to Lemma 3.31 and this is a contradiction with u ∈ I p,p . The fact that ∂u ∂r = 0 on C √ is a consequence of the following Lemma 3.41. We set Lemma 3.41. Assume that there exists u ∈ I p,p such that

F p = {u ∈ H 1 (A √ ,
E (u) = min{E (v); v ∈ I p,p } then E √ (u) = min{E √ (v); v ∈ F p }.
Conversely if there exists u 0 such that E √ (u 0 ) = min{E √ (v); v ∈ F p } then there exists u such that u is a minimizer of E in I p,p . Remark 3.42. Thanks to this lemma, in order to solve the problem (3.1) we can consider the simpler problem of minimizing E in F p . This problem is simpler because we prescribe constraints only on one boundary. The condition obtained on the boundary C √ is a Neumann homogeneous condition : ∂u ∂ν = 0 on C √ . This lemma comes from an idea of Shafrir and can be found in an other setting in the article of Sandier [START_REF] Sandier | The symmetry of minimizing harmonic maps from a two dimensional domain to the sphere[END_REF].

Let u ∈ I p,p we set :

u 1 (z) =    u(z) if √ < |z| < 1, u( z ) if < |z| < √ . , u 2 (z) =    u( z ) if √ < |z| < 1, u(z) if < |z| < √ .
We can see that

u 1 , u 2 ∈ H 1 (A, C) (A = A = {z ∈ C; < |z| < 1}).
Then because the map z → z is conformal we obtain

E (u 1 ) = 2 A √ |∇u| 2 , and E (u 2 ) = 2 A \A √ |∇u| 2 .
Hence 2E (u) = E (u 1 ) + E (u 2 ). Observe that if u ∈ I p,p then u 1 ∈ I p,p and u 2 ∈ I p,p . Let us assume that u is a minimizer of E in I p,p then E (u) = E (u 1 ) = E (u 2 ), indeed if it were not the case either u 1 or u 2 would satisfy E (u i ) < E (u). Now let v ∈ F p we have

E √ (v) ≥ E √ (u) Indeed define v 1 (z) =    v(z) if √ < |z| < 1, v( z ) if < |z| < √ .
We have v 1 ∈ I p,p and because u 1 is a minimizer of E it holds that

2E √ (v) = E (v 1 ) ≥ E (u 1 ) ≥ 2E √ (u)
then we can deduce that the restriction of u to

A √ minimizes E √ in F p .
Conversely if v ∈ F p minimizes E √ it is not difficult to see, thanks to the previous computation that

v 1 (z) =    v(z) if √ < |z| < 1, v( z ) if < |z| < √ .
minimizes E in I p,p . This proves the lemma and then Proposition 3.40.

Proposition 3.43. Let u be a solution of (3.1) with H u (z) = -|c| z 2 . If we set h(x, y) = |c| ln(x 2 + y 2 ), for (x, y) ∈ A then

X : A → R 3 , (x, y) → (u(x, y), h(x, y)) (3.34)
is a conformal immersion which parametrizes a minimal surface.

Proof. This is an application of Proposition 3.27. We only need to check that X is an immersion.

We have

∂ r X ∧ ∂ θ X = (∂ θ u 2 |c|, ∂ θ u 1 |c|, ∂ r u ∧ ∂ θ u). Note that ∂ θ u is never zero in A, since r 2 |∂ r u| 2 -|∂ θ u| 2 = c < 0 in A. Thus ∂ r X ∧ ∂ θ X is never zero in A.
Proposition 3.44. Let p > 0 be an integer. Let u be a minimizer of the Dirichlet energy E in I p,p . Then the minimal surface obtained by the process of the previous Proposition 3.43 is symmetric with respect to the plane {z = |c| ln( )} in R 3 .

Proof. This is a consequence of Proposition 3.40. Indeed if u is a minimizer of E in I p,p then

∂ r u = 0 on C √
and H u (z) = c z 2 with c < 0. This imply that the minimal surface given by X = (u, h) : A → R 3 , where h = 2 |c| ln(r) (with r = x 2 + y 2 ) intersects the plane {z = |c| ln( )} perpendicularly. Indeed the intersection between the minimal surface and this plane is the plane curve X( √ e iθ ), θ ∈ [0, 2π[ and the normal of the surface on this curve is given by

N = ∂ x X ∧ ∂ y X |∂ x X ∧ ∂ y X| = ∂ r X ∧ ∂ θ X |∂ r X ∧ ∂ θ X| taken at ( √ , θ) for θ ∈ [0, 2π[. But because of the fact that ∂ r u = 0 on C √ we obtain that
N is horizontal on the curve X( √ e iθ ), θ ∈ [0, 2π[ and hence the minimal surface parametrized by X intersects the plane {z = |c| ln( )} perpendicularly. Now we use the following classical symmetry result due to H.A. Schwarz (see [START_REF] Dierkes | Grundlehren der Mathematischen Wissenschaften[END_REF] p.128) if a minimal surface intersects some plane P perpendicularly, then P is a plane of symmetry of the surface.

In the case H u (z) = c z 2 (c < 0) the minimal surface obtained by Proposition 3.27 is a minimal surface bounded by two p-coverings of circles in parallel planes. This is due to the fact that |u| = 1 on ∂A and h = -ln r. In fact such a minimal surface gives rise to a solution of (3.1) with c < 0. Thus the problem of finding solution of (3.1) with c < 0 and the problem of finding minimal surfaces bounded by two p-coverings of circles in parallel planes are completely equivalent.

Proposition 3.45. Let X = (u, h) : A → R 2 be a conformal parametrization of a doubly connected minimal surface bounded by two p-coverings of circles in parallel planes then h = a ln r + b, for some (a, b) ∈ R 2 . And there exist two constants C 1 , C 2 such that u satisfies

           ∆u = 0 in A, |u| = C 1 a.e. on S 1 , |u| = C 2 a.e. on C , u ∧ ∂ ν u = 0
a.e. on ∂A.

(3.35)

Proof. The height function h is constant on each of the circles bounding the annular ring, h being harmonic we must have h = a ln r + b. Indeed h solve an equation of the following form

     ∆h = 0 in A, h = c 1 on S 1 , h = c 2 on C . (3.36)
This is a Dirichlet problem for the Laplacian and the solution is unique, of the form h = a ln r+b. Now we apply Proposition 3.25 to find that

H u (z) = -(∂ z h) 2 But ∂ z h = 1 2 (∂ r h -i 1 r ∂ θ h)e -iθ = b/2 r e -iθ , thus H u (z) = c z 2 with c = -b 2 /4 < 0. Now recall that z 2 H u (z) = r 2 |∂ r u| 2 -|∂ θ u| 2 -2ir ∂ r u, ∂ θ u . Then we can deduce that ∂ r u, ∂ θ u = 0 in A.
(3.37)

Moreover we have that |u| = C 1 on S 1 and |u| = C 2 on C because the surface is bounded by two circles in parallel planes. This implies that u, ∂ θ u = 0 on ∂A. We can then conclude from this and from (3.37) that u ∧ ∂ ν u = 0, on ∂A.

Thanks to a theorem of M.Shiffman and the equivalence between the problem of finding solutions of (3.1) with c < 0 and finding minimal surfaces bounded by two p-covering of circles in parallel planes we can prove that every solution of (3.1) in I 1,1, with c < 0 is radial.

Theorem 3.46 ([Shi56]

). Let S be a compact minimal surface in R 3 bounded by two plane curves Γ 1 , Γ 2 lying in parallel planes. If Γ 1 , Γ 2 are circles then the intersection of S by a plane parallel to the planes of Γ 1 , Γ 2 is again a circle. If the two circles Γ 1 and Γ 2 have a common axis of symmetry then the minimal surface S is a portion of catenoid.

Theorem 3.47. Let u be any (non necessary minimizing) solution of (3.1) in I 1,1 with c < 0 then there exists α ∈ S 1 such that

u(z) = α 1 + (r + r )e iθ . (3.38)
Let us recall that Berlyand and Mironescu proved in [BM] that the function u 1 (z) = 1 1+ (r+ r )e iθ is the only (modulo rotations) minimizer of E in I 1,1 .

If p > 1, the theorem of Shiffman and its proofs do not apply thus we can not conclude that every solution of (3.1) with c < 0 is radial. However we can use the same method as Berlyand and D.Golovaty in [START_REF] Golovaty | On uniqueness of vector-valued minimizers of the Ginzburg-Landau functional in annular domains[END_REF] to prove that if the annulus is thin enough then we have existence and uniqueness (modulo rotations) of a minimizer of E in I p,p .

Theorem 3.48. Let p ≥ 2. There exists p < 1 such that if > p then E has a unique (up to an arbitrary rotation), radially symmetric minimizer

u p (z) = α 1 + p (r p + p r p )e ipθ .
(3.39)

Proof. In order to prove this theorem we can follow step by step the proof of an analogous result for the Ginzburg-Landau energy of Berlyand and D.Golovaty in [START_REF] Golovaty | On uniqueness of vector-valued minimizers of the Ginzburg-Landau functional in annular domains[END_REF] taking ε = +∞.

In fact, contrarily to the case p = 1, if p > 1 and if the annulus is too thick then the radial solution is not minimizing anymore. Such similar phenomenon occurs in the study of the Ginzburg-Landau energy, but for the G.L energy even in degree one case the existence of minimizer depends on the capacity of the domain (cf. [START_REF] Berlyand | Nonexistence of Ginzburg-Landau minimizers with prescribed degree on the boundary of a doubly connected domain[END_REF], [BM], [START_REF]Ginzburg-Landau minimizers with prescribed degrees. Capacity of the domain and emergence of vortices[END_REF]).

Theorem 3.49. Let p ≥ 2. There exists p ≤ p such that if

< p then u p (z) = α 1+ p (r p + p r p )e ipθ is not a minimizer of E in I p,p .
Proof. Using Lemma 3.16 we have that for p ≥ 2 m(p, p) ≤ m(1, 1) + 2π(p -1).

(3.40)

We know that u 1 (z) = 1 1+ (r + r )e iθ is a minimizer of E in I 1,1 . Thus m(1, 1) = E(u 1 ) = 2π 1- 1+ .

Let u p (z) = 1 1+ p (r p + p r p )e ipθ then a direct computation leads to E(u p ) = 2πp

(3.41) ⇔ 2πp 1 -p 1 + p > 2π 1 - 1 + + 2π(p -1) ⇔ p(1 - 2 p 1 + p ) > (1 - 2 1 + ) + p -1 ⇔ 2 1 + > 2p p 1 + p ⇔ 1 + p > p p-1 (1 + ) ⇔ (p -1) p + p p-1 -1 < 0.
(3.42)

We set g p ( ) = (p -1) p + p p-1 -1. One can study the function g p for 0 ≤ ≤ 1 and show that

     g p (0) = -1 < 0, g p (1) = 2(p -1) > 0, g p ( ) = p(p -1)( p-1 + p-2 ) ≥ 0, ∀ ∈ [0, 1].
(3.43) Thus (3.43) proves that there exists p such that if 0 < < p then the solution u p is not minimizing. The fact that p ≤ p follows from the definition of p in Theorem 3.48. To prove that the sequence ( p ) p is increasing we show that g p+1 ( p ) < 0 for all p ∈ N, p ≥ 2. This will prove the monotonicity of the sequence p because g p is an increasing function on [0, 1] and by definition g p ( p ) = 0.

g p ( p ) = 0 ⇔ (p -1) p p + p p-1 p = 1.
Hence we deduce that

g p+1 ( p ) = p p+1 p + (p + 1) p p -1 = p p+1 p + (p + 1) p p -(p -1) p p -p p-1 p = p-1 p (p 2 p + 2 p -p)
Then we can study the sign of p 2 p + 2 p -p. We need to show that p < -1+

√

1+p 2 p to deduce that p 2 p + 2 p -p < 0. To this end we can prove that g p (

-1+ √ 1+p 2 p
) > 0. Simple but tedious computations lead to

g p ( -1 + 1 + p 2 p ) = ( 1 + p 2 + p)p p -( 1 + p 2 + 1) p ( 1 + p 2 + 1) p .
One can study the sign of the numerator and obtain that

( 1 + p 2 + p)p p -( 1 + p 2 + 1) p > ( 1 + p 2 + 1)(p p -( 1 + p 2 + 1) p-1 ) > ( 1 + p 2 + 1)(p p -(p + 2) p-1 ).
One shows that p p -(p + 2) p-1 ≥ 0 for all p ≥ 2 by showing that p ln p -(p -1) ln(p + 2) ≥ 0 for p ≥ 2 (we study the function l(x) = x ln x-(x-1) ln( x+2)). This proves the strict monotonicity of the sequence ( p ) p . The fact that 2 = √ 2 -1 is obtained by solving g 2 ( 2 ) = 0 which is equivalent to 2 2 + 2 2 -1 = 0.

Proposition 3.50. Let p = 2. Let 2 as in Theorem 3.49. Let us assume that ≥ 2 then m(2, 2) is attained.

Proof. Thanks to Theorem 3.3 we know that u 1 (z) = 1 1+ (r + r )e iθ is a minimizer of E in I 1,1 for all 0 < < 1 thus m(1, 1) = E(u 1 ) = 2π Indeed let (u (n) ) be a minimizing sequence for E in I 2,2 thanks to Lemma 3.18 we can deduce that u (n) converges weakly in H 1 to some u ∈ I r,s and such that E(u) = m(r, s). Furthermore Lemma 3.20 allows us to obtain that r < 2 and s < 2. We claim that r = s = 1. The argument is similar to the one of Proposition 3.39. Since u is a solution of (3.1), we have u ∈ I r,r for some r ∈ Z, u ∈ I r,s with 2 > r > 0 > s and u is holomorphic or u ∈ I r,s with 2 > s > 0 > r and u is antiholomorphic. We claim that the two last cases do not occur. Indeed if u ∈ I r,s with r > 0 > s and u is holomorphic for example. We then have

E(u) = π(r + |s|) = lim n→+∞ E(u n ) -π(2 -r + 2 + |s|) = m(2, 2) -4π + π(r -|s|)(3.46)
The last equalities are obtained by using Lemma 3.18 and 3.32. However because we always have m(1, 1) < 2π we obtain that m(2, 2) < 2π + 2π = 4π by Lemma 3.16. This implies, using (3.46) that π(r + |s|) < π(r -|s|) which is a contradiction. The reasoning for u antiholomorphic is the same.

Thus we obtain that u ∈ I r,r for some r ∈ Z, r < 2 and

E(u) = m(r, r) = lim n→+∞ E(u n ) -π(2 -r + 2 -r) = m(2, 2) -4π + 2rπ. (3.47)
Hence we deduce that d ≥ 1. Indeed since m(2, 2) < 4π, if r ≤ 0 then E(u) < 0 which is a contradiction. Thus we obtain that r = s = 1 and (3.45) holds. But this means that

E(v) > m(1, 1) + 2π, for all v ∈ I 2,2
and this is a contradiction with (3.44). Thus the proposition is proved.

Note that in the previous proposition we do not know if u 2 is a minimizer. We also do not know if a minimizer is unique (up to rotation). We obtain this information only requiring the stronger condition > 2 with 2 as in Theorem 3.48. Note also that we are not able to obtain similar result for the case p = q ≥ 3. Indeed the important ingredient in the proof of proposition 3.50 is that we know the exact value of m(1, 1) for all 0 < < 1. We do not have this information for m(2, 2) and hence we can not argue by induction. Theorems 3.48, 3.49 and Proposition 3.50 prove Theorem 3.6.

The case c > 0 : geometry of the helicoid

This case is more complicated than the previous one. This is because this time the imaginary part in the integral of (3.18) is not single-valued because it is precisely arg(z). Thus it is more difficult to make a clear link between the problem (3.1) with c > 0 and a minimal surface problem. However we can still lift a solution to the problem (3.1) in a minimal surface if we define the surface in an appropriated domain.

Proposition 3.51. Let u be a solution of (3.1) with H u (z) = c z 2 and c > 0. Then

X : [ , 1] × [0, 2π[→ R 3 , (r, θ) → (ũ(r, θ), -2 √ cθ) (3.48)
(where ũ(r, θ) = u(r cos θ, r sin θ)) is a conformal immersion which parametrizes a minimal surface.

We also have radial solutions of problem (3.1) with c > 0. A simple computation leads to Proposition 3.52. Let p ≥ 1 then for all α ∈ S 1 the functions

ũp (z) = α 1 -p (r p - p r p )e ipθ (3.49)
are (non-minimizing) solutions of (3.1). Besides we have

E(ũ p ) = 2πp 1+ p 1-p > E(u p ) = 2πp 1-p 1+ p .
The solution ũp when lifted to a minimal surface with Proposition 3.51 gives rise to an helicoid.

Non rotationally symmetric minimal surfaces bounded by two circles in parallel planes

We have seen that finding solutions to (3.1) with c < 0 (where c is the constant in the Hopf differential) is equivalent to finding minimal surfaces bounded by two (p-covering of) circles in parallel planes. We prove in this section the existence of non rotationally symmetric minimal surfaces with that property. They are obtained by bifurcation of a p-covering (p ≥ 2) of a catenoid.

Theorem 3.53. There exist non-rotationally symmetric immersed minimal surfaces bounded by two concentric p-coverings of circles in parallel planes. These surfaces are symmetric with respect to reflections around the planes P := {(x, y, z) ∈ R 3 ; z = 0} and P := {(x, y, z) ∈ R 3 ; y = 0}.

Proof. Let us parametrize a catenoid covered p times (p ≥ 2) by X(r, θ) = (cosh(pr) cos(pθ), cosh(pr) sin(pθ), pr),

for r ∈] -∞, +∞[, θ ∈ [-π, π].
This immersion provides us with a family of compact portions of catenoids. The parameter of that family is t such that r ∈ [-t, t] (t is the height of the compact portion of the catenoid). We want to parametrize this family on a fix domain thus we set

X t (r, θ) : [-1, 1] × [-π, π] → R 3 (r, θ)
→ (cosh(tpr) cos(pθ), cosh(tpr) sin(pθ), tpr).

(3.50)

Note that this parametrization is not conformal anymore if t = 1. Let N t = ∂rXt∧∂ θ Xt |∂rXt∧∂ θ Xt| be the unit normal vector to X t ,

N t = 1 cosh 2 (tpr)
(cosh(tpr) cos(pθ), cosh(tpr) sin(pθ), sinh(tpr) cosh(tpr)).

Let us fix 0 < α < 1. We denote by H the mean curvature operator from C 2,α to C 0,α . We look for minimal surfaces of the form

Y t = X t + uN t , with u ∈ C 2,α 0 ([-1, 1] × [-π, π], R), where C 2,α 0 ([-1, 1] × [-π, π], R)
is the set of C 2,α functions in the annulus which are null on the boundary ∂A . However for t fixed Y t is an immersion only if u is small enough (in the C 2,α norm), that is why we need to control the norm of u independently of t.

Lemma 3.54. If u C 2,α < 1 then Y t = X t + uN t defines an immersion for all t ∈ R * .

Proof. Let us drop the subscript t for this proof. Differentiating Y with respect to r and θ we find :

∂ r Y = ∂ r X + u∂ r N + ∂ r uN, ∂ θ Y = ∂ θ X + u∂ θ N + ∂ θ uN
We denote the first and second fundamental forms by

I = Edr 2 + 2F drdθ + Gdθ 2 , II = edr 2 + 2f drdθ + gdθ 2
It holds that ∂ r X t = pt(sinh(tpr) cos(pθ), sinh(tpr) sin(pθ), 1)

∂ θ X t = p(-cosh(tpr) sin(pθ), cosh(tpr) cos(pθ), 0).

Thus we find

E = ∂ r X t , ∂ r X t = t 2 p 2 (sinh 2 (tpr) + 1) = t 2 p 2 cosh 2 (tpr), F = ∂ r X t , ∂ θ X t = 0, G = ∂ θ X t , ∂ θ X t = p 2 cosh 2 (tpr).
In order to obtain the second fundamental form we also compute the second derivative and the normal vector to X t .

N t = 1 cosh 2 (tpr) (cosh(tpr) cos(pθ), cosh(tpr) sin(pθ), sinh(tpr) cosh(tpr))

∂ 2 r X t = t 2 p 2 (cosh(tpr) cos(pθ), cosh(tpr) sin(pθ), 0), ∂ 2 θ X t = -p 2 (cosh(tpr) cos(pθ), cosh(tpr) sin(pθ), 0) ∂ 2 rθ X t = tp 2 (-sinh(tpr) sin(pθ), sinh(tpr) cos(pθ), 0).

Thus e = ∂ 2 r X t , N t = t 2 p 2 , f = ∂ 2 rθ X t , N t = 0 g = ∂ 2 θ X t , N t = -p 2 .
We can write, using the fact that f = 0,

∂ r N = -e E ∂ r X, ∂ θ N = -g G ∂ θ X ∂ r Y ∧ ∂ θ Y = (1 -u e E )(1 -u g G )∂ r X ∧ ∂ θ X + (1 -u e E )∂ r X ∧ N + (1 -u g G )∂ θ X ∧ N = 1 -u Ge + Eg EG + u 2 eg EG ∂ r X ∧ ∂ θ X + (1 -u e E )∂ r X ∧ N + (1 -u g G )∂ θ X ∧ N.
Thus a sufficient condition for Y to be an immersion is 1 -u( Ge+Eg EG ) + u 2 eg EG = 0.

Since X is a minimal surface, we have Ge+Eg EG = 0 and the last condition is equivalent to

1 -u 2 1 cosh 4 (tpr) = 0.
Hence if u L ∞ < 1 then 1 -u 2 1 cosh 4 (tpr) = 0 for all t ∈ R and X + uN is an immersion. This result is also true if u C 2,α < 1.

We are interested in minimal surfaces which are normal on domain of the catenoid. Since we have

Y t|{-1}×[-π,π[ = X t|{-1}×[-π,π] , Y t|{1}×[-π,π[ = X t|{1}×[-π,π]
the surfaces Y t are bounded by two p coverings of circles in parallel planes. More specifically Y t and X t have same boundaries. We look for solutions of the following problem

F (t, u) := H(X t + uN t ) = 0, (3.51) with u ∈ V := {v ∈ C 2,α 0 ([-1, 1] × [-π, π], R); v C 2,α < 1}.
Thus if u is a solution of (3.51) for some t ∈ R then Y t is a minimal surface bounded by two coaxial circles in parallel planes. The function u = 0 is a trivial solution for all t ∈ R and we are looking for bifurcations from this trivial branch of solutions. Definition 3.55. We say that t * is a bifurcation point for F (from the trivial solution) if there is a sequence (t n , u n ) ∈ R × V with u n = 0 and F (t n , u n ) = 0 such that (t n , u n ) → (t * , 0). We set V := V ∩ E, where

E = {u ∈ C 2,α 0 ([-1, 1] × [-π, π], R); u(r, θ) = u(-r, θ) and u(r, -θ) = u(r, θ)}.
We want to find a non trivial branch of solutions u of (3.51) in V by applying the Crandall-Rabinowitz theorem (see [START_REF] Crandall | Bifurcation from simple eigenvalues[END_REF]). Let us recall the conditions which allow us to apply this theorem. We must prove that 1) There exists some t * and u * such that Ker(D u F (t * , 0)) = Vect(u * ).

2) For this t * , Im(D u F (t * , 0)) is of codimension one.

3

) Letting M := D u,t F (t * , 0) we have M u * / ∈ Im(D u F (t * , 0)).
By definition D u F (t, 0) is the Jacobi operator of the surface X t .

Step 1 :Computation of the Jacobi operator Here we give the computations which lead to the expression of the Jacobi operator. The formula for the Jacobi operator is

J t u := D u F (t, 0) = ∆ Xt u -2K t u (3.52)
where ∆ Xt is the Laplace-Beltrami operator of the surface X t and K t is the Gauss curvature of X t . Both quantities can be computed with the first and second fundamental forms. We find (see e.g. [START_REF] Dierkes | Grundlehren der Mathematischen Wissenschaften[END_REF]) that

K t = eg -f 2 EG -F 2 = - 1 cosh 4 (tpr) and ∆ Xt = 1 √ EG (∂ r (E -1 √ EG∂ r ) + ∂ θ (G -1 √ EG∂ θ )) = 1 tp 2 cosh 2 (tpr) ( 1 t ∂ 2 r + t∂ 2 θ ).
Thus we obtain that

J t u = 1 t 2 p 2 cosh 2 (tpr) ∂ 2 r u + t 2 (∂ 2 θ + 2p 2 cosh 2 (tpr) ) u.
We can easily see that J t : C 2,α 0 → C 0,α is a symmetric operator on the space L 2 ([-1, 1] × [-π, π], dA) with the usual inner product. Here dA denotes the area element defined by dA = t 2 p 2 cosh 2 (tpr)drdθ). Thus J t is a Fredholm operator with null index and if the condition 1) of the Crandall-Rabinowitz theorem is realized so is condition 2).

Step 2 : Decomposition in Fourier series and existence of Jacobi fields We are looking for non trivial solutions of J t v = 0

∂ 2 r v + t 2 (∂ 2 θ + 2p 2 cosh 2 (tpr) )v = 0 v(1, θ) = v(-1, θ) = 0 (3.53)
Finding a solution of (3.53) is equivalent to find a solution of

∂ 2 r v + ∂ 2 θ v + 2p 2 cosh 2 (pr) v = 0 v(t, θ) = v(t, θ) = 0 (3.54)
(indeed if we have a solution w(r, θ) of (3.54) then we set r = r t , r ∈ [-1, 1] and we obtain that v(r, θ) = w(tr, θ) is a solution of (3.53)).

Let us denote by

A t := [-t, t] × [-π, π] and A ∞ := R × [-π, π]. We denote by J the operator ∂ 2 r + ∂ 2 θ + 2p 2 cosh(pr)
. With a slight abuse of language we also call J the Jacobi operator on the surface X t . We consider the following eigenvalue problem :

∂ 2 r v + ∂ 2 θ v + ( 2p 2 cosh 2 (pr) + λ)v = 0 v(t, θ) = v(-t, θ) = 0. (3.55)
Finding a solution of (3.54) is equivalent to say that 0 is an eigenvalue of J in A t 0 . Now we use a decomposition in Fourier series :

v(r, θ) = n∈Z v n (r)e inθ .
Then v is solution of (3.55) if and only if v n is solution of

v n (r) + 2p 2 cosh 2 (pr) + (λ -n 2 ) v n = 0 v n (t) = v n (-t) = 0 (3.56)
for all n ∈ Z. Hence we consider another eigenvalue problem

w (r) + ( 2p 2 cosh 2 (pr) + µ)w = 0 w(t) = w(-t) = 0 (3.57)
The eigenvalue of (3.55) and (3.57) are linked by the following relation

λ = µ + n 2 .
(3.58)

More precisely the preceding relation means that if λ is an eigenvalue of problem (3.55) then there exists an integer n and a real number µ which is an eigenvalue of (3.57) such that λ = µ+n 2 . Conversely if µ is an eigenvalue of (3.57) then µ + n 2 is an eigenvalue of (3.55) for all integers n. Thus we deduce that the first eigenvalues of the two problems are the same λ 1 = µ 1 (with n = 0). Let us denote by g t the Gauss map of X t . If t is small enough then the area of g t (X t ) is strictly less than 2π (this area depends continuously of t and when t = 0 it is equal to 0). Then using the result of Barbosa Do-Carmo (see [START_REF] Barbosa | On the size of a stable minimal surface in R 3[END_REF]) we find that X t is stable for t small enough and the first eigenvalue of the Jacobi operator is non negative, i.e., λ 1 (A t ) > 0. However we claim that the first eigenvalue of the infinite catenoid covered p-times is -p 2 . This is because the function w(r) = 1 cosh(pr) is a positive function which satisfies

w (r) + ( 2p 2 cosh 2 (pr) -p 2 )w = 0 w(+∞) = w(-∞) = 0 (3.59)
thus w is the first eigenfunction of the operator J and λ 1 (A ∞ ) = µ 1 (A ∞ ) = -p 2 . We now use the continuity and strict monotonicity of the eigenvalues of the operator J with respect to variations of the domain. This fact is due to the min-max principle of Courant-Fischer for eigenvalue of a selfadjoint operator and we refer to [START_REF] Courant | Methods of mathematical physics[END_REF] p.407, or [START_REF] Chavel | Eigenvalues in Riemannian geometry[END_REF] p.18 for a proof. We obtain : there exist t 0 , t 1 , ...t p-1 such that

µ 1 (A t 0 ) = 0, µ 1 (A t 1 ) = -1, µ 1 (A t 2 ) = -4, ..., µ 1 (A t p-1 ) = -(p -1) 2 .
(3.60)

Thus there exist p possible instants of bifurcations. More precisely there exist t 0 , t 1 , ...t p-1 such that 0 is an eigenvalue of the Jacobi operator J t k on A t k . The corresponding eigenfunctions are sums of functions of the form v k (r)(A cos(kθ) + B sin(kθ)), where v k are simple eigenfunctions of (3.57) associated to the eigenvalue -k 2 . The case k = 0 is special. Indeed one can see that at t = t 0 , 0 is the first eigenvalue of the Jacobi operator J t 0 , and one can show that bifurcation occurs at this instant but to give another catenoid. This is known as the stable/unstable catenoid bifurcation. For t = t k , k = 0, k = 1, ..., (p -1) then 0 is an eigenvalue of multiplicity at least 2 in the space C 2,α 0 . However we can prove that for t = t 1 , zero is a simple eigenvalue of J t in the space V.

Step 3 : For t = t 1 , zero is a simple eigenvalue in V.

For t = t 1 because of the relation (3.58), we are able to rank the eigenvalues of J t 1 , denoted by λ 1 < λ 2 ≤ .... By definition of t 1 we have

λ 1 (A t 1 ) = µ 1 (A t 1 ) = -1.
Denoting the eigenvalues of (3.57) by µ 1 < µ 2 ≤ ... we claim that it holds that µ 2 (A t 1 ) > 0. A proof of this fact can be found in the paper of Shiffman [Shi56] b . We can see that there are two solutions of this equation r 1 (c), r 2 (c) with r 1 (c) < 0 < r 2 (c) ; and both r 1 (c) and r 2 (c) are increasing functions of c with r 1 (c) → -∞, r 2 (c) → 0 as c → -∞ and r 1 (c) → 0, r 2 (c) → +∞ as c → +∞. Thus every real number is covered exactly once by all the numbers r 1 (c), r 2 (c) the number 0 being covered by the case b = 0. Since the functions (3.61) have only two zeros, the Courant nodal's theorem allows us to say that 0 can not be an nth-eigenvalue of (3.57) with n ≥ 2. For t small we have µ 2 (A t ) > µ 1 (A t ) > 0. Hence because the eigenvalues µ n (A t ) are decreasing with respect to t we find that µ 2 (A t 1 ) > 0.

Thus we can deduce that λ 2 = µ 1 + 1 = 0 < µ 2 and we have the alternative

λ 3 = -1 + 4, λ 3 = µ 2 + 1 or λ 3 = µ 3 .
In any case we have λ 3 > 0 and for i ≥ 3, λ i > 0.

Hence the eigenfunctions associated to 0 are only of the form A 1 v 1 cos(θ) + B 1 v 1 sin(θ) with v 1 the first eigenfunction of J t 1 and A 1 , B 1 two constants. However the only eigenfunctions admissible in E are of the form A 1 v 1 cos(θ). We denote by u 1 (r, θ) := v 1 (r) cos(θ) v 1 cos(θ) L 2 . One can see that this function is in E. This comes from the fact that v 1 is the first eigenfunction associated to the first eigenvalue µ 1 = -1 of the problem (3.57). From a classical argument we obtain that v 1 (r) = v 1 (-r) for r ∈ [-1, 1]. This proves that for t = t 1 the Jacobi operator J t 1 is not invertible in the space E.

Step 4 : Verification of the Crandall-Rabinowitz condition.

Here we assume that t = t 1 . We still denote by u 1 = v 1 cos(θ) v 1 cos(θ) L 2 the eigenfunction associated to 0 in the space E. We set M := D u,t F (t 1 , 0), L := J t 1 . The operator M is obtained by differentiating J t with respect to t and taking t = t 1 . In order to apply the Crandall-rabinowitz theorem we must check that M u 1 / ∈ Im(L).

(3.62)

In order to prove this it suffices to prove that M u 1 , u 1 = 0 where •, • stands for the inner product in L 2 . Indeed if M u 1 is in Im(L) then we can write M u 1 = Lw for some w in E but then

M u 1 , u 1 = Lw, u 1 = w, L * u 1 , with L * the adjoint of L = w, Lu 1 , because L is symmetric = 0, because Lu 1 = 0 by definition.
We use an argument taken from [START_REF] Alama | On compound vortices in a two-component Ginzburg-Landau functional[END_REF] and [START_REF] Lamy | Bifurcation Analysis in a Frustrated Nematic Cell[END_REF]. We already mentioned that λ 1 (A t ) = µ 1 (A t ) is strictly decreasing with respect to t. Furthermore this quantity depends smoothly on t. If we denote by u t the eigenfunction in E associated to the eigenvalue λ 2 (A t ) = λ 1 (A t ) + 1 (cf.relation (3.58)) the function u t depends smoothly on t. This is due to the fact that we can rewrite the eigenvalue problem (3.53) on a fix domain with a new operator which depends analytically on t. We then use the Kato selection theorem (see e.g. the last paragraph p.386 in [START_REF] Kato | Perturbation theory for linear operators[END_REF] and Lemma 3.15 in [START_REF] Berger | Sur les premières valeurs propres des variétés riemanniennes[END_REF]) to obtain the fact that the eigenvalue and the eigenfunction are analytic. We have u t 1 = u 1 and λ 2 (A t ) = J t u t , u t . Thus

0 > λ 2 (A t 1 ) = d dt |t=t 1 J t u t , u t = [ d dt |t=t 1 J t ]u 1 , u 1 + J t 1 d dt |t=t 1 u t , u 1 + J t 1 u 1 , d dt |t=t 1 u t = [ d dt |t=t 1 J t ]u 1 , u 1 , because J t 1 is symmetric and J t 1 u 1 = 0 but [ d dt |t=t 1 J t ]u 1 = M u 1 by definition.
Hence we proved that M u 1 , u 1 < 0 and then (3.62) is satisfied.

We have all the ingredients to apply the bifurcation Theorem of [START_REF] Crandall | Bifurcation from simple eigenvalues[END_REF]. We obtain : Theorem 3.56. For any integer p ≥ 2 there exists an instant t * (p) such that (t * , 0) is a bifurcation point for F (cf. (3.51)). In addition the set of solutions of F = 0 near (t * , 0) is formed by two C 1 Cartesian curves which intersect each other transversely in 0 .

Step 5 : The bifurcating surfaces are not catenoids. Now in order to conclude the proof of Theorem 3.53 we must check that solutions Y t = X t + u(t)N t , with u(t) = 0 obtained via the bifurcation, are not portions of catenoids and that they are symmetric with respect to the planes P = {z = 0} and P = {x = 0}. The solution u of (3.51) inherits the symmetry of the space E and this implies the symmetries of the surfaces Y t . Indeed because of the symmetry of X t and N t if u ∈ E then Y t = X t + u(t)N t satisfies that its coordinates (y 1 , y 2 , y 3 ) are such that (y 1 (r, -θ), y 2 (r, -θ), y 3 (r, -θ)) = (y 1 (r, θ), -y 2 (r, θ), y 3 (r, θ)).

We prove now that Y t are not catenoids. Recall that X t and Y t have same boundaries. We know that there are at most two catenoids which pass through two given circles in parallel planes. One stable catenoid and one unstable catenoid. For t near t 1 , X t is the unstable catenoid because λ 1 (A t ) < 0. If t is near enough from t 1 then u(t) is small (that is u C 2,α is small). Thus for t near t 1 , Y t = X t + u(t)N t is in a small tubular neighborhood of X t . But for t near t 1 the stable catenoid is not contained in such small tubular neighborhood. Indeed note that when t = t 0 the stable and unstable catenoid coincide and for t near t 0 the stable catenoid is in a small tubular neighborhood of the unstable one, this is not the case here because t 1 > t 0 . Besides for t -t 1 small the surfaces Y t = X t + u(t)N t are on both sides of the unstable catenoid. Indeed making an expansion of Y t for t -t 1 small we find that :

Y t = X t 1 + (t -t 1 )[∂ t X |t=t 1 + u(t 1 )∂ t N |t=t 1 + ∂ t u |t=t 1 N t 1 ] + o(t -t 1 ).
Here ∂ t u |t=t 1 = u 1 is the Jacobi field found previously. Thus we know that u 1 = v 1 (r) cos(θ) v 1 (r) cos(θ) L 2 . We can check that ∂ t X |t=t 1 and ∂ t N |t=t 1 are both tangent to the surface X t 1 . Hence we can focus on the normal variation. Because the sign of function u 1 changes with θ we obtain that Y t = X t + u(t)N t is on both sides of the catenoid for t near t 1 . Thus we can deduce Theorem 3.53.

Remark 3.57. For p ≥ 2 we proved that bifurcation occurs at the first instant t 1 (cf. (3.60)) but we can show that we can apply the same arguments to obtain bifurcations at the instant t 2 , ..., t p-1 for p ≥ 3. Thus for p ≥ 2 we found p -1 branches of bifurcations.

From Theorem 3.53 and the equivalence between finding minimal surfaces bounded by two coaxial circles in parallel planes and finding solutions of (3.1) with c < 0 we obtain Theorem 3.8.

Y t = X t + u(t)N t N t X t

Two main questions remain open after the completion of this work.

Open problem 1 : Do the non radial solutions constructed in Section 6 minimize the energy in the class I p,p ? If the minimal surfaces obtained from these solutions by Proposition 3.43 were not symmetric with respect to some horizontal planes then we would be able to deduce that the solutions are not minimizers of E in I p,p because of Proposition 3.44. But we constructed minimal surfaces which are symmetric with respect to some horizontal planes hence the question remains open. If the non radial solutions of (3.1) do not minimize E in I p,p , do there exist minimizers ?

Open problem 2 : Can we construct (non-minimizing) non-radial solutions of (3.1) with c > 0 ? If yes what is the geometry of the minimal surfaces linked to these new solutions ?

Chapitre 4

Existence and non existence results for minimizers of Ginzburg-Landau energy with prescribed degrees Abstract : Let D = Ω \ ω ⊂ R 2 be a smooth annular type domain. We consider the simplified Ginzburg-Landau energy

E ε (u) = 1 2 D |∇u| 2 + 1 4ε 2 D (1 -|u| 2 ) 2
, where u : D → C, and look for minimizers of E ε with prescribed degrees deg(u, ∂Ω) = p, deg(u, ∂ω) = q on the boundaries of the domain. For large ε and for balanced degrees (i.e., p = q), we obtain existence of minimizers for domains with large capacity (corresponding to thin annulus). We also prove non-existence of minimizers of E ε , for large ε, if p = q, pq > 0 and if D is a circular annulus with large capacity. Our approach relies on similar results obtained for the Dirichlet energy E ∞ (u) = 1 2 D |∇u| 2 , on a previous existence result obtained by Berlyand and Golovaty and on a technique developed by Misiats.

Introduction and main results

We fix D = Ω \ ω ⊂ R 2 a smooth annular type domain : Ω and ω are smooth and bounded simply connected open sets s.t. ω ⊂ Ω ⊂ R 2 . In this article, some results are specific to the case where D is a circular annulus. In order to underline this specificity, when needed, we use the notation

A = B(0, 1) \ B(0, R) (with R ∈]0, 1[) instead of D.
We are interested in the existence or the non-existence of global minimizers of the Ginzburg-Landau type energy

E ε (u) = 1 2 D |∇u| 2 + 1 2ε 2 (1 -|u| 2 ) 2
in the connected components of

I := {u ∈ H 1 (D, C) | tr ∂D (u) ∈ H 1/2 (∂D, S 1 )} for large values of ε > 1.
Here, tr ∂D stands for the trace operator on ∂D and S 1 = {x ∈ C | |x| = 1}. We consider also the Dirichlet energy which corresponds formally to the case ε = +∞ :

E ∞ (u) = 1 2 D |∇u| 2 , u ∈ I.
For Γ ∈ {∂Ω, ∂ω} and for u ∈ I we let

deg Γ (u) = 1 2π Γ u ∧ ∂ τ u dτ .
Here :

• Each Jordan curve Γ is directly (counterclockwise) oriented.

• We let ν be the outward normal to Ω if Γ = ∂Ω or ω if Γ = ∂ω. Likewise we denote by τ the tangential vector on ∂Ω and ∂ω. We have that τ = ν ⊥ on ∂Ω and τ = -ν ⊥ on ∂ω.

• The differential operator ∂ τ = τ • ∇ is the tangential derivative and " • " stands for the usual scalar product in R 2 . We also use the standard notation "∂ ν " for the normal derivative

∂ ν = ν • ∇.
• The cross operator " ∧ " stands for the cross product in C, it is defined by (z

1 + ız 2 ) ∧ (w 1 + ıw 2 ) := z 1 w 2 -z 2 w 1 , z 1 , z 2 , w 1 , w 2 ∈ R.
• It is well known that deg Γ (u) is an integer see [START_REF]Ginzburg-Landau minimizers with prescribed degrees. Capacity of the domain and emergence of vortices[END_REF] (the introduction) or [START_REF]Degree theory : old and new, Topological nonlinear analysis[END_REF].

• The integral over Γ should be understood using the duality between H 1/2 (Γ) and H -1/2 (Γ) (see, e.g., [START_REF]Ginzburg-Landau minimizers with prescribed degrees. Capacity of the domain and emergence of vortices[END_REF] Definition 1).

• For u ∈ I, we write deg(u) = (deg ∂Ω (u), deg ∂ω (u)).

For P = (p, q) ∈ Z 2 , we are interested in the minimization of E ε for large ε > 1 in

I P = I p,q := {u ∈ I | deg(u) = (p, q)} .
For ε ∈]0, ∞] and P = (p, q) ∈ Z 2 , we denote

m ε (P ) = m ε (p, q) = inf I P E ε .
It turns out that the I P 's are the connected component of I (cf. the appendix of Boutet-de-Monvel and Gabber in [START_REF] Boutet De Monvel-Berthier | A boundary value problem related to the Ginzburg-Landau model[END_REF] and [START_REF] Brezis | Degree theory and BMO. I. Compact manifolds without boundaries[END_REF]). They are open and closed for the strong topology induced by the H 1 -norm. Hence if a minimizer of E ε in I p,q exists for some (p, q) ∈ Z 2 it satisfies the following Euler-Lagrange equations :

       -∆u = 1 ε 2 u(1 -|u| 2 ) in D |u| = 1 on ∂D u ∧ ∂ ν u = 0 on ∂D. (4.1)
These equations are obtained by making variations of the form [BM]).

u t = u + tϕ for t ∈ R, ϕ ∈ C ∞ 0 (D, R 2 ) and u t = ue itψ for t ∈ R, ψ ∈ C ∞ (D, R) (see Appendix C in
However the sets I P are not closed with respect to the weak convergence in H 1 (see Introduction in [BM]). This fact implies that, in general, the minimization problem m ε (P ) is not easy to handle since the direct minimization method fails. Namely in some cases m ε (P ) is not attained.

In contrast, for some other configurations, all minimizing sequences converge in H 1 -norm. Such questions are central in this article.

Remark 4.1. It is obvious that for p = q = 0 and ε ∈]0, ∞], m ε (0, 0) is attained and the minimizers are the constants of modulus 1. Thus we can focus on the case (p, q) = (0, 0).

In this article we obtained existence and non existence results for thin domains.

Definition 4.2. Since every annular domain is conformally equivalent to a circular annulus (see e.g. [Ahl78]), we can fix a conformal mapping

Φ : A = {x ∈ R 2 | R D < |x| < 1} → D.
for some R D ∈]0, 1[. -We call conformal ratio of D this number R D ∈]0, 1[. It is uniquely defined by the existence of a conformal mapping Φ. -When R D is "close to" 1, the domain D is thin. When R D is "close to" 0, the domain D is thick. -The conformal ratio is related to the H 1 -capacity of D (see [START_REF] Ahlfors | An introduction to the theory of analytic functions of one complex variable[END_REF] or Definition 2.1 in [BM])

via cap(D) = - 2π ln R D .
This article essentially contains two theorems. The first one is an existence result. Roughly speaking it states that for all p ∈ N * , under a hypothesis (H) (which expresses that the annulus is thin) and if ε is sufficiently large then m ε (p, p) is attained.

Theorem 4.3. Let D ⊂ R 2 be an annular type domain and let p ∈ N * . If

m ∞ (p, p) < m ∞ (p -1, p -1) + 2π (H)
then there exists ε p > 0 s.t. if ε p < ε ≤ +∞ then minimizing sequences for m ε (p, p) are compact (for the H 1 -norm). In particular m ε (p, p) is attained.

For (u ε ) ε>εp ⊂ I p,p a sequence of minimizers there is u ∞ ∈ I p,p a minimizer for m ∞ (p, p) s.t., up to a subsequence, we have :

u ε → ε→∞ u ∞ in C l (D), ∀l ∈ N.
Remark 4.4. 1. Since I -p,-p = {u | u ∈ I p,p } where u is the conjugate of u and since E ε (u) = E ε (u), it is easy to reformulate Theorem 4.3 for p < 0.

2. The condition (H) is theoretical. We are able to prove that this condition holds true under the following condition of capacity of the domain. There exists 0 < R p < 1 s.t. if the conformal ratio R D satisfies R p < R D < 1 then (H) holds. Actually we also obtain that we can take the same R p as in Theorem 4.5 below.

3. For 1 > R D > R p we have that the minimizers of m ∞ (p, p) are vortexless. This can be obtained thanks to Theorem 4.5 and thanks to the conformal invariance of the Dirichlet energy. Our result then implies that, for sufficiently large ε, the minimizers of m ε (p, p) are also vortexless.

The previous theorem is an "extension" to general annular type domains of a previous result of Berlyand and Golovaty :

Theorem 4.5 ([GB02]). Let p ∈ N * there exists a critical inner radius 0 < R p < 1 s.t. for R p < R < 1 and D = B(0, 1) \ B(0, R), m ε (p, p) is attained by a unique (up to a phase) radially symmetric minimizer for all 0 < ε < +∞.

Definition 4.6. In the previous theorem, the expression "up to a phase" means that if u is a minimizer, then ũ is a minimizer if and only if there exists α ∈ S 1 s.t. ũ = αu. Another way to explain this expression is to say that two minimizers have pointwise same moduli and the difference of their phases is a constant.

Remark 4.7. Theorem 4.5 may be easily extended to the case ε = ∞ (see Step 2 in the proof of Proposition 4.24).

Although Theorem 4.3 may be seen as an extension of Theorem 4.5, the methods used in their proofs are different. Condition (H) allows to make arguments in the spirit of concentrationcompactness phenomenon and bubbling analysis (see e.g. [START_REF] Brezis | Points critiques dans les problèmes variationnels sans compacité[END_REF]). See Section 4.3.3 for a detailed comparison between both theorems.

Note that in [START_REF] Farina | Uniqueness of vortexless Ginzburg-Landau type minimizers in two dimensions[END_REF] (Theorem 1.5), Farina and Mironescu have also extended Theorem 4.5, to general annular type domains. They proved that there is some explicit universal constant δ 0.045 such that if m ε (p, p) < δ then the infimum is attained and the minimizer is unique (up to a phase). Then using S 1 -valued test functions, and the conformal invariance of the Dirichlet energy, they obtained that if the annular domain is very thin, then the condition m ε (p, p) < δ holds. Their condition on the thinness of the annular domain is more restrictive than ours. However they obtained a more precise result : uniqueness of minimizer (up to a phase). We want to emphasize that the proof of uniqueness is a real challenge (existence is direct for δ < π).

Our second theorem is a non-existence result specific to the symmetric case D = A = B(0, 1) \ B(0, R) with R close to 1.

Theorem 4.8. Let p, q ∈ N * s.t. p = q. There are 0 < R min(p,q) < 1 and ε min(p,q) > 1 s.t. if R min(p,q) < R < 1, A = B(0, 1) \ B(0, R) and ε > ε min(p,q) then m ε (p, q) is not attained.

A technique to prove non existence of minimizers (or local minimizers) with prescribed degrees for the Ginzburg-Landau energy was devised by Berlyand, Golovaty and Rybalko in [START_REF] Berlyand | Nonexistence of Ginzburg-Landau minimizers with prescribed degree on the boundary of a doubly connected domain[END_REF]. They proved the non existence of minimizers of E ε in I 1,1 for thick annular domain. Then, perfecting this technique, Misiats proved the non existence of minimizers in some subsets of I p,q in [START_REF] Misiats | The necessary conditions for the existence of local Ginzburg-Landau minimizers with prescribed degrees on the boundary[END_REF]. The first non existence result for global minimizers of the Ginzburg-Landau energy with prescribed degrees p = q and pq > 0 was obtained by Mironescu in [START_REF]Size of planar domains and existence of minimizers of the ginzburg-landau energy with semistiff boundary conditions[END_REF] following the ideas of Berlyand, Golovaty, Rybalko and Misiats. It can be rephrased as follows :

Theorem 4.9. (Thm 4.16-[Mir14]) Let p, q ∈ N * , pq > 0 then there exists a critical value of the capacity C min(p,q) > 0 s.t. if cap(D) < C min(p,q) then m ε (p, q) is not attained for ε small.

Remark 4.10. Note that in the previous theorem the annulus is "thick", i.e., cap(D) is small, and ε is small. Hence we are in the opposite situation of Theorem 4.8. However the proofs of both theorems follow the same ideas. Note also that Theorem 4.9 still holds in the case p = q.

Our approach is similar to the one mentioned before. In particular we follow the strategy of Misiats in [START_REF] Misiats | The necessary conditions for the existence of local Ginzburg-Landau minimizers with prescribed degrees on the boundary[END_REF]. The new ingredient which allows us to obtain Theorem 4.8 is a non existence result for minimizers of E ∞ in I p,q with pq > 0 obtained in [HR] using the so-called Hopf quadratic differential.

Before proving both theorems (see Sections 4.3&4.4) we recall some classical results :

• In Section 4.2.1 we recall some basic results used to prove Theorems 4.3&4.8.

• In Sections 4.2.2&4.2.3 we list some results about the attainability or the non attainability of m ε (p, q) for ε ∈]0, ∞[ (Section 4.2.2) or ε = ∞ (Section 4.2.3).

4.2 Some known results and literature review 4.2.1 Bounds for m ε (p, q) and cost to move degrees

In the following for (p, q), (p , q ) ∈ Z 2 , we denote

|(p, q)| = |p| + |q| and |(p, q) -(p , q )| = |p -p | + |q -q |.
Proposition 4.11. Let P, P ∈ Z 2 . For 0 < ε < ε ≤ ∞ we have :

1. m ε (P ) ≤ π|P |, 2. m ε (P ) ≤ m ε (P ) + π|P -P |, 3. |m ε (P ) -m ε (P )| → 0 if ε ↑ ε.
Remark 4.12. Note that in the third assertion we may replace ε ↑ ε by ε → ε but in the following we only need ε ↑ ε.

Proof. The two first assertions of Proposition 4.11 are direct consequences of Proposition 4.13 below.

We prove the third assertion. For P ∈ Z 2 and ε ↑

ε ∈]0, ∞] we consider (u ε ) ε a minimizing sequence of m ε (P ) s.t. -∆u ε = u ε ε 2 (1 -|u ε | 2 ).
It is clear that such minimizing sequence always exists. Thanks to the maximum principle (see e.g. Proposition 2 [BBH93]), we have |u ε | ≤ 1. Since ε < ε we have

E ε (u ε ) ≥ m ε (P ) ≥ m ε (P ) = E ε (u ε ) -o ε (1)
where o ε (1) → 0 when ε → ε.

We denote

K(ε ) =    1 4ε 2 -1 4ε 2 if ε = ∞ 1 4ε 2 if ε = ∞ . It is clear that we have K(ε ) → 0 when ε → ε. Therefore we have K(ε )|D| ≥ K(ε ) D (1 -|u ε | 2 ) 2 = E ε (u ε ) -E ε (u ε ) ≥ m ε (P ) -m ε (P ) + o ε (1).
Here |D| is the measure of D. Since m ε (P )-m ε (P ) ≥ 0 we thus obtain that m ε (P )-m ε (P ) → 0 when ε ↑ ε.

E ε (v + ) ≤ E ε (u) + π + η, (4.2) E ε (v -) ≤ E ε (u) + π + η. (4.3)
The proof of Proposition 4.13 may be found in [START_REF] Santos | Local minimizers of the Ginzburg-Landau functional with prescribed degrees[END_REF] Lemma 7. The proposition can be used repeatedly in order to increase or decrease the degree by several units.

In order to drop η in (4.2) and (4.3) and to replace the less than or equal sign by a strict inequality, we need an extra-hypothesis about the behavior of u on the connected component of ∂D where the degree is modified.

Proposition 4.14. Let ε ∈]0, ∞], k ∈ N * and u ∈ I p,q be any smooth function which satisfies |u| ≤ 1 in D and ∂ ν |u| > 0, u ∧ ∂ ν u = 0 on ∂Ω.

1. Assume that there is

x 0 ∈ ∂Ω s.t. u ∧ ∂ τ u(x 0 ) > -u • ∂ ν u(x 0 ) then there exists v ∈ I p-k,q s.t. E ε (v) < E ε (u) + kπ.
2. Assume that there is

x 0 ∈ ∂Ω s.t. u ∧ ∂ τ u(x 0 ) < u • ∂ ν u(x 0 ) then there exists v ∈ I p+k,q s.t. E ε (v) < E ε (u) + kπ.
An analogous lemma can be stated considering the other boundary ∂ω.

Proposition 4.14 is proved in [RS14] (Lemma 1.2). Note that if u is a solution of the Ginzburg-Landau equation with the so-called semi-stiff boundary condition (4.1) then u ∧ ∂ ν u = 0 on ∂D and u is smooth with |u| ≤ 1 in D (Lemma 4.4 in [BM]). Besides ∂ ν |u| > 0 can be obtained from the Hopf lemma on the boundary.

One of the main tool in the study of the minimization of E ε in I p,q is the beautiful Price Lemma.

As it was explained, the degree deg : I → Z 2 is not continuous for the weak H 1 convergence, this lemma expresses the energetic cost to modify degrees for a weak H 1 -limit.

Lemma 4.15 (Price Lemma, see Lemma 1 in [START_REF]Ginzburg-Landau minimizers with prescribed degrees. Capacity of the domain and emergence of vortices[END_REF]). Let P ∈ Z 2 and (u n

) n ⊂ I P s.t. u n u in H 1 (D). Then lim inf n→+∞ E ∞ (u n ) ≥ E ∞ (u) + π|P -deg(u)|.
Using Sobolev embeddings it also holds that, for all ε > 0 :

lim inf n→+∞ E ε (u n ) ≥ E ε (u) + π|P -deg(u)|.

Some known Existence/Non Existence results : the case ε ∈]0, ∞[

The first non existence result is certainly the following.

Proposition 4.16. Let ε > 0, if (p, q) ∈ Z 2 are s.t. (p, q) = (0, 0) and pq ≤ 0, then m ε (p, q) is not attained.

Proof. The two following estimates are the starting point of the proof :

-the point-wise inequality |∇u| 2 ≥ 2|Jac u| (here Jac u = u x ∧ u y is the Jacobian of u) ; -the degree formula valid for u ∈ I (see e.g. (1.6) in [START_REF]Degree theory : old and new, Topological nonlinear analysis[END_REF]) :

D Jac u = π| deg ∂Ω (u) -deg ∂ω (u)|. (4.4)
By combining both previous estimates, if pq ≤ 0, then for all u ∈ I p,q , we obtain that

1 2 D |∇u| 2 ≥ π(|p| + |q|).
On the other hand, by Proposition 4.11.1 it holds that inf Ip,q

E ε ≤ π(|p| + |q|).
By associating both bounds, we obtain inf

Ip,q E ε = π(|p| + |q|).
Now we argue by contradiction and we assume that there exists ε > 0 s.t. m ε (p, q) is attained by u ε . Then we have

π(|p| + |q|) = 1 2 D |∇u ε | 2 = E ε (u ε ). Therefore D (1 -|u ε | 2 ) 2 = 0, i.e., u ε ∈ H 1 (D, S 1
). Since u ε is S 1 -valued we have Jac u ε = 0 and the degree formula (4.4) implies that p = q. This fact is in contradiction with (p, q) = (0, 0) and pq ≤ 0.

Our main results deal with the remaining cases : pq > 0. This condition means p, q > 0 or p, q < 0 and without lack of generality we may assume that p, q > 0 (since deg(u, Γ) = -deg(u, Γ) for Γ ∈ {∂Ω, ∂ω)}).

In a circular annulus A = B(0, 1) \ B(0, R), a natural candidate to be a minimizer for m ε (p, p) is the radial Ginzburg-Landau solution of degree p. The radial Ginzburg-Landau solution of degree p is a special solution of the semi-stiff problem

   -∆u = u ε 2 (1 -|u| 2 ) 2 in A |u| = 1, u ∧ ∂ ν u = 0 on ∂A. This solution is of the form u ε,p (x) = ρ ε,p (|x|) x |x| p (4.5) where ρ ε,p ∈ C ∞ ([R, 1], [0, 1]) is the unique solution of    -ρ - ρ r + p 2 ρ r 2 = ρ ε 2 (1 -ρ 2 ) in ]R, 1[ ρ(R) = ρ(1) = 1. (4.6)
As seen in the introduction, Berlyand and Golovaty proved a very precise existence result (see Theorem 2.13 in [START_REF] Golovaty | On uniqueness of vector-valued minimizers of the Ginzburg-Landau functional in annular domains[END_REF]) for the minimization of E ε in I p,p with p ≥ 1 in annuli A = B(0, 1) \ B(0, R) for R sufficiently close to 1.

For the special cases p = q = 1 and for an annular type domain D, by using a compilation of works of Berlyand, Golovaty, Mironescu and Rybalko (see e.g. [BM], [START_REF]Ginzburg-Landau minimizers with prescribed degrees. Capacity of the domain and emergence of vortices[END_REF], [START_REF] Berlyand | Nonexistence of Ginzburg-Landau minimizers with prescribed degree on the boundary of a doubly connected domain[END_REF]) we can state the following proposition : Proposition 4.17. Let D ⊂ R 2 be an annular type domain and let R D be the conformal ratio of D.

• If R D ≤ e 2 then m ε (1, 1) is attained for all ε.

• If R D > e 2 then there exists ε 0 > 0 s.t., for ε > ε 0 , m ε (1, 1) is attained and, for ε < ε 0 , m ε (1, 1) is not attained.

Some Existence/Non Existence results : the case ε = ∞

In the case of the Dirichlet energy, thanks to the conformal invariance of E ∞ , we may restrict the study to a ring A = B(0, 1) \ B(0, R) with R ∈]0, 1[.

As for the study of the minimization of the Ginzburg-Landau energy in a ring, a natural candidate to minimize the Dirichlet energy in I p,p is the radial harmonic map of degree p which solves the semi-stiff problem

   ∆u = 0 in A |u| = 1, u ∧ ∂ ν u = 0 on ∂A.
This solution is of the form

u ∞,p (x) = ρ ∞,p (|x|) x |x| p (4.7) where ρ ∞,p ∈ C ∞ ([R, 1], [0, 1]) is the unique solution of    -ρ - ρ r + p 2 ρ r 2 = 0 in ]R, 1[ ρ(R) = ρ(1) = 1. (4.8)
In an unpublished paper, Berlyand and Mironescu (see Lemma D.3 in [BM]) proved the following proposition that treats the case p = q = 1. Proposition 4.18. For all R ∈]0, 1[, the radial harmonic map of degree 1 is the unique (up to a phase) minimizer of m ∞ (1, 1).

Next, Hauswirth and Rodiac in [HR] considered the problem m ∞ (p, q) for p, q ∈ Z. They proved the following proposition : Proposition 4.19. Let p, q ∈ Z then we have

• If p = q and pq > 0, then m ∞ (p, q) is not attained. Without loss of generality we can assume that p > q > 0 and then it holds that m ∞ (p, q) = m ∞ (q, q) + π(p -q). • If p = q = 0, then there is 0 < R p < 1 s.t. for R p < R < 1, m ∞ (p, p) is attained and the radial harmonic map of degree p is the unique (up to a phase) minimizer of E ∞ in I p,p .

Remark 4.20. Note that the radius R p obtained by Hauswirth and Rodiac is the same as the radius obtained by Berlyand and Golovaty (see Theorem 4.5) and that if p > p then R p ≥ R p (see Step 1 in the proof of Proposition 4.24).

Existence Result

This section is dedicated to the proof of Theorem 4.3. We first study the behavior as ε n goes to some ε * ∈]0, +∞] of sequences (u n ) s.t. u n is almost minimizing for E εn . Then we derive a theoretical condition [Hyp. (H)] under which the compactness of minimizing sequences for E ε holds for large ε. Eventually we compare Hyp. (H) with the condition of Theorem 4.5.

The key argument

For (p, q) ∈ Z 2 we define By Proposition 4.11.1, there is u ∈ I P for some P ∈ Z 2 s.t., up to a subsequence, u n u in H 1 (D, C). Then P ∈ A P and u minimizes m ε * (P ). Moreover, if P = P then m ε * (P ) = m ε * (P ) + π|P -P |.

A (p,q) = (p , q ) ∈ Z 2 | pp ≥ 0,
Proof. Let P = (p, q) ∈ Z 2 , ε * ∈]0, ∞] and (ε n ) n be an increasing sequence s.t. ε n ↑ ε * or ε n = ε * for all n and a sequence (u n ) n ⊂ I P s.t. E εn (u n ) ≤ m εn (P ) + o n (1).
There exists u ∈ I P for some P ∈ Z 2 s.t., up to a subsequence, u n u in H 1 (D, C). By the Price Lemma (Lemma 4.15) we have

lim inf n E ∞ (u n ) ≥ E ∞ (u) + π|P -P |.
On the other hand, passing to a subsequence, we have |u n | → |u| in L 4 and thus :

1 4ε 2 n D (1 -|u n | 2 ) 2 -→ n→∞ 1 4ε 2 * D (1 -|u| 2 ) 2 if ε * < ∞ 0 if ε * = ∞.
By combining the two previous estimates we obtain :

lim inf n E εn (u n ) ≥ E ε * (u) + π|P -P |.
From Proposition 4.11.2&3 we deduce :

m ε * (P ) + π|P -P | = lim n m εn (P ) + π|P -P | ≥ lim n m εn (P ) = lim inf n E εn (u n ) ≥ E ε * (u) + π|P -P |.
(4.9)

Therefore we have u ∈ I P and m ε * (P ) ≥ E ε * (u). Consequently u minimizes E ε * in I P .

By contradiction we assume now that p ≥ 0 and that p > p.

Note that u satisfies the hypotheses of Proposition 4.14. Indeed u is a solution of (4.1), there exists

x 0 ∈ ∂Ω s.t. u∧∂ τ u(x 0 ) > 0 because deg ∂Ω (u) > 0, and -u(x 0 )•∂ ν u ( x 0 ) = -1 2 ∂ ν |u| 2 (x 0 ) < 0 thanks

Consequences of the key argument : existence of minimizers

The key argument [Lemma 4.21] describes what can happen to almost minimizing sequences (u n ) n for m εn (p, q) when ε n tends to ε * . Roughly speaking, if p, q > 0, u n converges weakly to some u in H 1 . We have that u ∈ I r,s with 0 ≤ r ≤ p, 0 ≤ s ≤ q, u minimizes E ε * in I r,s and the loss of energy is quantized : m ε * (r, s) = m ε * (p, q) -π(p -r + q -s). We can then show that a sharp inequality [Hyp. (H)] prevents minimizing sequences from falling in a class I r,s with r = p and s = p. ,C). This convergence contradicts the non compactness property of (u ε n ) n .

1 4ε 2 D (1-|u ε n | 2 ) 2 = 1 4ε 2 D (1-|u ε | 2 ) 2 . On the other hand lim n E ε (u ε n ) = m ε (p, p) = E ε (u ε ). Consequently lim inf n 1 2 D |∇u ε n | 2 = D |∇u ε | 2 which implies that u ε n → u ε in H 1 (D
Since

m ∞ (P 2 ) + π|(p -1, p -1) -P 2 | + 2π ≥ m ∞ (p -1, p -1) + 2π [Hyp. (H)] > m ∞ (p, p) [Prop. 4.11.3] = lim ε→∞ m ε (p, p) = lim ε→∞ lim inf n E ε (u n ε ) [Lemma 4.15] ≥ lim ε→∞ E ε (u ε ) + π|(p, p) -P 1 | ≥ m ∞ (P 2 ) + π|P 2 -P 1 | + + π|P 1 -(p, p)|.
Then we deduce that :

|(p -1, p -1) -P 2 | + 2 > |P 2 -P 1 | + |P 1 -(p, p)|.
By the triangle inequality we have :

|(p -1, p -1) -P 2 | + 2 > |P 2 -(p, p)|. Since P 2 = (p 2 , p 2 ) with p 2 ∈ [0, p -1], the last inequality means p -p 2 > p -p 2 .
This is clearly a contradiction and the proposition is proved.

By using the same strategy as in the proof of Proposition 4.22 we have :

Using Lemma 4.21 again we have

m ∞ (p -1, p -1) + 2π (H) > m ∞ (p, p) = m ∞ (P ) + 2π(p -p ).
Therefore we obtained m ∞ (p -1, p -1) > m ∞ (p , p ) + 2π|p -p -1|. This estimate is in contradiction with Proposition 4.11.2.

Consequently we have P = (p, p) and then m ∞ (p, p) is attained. Proof. We prove Proposition 4.24 in 3 steps.

Comparison with the work of Berlyand&Golovaty [GB02]

Step 1. The sequence of critical radii (R p ) p≥1 of Theorem 4.5 is non decreasing

The critical radius R p is defined by R p = max(α, β p ) with α ∈]0, 1[ which is a universal constant and β p ∈]0, 1[ which depends on p ≥ 1. In order to prove that (R p ) p≥1 is non decreasing, it suffices to prove the same for (β p ) p≥1 .

For p ≥ 1, the definition of β p consists in fixing β p ∈]0, 1[ s.t. for β p < R < 1 and for all ε > 0 we have

1 1 R -1 1 R t -1 ρ ε,p (t) -2 dt ≥ γ (4.13)
where ρ ε,p is defined in (4.5) and γ > 0 is a constant (the computations are made in [START_REF] Golovaty | On uniqueness of vector-valued minimizers of the Ginzburg-Landau functional in annular domains[END_REF] with γ = 4, note that there is a misprint in [START_REF] Golovaty | On uniqueness of vector-valued minimizers of the Ginzburg-Landau functional in annular domains[END_REF] and we should replace tρ ε,p (t) -2 by t -1 ρ ε,p (t) -2 ).

It holds that

ρ ε,p → ρ ∞,p in L ∞ ([R, 1]) (when ε → ∞) (4.14)
with ρ ∞,p defined in (4.7). This uniform convergence is obtained first with the H 1 convergence of u ε,p → u ∞,p (defined in (4.5)&(4.7)). Then using the radially symmetric structure of the function, the uniform convergence (4.14) follows directly.

Clearly, with the help of (4.14) and using the fact that ρ ε,p ≥ ρ ∞,p (see Lemma 4.32), the lower bound (4.13) holds for all ε > 0 if and only if

1 1 R -1 1 R t -1 ρ ∞,p (t) -2 dt ≥ γ. (4.15)
We are now in position to get that (β p ) p≥1 is non decreasing by proving that for all r ∈ [R, 1] and p ≥ 1 we have ρ ∞,p+1 (r) ≤ ρ ∞,p (r).

We fix r ∈ [R, 1] and we let

f r : [1, ∞[ → [0, 1] p → ρ ∞,p (r) = 1 1 + R p r p + R p r p .
The function f r is smooth and

f r (p) = ln(r) r p - R r p (1 + R p ) + ln(R) R r p -(Rr) p (1 + R p ) 2 .
Looking at the sign of f r (p) we see that

f r (p) ≤ 0 if √ R ≤ r ≤ 1. If R ≤ r ≤ √ R then letting r = sR with s ∈ [1, 1 √ R ] we have f r (p) = ln(s)(s p R p -s -p )(1 + R p ) + ln(R)(1 -R p )s p R p (1 + R p ) 2 .
Once again we obtain f r (p) ≤ 0.

Consequently the function f r is non increasing, i.e., ρ ∞,p+1 (r) ≤ ρ ∞,p (r). Thus the last inequality, with the help of definition of β p (see (4.13)), implies that

β p+1 ≥ β p . Therefore R p+1 ≥ R p .
Step 2.

For p ≥ 1, R p < R < 1 and D = B(0, 1) \ B(0, R), u ∞,p minimizes E ∞ in I p,p .
This step is a direct consequence of Theorem 4.5, Lemma 4.21 and (4.14). Indeed from Theorem 4.5, for ε > 0, u ε,p defined by (4.5)&(4.6) minimizes E ε in I p,p .

On the one hand, by (4.14), u ε,p → u ∞,p in L ∞ (B(0, 1) \ B(0, R)).

However, with the help of Lemma 4.21, up to a subsequence, when ε → ∞, u ε,p converges weakly in H 1 (B(0, 1) \ B(0, R)) to a minimizer of E ∞ in I P for some P ∈ A p,p .

By combining both previous claims we get that u ∞,p minimizes E ∞ in I p,p .

Step 3.

Conclusion

Note that for p = 1 m ∞ (1, 1) < 2π and thus the result of Proposition 4.24 is obvious.

We prove that if p ≥ 2, R p < R < 1 and

D = B(0, 1) \ B(0, R) then m ∞ (p, p) < m ∞ (p -1, p - 1) + 2π.
Once this is done, by conformal invariance, we get that if D is an annular type domain whose

conformal ratio satisfies R p < R D < 1 then we have m ∞ (p, p) < m ∞ (p -1, p -1) + 2π.
Let p ≥ 2, R p < R < 1 and D = B(0, 1) \ B(0, R). From Steps 1&2, we have for q ∈ {p -1, p} that m ∞ (q, q) is attained by u ∞,q . Consequently :

m ∞ (p, p) -m ∞ (p -1, p -1) = E ∞ (u ∞,p ) -E ∞ (u ∞,p-1 ) = 2π p 1 -R p 1 + R p -(p -1) 1 -R p-1 1 + R p-1 . Hence, for R ∈]0, 1[ E ∞ (u ∞,p ) -E ∞ (u ∞,p-1 ) < 2π ⇔ p(1 -R p )(1 + R p-1 ) -(p -1)(1 -R p-1 )(1 + R p ) < (1 + R p-1 )(1 + R p ) ⇔ Q p (R) := p -1 -pR -R p < 0 and E ∞ (u ∞,p ) -E ∞ (u ∞,p-1 ) = 2π ⇐⇒ Q p (R) = 0.
We can check that, for p ≥ 2 and

R ∈]0, 1[, Q p is decreasing and that Q p (1) = -2, Q p (0) = p-1. Therefore Q p admits a unique zero Rp in ]0, 1[ and for R ∈]0, 1[ we have Q p (R) < 0 ⇐⇒ Rp < R < 1.
We now prove that Rp ≤ R p . Let R p < R < 1. From Steps 1&2, for q ∈ {p -1, p} we have that m ∞ (q, q) is attained by u ∞,q for D = B(0, 1) \ B(0, R). Consequently, using Proposition 4.11.2 we have

E ∞ (u ∞,p ) -E ∞ (u ∞,p-1 ) ≤ 2π.
This inequality implies Q p (R) ≤ 0 (from the definitions of Q p and Rp ), and thus R ≥ Rp . Since R p < R < 1 is arbitrary this proves that Rp ≤ R p .

The inequality Rp

≤ R p expresses that, if R ∈]R p , 1[, then m ∞ (p, p) -m ∞ (p -1, p -1) < 2π
and this ends the proof of Proposition 4.24.

Remark 4.25. -Numerical computations. Berlyand and Mironescu obtained the existence of Ginzburg-Landau minimizers in I 1,1 for large ε without restriction on the capacity of the domain (cf. Corollary 5.5. in [BM]). In particular they proved that u ∞,1 minimizes E ∞ in I 1,1 for all R ∈]0, 1[ (cf. Proposition 5.2. in [BM]).

For us the first interesting configuration of degrees is

P = (2, 2). Since m ∞ (2, 2) ≤ E ∞ (u ∞,2 )
(the definition of u ∞,p is given in (4.7)), we obtain that (H) holds if we have : [START_REF] Golovaty | On uniqueness of vector-valued minimizers of the Ginzburg-Landau functional in annular domains[END_REF], the Hypothesis of Theorem 4.5 is artificial : the optimal thickness condition should depend on ε.

E ∞ (u ∞,2 ) = 4π 1 -R 2 1 + R 2 < 2π 1 -R 1 + R + 2π = m ∞ (1,
The formulation of Theorem 4.3 is not optimal in the sense given by Berlyand and Golovaty in Remark 2.14 of [START_REF] Golovaty | On uniqueness of vector-valued minimizers of the Ginzburg-Landau functional in annular domains[END_REF]. But it allows to have existence of minimizers for m ε (p, p) for a wider class of annular type domains :

• Theorem 4.3 holds for annular type domain while the work of Golovaty and Berlyand is specific to annuli.

• Proposition 4.24 means that if the hypothesis on the size of the annulus in Theorem 4.5 holds then hypothesis of Theorem 4.3 holds. Then it holds that, up to a subsequence,

Asymptotic behavior of minimizers as ε → +∞

u ε → u ∞ in C l ( D) for all l ∈ N, (4.17)
where u ∞ is a minimizer of E ∞ in I p,p .

The starting point of the proof of this proposition is the following :

E ∞ (ũ ∞ ) ≤ E ∞ (u ε ) ≤ E ε (u ε ) ≤ E ε (ũ ∞ ) = E ∞ (ũ ∞ ) + 1 4ε 2 D (1 -|ũ ∞ | 2 ) 2 .
Hence we see that (u ε ) ε is a minimizing sequence for m ∞ (p, p). By Proposition 4.23, along a subsequence we then have

u ε → u ∞ in H 1 (D) for some u ∞ for which m ∞ (p, p) is attained.
The C l loc convergence for all l ∈ N is obtained by classic elliptic estimates (see [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]).

We now prove that the convergence holds in C l (D) for all l ∈ N. To this end we adapt a strategy of Berlyand and Mironescu (Section 8 in [BM]).

We divide the proof into four steps :

Step 1. We have that |u ε | is uniformly close to 1 near ∂D for large ε Lemma 4.28. Let ρ ε := |u ε |. For all η > 0, there exist δ > 0 and ε 0 > 0 s.t. for all ε ≥ ε 0 and for all z s.t. dist(z, ∂D) < δ it holds that

ρ ε -1 L ∞ < η.
For the proof of this lemma we need the following reformulation of Berlyand&Mironescu (see Lemma 8.3 in [BM]) of a result of Brezis&Nirenberg :

Lemma 4.29 (Theorem A3.2. in [START_REF] Brezis | Degree theory and BMO. I. Compact manifolds without boundaries[END_REF]). Let (g n ) ⊂ VMO(∂D; S 1 ) be s.t. g n → g strongly in VMO(∂D). Then for each 0 < a < 1, there is some δ > 0, independent of n s.t.

a ≤ |ũ(g n )(z)| ≤ 1, if dist(z, ∂D) < δ .
Here ũ(g n ) denotes the harmonic extension of g n to D.

Proof of Lemma 4.28. Let u ε be a minimizer of E ε in I p,p for ε large enough. We write u ε = v ε + w ε with w ε which satisfies

   -∆w ε = 1 ε 2 u ε (1 -|u ε | 2 ) in D w ε = 0 on ∂D, (4.19)
and v ε the harmonic extension of tr ∂D u ε , i.e.,

∆v ε = 0 in D v ε = u ε on D.
(4.20)

To estimate ∇w ε L ∞ (D) we use the standard elliptic estimate Lemma 4.30

(Lemma A.2. in [BBH93]). Let w ∈ C 2 (D) satisfy ∆w = f in D w = 0 on ∂D. (4.21)
Then, for some constant C D depending only on D, we have :

∇w L ∞ (D) ≤ C D w 1/2 L ∞ (D) f 1/2 L ∞ (D) . (4.22)
Thanks to Lemma 4.30 we obtain (note that

w ε L ∞ (D) ≤ u ε L ∞ (D) + v ε L ∞ (D) ≤ 2) ∇w ε L ∞ (D) ≤ √ 2C D × 1 ε (4.23)
where C D is a constant depending only on D.

Thus, since w ε = 0 on ∂D we obtain that there exists a constant C D s.t.

|w ε (z)| ≤ C D 1 ε dist(z, ∂D). (4.24)
We note that, up to a subsequence, tr

∂D u ε → tr ∂D u ∞ strongly in H 1/2 (∂D) because u ε → u ∞ strongly in H 1 (D). Since H 1/2
→VMO in 1D we can apply Lemma 4.29 to obtain that for all η > 0 there exists δ and ε 0 s.t. for all ε ≥ ε 0

1 - η 2 ≤ |v ε | ≤ 1, if dist(z, ∂D) < δ .
Hence we find that

1 ≥ |u ε (z)| ≥ |v ε (z)| -|w ε (z)| ≥ 1 - η 2 -C D 1 ε dist(z, ∂D), if ε ≥ ε 0 and dist(z, ∂D) < δ ≥ 1 -η (4.25) if dist(z, ∂D) < δ := min{δ , ηε 0 C D },
and for all ε ≥ ε 0 .

Step 2. Lifting close to ∂D Now thanks to Lemma 4.28 we know that, for some δ > 0 and for sufficiently large ε, u ε does not vanish in

D + δ := {z ∈ D | dist(z, ∂Ω) < δ} nor in D - δ := {z ∈ D | dist(z, ∂ω) < δ}.
We set

ρ ε = |u ε | and ρ ∞ = |u ∞ |.
Up to a smaller value for δ we may assume that

|u ∞ | ≥ 1 -η in D + δ ∪ D - δ (because u ∞ is smooth in D, see Lemma 4.4 [BM]
).

Therefore we can write u ∞ = ρ ∞ e ıϕ , where ϕ is locally defined and ∇ϕ is globally defined. We can show that ϕ is solution of div(ρ 2 ∞ ∇ϕ) = 0 in D ± δ and ∂ ν ϕ = 0 on ∂D.

In D + δ we have that deg u ε u ∞ , ∂Ω = 0 and deg |u ∞ |u ε |u ε |u ∞ , ∂D + δ \ ∂Ω = 0.
We can thus find

ψ ε ∈ H 1 (D + δ , R) s.t. u ε = ρ ε e ı(ϕ+ψε) in D + δ (with ρ ε = |u ε |).
The same is true in D - δ . In D ± δ the Ginzburg-Landau equation is then equivalent to the following equations on ρ ε and ψ ε :

-∆ρ ε = 1 ε 2 ρ ε (1 -ρ 2 ε ) -ρ ε |∇(ϕ + ψ ε )| 2 in D ± δ ρ ε = 1 on ∂D, (4.26) -div(ρ 2 ε ∇ψ ε ) = div(ρ 2 ε ∇ϕ) in D ± δ ∂ ν ψ ε = 0 on ∂D. (4.27)
We can rewrite the equation on ψ ε in the following way :

∆ψ ε = div[(1 -ρ 2 ε )∇ψ ε -ρ 2 ε ∇ϕ] in D ± δ . (4.28) Step 3. ∇ψ ε is bounded in L 4 (D ± δ )
Fix z 0 ∈ ∂D. In order to simplify the proof we assume that z 0 = 0, D ⊂ {z ∈ C | Im(z) > 0} and ∂D ⊂ R in a neighborhood U of z 0 . These assumptions are not essential for carrying out the arguments below but make the redaction easier. Let r > 0 to be determined later s.t. B r := B(0, r) ⊂ U . Using a reflection argument, we extend ρ ε , ψ ε , ϕ and

F = (1-ρ 2 ε )∇ψ ε -ρ 2 ε ∇ϕ to B r \ D by setting : for z ∈ B r \ D ρε (z) = ρ ε (z), ψε (z) = ψ ε (z), φ(z) = ϕ(z) F (z) = [1 -ρ 2 ε (z)](∂ x ψε , -∂ y ψε ) -ρ 2 ε (z)(∂ x φ, -∂ y φ).
We can then show that ψε is a solution of

∆ ψε = div F (z) in B r . (4.29)
By standard elliptic estimates (see e.g. Theorem 7.1 in [START_REF] Giaquinta | An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs[END_REF]), we have

∇ ψε L 4 (Br) ≤ C 4 tr ∂Br ψε W 1-1 4 ,4 (∂Br)
+ F L 4 (Br) .

(4.30)

By scaling, the constant C 4 does not depend on r. We also have that

F L 4 (Br) ≤ 1 -ρε 2 L ∞ (Br) ∇ ψε L 4 (Br) + ρε 2 L ∞ (Br) ∇ φ L 4 (Br) . (4.31)
Thanks to Lemma 4.28 we can choose r small enough s.t. for ε large enough we have 1 -

ρε L ∞ (Br) < 1 2C 4 . Recalling that ρ ε ≤ 1 we obtain that ∇ ψε L 4 (Br) ≤ 2C 4 tr ∂Br ψε W 1-1 4 ,4 (∂Br) + 2C 4 ∇ φ L 4 (Br) . (4.32)
We can prove that, for a.e. r small enough we have tr ∂Br ψε is bounded in W 1-1 4 ,4 (∂B r ). Indeed, for a.e. r > 0 s.t. B r ⊂ U , tr ∂Br∩D ψε is bounded in H 1 (∂B r ∩ D). This is due to the coarea formula and to the fact that ψ ε is bounded in H 1 (D) (from the strong local convergence of u ε and since

|∇ψ ε | ≤ 1 ρε |∇u ε | + |∇ϕ| ≤ C|∇u ε | + |∇ϕ| in D ± δ ). Using the (continuous) Sobolev injection H 1 (∂B r ) → W 1-1
4 ,4 (∂B r ) we obtain the result. Furthermore ∇ φ L 4 (Br) is a finite constant which does not depend on ε. Thus (up to a subsequence) ∇ψ ε L 4 (Br∩D) is bounded for r small enough.

Repeating the previous argument we find that : for all z 0 ∈ ∂D there exist r z 0 > 0 and

M z 0 > 0 s.t. (up to a subsequence) ∇ψ ε L 4 (Br z 0 ∩D) ≤ M z 0 .
Thanks to the fact that ∂D is compact we deduce that there exist δ 1 > 0, a subsequence and M s.t., letting

D δ 1 = {z ∈ D | dist(z, ∂D) < δ 1 }, we have ∇ψ ε L 4 (D δ 1 ) ≤ M, for ε large enough. (M is independent of ε) Now since u ε → u ∞ in C l
loc for all l ∈ N we obtain that ∇ψ ε is bounded in L 4 (D + δ ) and in L 4 (D - δ ).

Step 4. Elliptic estimates and a bootstrap argument

We work in D + δ but the argument is the same for D - δ . First note that ∇ϕ is bounded in L ∞ (D + δ ). Indeed from Lemma 4.4 in [BM]

we see that u ∞ is in C ∞ (D) and |∇ϕ| = 1 ρ∞ |∇u ∞ | ≤ C|∇u ∞ | ≤ C in D ±
δ for some constants C, C > 0 . We use the equation satisfied by ρ ε (4.26), the fact that ∇ϕ is bounded in L ∞ (D + δ ) and the previous step to obtain that ∆ρ ε is bounded in L 2 (D + δ ). Hence the elliptic regularity implies that ρ ε is bounded in W 2,2 (D + δ/2 ). Indeed one can multiply ρ ε by a cut-off function χ ∈ C ∞ (D + δ ) s.t. χ ≡ 1 in D + δ/2 and χ = 0 on ∂D + δ \ ∂Ω. We then see that ∆(χρ ε ) is bounded in L 2 (D + δ ) and, since the boundary conditions are adapted to global regularity, we deduce that χρ ε is bounded in W 2,2 (D + δ ). Using the fact that χ ≡ 1 in D + δ/2 we obtain the result. Now since W 1,2 → cont L p for all 1 < p < +∞ we have that ∇ρ ε is bounded in L p (D + δ/2 ) for all 1 < p < +∞.

We now use the equation satisfied by ψ ε , (4.27) written as

∆ψ ε = - 2 ρ ε ∇ρ ε • ∇(ψ ε + ϕ) -∆ϕ. (4.33)
We note that 1/ρ ε , ∆ϕ and ∇ϕ are bounded in L ∞ (D + δ ) and we deduce that ∆ψ ε is bounded in L q (D + δ ) for all 1 < q < +∞. Hence using a similar argument as before with a cut-off function we can show that ψ ε is bounded in W 2,q (D + δ/2 ) for all 1 < q < +∞. In particular ∇ψ ε is bounded in W 1,q (D + δ/2 ) for all 1 < q < +∞. Using the fact that W 1,q ∩L ∞ is an algebra (see e.g. Proposition 9.4 in [START_REF]Functional analysis, Sobolev spaces and partial differential equations[END_REF]) we find that ∆ρ ε is bounded in W 1,q (D + δ/2 ) for all 1 < q < +∞ and thus ρ ε is bounded in W 3,q (D + δ/2 ). By a straightforward induction we obtain that ρ ε , ψ ε , are bounded in W m,q (D + δ/2 ) for all m ≥ 2, 1 < q < +∞.

Thanks to Sobolev injections for any l ∈ N and any 0 < γ < 1 we can choose m ≥ 1 and 1 < q < +∞ s.t. 1 -2 q > γ we then have W m,q → C l,γ (D + δ/2 ) and this embedding is compact. We thus have that, up to a subsequence, u ε = ρ ε e ı(ϕ+ψε) → u in C l,γ for some u as ε → ∞ in D + δ/2 . But by Lemma 4.27 we have u = u ∞ . Using the C l loc convergence, we can finally conclude that u ε → u in C l (D) for all l ∈ N.

Non Existence Result

This section is dedicated to the proof of Theorem 4.8. We fix p, q ∈ N * , p = q. For the simplicity of the presentation we assume that p > q. The case p < q is similar.

We adapt here the strategy of Misiats (used to prove Theorem 2 in [START_REF] Misiats | The necessary conditions for the existence of local Ginzburg-Landau minimizers with prescribed degrees on the boundary[END_REF]).

We let d := p -q ∈ N * and through all this section we work with a circular annulus A := B(0, 1) \ B(0, R) where R ∈]0, 1[. We prove that for R sufficiently close to 1 and for large ε, m ε (p, q) is not attained.

Strategy of the proof

By Theorem 4.5, there is R

(1)

q (R (1)
q is independent of ε) s.t. m ε (q, q) is attained by the radial Ginzburg-Landau solution u ε = ρ ε e ıqθ (here ρ ε = ρ ε,q depends also on q see (4.5)&(4.6)).

Because ρ ε > 0 in A, it is easy to see that

I p,q = {ρ ε w | w ∈ I p,q }. Thus we have m ε (p, q) = inf w∈Ip,q E ε (ρ ε w). (4.34)
By Lemma 21 in [START_REF] Berlyand | Solutions with vortices of a semi-stiff boundary value problem for the Ginzburg-Landau equation[END_REF], it holds that for w ∈ I

E ε (ρ ε w) = E ε (u ε ) + L ε (w) (4.35) with L ε (w) = 1 2 A ρ 2 ε |∇w| 2 -q 2 ρ 2 ε |∇θ| 2 |w| 2 + 1 2ε 2 ρ 4 ε (1 -|w| 2 ) 2 . ( 4
m ε (p, q) = E ε (u ε ) + inf w∈Ip,q
L ε (w). (4.37)

We argue by contradiction and we assume that

there is ε = ε n ↑ ∞ s.t. m ε (p, q) is attained by ρ ε w ε .
Our strategy consists in proving that for R sufficiently close to 1, we have

L ε (w ε ) > dπ. (4.38)
Estimates (4.37) and (4.38) imply that m ε (p, q) > m ε (q, q) + dπ, which is in contradiction with Proposition 4.11.2.

The key argument is to find a lower bound of L ε (w ε ) which can be written as a sum of infinitely many infima of functionals (see (4.42)). These functionals have the form

|a k | 2 F k (•)
where the a k 's are the Fourier coefficients of tr S 1 (we -ıqθ ). The F k 's are defined in H 1 (]R, 1[, C) and we impose Dirichlet boundary condition for r = 1 whereas we let the other boundary r = R free (see (4.41)). Note that since the boundary r = R is free we obtain homogeneous Neumann boundary conditions for r = R.

By using some properties of (a k ) k ∈ C Z we apply Lemma 3 in [START_REF] Misiats | The necessary conditions for the existence of local Ginzburg-Landau minimizers with prescribed degrees on the boundary[END_REF] (see Proposition 4.31.3 below) in order to obtain that for large ε we have (4.38).

Asymptotic analysis of

v ε = w ε e -ıqθ
Here we prove that tr S 1 (w ε e -ıqθ ) → 1 in L 2 (S 1 ).

By Lemma 4.21, up to a subsequence, there is P ∈ A (p,q) and u ∞ ∈ I P s.t.

ρ ε w ε u ∞ in H 1 . Moreover u ∞ minimizes E ∞ in I P , m ∞ (p, q) = m ∞ (P ) + π|P -(p, q)|, (4.39)
and we have P = (q , q ) for some 0 ≤ q ≤ q from Proposition 4.19. However for R > R

(1) q we have that q = q. Indeed, recall that for R > R

(1) q , m ∞ (q, q) is uniquely attained by the radial harmonic map and, according to the discussion in Section 3.3 it holds that for all 0 ≤ r < q we have m ∞ (q, q) < m ∞ (r, r) + 2π(q -r).

But if q < q then we find that (using Lemma 4.21)

m ∞ (p, q) = m ∞ (q , q ) + π(p -q) + 2π(q -q ) < m ∞ (q, q) + π(p -q)
which is in contradiction with Proposition 4.19.

Consequently, up to a constant in S 1 , we have that u ∞ = u ∞,q (defined in (4.7)) where u ∞,q (re ıθ ) = 1 1 + R q r q + R q r q e ıqθ .

We now write w ε ∈ I p,q as w ε = v ε e ıqθ with v ε ∈ I d,0 .

From the previous arguments we know that ρ ε w ε = ρ ε v ε e ıqθ u ∞,q = ρ q e ıqθ in H 1 (D) (here we write ρ q instead of ρ ∞,q ). Moreover, from Lemma 4.27, we have ρ ε e ıqθ → ρ q e ıqθ in H 1 (D). Consequently v ε 1 in H 1 (D) and therefore tr S 1 v ε → 1 in L 2 (S 1 ).

Reformulation of L ε (w ε ) and a first lower bound of L ε (w ε )

The argument is based on the Fourier expansion of tr S 1 v ε :

tr S 1 v ε (e ıθ ) = k∈Z a ε k e ıkθ = k∈Z a k e ıkθ .
The Fourier coefficients of tr S 1 v ε satisfy the following conditions :

Proposition 4.31. 1.

k∈Z k|a k | 2 = d. 2. k∈Z * |a k | 2 → 0 when ε → ∞.
3. Let k 0 ∈ N * , then there is C 1 (depending only on k 0 ) and a sequence c ε > 0 (depending only on k 0 and ε) s.t. c ε → 1 when ε → ∞ satisfying for k = 1, ..., k 0 .

|a k | ≤ c ε |a -k | + C 1 l∈Z * |a l | 2 .
Proof. The first assertion is the degree formula written with Fourier coefficients. The second assertion comes from the convergence tr S 1 v ε → 1 in L 2 . The third assertion is Corollary 2 in [START_REF] Misiats | The necessary conditions for the existence of local Ginzburg-Landau minimizers with prescribed degrees on the boundary[END_REF]. We can use it by noting that Lemma 3 in [START_REF] Misiats | The necessary conditions for the existence of local Ginzburg-Landau minimizers with prescribed degrees on the boundary[END_REF] holds.

We go back to the L ε functional. Writing w ε = v ε e ıqθ we have

L ε (ve ıqθ ) = 1 2 A ρ 2 ε q 2 |∇θ| 2 |v| 2 + |∇v| 2 + 2q∇θ • (v ∧ ∇v) - -q 2 ρ 2 ε |∇θ| 2 |v| 2 + 1 2ε 2 ρ 4 ε (1 -|v| 2 ) 2 = 1 2 A ρ 2 ε |∇v| 2 + 2qρ 2 ε ∇θ • (v ∧ ∇v) + 1 2ε 2 ρ 4 ε (1 -|v| 2 ) 2 =: Lε (v) + 1 4ε 2 A ρ 4 ε (1 -|v| 2 ) 2 .
We now focus on the Lε functional and we prove that for sufficiently large ε and for R sufficiently close to 1, we have

L ε (w ε ) ≥ Lε (v ε ) > dπ. (4.40)
To prove (4.40) we switch to polar coordinates (with a little abuse of notation) and we write

v ε (r, θ) = k∈Z a k f k (r)e ıkθ , r ∈]R, 1[, θ ∈]0, 2π[ where f k ∈ H 1 (]R, 1[, C) is s.t. f k (1) = 1.
Note that the map ρ ε depends only on r ∈]R, 1[. Therefore we have the following expansion :

Lε k∈Z a k f k (r)e ıkθ = π k∈Z |a k | 2 1 R ρ 2 ε r|f k | 2 + k 2 + 2qk r |f k | 2 .
For k ∈ Z, and f ∈ H 1 (]R, 1[, C), we let

F k (f ) = 1 R ρ 2 ε r|f | 2 + k 2 + 2qk r |f | 2 and m k = inf F k (f ) | f ∈ H 1 (]R, 1[, C) s.t. f (1) = 1 . (4.41)

Lower bound of Lε (v ε )

It is clear that we have

L ε (w ε ) ≥ Lε k∈Z a k f k (r)e ıkθ ≥ π k∈Z |a k | 2 m k . (4.42)
In order to get a lower bound for m k we use the following lemma :

Lemma 4.32. For ε > 0 we have ρ ε ≥ ρ q , where ρ q (r) = 1 1 + R q r q + R q r q .

Proof. Let ε > 0 and let U = {x ∈ A | ρ ε (x) < ρ q (x)}. We argue by contradiction and we assume that U = ∅. Note that U is a smooth open set and that tr ∂U (ρ ε e ıqθ ) = tr ∂U (ρ q e ıqθ ).

By the minimality of ρ q e ıqθ we have

E ∞ (ρ q e ıqθ , U ) ≤ E ∞ (ρ ε e ıqθ , U ).
On the other hand, by the definition of U and because 0 ≤ ρ ε , ρ q ≤ 1 we have

U (1 -ρ 2 q ) 2 < U (1 -ρ 2 ε ) 2 . Consequently E ε (ρ q e ıqθ , U ) < E ε (ρ ε e ıqθ , U )
and this is in contradiction with the minimality of ρ ε e ıqθ .

From Lemma 4.32, for f ∈ H

1 (]R, 1[, C) F k (f ) ≥        1 R ρ 2 q r|f | 2 + k 2 + 2qk r |f | 2 if k 2 + 2qk > 0 1 R ρ 2 q r|f | 2 + k 2 + 2qk r |f | 2 if k 2 + 2qk ≤ 0.
We let

ρ min = min [R,1] ρ q = 2R q/2 1 + R q .
In order to get (4.40), it suffices to replace the minimization problem m k [defined in (4.41)] by mk where :

• for k ≤ 0 and k 2 + 2qk > 0 mk = ρ 2 min inf 1 R r|f | 2 + k 2 + 2qk r |f | 2 f ∈ H 1 (]R, 1[, C) s.t.f (1) = 1 • for k ≤ 0 and k 2 + 2qk ≤ 0 mk = ρ 2 min inf 1 R r|f | 2 + k 2 + 2qk rρ 2 min |f | 2 f ∈ H 1 (]R, 1[, C) s.t.f (1) = 1 • for k > 0, mk = 1 (1 + R q ) 2 m(1) k + 2R q m(2) k + R 2q m(3) k where m(1) k = inf 1 R r 2q+1 |f | 2 + r 2q-1 (k 2 + 2qk)|f | 2 f ∈ H 1 (]R, 1[, C) s.t. f (1) = 1 , m(2) k = inf 1 R r|f | 2 + k 2 + 2qk r |f | 2 f ∈ H 1 (]R, 1[, C) s.t. f (1) = 1 , m(3) k = inf 1 R r -2q+1 |f | 2 + r -2q-1 (k 2 + 2qk)|f | 2 f ∈ H 1 (]R, 1[, C) s.t. f (1) = 1 .
We first study the cases k ≤ 0. According to the definition of mk we divide the presentation in two parts :

k 2 + 2qk > 0 and k 2 + 2qk ≤ 0.
It is clear that k 2 + 2qk ≤ 0 ⇔ k = -2q, ..., 0. We now treat the case k 2 + 2qk > 0 and k ≤ 0, i.e., k < -2q.

Case I. k < -2q

If k < -2q, it is obvious that mk > 0, (4.43) and this estimate is sufficient for our argument.

Case II. k = -2q, ..., 0

In that case we claim that k 2 + 2qk ≥ -q 2 . Therefore, by a Poincaré type inequality, there is 1 > R

(2)

q ≥ R (1) q s.t. for R (2) q < R < 1 inf f ∈H 1 (]R,1[,C) s.t.f (1)=1 1 R r|f | 2 + k 2 + 2qk rρ 2 min |f | 2 > -∞.
Hence, by direct minimization, the infimum is attained. One can prove that the minimizer of mk is unique and, letting α := k 2 + 2qk ρ 2 min , it satisfies :

   -(rf ) + α r f = 0 for r ∈]R, 1[ f (1) = 1 and f (R) = 0.
By solving the ordinary differential equation we get that

f 0 (r) = A cos( √ -α ln r) + B sin( √ -α ln r).
With the boundary conditions we obtain

f 0 (r) = cos( √ -α ln r) + tan( √ -α ln R) × sin( √ -α ln r).
By using an integration by part it comes that inf

f ∈H 1 (]R,1[,C) s.t.f (1)=1 1 R r|f | 2 + k 2 + 2qk rρ 2 min |f | 2 = f 0 (1)f 0 (1) -f 0 (R)f 0 (R) = f 0 (1) = √ -α tan( √ -α ln R).
Thus, if k = -2q, ..., 1, then we have

mk = ρ min -k 2 -2qk × tan -k 2 -2qk ρ min × ln R .
Consequently, we have for k = -2q, ..., -1

   mk ≥ (k 2 + 2qk)(1 -R) + O[(1 -R) 2 ] m0 = 0.
Thus there is 1 > R

(3)

q ≥ R (2)
q (depending on q) s.t. for 1 > R > R

(3) q we have for k = -2q, ..., -1

   mk ≥ (k 2 + 2qk -δ)(1 -R) m0 = 0. (4.44)
where δ > 0 is a sufficiently small number (we can take for example δ = 10 -6 ).

Case III. k > 0

We study the minimization problems m(l) k for l = 1, 2, 3 and for k > 0.

For l = 1, 2, 3, we have, letting

α = k 2 + 2qk, m(l) k = inf f ∈H 1 (]R,1[,C) s.t.f (1)=1 1 R r β l +1 |f | 2 + r β l -1 α|f | 2 with β l =          2q if l = 1 0 if l = 2
By direct minimization, we can see that m

(l)
k admits a solution. Moreover a solution f l satisfies

   -(r β l +1 f ) + αr β l -1 f = 0 for r ∈]R, 1[ f (1) = 1 and f (R) = 0.
From the ordinary differential equation we get that

f l (r) = A l r s l + B l r t l , A l , B l ∈ C with s l = -β l + β 2 l + 4α 2 and t l = -β l -β 2 l + 4α 2 .
Note that

s l t l = -α and s l -t l = β 2 l + 4α. (4.45)
For the simplicity of the presentation we drop the subscript l.

From the boundary conditions we have

   A + B = 1 AsR s + BtR t = 0 ⇔      A = tR t-s tR t-s -s B = s s -tR t-s .
As for the previous cases it holds that m(l)

k = f l (1) = A l s l + B l t l = s l t l R t l -s l t l R t l -s l -s l + s l t l s l -t l R t l -s l = s l t l (1 -R t l -s l ) s l -t l R t l -s l [by (4.45)] = -α(1 -R - √ β 2 l +4α ) s l -t l R - √ β 2 l +4α
.

In order to handle the expression of m(l) k , we note that for γ ∈ R we have

R γ = 1 -γ(1 -R) + O[(1 -R) 2 ], as R goes to 1.
Therefore, for fixed k ≥ 0, we have (recall that s l -

t l = β 2 l + 4α) m(l) k = -α 1 -1 + β 2 l + 4α(1 -R) + O[(1 -R) 2 ] s l -t l + t l β 2 l + 4α(1 -R) + O[(1 -R) 2 ] = α β 2 l + 4α(1 -R) + O[(1 -R) 2 ] β 2 l + 4α + t l β 2 l + 4α(1 -R) + O[(1 -R) 2 ] = α(1 -R) + O[(1 -R) 2 ].
Consequently, for k ∈ {1, ..., 2q}, we get

mk = (k 2 + 2qk)(1 -R) + O[(1 -R) 2 ]. Thus there is 1 > R (4) q ≥ R (3) q (depending on q) s.t. for 1 > R > R (4) q and k ∈ {1, ..., 2q} we have mk ≥ (k 2 + 2qk -δ)(1 -R), (4.46)
where δ > 0 is the same sufficiently small number ; as in (4.44) we may take δ = 10 -6 . We may also assume that

1 -2q(1 -R) > 0. (4.47)
On the other hand, by noting that q 2 + α = (q + k) 2 and that q, k ≥ 0, we have for fixed R, when k → ∞ :

m(1) k = (k 2 + 2qk)(1 -R 2(q+k) ) kR 2(q+k) + 2q + k = (k + 2q)(1 + o k (1)), (4.48) m(2) k = k 2 + 2qk(1 -R 2 √ k 2 +2qk ) 1 + R 2 √ k 2 +2qk = (k + q)(1 + o k (1)), (4.49) m(3) k
= (k 2 + 2qk)(1 -R 2(q+k) ) k + (2q + k)R 2(q+k) = (k + 2q)(1 + o k (1)). (4.50)
From (4.48), (4.49) and (4.50), we can prove that for 1 > R > R (4) q there is K R ≥ 2q + 2 (depending on R and q) s.t. for k ≥ K R we have that for l = 1, 2, 3 :

m(l) k ≥ k + 1 4 . (4.51)
Consequently from (4.51) we have for k

≥ K R mk = 1 (1 + R q ) 2 m(1) k + 2R q m(2) k + R 2q m(3) k ≥ k + 1 4 . ( 4

Last computations and conclusion

We are now in position to prove (4.40).

On the one hand we have, with (4.42), (4.44) (4.46), (4.52) and Proposition 4.31.1 :

Lε (v ε ) π -d ≥ k∈Z |a k | 2 ( mk -k) ≥ k≤-2q-1 |a k | 2 ( mk + |k|) + -1 k=-2q |a k | 2 (k 2 + 2qk -δ)(1 -R) + |k| + + 2q k=1 |a k | 2 (k 2 + 2qk -δ)(1 -R) -k + K R -1 k=2q+1 |a k | 2 ( mk -k) + + k≥K R |a k | 2 4 = S 1,2q + S 2q+1,K R -1 + S K R ,∞ .
Where

S 1,2q = 2q k=1 |a k | 2 (k 2 + 2qk -δ)(1 -R) -k + + |a -k | 2 (k 2 -2qk -δ)(1 -R) + k , S 2q+1,K R -1 = K R -1 k=2q+1 k(|a -k | 2 -|a k | 2 ) + |a k | 2 mk + |a -k | 2 m-k , S K R ,∞ = k≥K R |a k | 2 4 + |a -k | 2 ( m-k + k).
From (4.43) we have for k ≥ K R > 2q that m-k > 0, then

S K R ,∞ ≥ 1 4 k≥K R {|a k | 2 + |a -k | 2 }. (4.54)
By Proposition 4.31.3, there are C 1 > 0 and c ε > 0 s.t. c ε → ε→∞ 1 and for k ∈ {1, ..., K R } we have

|a k | 2 ≤ c 2 ε |a -k | 2 + 2c ε |a -k |C 1 l∈Z * |a l | 2 + C 2 1 l∈Z * |a l | 2 2 [Proposition 4.31.2] ≤ c 2 ε |a -k | 2 + o l∈Z * |a l | 2 .
Consequently, for k ∈ {1, ..., K R } we have

|a -k | 2 -|a k | 2 ≥ |a -k | 2 (1 -c 2 ε ) + o l∈Z * |a l | 2 [c ε → 1 & Proposition 4.31.2] = o l∈Z * |a l | 2 as ε → ∞. (4.55)
We thus obtain

S 1,2q = 2q k=1 |a k | 2 (k 2 + 2qk -δ)(1 -R) -k +|a -k | 2 (k 2 -2qk -δ)(1 -R) + k = (1 -R) 2q k=1 (|a k | 2 + |a -k | 2 )(k 2 -δ) + + [1 -2q(1 -R)] 2q k=1 k(|a -k | 2 -|a k | 2 ) ≥ (1 -R) 2q k=1 (|a k | 2 + |a -k | 2 )(k 2 -δ) + o l∈Z * |a l | 2 . (4.56)
In the last inequality we used (4.47) and (4.55). Clearly, from (4.44)&(4.53), there is 1 4 > η > 0 (independent of ε) s.t. we have

   mk , m-k > η for k ∈ {2q + 1, ..., K R -1} (1 -δ)(1 -R) > η
and consequently (with (4.55))

S 2q+1,K R -1 ≥ η K R -1 k=2q+1 {|a k | 2 + |a -k | 2 } + o l∈Z * |a l | 2 when ε → ∞. (4.57)
Therefore, by combining (4.54), (4.56) and (4.57) we obtain

Lε (v ε ) π -d ≥ S 1,2q + S 2q+1,K R -1 + S K R ,∞ ≥ η l∈Z * |a l | 2 + o l∈Z * |a l | 2
> 0, for sufficiently large ε.

This last result ends the proof of Theorem 4.8.

Comments and perspectives

In order to prove our results we have made several restrictions on the parameter ε, on the capacity of the domain and on the form of the domain (for Theorem 4.8). We discuss here why these restrictions appear and give some heuristics on the necessity of these restrictions.

In Theorem 4.3 we assumed that the annular domain is "thin" (with large capacity) and that ε is large. In view of Theorem 4.9 of Mironescu (see [START_REF]Size of planar domains and existence of minimizers of the ginzburg-landau energy with semistiff boundary conditions[END_REF]) we know that if the annular domain is "thick" and if ε is small then minimizers of m ε (p, p) do not exist (for p ∈ N * ). However it is an open question to know if minimizers do exist for ε large when the annular domain has small capacity for p > 1. This is indeed the case for p = 1, but for p > 1 even for the Dirichlet energy E ∞ this is not known.

In Theorem 4.8 we also assumed that the annulus is "thin". The main reason for that is the following : in order to prove non existence of minimizers of E ε we want to show that for every v ∈ I p,q E ε (v) > m ε (q, q) + π(p -q)

if p > q. However it is easier to compute the difference E ε (v) -m ε (q, q) when the infimum m ε (q, q) is attained. Indeed we can then use a decomposition Lemma (see (4.38)). For example when m ε (1, 1) is not attained we know that m ε (1, 1) = 2π thanks to the Price Lemma 4.15. Thus in order to prove non existence of minimizers in I p,1 for p > 1 one could try to show that

E ε (v) > 2π + π(p -1)
for all v ∈ I p,1 .

Other technical reasons appear in the process of the proof of Theorem 4.8. In [START_REF] Misiats | The necessary conditions for the existence of local Ginzburg-Landau minimizers with prescribed degrees on the boundary[END_REF] the author was able to get rid of the technical restrictions on the size of the domain. Its argument does not apply in our case, this is mainly due to the fact that |u ε | does not converge to 1 (or to a constant) when ε → +∞. The restriction on the shape of the domain in Theorem 4.8 also comes from the fact that |u ε | does not converge to a constant as ε → +∞. More precisely we used in a crucial way that ρ q ε > ρ q = |u q ∞ | in the proof of the Theorem. We also used that ρ q ε only depends on r in order to use a decomposition in Fourier series. We did not obtain analogous results in the case of a general annular domain. However we believe that Theorem 4.8 holds for all annular type domain regardless of the shape or of the size.

Here tr |∂Ω u denotes the trace operator. If u ∈ I then g = tr |∂Ω u ∈ W 1-1 n ,n (∂Ω, S n-1 ). It holds that (cf. example 2 in [START_REF] Brezis | Degree theory and BMO. I. Compact manifolds without boundaries[END_REF])

W 1-1 n ,n (∂Ω, S n-1 ) → VMO(∂Ω, S n-1 ).
Since Brézis and Nirenberg developed a degree theory for maps in VMO (cf. [START_REF] Brezis | Degree theory and BMO. I. Compact manifolds without boundaries[END_REF], [START_REF]Degree theory and BMO. II. Compact manifolds with boundaries[END_REF]), we can conclude that maps in the space I have a well-defined (topological) degree on the boundary of Ω. This degree generalizes the classical Brouwer degree defined for maps in C 0 (∂Ω, S n-1 ). We denote the degree by deg(u, ∂Ω) or deg(g, ∂Ω) and we will come back to the definition and properties of this quantity in the next section. For d ∈ N * , we let

I d = {u ∈ I; deg(u, ∂Ω) = d}.
For ε > 0, we consider the simplified n-Ginzburg-Landau energy

E ε (u) = 1 n Ω |du| n + 1 4ε n Ω (1 -|u| 2 ) 2 and the n-Dirichlet energy E ∞ (u) = 1 n Ω |du| n .
Here du denotes the differential of u (this is a n × n matrix) and |du| = tr(du t du) = n i=1 |∂ x i u| 2 is the Hilbert-Schmidt norm of this matrix.

In dimension n = 2, the simplified Ginzburg-Landau equations with Dirichlet boundary conditions have been extensively studied (see [START_REF] Bethuel | Progress in Nonlinear Differential Equations and their Applications[END_REF] and the references therein). In [START_REF] Berlyand | Symmetry breaking in annular domains for a ginzburg-landau superconductivity model[END_REF], [START_REF]Ginzburg-Landau minimizers with prescribed degrees. Capacity of the domain and emergence of vortices[END_REF] the authors suggested to relax the Dirichlet boundary condition by prescribing only the modulus of the function u on the boundary and its topological degree. The Ginzburg-Landau equations are used in a physical context to describe superconductivity phenomenon. In this context, both quantities have physical meaning : |u| 2 represents the density of Cooper pair of electrons and deg(u, ∂Ω) is the vorticity of superconducting currents. They proposed that as an intermediate model between the simplified Ginzburg-Landau (G.L in short) equations and the full Ginzburg-Landau equations with magnetic field (see [START_REF] Sandier | Progress in Nonlinear Partial Differential Equations Birkhauser[END_REF] for more on the G.L equations with magnetic field). Their model was called semi-stiff because it is somehow intermediate between the Dirichlet problem (referred as soft in the literature) and the homogeneous Neumann problem (called stiff). There is now an abundant literature concerning the semi-stiff boundary conditions in 2D

[BV02], [GB02], [BM06], [BGR06], [BM], [BR10], [DS09], [BMRS14], [LP14]
, [START_REF]Size of planar domains and existence of minimizers of the ginzburg-landau energy with semistiff boundary conditions[END_REF]. The difficulty of this model is that it leads to a problem of lack of compactness. Indeed the degree is not continuous with respect to weak W 1-1 n ,n convergence. Thus it is not direct to prove that the G.L equations with semi stiff boundary conditions have non-trivial solutions.

There has been some interests in the literature for the study of the simplified n-Ginzburg-Landau energy with Dirichlet boundary condition (see, e.g. [START_REF] Strzelecki | Asymptotics for the minimization of a Ginzburg-Landau energy in n dimensions[END_REF], [GSZ]). Indeed in dimension n = 2 it is shown in [START_REF] Bethuel | Progress in Nonlinear Differential Equations and their Applications[END_REF] that solutions of the Ginzburg-Landau equations converge (in an appropriate sense) to harmonic maps. A possible generalization of harmonic maps to higher dimensions is the notion of n-harmonic maps, i.e. critical points of the n-Dirichlet energy. As the Dirichlet energy for n = 2, the n-Dirichlet energy is conformally invariant. The papers [START_REF] Strzelecki | Asymptotics for the minimization of a Ginzburg-Landau energy in n dimensions[END_REF], [START_REF] Hong | Asymptotic behavior for minimizers of a Ginzburg-Landau-type functional in higher dimensions associated with n-harmonic maps[END_REF], [START_REF] Han | Degenerate elliptic systems and applications to Ginzburg-Landau type equations. I[END_REF], [GSZ] study the convergence of solutions of the n-Ginzburg-Landau solutions to n-harmonic maps.

In this paper we relax the Dirichlet boundary condition and investigate the existence of critical points of the n-G.L energy with semi-stiff boundary conditions. In other words we are looking for solutions of the following equations :

       -∆ n u = 1 ε n (1 -|u| 2 )u in Ω, |u| = 1 on ∂Ω, ∂ ν u |u| = 0 on ∂Ω.
(5.1)

Here ∆ n stands for the n-Laplacian : ∆ n u = div |du| n-2 du and ν stands for the outer unit normal to ∂Ω. Several difficulties arise in the study of equation (5.1), the most important one being the lack of compactness of the problem. In order to produce non-trivial solutions the first idea is to minimize the energy E ε in the class I d for d = 0 (the sets I d are open for the topology induced by the W 1,n norm cf. Appendix of [START_REF] Boutet De Monvel-Berthier | A boundary value problem related to the Ginzburg-Landau model[END_REF]). However we show in Section 5.3 that the energy does not attain its infimum on these subsets.

Theorem 5.1.

The infimum of E ε in I d is never attained for d ∈ Z * , ε > 0.
A natural question is then : do there exist minimax critical points of the n-G.L energy ? In the euclidean ball, we can show that there exists a rotationally symmetric solution of the n-G.L energy and that solution is a solution of (5.1) (see e.g. [START_REF] Lei | Uniqueness for radial Ginzburg-Landau type minimizers[END_REF]). For n = 2, in [START_REF] Berlyand | Minimax critical points in Ginzburg-Landau problems with semi-stiff boundary conditions : existence and bubbling[END_REF], the authors have proved the existence of minimax critical points of E ε with prescribed degrees one in general simply connected domains (for large ε). This paper was complemented by a result of Lamy and Mironescu in [START_REF] Lamy | Existence of critical points with semi-stiff boundary conditions for singular perturbation problems in simply connected planar domains[END_REF] which show that there also exist solutions of the semi-stiff problem for small ε in simply connected domains. In this chapter for n ≥ 3 we prove that there exist minimax critical points of a perturbed energy

E α ε = 1 n Ω |du| n+α + 1 4ε n Ω (1 -|u| 2 ) 2 in the space I α = {v ∈ W 1,n+α (Ω, R n ); | tr |∂Ω v| = 1}.
We can show that the energies E α ε satisfy the Palais-Smale conditions and critical points are obtained using the Mountain Pass Theorem of Ambrosetti-Rabinowitz. Besides they have a degree equals to one on the boundary ∂Ω.

Theorem 5.2. There exists α 0 > 0 such that, for all 0 < α < α 0 , there exist minimax critical points of E α ε which satisfy

       -n+α n ∆ n+α u α = 1 ε n 1 -|u α | 2 u α in Ω, |u α | = 1 on ∂Ω, ∂ ν uα |uα| = 0 on ∂Ω (5.2)
and deg(u α , ∂Ω) = 1.

A natural next step would be to study the asymptotic behavior of the solutions u α obtained as α goes to 0. However this analysis would require a blow-up analysis together with global regularity theory and estimates for solutions u α . The idea of using a perturbed energy in the study of variational problems with lack of compactness which lie in a "limiting case" of the Sobolev injections is due to Sacks and Uhlenbeck in [START_REF] Sacks | The existence of minimal immersions of 2-spheres[END_REF]. It has been used several times in other variational problems (see e.g. [START_REF] Struwe | On a free boundary problem for minimal surfaces[END_REF], [START_REF] Fraser | On the free boundary variational problem for minimal disks[END_REF]).

Our paper is organized as follows. In Section 2, we recall the definition and properties of the boundary degree for maps in I. In Section 3 we consider the minimization of the n-Ginzburg-Landau energy and the n-Dirichlet energy in the sets I d . Section 4 is devoted to the existence of minimax critical points of the perturbed energies E α ε . We then conclude by giving some comments on the perspective of studying the asymptotic behavior of these critical points as α goes to zero.

Definition and properties of the degree

We recall here the definition and the basic properties of the degree used in the following. We refer to [START_REF] Brezis | Degree theory and BMO. I. Compact manifolds without boundaries[END_REF], [START_REF]Degree theory and BMO. II. Compact manifolds with boundaries[END_REF], [START_REF] Mironescu | Sobolev maps on manifolds : degree, approximation, lifting[END_REF], and [START_REF]Degree theory : old and new, Topological nonlinear analysis[END_REF] for more on the degree theory.

Let X, Y be two (n -1)-dimensional smooth riemannian manifolds. We assume that they are compact, closed and connected. The case Y = S n-1 is of special interest for us and we can think of X = Y = S n-1 in a first time. For a smooth (C 1 is sufficient) map we can define its topological degree by Definition 5.3. Let σ be a (n -1) differential form on Y . Let u : X → Y be a smooth map. Then the degree of u is

deg(u, X) = 1 Y σ X u * σ, (5.3)
where u * σ denotes the pull-back of σ by u.

We can show that the degree is an integer. It is independent of the chosen differential form and it counts the number of times Y is covered by u(X) taking into account algebraic multiplicity. We can extend the definition of the degree to continuous maps form X to Y thanks to the following fact

: if u, v ∈ C 1 (X, Y ) are such that u -v L ∞ (X) is small then deg(u, X) = deg(v, X).
The degree of a continuous map is then defined by approximation by smooth maps (recall that

C ∞ (X, Y ) is dense in C 0 (X, Y )).
In the following we take Y = S n-1 , we set

ω = dx 1 ∧ ... ∧ dx n
the volume form on R n , and

σ = n j=1 (-1) j-1 x j dx 1 ∧ ... ∧ d x j ∧ ... ∧ dx n .

Definition and properties of the degree 123

Here d x j means that we omit the term dx j in the wedge product. The differential form σ is the classical volume form on S n-1 and it holds that ω = 1 n dσ, where d denotes the exterior differential. We can rewrite the formula (5.3) as

deg(u, X) = 1 |S n-1 | X Jac u(x)dσ(x), (5.4)
where Jac u(x) is the Jacobian of u computed using geodesic normal coordinates at x and geodesic normal coordinates at u(x). We also have

deg(u, X) = 1 |S n-1 | X det(u, u τ 1 , u τ 2 , ..., u τ n-1 ), (5.5)
with (τ 1 (x), ..., τ n (x)) an orthonormal basis of the tangent space T u(x) S n-1 . Now, suppose that X = ∂Ω, with Ω a smooth bounded domain of R n . Let ũ be any C 1 extension to Ω of u. Then using the Stokes's Theorem we obtain

deg(u, X) = 1 |S n-1 | X u * σ = 1 |S n-1 | Ω d[ũ * σ] = 1 |S n-1 | Ω ũ * dσ = n |S n-1 | Ω ũ * ω. But since |B n | = S n-1
n we obtain Lemma 5.4. Let u : X → S n-1 a C 1 map, with X = ∂Ω and Ω a smooth bounded domain in R n . Let ũ be any C 1 extension of u to Ω then

deg(u, ∂Ω) = 1 |B n | Ω Jac u(x)dx = 1 |B n | Ω u x 1 ∧ ... ∧ u xn dx, (5.6)
where Jac u(x) = det (du(x)), and dx is the Lebesgue measure on R n .

Note that this formula makes sense for maps u ∈ W 1,n (Ω, R n ). In the case n = 2, Boutetde-Monvel and Gabber defined a degree for maps u ∈ W 1-1 2 ,2 (S 1 , S 1 ) = H 1/2 (S 1 , S 1 ) (see [START_REF] Boutet De Monvel-Berthier | A boundary value problem related to the Ginzburg-Landau model[END_REF]). In the case n = 2 formula (5.3) becomes

deg(u, S 1 ) = 1 2π S 1 u ∧ u τ dτ = u, u τ H 1/2 ,H -1/2 .
(5.7)

The right hand-side shows that we can interpret the degree formula using the H 1/2 , H -1/2 duality. Then the formula makes sense for maps u ∈ H 1/2 . Furthermore Boutet-de-Monvel and Gabber proved that C ∞ (S 1 , S 1 ) is dense in H 1/2 (S 1 , S 1 ). Hence, via an approximation argument, we can see that the quantity u, u τ H 1/2 ,H -1/2 is an integer for u ∈ H 1/2 (S 1 , S 1 ).

Here is a slightly different approach used to define a degree for maps u ∈ W 1-1 n ,n (X, S n-1 ) for n ≥ 2 and X = ∂Ω with Ω a smooth simply connected domain in

R n . Let ũ ∈ W 1,n (Ω, R n ) be any extension of u. We can set deg(u, ∂Ω) = 1 |B n | Ω Jac ũ(x)dx, (5.8)
since the right hand side makes sense for maps in W 1,n (Ω, R n ). Then we use the fact that

C 1 (X, S n-1 ) is dense in W 1-1 n ,n (X, S n-1 ) and thus C 1 (Ω, R n ) ∩ {v ∈ W 1,n (Ω, R n ); tr |∂Ω u ∈ S n-1 } is dense in {v ∈ W 1,n (Ω, R n ); tr |∂Ω u ∈ S n-1 }. Moreover the quantity 1 |B n | Ω Jac u(x)
is continuous with respect to the W 1,n -norm. Hence the quantity (5.8) is an integer and the degree of maps u ∈ W 1-1 n ,n (X, S n-1 ) is well-defined. Note that we the degree is naturally defined for maps in Sobolev spaces W 1-s,n (X, S n-1 ) for sn > n -1 because of the Sobolev injections W s,n (X, S n-1 ) → C 0 (X, S n-1 ). The case s = 1 -1 n is thus a limiting case. Moreover we can show (see e.g. [START_REF] Mironescu | Sobolev maps on manifolds : degree, approximation, lifting[END_REF]) that we can not define a reasonable notion of degree for maps in W s,n (X, S n-1 ) for sn < n -1 because we can construct a sequence of C 1 maps of degree 1 which converge strongly in W 1-s,n to a constant. As stated in introduction we can also define the degree for maps u ∈ W 1-1 n ,n (X, S n-1 ) via the injection W 1-1 n ,n (X, S n-1 ) → VMO(X, S n-1 ) and the definition of the degree for maps in VMO (see [START_REF] Brezis | Degree theory and BMO. I. Compact manifolds without boundaries[END_REF], [START_REF]Degree theory and BMO. II. Compact manifolds with boundaries[END_REF]). The two definitions coincide. To sum up we can state Proposition 5.5 (Theorem 2.2 in [START_REF] Mironescu | Sobolev maps on manifolds : degree, approximation, lifting[END_REF]). Let X be a (n -1) dimensional smooth riemanian manifold closed compact and connected. There is a degree ( equivalently the degree of smooth maps is continuous with respect to the W s,p -norm ) in W s,n (X, S n-1 )if and only if sp ≥ n -1.

Recall that we set I = {u ∈ W 1,n (Ω, R n ); | tr |∂Ω u| = 1} and I d = {u ∈ I; deg(u, ∂Ω) = d}. As in C 0 the degree can be used to characterize the arc-wise connected components of the space I (see the introduction of [START_REF] Brezis | Degree theory and BMO. I. Compact manifolds without boundaries[END_REF]).

Proposition 5.6. The sets I d are the connected components of I. They are open and close in I for the topology induced by the W 1,n -norm. We have

I = d∈Z I d .
The latter proposition shows that if a minimizer of E ε (or E ∞ ) in some I d exists then it is a local minimizer of E ε in I and thus a critical point of E ε in I. However we can not produce minimizer of E ε in I d using the direct method of calculus of variations because of the following Proposition 5.7. The degree is not continuous with respect to the weak W 1-1 n ,n (∂Ω, S n-1 ) convergence.

Proof. Here we take Ω

= B n . Let a ∈ B n , we set ψ a (x) = a + (1 -|a| 2 ) a-x |a-x| 2 and M a (x) = ψ a (x) |ψ a (x)| 2 .
(5.9)

M a is called a Möbius transform. It is a conformal map (i.e., its differential is a similitude), and it maps the B n onto B n and S n-1 onto S n-1 . These maps play an important role in this work. Since M a is conformal we can show that

B n |dM a (x)| n dx = n n 2 B n Jac u(x)dx = n n 2 |B n | (cf. Lemma 5.8). Now let (a k ) ⊂ B n such that lim n→+∞ a n = a ∈ S n-1 .
Then it holds that M an converges weakly to a constant map identically equals to a in W 1,n . This means that M an |S n-1 converges weakly in W 1-1 n ,n to a constant. But since M an is a bijection from S n-1 to S n-1 we have deg(M an , S n-1 ) = 1 for all n ∈ N, however deg(a, S n-1 ) = 0.

Hence finding critical points of E ε or E ∞ falls into the category of problems with lack of compactness.

Minimization with prescribed degrees

In this section we prove that in general there are no minimizers of E ε or E ∞ with prescribed degrees. First we obtain a lower bound of the energy in the class

I d . Lemma 5.8. Let Ω be as before. Let u ∈ W 1,n (Ω, R n ) then Ω |du| n ≥ n n 2 | Ω Jac u|.
(5.10)

Hence if u ∈ I d , thanks to formula (5.6), it holds that

E ε (u) ≥ E ∞ (u) = 1 n Ω |du| n ≥ n n 2 -1 |B n |d.
(5.11)

Moreover equality holds in the previous inequality if and only if u is conformal.

Proof. We have Jac u = u x 1 ∧ ... ∧ u xn . Hence thanks to the Hadamard inequality it comes

| Jac u| ≤ |u x 1 ||u x 2 |...|u xn | (5.12)
with equality if and only if (u x 1 , ..., u xn ) is an orthogonal family. But it also holds that

|u x 1 ||u x 2 |...|u xn | ≤ 1 n n i=1 |u x i | 2 n 2 = 1 n n/2 |du| n .
(5.13) Indeed let us show (5.13). If one of the u x i 's vanishes then the inequality is true. Assume that none of the u x i 's vanishes then :

(5.13)

⇔ n i=1 ln |u x i | ≤ n 2 ln 1 n n i=1 |u x i | 2 ⇔ n i=1 ln |u x i | 2 ≤ n ln 1 n n i=1 |u x i | 2 .
Thanks to the concavity of the logarithm it holds that ln 1 n

n i=1 |u x i | 2 ≥ 1 n n i=1 ln |u x i | 2
with equality of and only if |u x i | = |u x j | for all i = j. This proves that | Jac u| ≤ 1 n n/2 |du| n with equality if and only if (u x 1 , ..., u xn ) is an orthogonal family such that |u

x 1 | = |u x 2 | = ... = |u xn |.
These two conditions mean precisely that du(x) is a similitude and then u conformal.

Next we prove that we can construct test maps which show that the lower bound previously obtained is in fact the infimum of the energies E ε . For all 0 < ε ≤ +∞, we set

m ε (d, Ω) = inf{E ε (v); v ∈ I d }.
It is obvious that we have m ε (0, Ω) = 0. Lemma 5.9. Let 0 < ε ≤ +∞. Let δ > 0 and u ∈ I d . Then there exists v ∈ I d such that

E ε (v) ≤ E ε (u) + n n 2 -1 |B n ||d -d | + δ.
(5.14)

In particular it holds that

m ε (d ) ≤ m ε (d) + n n 2 -1 |B n ||d -d |, for all (d, d ) ∈ Z. (5.15)
As a consequence of Lemma 5.8 and Lemma 5.9 we can state Theorem 5.10. Let d ∈ Z * and Ω be a smooth bounded simply connected domain of R n . Then

m ε (d, Ω) = m ∞ (d, Ω) = n n 2 -1 |B n ||d -d |.
(5.16)

Besides i) If ε < +∞ then m ε (d, Ω) is never attained. ii) The infimum m ∞ (d, Ω
) is attained if and only if Ω is a euclidean ball and |d| = 1. In this case the minimizers are exactly the conformal transformations of the ball (i.e., the Möbius maps).

Before proving this theorem let us state a Theorem due to Liouville which describes all the conformal transformations of the Euclidean space R n for n ≥ 3.

Theorem 5.11 (Liouville). Every conformal transformation of a domain of Euclidean space of dimension n ≥ 3 is a composite of isometries, dilations and inversions.

A proof of this Theorem, under some additional smoothness assumption can be found in [START_REF] Dubrovin | Modern geometry-methods and applications[END_REF] Theorem 15.2 p. 138. The smoothness assumption can be removed (see [START_REF] Iwaniec | Geometric function theory and non-linear analysis[END_REF] Chapter 5).

Proof of Theorem 5.10. Thanks to Lemma 5.8 we have m

ε (d, Ω) ≥ m ∞ (d, Ω) ≥ |d|n n 2 -1 |B n |.
However using the fact that m ε (0, Ω) = 0 and Lemma 5.9 we obtain that m

ε (d, Ω) ≤ |d|n n 2 -1 |B n |. Hence m ε (d, Ω) = m ∞ (d, Ω) = |d|n n 2 -1 |B n |. Now let us assume that there exists u ∈ I d such that E ε (u) = m ε (d, Ω
). Using Lemma 5.8 again we obtain that u is conformal and that 1 4ε n Ω (1 -|u| 2 ) 2 = 0. That is |u| ≡ 1 a.e. in Ω. However the conformal maps between open sets of R n are given by the Liouville Theorem, and this condition is not compatible with |u| = 1 a.e. in Ω unless u is a constant. This proves m ε (d, Ω) is never attained if d = 0. The same argument shows that if m ∞ (d, Ω) is attained by a map u, then u is a conformal map. But by using the Liouville Theorem again there exists a conformal map u : Ω → B n such that u(∂Ω) ⊂ S n-1 if and only if Ω is an euclidean ball. Since all the non-constant conformal maps given by the Liouville Theorem are bijective, they have a degree equal to ±1 (the sign depending on the fact that the map preserves or reverses the orientation).

In order to conclude the Theorem we need to give a proof of Lemma 5.9 :

Proof of Lemma 5.9. Let x 0 ∈ ∂Ω. The idea is to replace u in a small neighbourhood of x 0 by a map which is almost conformal. This map must have a very concentrated energy and its image must be almost all B n . We first show that there exists

v ∈ I d+1 such that E ε (v) ≤ E ε (u) + n n 2 -1 |B n | + δ,
for δ small. Without loss of generality we can assume that x 0 = (0, ..., 0, 1) =: e n and ν(x 0 ) = e n where ν denotes the unit outer normal to ∂Ω. This is always possible, up to translating and rotating Ω. By density of smooth maps in the space I we can also suppose that u is smooth and, up to multiply u by a suitable rotation, we can suppose that u(x 0 ) = e n .

Step1 : The case where ∂Ω is flat near x 0 .

We first assume that there exists σ small enough such that B(x 0 , 2σ) ∩ ∂Ω = {x ∈ R n ; x n = 1} ∩ B(x 0 , 2σ). We also assume that Ω ⊂ {x ∈ R n ; x n < 1}. We consider the following conformal maps

I σ : R n \ 0 → R n x → σ(x-en) |x-en| 2 P : R n \ { 3 2 e n } → R n x → -e n - x-3 2 en |x-3 2 en| 2 . It holds that i) P maps the half-space {x ∈ R n ; x n < 1} onto B n ii) P (e n ) = e n , dP (e n ) =       -4 0 • • • 0 0 4 • • • 0 . . . 0 . . . . . . 0 0 • • • 4      
iii) I σ and P reverse the orientation so that ψ σ := P • I σ preserves the orientation.

We let τ = σ + σ 1+ 1 2(n-1) . In the sequel we use the notation B(x 0 , r) = B r for r > 0. We then set

v σ =          ψ σ (x) if x ∈ B σ ∩ Ω L σ (x) if x ∈ (B τ \ B σ ) ∩ Ω u(x) if x ∈ Ω \ B τ .
(5.17)

Here L σ is defined such that L σ (x) = ψ σ (x), for all x ∈ ∂B σ ∩Ω, L σ (x) = u(x) for all x ∈ ∂B τ ∩Ω and |L σ (x)| = 1 for x ∈ ∂Ω ∩ (B τ \ B σ ). We take L σ (x) = r -σ τ -σ |u(τ θ)| + τ -r τ -σ |ψ σ (σθ)| r-σ τ -σ u(τ θ) + τ -r τ -σ ψ σ (σθ) | r-σ τ -σ u(τ θ) + τ -r τ -σ ψ σ (σθ)| .
In this expression r = |x -e n | and θ = x-en |x-en| . We can see that L σ is well-defined for σ small enough. Indeed by using Taylor's expansion Theorem we have

u(τ θ) = u(x 0 ) + O(τ ) = e n + O(σ) ψ σ (σθ) = ψ σ (x 0 ) + O(σ) = e n + O(σ).
Hence, for σ small enough

r -σ τ -σ u(τ θ) + τ -r τ -σ ψ σ (σθ) = u(0) + O(σ) = 0.
Computation of the energy of v σ . We have that

E ε (v σ ) = E ε (u) -E ε (u, B τ ∩ Ω) + E ε (L σ , (B τ \ B σ ) ∩ Ω) + E ε (ψ σ , B σ ∩ Ω).
(5.18) * Since ψ σ is conformal, its n-Dirichlet energy is equal to the area of its image counted with multiplicity. The map I σ sends B σ ∩ Ω onto {x ∈ R n ; x n < 1} \ B σ because I σ is an inversion. Now, using the fact that dP (e n ) is a reflection multiplied by a dilatation, for σ small we obtain that the image of {x ∈ R n ; x n < 1} \ B σ is closed to B n \ B(e n , 4σ) (it is closed up to order σ 2n ).

We thus obtain that

E ∞ (ψ σ , B σ ∩ Ω) = n n 2 -1 |B n | -C|B N |σ n + O(σ n+1 ). (5.19) * We now estimate E ε (L σ , (B τ \ B σ ) ∩ Ω).
To this end we let

R σ = r-σ τ -σ u(τ θ) + τ -r τ -σ ψ σ (σθ) | r-σ τ -σ u(τ θ) + τ -r τ -σ ψ σ (σθ)|
and we compute the radial and the tangential parts of the differential of L σ . It holds that

∂ r L σ = 1 τ -σ (|u(τ θ)| -|ψ σ (σθ)|) R σ + r -σ τ -σ |u(τ θ)| + τ -r τ -σ |ψ σ (σθ)| ∂ r R σ .
Using the Taylor's expansions we obtain that |u(τ

θ)| -|ψ σ (σθ)| = O(σ). Note that for V ∈ C ∞ (Ω, R n \ {0}) we have |∂ r V |V | | ≤ 2 |∂ r V | |V | .
Hence we obtain that

|∂ r R σ | ≤ O(σ) C(τ -σ) since | r-σ τ -σ u(τ θ) + τ -r τ -σ ψ σ (σθ)| ≥ C, for some C > 0 independent of σ for σ small enough. We can conclude that |∂ r L σ | = O(σ) (τ -σ)
and

(Bτ \Bσ)∩Ω |∂ r L σ | n = O(σ n ) σ n+ n 2(n-1) (τ n -σ n ) = O(σ n ) σ 1+ 1 2(n-1) σ 1 2(n-1) = O(σ n-1 ). Similar computations lead to |∇ tan L σ | = O(σ), where ∇ tan L σ denotes the tangential part of the differential of L σ . Thus (Bτ \Bσ)∩Ω |∇ tan L σ | n = O(σ n ) and (Bτ \Bσ)∩Ω |dL σ | n = O(σ n-1 ).
* Note that the potential term is easy to control. Indeed we have that

|v σ | ≤ 1 in Ω ∩ (B τ \ B σ ). Hence 1 4ε n Ω (1 -|v σ | 2 ) 2 = 1 4ε n Ω (1 -|u| 2 ) 2 - 1 4ε n Bτ ∩Ω (1 -|u| 2 ) 2 + 1 4ε n Bτ ∩Ω (1 -|v σ | 2 ) 2 ≤ 1 4ε n Ω (1 -|u| 2 ) 2 + O(σ n ) since we have | Bτ ∩Ω (1 -|v σ | 2 ) 2 | ≤ Cτ n = O(σ n ).
To sum up we obtained that

E ε (v σ ) ≤ E ε (u) + n n 2 -1 |B n | + O(σ).
Computation of the degree of v σ . Now we show that deg(v σ , ∂Ω) = deg(u, ∂Ω) + 1 for σ small enough.

Recall that deg(v σ , ∂Ω) = 1 |B| n Ω Jac v σ . Thus deg(v σ , ∂Ω) = 1 |B n | Ω\Bτ Jac u + Bσ∩Ω Jac Φ σ + Bτ \Bσ Jac L σ = deg(u, ∂Ω) -o σ (1) + 1 |B n | (|B n | -C|B n |σ n ) + o σ (1) = deg(u, ∂Ω) + 1 + o σ (1).
Here we used the fact that

| Bτ \Bσ Jac L σ | ≤ 1 n n/2-1 E ∞ (L σ , (B τ \ B σ ) ∩ Ω) = o σ (1)
. Since the degree is an integer, for σ small enough we obtain the result.

Step 2 : The case where ∂Ω is not flat near x 0 .

We still assume that x 0 = e n and ν(x 0 ) = e n . The tangent space to ∂Ω at x 0 is then given by T x 0 ∂Ω = {x ∈ R n ; x n = 1} =: H. Since we assumed that ∂Ω is smooth there exists τ 0 > 0 small and a C 1 diffeomorphism Φ : B τ 0 ∩ H → Ω such that * Φ(e n ) = e n and * dΦ(e n ) = Id. We let B - τ := B τ 0 ∩ H and V τ := Φ(B - τ ), for 0 < τ < τ 0 . There exists a constant

C > 0 such that if τ < τ 0 2 then Φ -Id C 1 (B - τ ) ≤ Cτ Φ -1 -Id C 1 (Vτ ) ≤ Cτ.
(5.20)

Let v : V τ → R n smooth, we set ṽ = v • Φ : B - τ → R n .
We then have, using (5.20) :

E ∞ (v, V τ ) = 1 n Vτ |d(ṽ • Φ)| n = 1 n Vτ |dṽ(Φ(x)).dΦ(x)| n dx = 1 n Vτ |dv(Φ(x))| n dx + O(τ ) 1 n Vτ |dv(Φ(x))| n dx
(5.21)

Note that Vτ |dv(Φ(x))| n dx ≤ Cτ n since v is smooth. We use the following change of variable y = Φ(x), we then obtain :

E ∞ (v, V τ ) = 1 n B - τ |dv(y)| n | Jac Φ(y)| n dy + O(τ n+1 ). But | Jac Φ(y)| = 1 + O(τ ) for y ∈ B - τ , hence E ∞ (v, V τ ) = 1 n B - τ |dv(y)| n dy + O(τ n+1 ).
By using a similar argument we can show that

Vτ 1 -|v(x)| 2 2 dx = B - τ 1 -|v(y)| 2 2 dy + O(τ n+1 ). (5.22) Let u ∈ I d , since C ∞ ∩ I d is dense in I d we can assume that u is smooth. Then E ε (u) = E ε (u, V τ ) + E ε (u, Ω \ V τ ) = E ε (ũ, B - τ ) + E ε (u, Ω \ V τ ) + O(τ n+1 ).

But using

Step 1, we can find χτ :

B - τ → R n such that E ε ( χτ , B - τ ) = |B n |n n 2 -1 + O(τ ) (
χτ contains the "bubble" and the interpolation part). For τ = σ + σ 1+ 1 2(n-1) and σ small we then set χ τ = χτ • Φ -1 and

v σ =    χ τ if x ∈ Ω ∩ V τ u if x ∈ Ω \ V τ .
We thus have

E ε (v σ ) ≤ E ε (u) + n n 2 -1 |B n | + O(σ)
and deg(v σ , ∂Ω) = deg(u, ∂Ω) + 1 + O(σ). thus for σ small enough v σ satisfied the desired properties.

Since this is a local construction we can repeat it near several different points in order to increase the degree by several units. We can also decrease the degree by using maps which reverse the orientation.

Mountain Pass Approach for a perturbed problem

Here we assume that Ω is a smooth simply connected domain C 1 diffeomrophic to the unit euclidean ball B n . Since there are no minimizers of E ε in I d for d ∈ Z * we can wonder if there exist non-minimizing solutions of (5.1). In this section we show that there exist minimax critical points of a perturbed energy which have degree one on the boundary. Let α > 0 small, we consider the following energy

E α ε = 1 n Ω |du| n+α + 1 4ε n Ω (1 -|u| 2 ) 2 .
This energy is well defined in the space I α = {v ∈ W 1,n+α (Ω, R n ); | tr |∂Ω v| = 1}. Using the Sobolev injections we have that I α → C 0,β (Ω, R n ), with β = 1 -n n+α . Hence the degree on ∂Ω for maps of I α is well-defined. Furthermore we can see that E α ε attains its infimum on each class I d ∩ I a , due to the fact that the Sobolev injection I α → C 0,β (Ω, R n ) is compact. However we can not expect that these local minimizers converge to some non-trivial solutions of (5.1) since there are no minimizers of E ε in I d , d = 0. That is why we develop a mountain pass approach to find unstable solutions of a perturbed equation which may converge to some non trivial solution of (5.1). First The space I α enjoys a nice differentiable structure.

Lemma 5.12. For all α > 0, I α is a reflexive Banach manifold. The tangent space T u I α at a given point u ∈ I α is given by

T u I α = {ϕ ∈ W 1,n+α (Ω, R n ); ϕ(x) ∈ T u(x) S n-1 , ∀x ∈ ∂Ω}.
Moreover its tangent bundle is endowed with a Finsler structure and hence it is a Finsler manifold.

The space I 0 := I ∩ C 0 (Ω, R n ) is a non-reflexive Banach manifold with tangent spaces

T u I 0 = {ϕ ∈ W 1,n ∩ C 0 (Ω, R n ); ϕ(x) ∈ T u(x) S n-1 , ∀x ∈ ∂Ω}.
For the definition of a Finsler manifold we refer to [START_REF] Struwe | Variational methods[END_REF] p.85. We let V α = W 1,n+α (Ω, R n ), for α > 0 and V 0 = W 1,n ∩ C 0 (Ω, R n ). Note that E α ε is C 2 in V α , for α ≥ 0 and its differential is defined by

dE α ε (u).v = n + α n Ω |du| n+α-2 du, dv + 1 ε n Ω (1 -|u| 2 )u, v , ∀ u, v ∈ V α .
Hence E α ε is differentiable on I α and its differential, still denoted by dE a ε , is a map dE α ε : I α → T * I α defined by

dE α ε (u).v = n + α n Ω |du| n+α-2 du, dv + 1 ε n Ω (1 -|u| 2 )u, v , for all u ∈ I α , v ∈ T u I α .
We define a projection

π α (•, •) : I α × V α → T I α (u, v) → v -η α [u u, v ] |∂Ω (5.23)
where

η α : W 1-1 n+α ,n+α (Ω, R n ) → W 1,n+α (Ω, R n
) is a continuous linear extension operator. For α ≥ 0, we denote by • 1,n+α the usual norm in W 1,n+α (Ω, R n ). We let • p be a L p norm of a map (p ≥ 1). We also denote by • 1-1 n+α ,n+α the norm in W 1-1 n+α ,n+α (∂Ω, R n ). Recall that this norm is equal to

u n+α 1-1 n+α ,n+α = ∂Ω |u| n+α dx + ∂Ω ∂Ω |u(x) -u(y)| n+α |x -y| 2(n-1)+α .
As in [START_REF] Struwe | On a free boundary problem for minimal surfaces[END_REF] Lemma 2.1 we have :

Lemma 5.13. Let α ≥ 0. i) If u, v ∈ W 1-1 n+α ,n+α ∩ L ∞ (∂Ω, R n ), then u • v ∈ W 1-1 n+α ,n+α ∩ L ∞ (∂Ω, R n ) and u • v 1-1 n+α ,n+α ≤ u ∞ v 1-1 n+α ,n+α + v ∞ u 1-1 n+α ,n+α . ii) If u ∈ W 1-1 n+α ,n+α (∂Ω, R n ) and σ ∈ C 1 (R n ) then σ • u 1-1 n+α ,n+α ≤ σ • u n+α + (dσ) • X ∞ u 1-1 n+α ,n+α .
The sequel follows closely [START_REF] Struwe | On a free boundary problem for minimal surfaces[END_REF] (see also [START_REF] Fraser | On the free boundary variational problem for minimal disks[END_REF]). We represent T u I α = π α (u, V α ) and we define d S E α ε :

I α → V * α by letting d S E α ε (u), v Vα = dE α ε , π α (u, v) TuIα .
Using Lemma 5.13 we can see that the projection π α is lipschitz continuous. Thus we can see with the previous representation that I α is a C 1,1 Finsler manifold and

E α ε is C 1,1 in I α . Definition 5.14. Let α ≥ 0. A map u is a critical point of E α ε in I α if d S E α ε (u) = 0.
By using the previous definition and an integration by parts we can see that u is a critical point of

E α ε if        -n+α n ∆ n+α u = 1 ε n 1 -|u| 2 u in Ω, |u| = 1 on ∂Ω, ∂ ν u |u| = 0 on ∂Ω.
(5.24)

We thus obtain a Dirichlet boundary condition for the modulus of u and a homogeneous Neumann boundary condition for the "phase" u |u| . The condition ∂ ν u |u| = 0 can also be interpreted as a free boundary condition : the normal derivative of u is orthogonal to the sphere on the boundary of Ω.

Lemma 5.15. For α > 0. Let (u m ) m be a sequence in I α such that u m u in W 1,n+α and u m → u uniformly in Ω as m → +∞. Then

(u m -u) -π α (u m , u m -u) 1,n+α → 0, m → +∞.
Proof. Let (u m ) m be as in the statement of the lemma. Since u m converges weakly to u in W 1,n+α , we can see that u ∈ I α (i.e., |u| = 1 on ∂Ω). We assume that m is large enough so that u m -u ∞ < ρ S n-1 , where ρ S n-1 is the injectivity radius of S n-1 . Let x ∈ S n-1 and T be the geodesic distance between u k (x) and u(x) on S n-1 . Let (Y t ) 0≤t≤T be the unique geodesic line joining Y 0 = u k (x) and Y T = u(x) in the geodesic ball B ρ S n-1 (Y 0 ). We assume that this geodesic is arc-length parametrized. Since d dt | t=0 Y t ∈ T Y 0 S n-1 and u k (x) = Y 0 is normal to the sphere, it holds that Y 0 , d dt | t=0 Y t = 0. Thus we deduce that

Y 0 Y 0 , Y 0 -Y T = - T 0 t 0 Y 0 Y 0 , Y 0 - d 2 dt 2 Y t dsdt =: σ(Y 0 , Y T ).
Since the sphere S n-1 is smooth, using the properties of the exponential map we can see that

σ is differentiable with respect to (Y 0 , Y T ) ∈ S n-1 × S n-1 such that |Y 0 -Y T | < ρ S n-1 . Besides we have that |dσ(Y 0 , Y T )| ≤ C|Y 0 -Y T |
with C a constant which depends only on S n-1 . We then obtain, using Lemma 5.13

(u m -u) -π α (u m , u m -u) n+α 1,n+α = η α [u m u m , u m -u >] |∂Ω n+α 1,n+α ≤ C u m u m , u m -u n+α 1-1 n+α ,n+α = C σ • (u m , u) n+α 1-1 n+α ,n+α ≤ C u m -u n+α ∞ u m n+α 1-1 n+α ,n+α + u n+α 1-1 n+α ,n+α
→ 0 as m goes to ∞.

Before proving that E α ε satisfy the Palais-Smale condition we state a useful inequality :

Lemma 5.16. Let α > 0, there is a constant C > 0 such that

|v| n+α-2 v -|u| n+α-2 u • (v -u) ≥ C|v -u| n+α for all (u, v) ∈ R n 2 .
Proposition 5.17. For all α > 0, E α ε satisfies the Palais-Smale condition in

I α : if (u m ) m is a sequence in I α such that E α ε (u m ) ≤ C and d S E α ε (u m ) → 0 in V * α then, up to a subsequence, it holds that u m → u strongly in W 1,n+α (Ω, R n ).
Proof. We have that (u m ) is uniformly bounded in W 1,n+α . This comes from the fact that du m n+α n+α ≤ nE α ε and u m|∂Ω ∞ ≤ 1 (these two facts imply that u m n+α is bounded). Thus, up to a subsequence, we can assume that u m u in W 1,n+α (Ω, R n ) and u m → u uniformly in Ω, for some u ∈ W 1,n+α (Ω, R n ). Using Lemma 5.16 it comes

C Ω |du m -du| n+α ≤ Ω |du m | n+α-2 du m -|du| n+α-2 du, du m -du ≤ Ω |du m | n+α-2 du m , du m -du + o(1) ≤ Ω |du m | n+α-2 du m , du m -du + 1 ε n Ω (|u m | 2 -1)u m , u m -u + o(1) = d S E α ε (u m ), u m -u Vα + dE α ε (u m ), (u m -u) -π α (u m , u m -u) W 1,n+α + o(1) ≤ C (u m -u) -π α (u m , u m -u) 1,n+α + o(1) → 0 by Lemma 5.15.
Since E α ε satisfy the Palais-Smale condition we can try to apply the Mountain Pass Theorem of Ambrosetti-Rabinowitz in order to find critical points (see e.g. [START_REF] Struwe | Variational methods[END_REF] and [START_REF] Mawhin | Critical point theory and Hamiltonian systems[END_REF]).

Theorem 5.18. Let K 0 ⊂ K two compact metric spaces. Let X be a Banach space or a C 1,1 Finsler manifold and J ∈ C 1 (X, R). Let χ ∈ C 0 (K 0 , X) be a fixed map. We define

c := inf{max K J • F ; F ∈ M }, where M = {F ∈ C 0 (K, X); F = χ on K 0 } (5.25)
Assume that c > c 1 := max a∈K 0 J (χ(a)) .

(5.26)

Then there exists a sequence (x k ) ⊂ X such that J(x k ) → c and dJ(x k ) → 0 as k → +∞.

(5.27)

If J satisfies (5.26) for some K 0 , K we say that J has a Moutain Pass geometry. Under this condition the theorem produces a Palais-Smale sequence, i.e., a sequence such that (5.27) holds. Assume furthermore that the functional J satisfies the following Palais-Smale condition each sequence (x k ) satisfying (5.27) contains a convergent subsequence .

(PS)

Then the sequence (x k ) in (5.27) converges to x which is a critical point of J, such that J(x) = c. Contrarily to the energy E ε , thanks to Proposition 5.17 the perturbed energies E α ε do satisfy the Palais-Smale condition.

In the rest of this section we prove that E α ε possesses the Mountain Pass geometry for α small. Then applying Theorem 5.18 we will obtain critical points of E α ε in I α . We choose K 0 = ∂Ω, K = Ω (K 0 ⊂ K). We let J = E α ε and X = I α . We then define a map χ r : K 0 → X (for r > 0 small enough to be chosen later) by the following way. Let Φ : Ω → B n a C 1 -diffeomorphism such that Φ(∂Ω) = S n-1 .

Proposition 5.19. Let Φ as before. For all a ∈ ∂Ω, there exists

Φ a : Ω → B n a C 1 -diffeomorphism and β ∈ R such that i) Φ a (a) = Φ(a)
ii) dΦ a (a) = βR a where R a ∈ O n (R). Furthermore the map a → Φ a is continuous.

In order to prove this proposition we first prove a similar proposition which concerns only diffeomorphisms from ∂Ω to S n-1 .

Proposition 5.20. Let ϕ : ∂Ω → S n-1 a C 1 diffeomorphism. For all a ∈ ∂Ω there exists a C 1 diffeomorphism ϕ a and β ∈ R such that i) ϕ a (a) = ϕ(a) ii) dϕ a (a) = βO a , where O a : T a ∂Ω → T ϕa(a) S n-1 is a linear map which conserves the inner product. Furthermore the map a → ϕ a is continuous.

The first step to prove this proposition is the following lemma which is just the polar decomposition of matrices adapted to a manifold setting.

Lemma 5.21. Let ϕ : ∂Ω → S n-1 a C 1 diffeomorphism. For all a ∈ ∂Ω there exist linear maps S a : T a ∂Ω → T a ∂Ω and O a : T a ∂Ω → T ϕ(a) S n-1 such that i) dϕ(a) = O a S a ii) O a preserves the inner product iii) the maps a → S a and a → O a are continuous.

Proof of Lemma 5.21. We let L a := dϕ(a), this is a linear map L a : T a ∂Ω → T ϕ(a) S n-1 . Thanks to the riemmanian structures on T a ∂Ω and T ϕ(a) S n-1 (inherited from the euclidean structure of R n ) we can identify

T * a T a ∂Ω, T * ϕ(a) S n-1 T ϕ(a) S n-1 .
Hence the adjoint operator L * a : T * ϕ(a) S n-1 → T * a ∂Ω induces a map L * a : T ϕ(a) S n-1 → T a ∂Ω such that, for all w ∈ T ϕ(a) S n-1 and all v ∈ T a ∂Ω we have

L * a L a v, w Ta∂Ω = L a v, L a w T ϕ(a) S n-1 .
Thus we obtain that L * a L a is a symmetric endomorphism of T a ∂Ω. It is also positive definite since L a is bijective. We can then define L * a L a : T a ∂Ω → T a ∂Ω by the usual way and L * a L a is also a symmetric positive definite endomorphism. We set S a := L * a L a and O a = L a S -1 a . If we denote by Õ * a the linear map obtained from the adjoint of O a and the identification of the tangent spaces and their dual we have :

O a Õ * a = Id |T ϕ(a) S n-1 Õ * a O a = Id |Ta∂Ω .
Thus O a preserves the inner product. Since ϕ is C 1 we have that a → L a is continuous. We can also check that taking the adjoint of an operator and taking the square root of a symmetric definite poisitive operator are continuous operations. Hence a → O a is continuous.

Proof of Proposition 5.20. Let a ∈ ∂Ω and S a given by Lemma 5.21. Since a → S a is continuous, the map a → λ 1 a is continuous. Here λ 1 a denotes the smallest eigenvalue of S a . Since ∂Ω is compact we obtain that there exits β > 0 such that ∀a ∈ ∂Ω, λ 1 a > β.

Now let a ∈ ∂Ω, let 0 < η < ρ 4 where ρ is the injectivity radius of ∂Ω. For p in the geodesic ball B(a, η) we denote by γ a→p the unique geodesic from a to p. We also set u p = γ a→p (0) ∈ T a ∂Ω. If η is small enough in B(a, η) we can write

d dt [(β(1 -λ(t))I + λ(t)S a ) tv] = βI + λ (t) + tλ(t) (S a -βI) v.
Thanks to the choice of β we have S a -βI > 0, i.e., S a -βI is a positive definite symmetric matrix, and thus [βI + (λ (t) + tλ(t)) (S a -βI)] is invertible. That proves that ϕ a is a diffeomorphism.

Now we need to extend the ϕ a given by Proposition 5.20.

Proof of Proposition 5.19. Let Φ : Ω → B n a C 1 diffeomorphism and ϕ = Φ |∂Ω . Let 0 < η < ρ 4 , where ρ is the injectivity radius of ∂Ω. Let a ∈ ∂Ω and a 1 , a 2 ∈ ∂Ω such that d g (a, a 1 ) = η) and d g (a, a 2 ) = 2η, here d g denotes the geodesic distance on ∂Ω. We set δ 1 = d eucl (a, a 1 ) and δ 2 = d eucl (a, a 2 ). For p ∈ B eucl (a, δ 2 ) ∩ Ω we let u p be the vector whose origin is a and whose end is p. We denote by u T p the projection of u p on T a ∂Ω and u N p the norm of the projection of u p on ν a the normal at ∂Ω at a. Likewise we denote by v p the vector ------→ Φ(a)Φ(p), v T p its projection on T Φ(a) S n-1 and v N p the norm of its projection on ν Φ(a) the unit inner normal to S n-1 at Φ(a). This can be seen by using a change of variable, the fact that the n-energy is conformally invariant and the fact that dΦ a (a) is a similitude from Lemma 5.19. If r is small enough then the Möbius map M (1-r)Φ(a) is very concentrated and hence Φa(Bσ(a))

|dM (1-r)Φ(a) (x)| n dx n n 2 |B n |.
Furthermore we have that χ r a converges to a constant of modulus one in L p for all p < +∞ as r → 0. Thus the potential term goes to 0 as r → 0. We can also check that this term goes to 0 as ε → +∞.

Lemma 5.23. There exists r 0 > 0 sufficiently small such that if 0 < r < r 0 then : for all γ ∈ M , there exists a point a γ ∈ Ω such that

1 |B n | Ω γ(a γ )(x)dx = 0 ∈ B n .
Proof. For γ ∈ M (i.e., γ ∈ C 0 (Ω, I α ) such that γ |∂Ω = χ r ) we consider

G γ : Ω → B n a → 1 |B n | Ω γ(a)(x)dx.
For all a ∈ ∂Ω, G γ (a) = 1 |B n | Ω χ r (a)(x)dx. We recall that χ r (a)(x) = M(1-r)Φ(a) • Φ a (x). As r → 0, we have χ r → Φ(a) in all L p , p ≥ 1. Thus there exists r 0 > 0 such that if 0 < r < r 0 G γ |∂Ω -Φ |∂Ω L ∞ is small. Hence by a topological argument (similar to the Brouwer fix point Theorem) we deduce that there exists

a γ ∈ Ω such that G γ (a γ ) = 1 |B n | Ω γ(a γ ) = 0.
Lemma 5.24. There exists

c 2 > n n 2 -1 |B n | such that inf{max a∈Ω E ε (γ(a)) ; γ ∈ M } ≥ c 2 > n n 2 -1 |B n |
for all α > 0, r 0 > r > 0, ε > 0.

k ) ⊂ Ω such that i) E ε (γ k (a k )) → n n 2 -1 |B n | ii) 1 |B n | Ω γ k (a k ) = 0. We let γ k (a k ) =: u k ⊂ I 1 . Since (u k ) is bounded in W 1,
n (because of i) , up to a subsequence, we have that u k u ∞ in W 1,n and u k → u ∞ in L p for all p ≥ 1. We thus obtain from ii) that

Ω u ∞ = 0. However using i) again it holds that E ε (u ∞ ) ≤ n n 2 -1 |B n | and 1. Ω |du k | n → n n 2 -1 |B n |, 2. Ω (1 -|u k | 2 ) 2 → Ω (1 -|u ∞ | 2 ) 2 = 0. Thus we must have |u ∞ | = 1 a.e.
in Ω and Ω u ∞ = 0 which is a contradiction.

Lemma 5.25. There exists α 0 > 0 and r 1 small enough such that for all 0 < α < α 0 and 0 < r < r 1 we have c 1 (α, r, ε) > c(α, r, ε).

Proof. From the two previous lemma we have c 1 (α, r, ε) = n . Hence we obtain

1. c 1 (α, r, ε) < 2 c 2 -n n 2 -1 |B n | 3 and 2. c > 2 c 2 -n n 2 -1 |B n | 3 . Thus c 1 (α, r, ε) > c(α, r, ε).
Applying the previous results of this section we can conclude that Theorem 5.26. There exists α 0 > 0 small enough such that for all 0 < α < α 0 , E α ε has a minimax critical point in I α .Moreover deg(u α , ∂Ω) = 1 and E α ε (u α ) = c(α, r, ε) with c(α, r, ε) defined previously.

Convergence of critical points of the perturbed problem

In order to obtain non-trivial critical points of E ε in I it is natural to study the asymptotic behavior, as α goes to 0, of solutions of the perturbed problem obtained in the previous section. In this section we would like to indicate some elements on the convergence of the solutions u α .

Regularity of the perturbed problem

First we state a result which proves interior regularity of solutions of the perturbed and initial problem along with some estimates.

Proposition 5.27. Let p > 1, v ∈ W 1,p (B r (x), R n ) such that ∆ p v = C(1 -|v| 2 )v, with C a constant. Then 1) v ∈ C 1,β (B r (x), R n ) for any β ∈]0, 1[ and there exists a constant C 1 = C(n, p) such that sup B r 2 (x) |du| p ≤ C 1 1 |B r (x)| Br(x) |du| p .
2) Furthermore for all β ∈]0, 1[, there exists γ ∈]0, 1[ and

C 2 = C 2 (β, n, p) such that u C 1,γ B r 2 (x) ≤ C 2 u C 0,β (Br(x)) .
This proposition can be proved using Theorem 2.3 in [START_REF] Mou | Regularity for n-harmonic maps[END_REF] (see alos Theorem 3.1 in [START_REF] Hardt | Mappings minimizing the L p norm of the gradient[END_REF]).

Note also that we can apply a maximum principle to solutions of (5.2) and obtain :

Proposition 5.28. let α ≥ 0. If u ∈ W 1,n+α satisfies (5.2) then |u| ≤ 1 in Ω.
The proof of this proposition is an adaptation of the proof of Lemma 6 in [START_REF] Strzelecki | Asymptotics for the minimization of a Ginzburg-Landau energy in n dimensions[END_REF] (see also Lemma 5.3.1 in [GSZ]).

Now we consider a sequence (α m ) m∈N , α m > 0 and α m → 0. We let u m = u αm be a solution of the perturbed problem (5.2) with α = α m obtained by the min-max scheme.

Proposition 5.29. Let (u m ) m be as before. Then there exists u ∈ W 1,n (Ω, R n ) such that, up to a sequence, it holds that i)

u m u in W 1,n (Ω, R n ), ii) u m → u in C 1 loc (Ω, R n ) (and thus in W 1,n loc (Ω, R n )).
Proof. i) The first point comes from the fact that u m is bounded in W 1,n (Ω, R n ). Indeed using the maximum principle 5.28 we have that

|u m | ≤ 1 in Ω. Thus (u m ) is bounded in L n (Ω, R n ).
Besides, by using Hölder inequality it comes

Ω |du m | n ≤ Ω |du m | n+αm n n+αm |Ω| αm n+αm ≤ C [E αm ε (u m )] n n+αm |Ω| αm n+αm ≤ C c(α m , r, ε) n n+αm |Ω| αm n+αm
where C is constant and c(α m , r, ε) is defined by (5.29). Moreover we can see that c(α m , r, ε) → n 2 -1|B n | + 1 4ε n |Ω| as m → +∞ and r → 0. This concludes the proof.

ii) In order to prove the second point we use Proposition 5.27. Let x ∈ Ω and R > 0 be small such that

B R = B R (x) ⊂ Ω we obtain sup B R 2 |du m | αm ≤ C B R |du m | αm ≤ C .
Then, up to a subsequence, u m converges uniformly to u. We then use the second point of Proposition 5.27 to deduce that u m is uniformly bounded in C 1,γ (B R

2

) and thus, up to a subsequence, we have that

u m → u in C 1 (B R 2
). Using a covering argument we can show that for every compact

K ⊂ Ω it holds that u m → u in C 1 (R, R n ).
The last proposition shows that if the strong convergence does not hold for a sequence of critical of the perturbed problem (u m ) m it is due to boundary effect. The potential loss of compactness takes place at the boundary.

Proposition 5.30. Let (u m ) m be as before. Assume that there exists K ∈ R + such that sup Proof. We have seen in the previous proposition that, up to a subsequence, u m u in W 1,n (Ω, R n ), and du m → du a.e. in Ω. Furthermore by using the Lebesgue dominated convergence we obtain that E ∞ (u m ) → E ∞ (u). But for a weakly convergent sequence u m we have that u m → u strongly in W 1,n if and only if E ∞ (u m ) → E ∞ (u) (see e.g [START_REF] Evans | Weak convergence methods for nonlinear partial differential equations[END_REF] the remark below Theorem 1.1.1). Since the degree is continuous with respect to the strong convergence in W 1,n it holds that deg(u, ∂Ω) = 1. It remains to show that u is a solution of (5.1). We observe that div(|du| n-2 du) = 1 ε n (1-|u| 2 )u in Ω by the strong convergence in W 1,n . It also holds that |u| = 1 on ∂Ω since tr |∂Ω u converges almost everywhere. Now since sup x∈Ω |du m (x)| ≤ K, we deduce that (u m ) m is equicontinuous and um |um| converges to u |u| strongly in W 1,n (U, R n ) where U ⊂ Ω is a neighborhood of ∂Ω. We can check that the maps um |um| have a well-defined normal derivative and the normal derivative operator is continuous with respect to the strong W 1,n convergence. We thus find that ∂ ν u |u| = 0.

In order to pursue the asymptotic analysis of the sequence (u m ) a possibility is to perform a blow-up analysis. The last proposition shows that if the strong convergence (in W 1,n ) of the u m fails then sup x∈Ω |du m (x)| → +∞. The natural strategy (cf. [START_REF] Sacks | The existence of minimal immersions of 2-spheres[END_REF], [START_REF] Mou | Bubbling phenomena of Palais-Smale-like sequences of m-harmonic type systems[END_REF], [START_REF] Brezis | Convergence of solutions of H-systems or how to blow bubbles[END_REF]) is then to rescale the maps u m near the blowing-up points and prove that the properly rescaled maps converge to non-trivial maps which satisfy some equation. Such a maps is called a bubble. The idea is to prove a theorem of (exact) energy decomposition in terms of the energy of the bubbles, in the spirit of Theorem 8.13 in [START_REF] Berlyand | Minimax critical points in Ginzburg-Landau problems with semi-stiff boundary conditions : existence and bubbling[END_REF] (cf. also [START_REF]Unstable solutions of two-dimensional geometric variational problems[END_REF], [START_REF]Two-dimensional geometric variational problems[END_REF] for similar theorem in the context of harmonic maps, and [START_REF] Mou | Bubbling phenomena of Palais-Smale-like sequences of m-harmonic type systems[END_REF] for bubbling analysis of n-harmonic maps in a manifold without boundary). In order to perform this bubbling analysis, the usual tools we need are : global regularity and estimates up to the boundary for solutions of (5.2) and the fact that non-trivial bubbles have a uniform lower bound for their energy. In the next subsection we say a few words about bubbles in our problem.

The bubbles

Definition 5.31. We call a bubble a map w : H n → B n , where

H n = {x ∈ R n ; x n > 0} such that      ∆ n w = 0 in H n , |w| = 1 on ∂H n , ∂ ν w |w| = 0 on ∂H n .
(5.31) Since the half-space H n and the ball B n are conformally equivalent, a bubble can also be viewed as a map w :

B n → B n such that      ∆ n w = 0 in B n , |w| = 1 on S n-1 , ∂ ν w |w| = 0 on S n-1 .
(5.32) This definition is motivated by the fact that, if a sequence (u αm ) does not converge strongly in W 1,n to some u, then after some rescaling a piece of the maps (u αm ) converges to w which satisfies equation (5.31). We first give some properties of the bubbles.

Lemma 5.32. Let u ∈ I such that (5.32) holds. We set

T u := |du| n-2 du t du - 1 n |du| n Id
(T u is the stress energy tensor associated to the energy E ∞ ). Then it holds that i) div (T u ) = 0.

ii) If x ∈ ∂Ω then T u (x) : T x ∂Ω → T x ∂Ω and T u (x) : N x ∂Ω → N x ∂Ω (N x ∂Ω stands for the normal space to ∂Ω. iii) T u = 0 is equivalent to u conformal.

Proof.

i) The first point can be proved by direct computation.

ii) Since ∂ ν u |u| = 0 we have that ∂ ν u is parallel to u on the boundary. Thus du(x).ν(x) = ∂ ν u(x) is normal to S n-1 and for x ∈ ∂Ω, du(x) : N x ∂Ω → N u(x) S n-1 . Now we also have that |u| = 1 on ∂Ω, and this implies that u(x), ∂ τ u(x) = 0 for all x ∈ ∂Ω and τ ∈ T x ∂Ω. This means that du(x).τ ∈ T u(x) S n-1 for all τ ∈ T x ∂Ω and du(x) : T x ∂Ω → T u(x) S n-1 . Hence we can prove that du t du(x) : N x ∂Ω → N x ∂Ω for all x ∈ ∂Ω. Indeed for all τ ∈ T x ∂Ω we have that du t du(x) .ν, τ = du(x).ν, du(x).τ = 0. Since du t du is a symmetric tensor we also have that du t du(x) : T x ∂Ω → T x ∂Ω for x ∈ ∂Ω.

iii) We recall that u is conformal if and only if its differential is a similitude, i.e., a product of dilations and rotations. Thus u is conformal if and only if du t du = |du| 2 Id and this is equivalent to T u = 0.

In dimension n = 2 the last lemma suffices to characterize all the bubbles. Indeed we have :

Lemma 5.33. Let n = 2 and M :

B 2 → M 2 (R) such that M is symmetric, M ∈ L 1 (B 2 ) and tr(M ) = 0. Assume furthermore that i) div(M ) = 0 in B 2 . ii) For all x ∈ ∂, B 2 M (x) : T x ∂Ω → T x ∂S 1 . Then M ≡ 0 in Ω.
Proof. We use polar coordinates (r, θ). Since div(M ) = 0, and B 2 is simply connected, using the Poincaré Lemma we deduce that there exist f, g :

B 2 → R such that -1 r ∂ θ f ∂ r f -1 r ∂ θ g ∂ r g
.

By using the symmetry of M and the fact that tr(M ) = 0 we obtain

   ∂ r f = -1 r ∂ θ g ∂ r g = 1 r ∂ θ f in B 2 .
Differentiating these expressions it comes that ∆f = ∆g = 0 in B 2 . We then use the boundary conditions. Since M (x)e r ∈ Vect(e r ) and M (x)e θ ∈ Vect(e θ ) on the boundary we find that ∂ θ g = ∂ r f = 0 on ∂B 2 . Finally we conclude that f, g are constants and thus M ≡ 0 in B 2 .

Another proof of the previous lemma can be found in [START_REF] Berlyand | Minimax critical points in Ginzburg-Landau problems with semi-stiff boundary conditions : existence and bubbling[END_REF] where the authors uses the socalled Hopf differential to prove the result. In that article the authors deduce that all the bubbles are conformal and then they find that these are exactly product of Möbius maps. We know that in dimension n ≥ 3 the Möbius maps satisfy (5.32). They all have the same n-Dirichlet energy equals to n n 2 -1 |B n |. However we do not know if there exist other type of bubbles, i.e., if there exist non-conformal solutions of (5.32). It would be interesting to answer this question. Definition 6.1. A function h in H 1 (Ω) is stationary harmonic if div(T h ) = 0 in Ω in the sense of distributions, where T h is the stress-energy tensor associated to the Dirichlet energy, defined by

T h = 1 2 (∂ y h) 2 -(∂ x h 2 ) -∂ x h∂ y h -∂ x h∂ y h 1 2 (∂ x h) 2 -(∂ y h 2 ) . (6.1)
Equivalently h is stationary harmonic in Ω if

ω h := (∂ x h) 2 -(∂ y h) 2 -2i∂ x h∂ y h is holomorphic in Ω. (6.2) Equation (6.1) means that ∂ x (T h ) i1 + ∂ y (T h ) i2 = 0 for i = 1
, 2 in the sense of distributions. Let us denote by H -1 (Ω) the dual of the Sobolev space H 1 0 (Ω). The aim of this paper is to describe the local regularity of Radon measures µ which satisfy the following conditions :

µ ∈ H -1 (Ω), (6.3)
there exists a function h µ such that ∆h µ = µ in Ω, (6.4) and h µ is stationary harmonic. (6.5) Note that if h µ is a solution of (6.4) then h µ ∈ H 1 (Ω) and condition (6.5) is well-defined. Indeed we can see that there exists a solution of (6.4) in H 1 0 (Ω) using the Lax-Milgram theorem. Then all the solutions are in H 1 (Ω) since the difference between two solutions is harmonic in Ω end hence belongs to H 1 (Ω).

We will discuss the physical motivations of this problem in the next section. Before we would like to mention that there are some results in the literature about regularity for inner variational equations in different context, see e.g. [START_REF]Lipschitz regularity for inner-variational equations[END_REF], [START_REF] Le | Regularity and nonexistence results for some free-interface problems related to Ginzburg-Landau vortices[END_REF]. Now we wish to examine in slightly more details the condition (6.5) and some of its direct consequences. One can show that if h is harmonic (∆h = 0) then h is stationary harmonic but the converse is not true in general. It is true if h is regular. Indeed using the same techniques as in [START_REF] Sandier | Progress in Nonlinear Partial Differential Equations Birkhauser[END_REF] Chapter 13 we can prove that if µ is in L p for some p > 1 then a solution of (6.3), (6.4),(6.5) is harmonic, i.e., µ = 0. For the proof of these facts and other properties of stationary harmonic functions we refer to the Appendix.

Another direct consequence of condition (6.5) is that ∇h µ ∈ L ∞ loc and then h µ is locally lipschitz continuous. This is due to the fact that |∇h µ | 2 = |ω hµ | 2 and ω hµ is holomorphic in Ω. In particular h µ and |∇h µ | are continuous. The fact that ω hµ is holomorphic also gives us the following : Proposition 6.2. Let h µ which satisfies that ω

hµ = (∂ x h µ ) 2 -(∂ y h µ ) 2 -2i∂ x h µ ∂ y h µ is holo- morphic.
Then the zeros of ω hµ are isolated in Ω. If Ω is compact there is a finite number of such critical points in Ω.

In the present paper we are interested in describing the properties of Radon measures µ which satisfy hypothesis (6.3), (6.4), (6.5). Let us recall that the support of a measure µ is the complement of the largest open set A such that µ(A) = 0. Our first result describes the local regularity of the measure µ in the neighborhood of point z 0 which belongs to the support of µ and such that ω hµ (z 0 ) = 0. Note that we can always assume that h µ (z 0 ) = 0 because adding a constant to h does not change the hypothesis (6.3), (6.4), (6.5). Note also that near a point z 0 which does not belong to the support of µ the function h µ is a harmonic function.

Theorem 6.3. Let z 0 ∈ supp µ, with (h µ , µ) which satisfy assumptions (6.3), (6.4), (6.5) and such that ω hµ (z 0 ) = 0. We assume that h µ (z 0 ) = 0. Then there exist a neighborhood V of z 0 and a harmonic function H in V such that

h µ = |H|, in V or h µ = -|H|, in V (6.6) supp µ V = {z ∈ V ; H(z) = 0}. (6.7)
Furthermore we have that ∇H(z 0 ) = 0 and the set {z ∈ V ; H(z) = 0} is a smooth simple curve diffeomorphic to a straight line (see Figure 6.1). Near a point z 0 such that ω hµ (z 0 ) = 0 the behavior of h µ and the geometry of the support of µ is a little bit more complicated. Nevertheless if z 0 is a zero of even order of ω hµ the situation is similar.

Theorem 6.4. Let z 0 ∈ supp µ, with (h µ , µ) which satisfy assumptions (6.3), (6.4), (6.5), and such that z 0 is a zero of even order of ω hµ . We assume that h µ (z 0 ) = 0. Then there exist a neighborhood V of z 0 , a harmonic function H in V and a function θ : V → {±1} such that the function θH being continuous and ∇H(z 0 ) = 0. Besides the support of µ V is a union of smooth curves included in {z ∈ V ; H(z) = 0} which end at z 0 .

h µ (z) = θ(z)H(z) in V. (6.8) 0 D 1 D 2 V
Furthermore the function H 1 is such that : there exist an unique integer n ≥ 1, a small number r > 0 and a biholomorphism Φ : B(0, r) → V such that Φ(0) = z 0 and

H 1 • Φ(z) = Re(z n+ 1
2 ), for z ∈ B(0, r). (6.10)

Thanks to the property satisfied by the function H 1 in the previous theorem we can obtain a description of the set of zeros of H 1 similar to Theorem 6.5 (see Figure 6.3).

Theorem 6.7. let H 1 be as in the previous Theorem 6.6. Then there exist 2n+1 analytic curves

γ k :] -1, 1[→ V, (k = 1, 2, ..., 2n + 1) such that γ k (0) = z 0 and 1) {z ∈ R 2 ; H 1 (z) = 0} ∩ V = ∪ 2n+1 k=1 γ k 2) ang(γ k , γ k+1 ) = 2π
n+1 , k = 1, ..., 2n + 1, where γ 2n+n denotes γ 1 and ang(γ k , γ k+1 ) is the angle between γ k and γ k+1 at z 0 .

3) There exists an analytic diffeomorphism φ :

V → B(0, 1) such that φ • γ k (t) = t exp(iθ k ) where t ∈] -1, 1[, k = 1, ..., 2n + 1, and θ k = π 2n+1 + 2(k-1)π 2n .
In order to conclude this introduction we would like to comment on the hypothesis (6.3), (6.4), (6.5). First note that the fact that h µ is in H 1 (or equivalently that µ ∈ H -1 ) is essential to assume (6.5) since we take the divergence of the tensor T µ in the sense of distributions we must have that its coefficients are in L 1 loc . Then we want to give an example which shows that (6.5)

z 0 suppµ V Figure 6
.3: Illustration of Theorem 6.6.

does not necessarily imply that µ is a Radon measure. The example is the following : one can take h defined on [0, 1] such that h(0) = 0 and

h (x) =    +1, if x ∈] 1 n+1 , 1 n [ with n even -1, if x ∈] 1 n+1 , 1 n [ with n odd.
We then have that h ∈ H 1 ([0, 1]), and

h satisfies ω h = |h (x)| 2 = 1 is holomorphic. But ∆h = +∞ n=2 δ 1 n is not a Radon measure.
The paper is organized as follows : In Section 2 we explain the physical motivations for studying this problem. Section 3 is devoted to the description of the measure µ near a point z 0 such that ω hµ (z 0 ) = 0. In Section 4 we discuss the case of a zero of even order of ω hµ and in Section 5 the case of a zero of odd order of ω hµ .

6.2 Physical motivations of the problem 6.2.1 Connections to Ginzburg-Landau vortices without magnetic field.

The conditions (6.3), (6.4), (6.5) are motivated by the problem of describing limiting vorticities for the critical points (u ε ) ε of the Ginzburg-Landau energy without magnetic field

E ε (u) = 1 2 Ω |∇u| 2 dx + 1 4ε 2 Ω (1 -|u| 2 ) 2 dx. (6.11)
Here u is a complex-valued function called the order parameter and its isolated zeros are called vortices. The Ginzburg-Landau theory is a model for describing the superconductivity. The Ginzburg-Landau system without magnetic field was studied by Béthuel-Brézis-Hélein in [START_REF] Bethuel | Progress in Nonlinear Differential Equations and their Applications[END_REF]. Later on Sandier-Serfaty in [START_REF] Sandier | Progress in Nonlinear Partial Differential Equations Birkhauser[END_REF] studied the Ginzburg-Landau system with magnetic field which is a more physically relevant model. The vortices are important features of the model. They correspond to small regions in the superconducting sample where the superconductivity is destroyed. Let Ω be a bounded domain in R 2 . We consider a family (u ε ) ε>0 of solutions of

-∆u ε = u ε ε 2 (1 -|u ε | 2 ) in Ω. (6.12)
We assume that |u ε | ≤ 1 in Ω and

E ε (u ε ) < C 0 ε α-1 , α > 2 3 (6.13)
for every ε > 0. We let j ε = iu ε , ∇u ε where ., . denotes the inner product in C identified with R 2 . We also let µ ε = curl j ε . Here j ε describes superconducting currents and µ ε is the vorticity of these currents. A direct calculation shows that div j ε = 0 hence we can write j ε = ∇ ⊥ h ε for some function h ε . Furthermore this function satisfies the following equation

∆h ε = µ ε in Ω ∂ ν h ε = j ε , τ on ∂Ω. (6.14)
Here ν is the outward pointing normal to ∂Ω and τ = ν ⊥ . By the solution to (6.14) we mean the solution with zero average in Ω. We split h ε into two pieces : let us define h 0 ε and h 1 ε by

-∆h 1 ε = µ ε in Ω, h 1 ε = 0 on ∂Ω. , h 0 ε = h ε -h 1 ε .
We recall the following result which describes the behavior of the vorticity measure as ε goes to 0 (see [START_REF] Sandier | Limiting vorticities for the Ginzburg-Landau equations[END_REF] and [START_REF] Sandier | Progress in Nonlinear Partial Differential Equations Birkhauser[END_REF]).

Theorem 6.8 (Theorem 13.2 in [START_REF] Sandier | Progress in Nonlinear Partial Differential Equations Birkhauser[END_REF]). A) Let {u ε } ε>0 be solutions of (6.12). Then for any ε > 0, there exists a measure ν ε of the form 2π i d ε i δ a ε i where the sum is finite, a ε i ∈ Ω and

d ε i ∈ Z for every i, such that, letting n ε = i |d ε i |, n ε ≤ C E ε (u ε , B ε ) | log ε| , (6.15)
where B ε is a union of balls of total radius less than Cε 2/3 , and such that

µ ε -ν ε W -1,p (Ω) µ ε -ν ε (C 0 (Ω)) * → 0, (6.16)
for some p ∈ (1, 2). B) Let {ν ε } ε be any measures of the form 2π i d ε i δ a ε i satisfying (6.16), let n ε = i |d ε i |, and let {M ε } ε be positive real numbers such that {h 0 ε /M ε } ε converges in L 1 loc (Ω) to a function H 0 . Then H 0 is harmonic and, possibly after extraction, one of the following holds. 0) n ε = 0 for every ε small enough and then µ ε tends to 0 in W -1,p (Ω). 1) n ε = o(M ε ) is nonzero for ε small enough, and then µ ε /n ε converges in W -1,p (Ω) to a measure µ such that µ∇H 0 = 0, hence the support of µ is contained in the set of critical points of H 0 . 2) M ε ∼ λn ε , with λ > 0, and then µ ε /M ε converges in W -1,p (Ω) to a measure µ, and h ε /M ε converges in W 1,p loc (Ω) to a solution of ∆h µ = µ in Ω. Moreover the symmetric 2-tensor T µ with coefficients T ij given by

T ij = -∂ i h µ ∂ j h µ + 1 2 |∇h µ | 2 δ ij (6.17)
is divergence-free in finite part (see Definition 6.9 below). 3) M ε = 0(n ε ), and then µ ε /n ε converges in W -1,p (Ω) to a measure µ, and h ε /n ε converges in W 1,p loc (Ω) to the solution of

∆h µ = µ in Ω h µ = 0 on ∂Ω. (6.18)
Moreover the symmetric 2-tensor T µ with coefficients T ij given by (6.17) is divergence-free in finite part.

In cases 2) and 3), if µ ∈ H -1 (Ω) then solutions of ∆h µ = µ are in H 1 loc (Ω). Thus T µ is in L 1 loc (Ω) and we have that div(T µ ) = 0 in the sense of distributions. In other words h µ is stationary harmonic.

Hence we can see that the limiting vorticity in cases 2), 3), with the additional hypothesis that µ ∈ H -1 (Ω) satisfies condition (6.4), (6.5). Understanding the limiting measure µ will in turn give qualitative information on the behavior of vortices.

We now recall the definition of the notion of divergence-free in finite part taken from [START_REF] Sandier | Progress in Nonlinear Partial Differential Equations Birkhauser[END_REF]. Definition 6.9. Assume X is a vector field in Ω. We say that X is divergence-free in finite part if there exists a family of sets {E δ } δ>0 such that 1. For any compact K ⊂ Ω, we have lim

δ→0 cap 1 (K ∩ E δ ) = 0. 2. For every δ > 0, X ∈ L 1 (Ω \ E δ ). 3. For every ζ ∈ C ∞ c (Ω), Ω\F δ X • ∇ζ = 0 where F δ = ζ -1 (ζ(E δ )). If T is a 2-tensor with coefficients {T ij } 1≤i,j≤2
, we say that T is divergence free in finite part if the vectors T i = (T i1 , T i2 ) are, for i = 1, 2.

In this definition we denoted by cap 1 the 1-capacity of a set E ⊂ R 2 and we recall from Evans-Gariepy [EG92] that the p-capacity (1 ≤ p < 2) of a set E is defined as

cap p (E) = inf{ R 2 |∇ϕ| p ; ϕ ∈ L p * (R 2 ), ∇ϕ ∈ L p (R 2 ), A ⊂ int(ϕ ≥ 1)},
However because of the condition (6.5) we have

Ω 1 2 (∂ y h 2 -∂ x h 2 )∂ x ϕ 1 -(∂ x h∂ y h)∂ y ϕ 1 = 0 Ω (-∂ x h∂ y h)∂ x ϕ 2 + 1 2 (∂ x h 2 -∂ y h 2 )∂ y ϕ 2 = 0.
Hence we can rewrite

Ω ∇ ⊥ h ⊗ ∇ ⊥ h, Dϕ = Ω 1 2 (∂ x h 2 + ∂ y h 2 )∂ x ϕ 1 + Ω 1 2 (∂ x h 2 + ∂ y h 2 )∂ y ϕ 2 .
We then set p = 1 2 |∇h| 2 ∈ L 1 (Ω) and we obtain that for all ϕ ∈ C ∞ c (Ω, R 2 ) we have

Ω ∇ ⊥ h ⊗ ∇ ⊥ h, Dϕ = - Ω p div(ϕ).
Thus v = ∇ ⊥ h is a weak solution of stationary Euler system, with pressure p = 1 2 |∇h| 2 and with vorticity equal to curl ∇ ⊥ h = ∆h = µ.

The previous Proposition 6.11 combined with Theorems 6.3, 6.4, 6.6, implies that if v is a weak solution of the Euler system (6.21) such that v = ∇ ⊥ h with h which satisfies hypothesis (6.3), (6.4), (6.5) then v is a vortex sheet solution of (6.21). The vortex sheet problem consists in finding a solution (v, p) of (6.19) such that the initial data v |t=0 = v 0 satisfies that div(v 0 ) = 0 and ω 0 = curl v 0 = δ Σ with Σ a compact smooth curve in R 2 . The existence of global solution of vortex sheet solutions of the Euler equation is due to J.M Delort in [START_REF] Delort | Existence de nappes de tourbillon en dimension deux[END_REF]. Note that in his paper an important assumption for the proof of the existence of global solution of vortex sheet solutions is that the initial data ω 0 = curl v 0 = µ is a positive (or negative) measure. However in cases of theorems 6.4, 6.6, it can happen that µ has no sign. For example setting h(r, ϕ) = θ(ϕ)r 2 cos(2ϕ) with θ

(ϕ) = +1, if -3π 4 ≤ ϕ ≤ π 4 and θ(ϕ) = -1 if ϕ ∈ [-π, π] \ [-3π 4 , π 4
]. Then one can check that ∆h is a measure with no fixed sign.

Let us mention that not all stationary solutions of the Euler system (6.21) can be written as v = ∇ ⊥ h with h which satisfies (6.3),(6.4),(6.5). For example we take v = (-y, x) for x, y ∈ B(0, 1). Then we can check that v is a solution of (6.21) with p = 1 2 (x 2 + y 2 ). We can write v = ∇ ⊥ h with h = 1 2 (x 2 +y 2 ). But h satisfies that ω h = (x-iy) 2 and it is not holomorphic. Hence (6.5) is not satisfied. Note that in this case ∆h = 1 in B(0, 1). Such a solution is called a vortex patch.

Connections to system of point vortices

A system of N -point vortices in evolution is described by the following system of ordinary differential equations

dz i dt (t) = ∇ ⊥   N j=1,j =i d j ln |z -z j (t)|   (z i (t)), ∀i = 1, ...,
d j z i -z j |z i -z j | 2 = 0, ∀i = 1, ..., N.
(6.24)

A natural question is the following : What are the limiting vorticities of a stationary system of point vortices when the number of points tends to infinity ?

Let us reformulate precisely this question. Let Ω be a bounded domain, we are interested in Radon measure µ which satisfies the following conditions :

∀ε > 0, ∃N ε ∈ N, (z ε i ) 1≤i≤Nε ∈ Ω, d ε i ∈ Z s.t. µ -2π N ε i=1 d ε i δ z ε i (C 0 (Ω)) * < ε (6.25) (z ε i ) 1≤i≤Nε , (d ε i ) 1≤i≤N ε
define a stationary system of point vortices. (6.26)

The limiting vorticities of a stationary system of point vortices are described by a result analog to Theorem 6.8 : Theorem 6.12. Let Ω be a bounded domain. Let µ be a Radon measure in Ω which satisfies (6.25) and (6.26). There exists a function u ∈ L 1 loc (Ω) such that 1) ∆u = µ 2) the tensor

T u = 1 2 (∂ y u) 2 -(∂ x u) 2 -∂ x u∂ y u -∂ x u∂ y u 1 2 (∂ x u) 2 -(∂ y u) 2 is divergence-free in finite parts. Furthermore if µ is in H -1 (Ω) then u is in H 1 (Ω)
and div(T u ) = 0 in the sense of distributions. That is u satisfies the conditions (6.3), (6.4), (6.5).

Thanks to the previous theorem we see that studying the conditions (6.3), (6.4), (6.5) can be useful to obtain information about the vorticity of a stationary system of point vortices when the number of vortices tends to infinity. The rest of this subsection is devoted to the definitions needed in the statement of Theorem 6.12 and its proof. The definitions and some results are taken from [START_REF] Sandier | Progress in Nonlinear Partial Differential Equations Birkhauser[END_REF] Chapter 13.

In this section we use an equivalent definition of divergence-free in finite part : Definition 6.13. Let X be a vector field in Ω, and z 1 , ...z N in Ω such that X ∈ C 0 (Ω \ {z 1 , ..., z N }). We say that X is divergence free in finite part if 1. div(X) = 0 in D (Ω \ {z 1 , ..., z N }). 2. ∂B(z i ,δ) X.ν i = 0, ∀ i = 1, ..., N, ∀δ > 0 where ν denotes the outward unit normal to ∂B(z i , δ).

The equivalence between the two previous definitions can be proved using the coarea formula.

Definition 6.14. We say that u is weakly stationary harmonic if T u is divergence free in finite part.

Example 6.1. u(z) = ln |z| is weakly stationary harmonic in R 2 .

Proof. Let z = x + iy. We have that ln |z| is harmonic in R 2 \ {0} and smooth in R 2 \ {0}.

Then it is stationary harmonic in R 2 \ {0}, that is div(T u ) = 0 in R 2 \ {0}, with T u = (∂ x u) 2 -(∂ y u) 2 2∂ x u∂ y u 2∂ x u∂ y u (∂ y u) 2 -(∂ x u) 2 .
We want to show the second condition in the previous definition. Let δ > 0 we have

∂ x u 2 -∂ y u 2 = x 2 -y 2 |z| 2 and 2∂ x u∂ y u = xy |z| 2 .
The outward unit normal to ∂B(0, δ) is ν = z |z| .Hence ∂B(0,δ)

(∂ x u 2 -∂ y u 2 )ν 1 + (2∂ x u∂ y u)ν 2 = ∂B(0,δ) x(x 2 + y 2 ) |z| 3 = δ 2π 0 cos(ϕ)dϕ = 0.
The integral of the other component of T h is computed the same way and we also find that it is equal to 0. Thus u is weaky stationary harmonic.

We can associate to a system of point vortices (6.24) the measure N i=1 d i δ z i , where we denoted by δ z 0 the Dirac mass in z 0 . Let us consider the particular solution of Remark : Note that if u is not in H 1 (Ω) then it does not make sense to say that u is stationary harmonic that is why we need the notion of weak stationary harmonicity.

Proof. We use Definition 6.13. Again away from the points z 1 , ..., z N , u is harmonic and smooth. Thus it is stationary harmonic. Near z 1 we have

u(z) = α 1 ln |z -z 1 | + H 1 (z)
where

H 1 (z) := 1 M N N i=2 d i ln |z -z i | is harmonic near z 1 (
in a neighborhood of z 1 which contains only z 1 and no other z i ) and α 1 is a constant. Without loss of generality we can assume that α 1 = 1 and z 1 = 0. We then have :

∂ x u 2 -∂ y u 2 = x |z| + ∂ x H 1 (z) 2 - y |z| + ∂ y H 1 (z) 2 = x 2 -y 2 |z| 2 + (∂ x H 2 1 -∂ y H 2 1 ) + 2 x |z| ∂ x H 1 - y |z| ∂ y H 1 2∂ x u∂ y u = 2 xy |z| 2 + 2∂ x H 1 ∂ y H 1 + 2 x |z| ∂ y H 1 + y |z| ∂ x H 1 . Thus ∂B(0,δ) (∂ x u 2 -∂ y u 2 )ν 1 + (2∂ x u∂ y u)ν 2 = ∂B(0,δ) x 2 -y 2 |z| 2 ν 1 + 2 xy |z| 2 ν 2 + ∂B(0,δ) (∂ x H 2 1 -∂ y H 2 1 )ν 1 + (2∂ x H 1 ∂ y H 1 )ν 2 + ∂B(0,δ) 2 x |z| ∂ x H 1 - y |z| ∂ y H 1 ν 1 + 2 x |z| ∂ y H 1 + y |z| ∂ x H 1 ν 2 .
The first term in this sum is zero because ln |z| is weakly stationary harmonic. The second term is also zero because H is harmonic, smooth, and hence stationary harmonic and weakly stationary harmonic. For the third term we can use the fact that the normal on ∂B(0, δ)

is ν = z-z 1 |z-z 1 | to prove that it is equal to 2 ∂B(0,δ) ∂ x H 1 .
Hence if u is weakly stationary harmonic this term must be equal to zero for all δ. Then dividing this quantity by δ and letting δ go to 0 we find that ∂ x H 1 (z 1 ) = 0. With the same method applied to the other component of T u we obtain

∂B(0,δ) (2∂ x u∂ y u)ν 1 + (∂ y u 2 -∂ x u 2 )ν 2 = 2 ∂B(0,δ) ∂ y H 1 .
Thus if u is weakly stationary harmonic we find that ∇H 1 (0) = 0. By repeating this argument near each z i , we obtain that if u is weakly stationary harmonic then z 1 , ..., z N form a stationary system of point vortices :

N j=1,j =i d j z i -z j |z i -z j | 2 = 0 ∀i = 1, ..., N.
We now prove Theorem 6.12. Let µ be a Radon measure which satisfies (6.25), (6.26). We set

u N ε := 1 M N ε N i=1 d ε i ln |z -z ε i | (6.29) with M N ε = N ε i=1 |d ε i |.
We want to prove that u N ε converges to a function u when ε goes to 0 such that u satisfies ∆u = µ and u is weakly stationary harmonic. However we need to have a notion of convergence which preserves the notion of weak stationary harmonicity. This is the object of the following definition. Definition 6.16 ([SS07]). We say (with some abuse of notation) that a sequence (

X n ) n in L 1 (Ω) converges in L 1 δ (Ω) to X if X n → X in L 1 loc (Ω)
except on a set of arbitrarily small 1-capacity, or precisely if there exists a family of sets (E δ ) δ>0 such that for any compact K ⊂ Ω, We define similarly the convergence in L 2 δ by replacing L 1 by L 2 in the above.

Proposition 6.17 ([SS07]). Assume (X n ) n∈N is a sequence of divergence-free in finite part vector fields which converges to X in L 1 δ (Ω). Then X is divergence free in finite part.

Corollary 6.18. Assume that u N is a sequence of weakly stationary harmonic functions such that u N converges to u in L 2 δ (Ω) and ∇u N converges in L 2 δ (Ω) then u is weakly stationary harmonic.

Thus to prove Theorem 6.12 we only need to prove that the functions u N ε converge in L 2 δ (Ω) to a function u such that ∇u N ε converge to ∇u in L 2 δ (Ω).

Proposition 6.19. Let Ω be a bounded open set in R 2 . Let µ be a Radon measure in Ω such that (6.25),(6.26) hold. Let

u N ε = 1 M N ε N i=1 d ε i ln |z -z ε i | then there exists u such that u N ε converge in L 2 δ (Ω) to u and ∇u N ε converge to ∇u in L 2 δ (Ω).
Proof. We let

µ N ε := 2π M N ε N ε i=1 δ z ε i . (6.31)
Since Ω is bounded the measure µ N ε has compact support and we can then write

u N ε = ln |z| * µ N ε (6.32)
where * denotes the convolution product. Then for all ϕ in C ∞ c (R 2 ) we have

u N ε , ϕ = ln |z| * µ N ε , ϕ = µ N ε , ln |z| * ϕ .
Now we let ε go to 0, by hypothesis µ N ε converges to µ in (C 0 (Ω)) * . Hence

u N ε , ϕ → µ, ln |x| * ϕ .
This proves that u N ε converges to some u in the sense of distributions.

In the rest of the proof we drop the subscript ε and consider the limit N → +∞ (if N ε stays bounded the proof is immediate). We follow closely the proof of Proposition 13.2 in [START_REF] Sandier | Progress in Nonlinear Partial Differential Equations Birkhauser[END_REF]. We choose a bounded open set Ω such that Ω ⊂⊂ Ω . We can define µ N , µ, u N and u in Ω (using formulas (6.32), (6.31) for u N and µ N valid in R 2 and passing to the limit in Ω ). In Ω we set

v N = u N -u, α N = µ N -µ. (6.33)
We then have ∆v N = α N in Ω . (6.34)

It holds that lim

N →+∞ α N C 0 (Ω ) * = 0. But since we have W 1,q (Ω ) → C 0 (Ω ) for q > 2 we also have C 0 (Ω ) * → W -1,p (Ω ) for p < 2. Thus we obtain lim

N →+∞ α N W -1,p (Ω ) = 0 for p < 2. Now we let δ N = α N W -1,p (Ω ) α N C 0 (Ω ) * + 1 1/2 , F N = {x ∈ Ω; |v N | ≥ δ N }. (6.35)
We have the following bound on the p-capacity of F

N (cf. [EG92] p.158) cap p (F N ) ≤ C v n p W 1,p (Ω) δ p N . (6.36)
We note note that by elliptic regularity theory v N W 1,p (Ω) ≤ C α N W -1,p (Ω ) because of (6.34) and because Ω ⊂⊂ Ω . Thus from (6.35) and (6.36) we find that

cap p (F N ) ≤ C α N p/2 W -1,p (Ω ) ( α N C 0 (Ω ) * + 1) p/2 ,
and therefore tends to 0 as N goes to infinity. This implies in turn that lim

N →+∞ cap 1 (F N ) = 0.
Now we use a cut-off function ϕ ∈ C ∞ (Ω ) such that |ϕ(x)| ≤ 1 for all x ∈ Ω , ϕ ≡ 1 in Ω and ϕ = 0 on ∂Ω . We also set FN = {x ∈ Ω ; |ϕv N | ≥ δ N }.

We have that F N ⊂ FN and FN ∩ Ω = F N since ϕ ≡ 1 in Ω. We use the following truncated function :

ϕv N = ϕv N if |ϕv N | ≤ δ N , δ N if |ϕv N | > δ N . (6.37)
From a property of Sobolev functions (see e.g. Lemma 7.7 in [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]), we have ∇(ϕv N ) = 0 almost everywhere in FN . We thus obtain :

Ω\F N |∇v N | 2 ≤ Ω \ FN |∇(ϕv N )| 2 ≤ Ω ∇(ϕv N ) • ∇(ϕv N ) ≤ Ω -∆(ϕv N )ϕv N .
The last inequality being true since ϕ = 0 on ∂Ω . Using the Leibniz formula we obtain that

∆(ϕv N ) = ∆ϕv N + 2∇ϕ • ∇v N + ϕ∆v N . Hence Ω\F N |∇v N | 2 ≤ Ω |∆ϕv N (ϕv N )| + Ω 2|∇ϕ||∇v N ||ϕv N | + Ω |ϕv N |dα N
where we used the fact that ∆v N = α N in Ω . Now we use Hölder inequality to obtain that

Ω\F N |∇v N | 2 ≤ Cδ N Ω |v N | p 1/p + Cδ N Ω |∇v N | p 1/p + δ N α N C 0 (Ω) * ≤ Cδ N v N W 1,p (Ω ) + α N C 0 (Ω ) * . Thus lim N →+∞ ∇v N L 2 (Ω\F N ) = 0. (6.38)
We can also see, from the definition of F N and because Ω is bounded that lim

N →+∞ v N L 2 (Ω\F N ) = 0. (6.39)
We conclude as in [START_REF] Sandier | Progress in Nonlinear Partial Differential Equations Birkhauser[END_REF]. Since lim n→+∞ cap 1 (F N ) = 0, there is a subsequence, still denoted by {n}, such that n cap 1 (F N ) < +∞. We define

E δ = N > 1 δ F N .
Then cap 1 (E δ ) tends to zero as δ goes to zero since it is bounded above by the tail of a convergent series. Moreover, for any δ > 0 we have F N ⊂ E δ when N is large enough and therefore (6.38) and (6.39) imply that lim

N →+∞ v N L 2 (Ω\E δ ) = lim N →+∞ ∇v N L 2 (Ω\E δ ) = 0.
This proposition proves point 1) and 2) of Theorem 6.12. The next proposition shows that if we add the hypothesis that µ is in H -1 (Ω), then u weakly stationary harmonic implies u stationary harmonic.

Proposition 6.20

(Proposition 13.1 in [SS07]). Assume that X is divergence-free in finite part in Ω and that X is in L 1 (Ω \ E). Then for every ζ ∈ C ∞ c (Ω), Ω\F X • ∇ζ = 0, where F = ζ -1(ζ(E)).
In particular if X is in L 1 (Ω), then F = ∅ in the above and therefore div X = 0 in D (Ω).

If µ is in H -1 (Ω) we have seen in the introduction that u is in H 1 (Ω) and T u is in L 1 (Ω). Thanks to the previous proposition u weakly stationary harmonic implies u stationary harmonic.

Local behavior near a regular point of h µ

Let us recall that we consider a couple (µ, h µ ) which satisfies

h µ ∈ H 1 (Ω) (6.40) ∆h µ = µ, in Ω (6.41)
where µ is a Radon measure and

ω hµ = (∂ x h µ ) 2 -(∂ y h µ ) 2 -2i∂ x h µ ∂ y h µ is holomorphic in Ω (6.42)
In this section we drop the subscript µ when there is no possible confusion. We denote by B r = B(z 0 , r) = {z ∈ C; |z -z 0 | < R} the ball of center z 0 and of radius r. The starting point of the proof of Theorem 6.3 is the following : Lemma 6.21. Let h which satisfies (6.40), (6.41), (6.42). Let z 0 ∈ Ω such that ω h (z 0 ) = 0.

Then there exist R > 0, a function θ : B R → {±1} and a harmonic function

H : B R → R such that ∂ x h(z) -i∂ y h(z) = θ(z) [∂ x H(z) -i∂ y H(z)] , ∀ z ∈ B R . (6.43) Proof. It holds that ω h = (∂ x h) 2 -(∂ y h) 2 -2i∂ x h∂ y h = 4(∂ z h) 2 = (∂ x h -i∂ y h) 2 . Thus (∂ x h-i∂ y h) 2 is a holomorphic function in Ω. If z 0 is such that ω h (z 0 ) = 0 then f := (∂ x h- i∂ y h) 2
satisfies that f is holomorphic in Ω and f(z 0 ) = 0. This implies that in a neighborhood U of z 0 where f (z) does not vanish there exists a function g : U → C such that g 2 = f in U . We can hence deduce that there exists θ : U → {±1} such that

∂ x h -i∂ y h = θ(z)g(z) in U.
From now on we take U = B(z 0 , R) =: B R for R sufficiently small. We then set H(z) := Re z z 0 g(s)ds. This is well defined since B R is simply connected. The function H satisfies the following properties :

1) H vanishes at z 0 .

2) H is harmonic in B R because it is the real part of an holomorphic function.

3) 2∂ z H(z) = g(z) or equivalently ∂ x H -i∂ y H = g Thus ∂ x h -i∂ y h = θ(∂ x H -i∂ y H), in B R . (6.44) Besides we have that ∇H(z 0 ) = 0 since |∇H(z 0 )| 2 = |ω h (z 0 )| 2 = 0.
We set :

B + R := {z ∈ B R ; θ(z) = +1} B - R := {z ∈ B R ; θ(z) = -1}. (6.45)
Idea of the proof of Theorem 6.3 : The strategy of the proof is the following : we first show that the function θ is in BV (B R ). Hence B + R and B - R are sets of finite perimeter in B R . It turns out that the support of µ B R is equal to the essential boundary of B + R minus the (topological) boundary ∂B R . Then we use a theorem of structure of sets of finite perimeter in R 2 due to Ambrosio-Caselles-Morel-Masnou in [START_REF] Ambrosio | Connected components of sets of finite perimeter and applications to image processing[END_REF] (see also [START_REF] Fleming | Functions with generalized gradient and generalized surfaces[END_REF]) to decompose the essential boundary of B + R as a disjoint union of Jordan curves. Because of the relation (6.43) we are able to show that these Jordan curves are unions of some part of the boundary ∂B R and of level curves of the harmonic function H. Since µ is a Radon measure we prove that there can not be an infinite number of level curves of H in the support of µ near z 0 (otherwise µ(B R ) = +∞). Then we can take a smaller open set V containing z 0 such that the support of µ V is the set of zeros of H. In V we can use the fact that ∇(h -θH) = 0 or use the maximum principle to obtain that h = +|H| or h = -|H|. Lemma 6.22. Let h which satisfies (6.40), (6.41), (6.42). Let R > 0 be small enough, θ : B R → {±1} and H : B R → R such that (6.43) holds. Then θ is in BV (B R ).

Proof. We set g = ∂ x H -i∂ y H. Since H is harmonic it holds that g is holomorphic. Since z 0 is not a zero of the function f = (∂ x h -i∂ y h) 2 we have that g does not vanish in B R . Then we can write θ(z) = ∂ x h(z) -i∂ y h(z) g(z) .

We obtain that θ is in L 1 (B R ) since h is in H 1 (Ω) and g is in C ∞ (B R ).

Furthermore we can differentiate θ in the sense of distributions using the Leibniz rule since g ∈ C ∞ (B R ). We obtain

∂ x θ = (∂ 2 xx h -i∂ 2 yx h)g -∂ x g(∂ x h -i∂ y h) g 2 , ∂ y θ = (∂ 2 xy h -i∂ 2 yy h)g -∂ y g(∂ x h -i∂ y h) g 2 .
Summing these two equalities it comes

∂ x θ + i∂ y θ = ∆h g - (∂ x g + i∂ y g)(∂ x h -i∂ y h) g 2 .
But since g is holomorphic in B R it holds that ∂ z g = 1 2 [∂ x g + i∂ y g] = 0. Hence ∂ x θ + i∂ y θ = ∆h g . Now ∆h = µ is a Radon measure and we can write ∂ x θ = Re( 1 g )∆h , ∂ y θ = Im( 1 g )∆h. Let us denote by ., . the duality bracket for distributions. For all ϕ ∈ C 1 c (B R , R 2 ) with |ϕ| ≤ 1, we have

B R θ div ϕ = -∂ x θ, ϕ 1 -∂ y θ, ϕ 2 = - B R Re( 1 g )ϕ 1 dµ - B R Im( 1 g )ϕ 2 dµ.
Hence we obtain

| B R θ div ϕ| ≤ 1 g L ∞ µ(B R ) < +∞
which means that θ is in BV (B R ) by definition.

Recall that a set E ⊂ Ω is a set of finite perimeter in Ω if its characteristic function χ E is in BV (Ω). We have

χ B + R = 1 2 (1 + θ), χ B - R = 1 2 (1 -θ).
Hence B + R and B - R are sets of finite perimeter in B R . We need several definitions and results from the theory of sets of finite perimeter we recall these notions now and we refer the reader to the books [START_REF] Evans | Measure theory and fine properties of functions[END_REF], or [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF] for the proof of these results. Theorem 6.23 ([EG92] p.167). Let E be a set of locally finite perimeter in Ω, then there exists a Radon measure on Ω denoted by ∂E and a ∂E -measurable function ν E : Ω → R such that 1) |ν E (x)| = 1 ∂E -a.e. , and 2)

E div ϕdx = Ω ϕ • ν E d ∂E for all ϕ ∈ C 1 c (Ω, R n ).
We present two notions of "boundary" of sets of finite perimeter : Definition 6.24. Let E be a set of locally finite perimeter in R n and x ∈ R n . We say that x ∈ ∂ E, the reduced boundary of E, if i) ∂E (B(x, r)) > 0 for all r > 0, ii) lim ii) H n-1 (∂ E \ ∂ E) = 0.

We will also use the following theorem Theorem 6.28 (Gauss-Green formula [START_REF] Evans | Measure theory and fine properties of functions[END_REF] p.209). Let E ⊂ R n have locally finite perimeter. i) Then H n-1 (∂ E ∩ K) < +∞ for each compact set K ⊂ R n . ii) Furthermore, for H n-1 a.e. x ∈ ∂ E, there is a unique measure theoretic unit outer normal ν E (x) such that where in the last equalities we set ψ := (-∂ y Hϕ, ∂ x Hϕ). We then use Theorem 6.28 to obtain We can now describe the support of the measure µ B R in terms of the boundary of B + R .

E div(ϕ)dx = ∂ E ϕ • ν E dH n-
∂ B + R ψ • ν B + R dH 1 - ∂ B - R ψ • ν B - R dH 1 = 0. But ∂ B + R = ∂ B - R and ν B + R = -ν B - R because B + R = B R \ B - R .
Lemma 6.30. Let h satisfy the hypothesis (6.40), (6.41), (6.42). Let θ, H, B R , B + R as in Lemma 6.21. Then the support of µ B R is ∂ B + R \ ∂B R and we have

µ B R = -2∇H • ν B + R H 1 ∂ B + R \∂B R . Proof. It holds that ∆h, ϕ = - B R ∇h • ∇ϕ, ∀ϕ ∈ C ∞ c (B R ) (h ∈ H 1 (Ω)) = - B + R ∇H • ∇ϕ + B - R ∇H • ∇ϕ (∇h = ±∇H in B ± R )
Now we use the fact that H is harmonic in B R (∆H = 0 in B R ) and the Gauss-Green formula 6.28 to obtain ∆h, ϕ = - We study in more details ∂ B + R \∂B R . In particular since ∇h and ν B + R are parallel on ∂ B + R \∂B R we expect H to be constant on the connected components of this set. In order to prove this fact we need more definitions and more results from geometric measure theory, these can be found in the article [START_REF] Ambrosio | Connected components of sets of finite perimeter and applications to image processing[END_REF]. Definition 6.32. A curve Γ ⊂ R 2 is rectifiable if H 1 (Γ) < ∞. Lemma 6.33 (Lemma 3 in [START_REF] Ambrosio | Connected components of sets of finite perimeter and applications to image processing[END_REF]). Let C ⊂ R n be a compact connected set with H 1 (C) < ∞. Then for any pair of distinct points x, y ∈ C there exists a Lipschitz one-to-one map γ : [0, 1] → C such that γ(0) = x and γ(1) = y.

∂ B + R \∂B R ϕ∇H • ν B + R dH 1 + ∂ B - R \∂B R ϕ∇H • ν B - R dH 1 = -2 ∂ B + R \∂B R ϕ∇H • ν B +
A consequence of this lemma is that any rectifiable Jordan curve admits a Lipschitz re-parametrization.

In order to state the next theorem, following [START_REF] Ambrosio | Connected components of sets of finite perimeter and applications to image processing[END_REF], we introduce a formal Jordan curve J ∞ whose interior is R n and a formal Jordan curve J 0 whose interior is empty. We denote by S the set of Jordan curves and formal Jordan curves. We then have the following description of the essential boundary of sets of finite perimeter in R 2 . Theorem 6.34 (Corollary 1 in [START_REF] Ambrosio | Connected components of sets of finite perimeter and applications to image processing[END_REF]). Let E be a subset of R 2 of finite perimeter. Then there is a unique decomposition of ∂ E into rectifiable Jordan curves {C + i , C - k : i, k ∈ N} ⊂ S, such that i) Given int(C + i ), int(C + k ), i = k, they are either disjoint or one is contained in the other ; given int(C - i ), int(C - k ), i = k, they are either disjoint or one is contained in the other. Each int(C - i ) is contained in one of the int(C + k ). ii) P (E) = i H 1 (C + i ) + k H 1 (C - k ). iii) If int(C + i ) ⊂ int(C + j ), i = j, then there is some rectifiable Jordan curve C - k such that int(C + i ) ⊂ int(C - k ) ⊂ int(C + j ). Similarly if int(C - i ) ⊂ int(C - j ), i = j, then there is some rectifiable Jordan curve C + k such that int(C - i ) ⊂ int(C + k ) ⊂ int(C - j ). iv) Setting L j = {i; int(C - i ⊆ int(C + j )}, the sets Y j = int(C + j ) \ ∪ i∈L j int(C - i ) are pairwise disjoint, indecomposable and E = ∪ j Y j .

We are now able to prove : Lemma 6.35. Let θ be such that (6.43) holds, θ ∈ BV (B R ). Let B + R as before. There exist (possibly infinitely many) disjoint rectifiable Jordan curves γ i such that

∂ B + R = +∞ i=1 γ i .
Proof. We must check that B + R is a set of finite perimeter in R 2 (not just in B R ) in order to apply Theorem 6.34. To this end we set

θ := θ if x ∈ B R , -1 if x ∈ R 2 \ B R .
We also set χ B + R = 1 2 (1 + θ), this is the characteristic function of B + R in R 2 . We must prove that χ B + R ∈ BV (R 2 ). First we note that χ B + R ∈ L 1 (R 2 ) because it is bounded in B R and it is null in R 2 \ B R . Second for all ϕ ∈ C ∞ c (R 2 , R 2 ) we have

R 2 χ B + R div(ϕ) = 1 2 R 2 θ div(ϕ)
since ϕ has compact support in R 2 . Thus

R 2 χ B + R div(ϕ) = 1 2 B R θ div(ϕ) - R 2 \B R div(ϕ) = 1 2 B + R div(ϕ) - 1 2 B - R div(ϕ) - 1 2 R 2 \B R div(ϕ).
Now we claim that B + R , B - R and R 2 \ B R have locally finite perimeter in R 2 . This is obvious for R 2 \ B R because B R is a smooth open set with finite perimeter in R 2 . For B + R , B - R thanks to a deep criterion (cf. Theorem 1 p.222 of [START_REF] Evans | Measure theory and fine properties of functions[END_REF]) we must only check that for all K compact subset of R 2 H 1 (K ∩ ∂ B + R ) < +∞.

But H 1 (K ∩ ∂ B + R ) ≤ H 1 (B R ∩ ∂ B + R ) < +∞ because B + R has finite perimeter in B R by definition, and the same is true for B - R . We can thus apply the Gauss-Green formula 6.28 to obtain

R 2 θ div(ϕ) = ∂ B + R ϕ • ν B + R dH 1 - ∂ B - R ϕ • ν B - R dH 1 - ∂B R ϕ • ν B R dH 1 . Hence for all ϕ ∈ C ∞ c (R 2 , R 2 ) we have | R 2 θ div(ϕ)| ≤ 2H 1 (∂ B + R ) + H 1 (∂B R ) < +∞.
This proves that χ B + R is in BV (R 2 ). We can thus apply the Theorem 6.34 to obtain the lemma.

In order to pursue the proof of the main result we need the following version of the coarea formula : where d E f x is the tangential differential of f at x ∈ E, C k d E f x is the k-dimensional coarea factor, H 0 is the 0-dimensional Hausdorff measure (this is the counting measure) and for the definitions of these notions we refer to [AFP00] Chapter 2.

We can apply the previous theorem with the function f : R 2 → R, x → |x| (we have that |d E f x | ≤ 1 for this f and all E countably H 1 -rectifiable subset of R 2 ). We then find that for all rectifiable Jordan curves γ we have, for R > ρ > 0

H 1 (γ ∩ (B R \ B R-ρ )) ≥ R R-ρ
H 0 (γ i ∩ C t )dt (6.51)

where C t = {z ∈ R 2 ; |z| = t}. We then obtain : Lemma 6.37. Under the same assumptions as in Lemma 6.35. There exist 0 < R < R and (possibly infinitely many) connected rectifiable simple curves Γ j such that Proof. We use the formula (6.51), and the fact that H 0 (γ i ∩ C t ) is finite for almost every t ∈ [R -ρ, R] . We choose R such that H 0 (γ i ∩ C R ) < +∞ and we have that for all i ∈ N there exists k i ∈ N and k i intervals of R such that γ i ∩ B R = γ i (I 1 ) ∪ ... ∪ γ i (I k i ) with I j =]a j , b j [ and γ i (a j ), γ i (b j ) ∈ ∂B R for j = 1, ..., k i . Hence γ i ∩ B R is a finite union of connected rectifiable simple curves. We define B + R = B + R ∩ B R and we find that

∂ B + R \ ∂B R = +∞ j=1 Γ j . ( 6
supp µ B R = ∂ B + R \ ∂B R = supp µ B R ∩ B R = +∞ i=1 γ i ∩ B R = +∞ j=1 Γ j
with Γ j connected rectifiable simple curves.

We are now in position to prove Theorem 6.3.

Proof of Theorem 6.3. Let h which satisfies (6.40), (6.41),(6.42). Let θ, H be defined by (6.43).

Let R > 0 be as in Lemma 6.37. From now on we denote by B the ball B R . We also denote by B + = {z ∈ B; θ(z) = +1}. Let {Γ j } j∈N simple connected rectifiable given by Lemma 6.37. The next claim states that each connected component of ∂ B + \ ∂B is a connected component of some level curve of the function H in B.

Claim 1. For all i ∈ N, there exists c i ∈ R such that

Γ i = {z ∈ R 2 ; H(z) = c i } ∩ B.
1) We first show that for all i ∈ N there exists c i ∈ R such that Γ i ⊂ {H = c i } ∩ B, where {H = c i } is a short for {z ∈ R 2 ; H(z) = c i }. Indeed let x, y ∈ Γ i , x = y, thanks to Lemma 6.33 we can find a bijective lipschitz map f : [0, 1] → Γ i such that f (0) = x and f (1) = y. We then have

H(y) -H(x) = 1 0 d dt (H • f )(t)dt because H • f ∈ W 1,1 ([0, 1], R), (that is H • f is absolutely continuous).
To prove the absolute continuity we use that H ∈ C ∞ (B) and f ∈ W 1,∞ ([0, 1], B). Hence ( we obtain that H • f ∈ W 1,∞ ([0, 1]) ⊂ W 1,1 ([0, 1]) (see e.g. Proposition 9.5 p. 270 of [START_REF]Functional analysis, Sobolev spaces and partial differential equations[END_REF]). Thus

H(y) -H(x) = 1 0 ∇H(f (t) • f (t)dt
where f (t) denotes the derivative of f which exists for L 1 -almost every t ∈ [0, 1] (because Lipschitz functions are differentiable almost everywhere). But f (t) is tangent to Γ i and ∇H(f (t))

is orthogonal to f (t) for almost every t ∈ [0, 1]. Indeed thanks to Lemma 6.29, we have that ∇H parallel to ν B + H 1 -a.e. Hence we obtain that ∇H(f (t)) • f (t) = 0 a.e. and H(y) = H(x). This shows that Γ i ⊂ {H = c i } ∩ B.

2) We show that Γ i = {H = c i } ∩ B using the following Lemma 6.38. We use the fact that since ∇H does not vanish in B the level curves {H = c i } ∩ B are diffeomorphic to straight line (this is a consequence of the implicit function theorem or can be seen in Theorem 6.5) if R is small enough. Hence they are connected. We then apply Lemma 6.38 to Γ i and {H = c i } ∩ B. These two curves are rectifiable, connected and simple, and we have Γ i ⊂ {H = c i } ∩ B and Γ i ∩ ∂B = {H = c i } ∩ ∂B by continuity of H.

Lemma 6.38. Let B be a ball of radius R. Let γ and γ be two connected rectifiable simple curves. We also denote by γ, γ : [0, 1] → R 2 some Lipschitz parametrization of these curves. We suppose that γ, γ are homeomorphism from [0, 1] onto their image. Assume that i) γ(]0, 1[) ⊂ B and γ(0), γ(1) ∈ ∂B, ii) γ(]0, 1[) ⊂ B and γ(0), γ(1) ∈ ∂B, iii) γ([0, 1]) ⊂ γ([0, 1]).

Then γ = γ.

We postpone the proof of this lemma at the end of the section. Now that we know the geometry of the curves Γ i we can prove that there exists only a finite number of such curves in a sufficiently small ball.

Claim 2. Let ρ > 0 small enough such that Γ i ∩ B(z 0 , ρ) = {H = c i } ∩ B(z 0 , ρ) is diffeomorphic to an open segment for all i ∈ N such that Γ i = ∅. Then there exists a finite number of curves Γ i such that Γ i ∩ B(z 0 , ρ) = ∅.

With ρ as in the statement of the claim we let B ρ = B(z 0 , ρ). Since θ ∈ BV (B R ) we also have θ ∈ BV (B ρ ).Thus using the same notations as before we have

+ ∞ > H 1 (∂ B + ρ \ ∂B ρ ) = H 1 (supp(µ Bρ )) = H 1 ( +∞ i=1 Γ i ∩ B ρ ) ≥ ρ 0 H 0 ( +∞ i=1 Γ i ∩ C t )dt
where in the last equality we used the coarea formula (Theorem 6.36), and we let C t = {z ∈ C; |z| = t}. The coarea formula also tells us that for almost every t ∈ [0, ρ] we have H 0 ( +∞ i=1 Γ i ∩ C t ) < +∞. But if ρ is small enough then every level curves of the harmonic function H meet the boundary of the ball B ρ . This is a consequence of the maximum principle.

As a consequence we have that H 0 ( +∞ i=1 Γ i ∩ C t ) is exactly two times the number of curves Γ i inside B t . Then the number of curves Γ i is finite inside B ρ .

We can now conclude the proof of Theorem 6.3. The last claim proved that there exists a finite number of Γ i near Z 0 := {z ∈ B; H(z) = 0}. Thus there exists η > 0 such that dist(Z 0 , Γ i ) > η for all i ∈ N such that Γ i is not included in Z 0 . We then set V := B(z 0 , η 2 ). Because of the definition of η we obtain that supp(µ V ) = {z ∈; H(z) = 0}.

Note that Z 0 is a smooth connected rectifiable curve near z 0 (since ∇H(z 0 ) = 0). We also set as usual V + = {z ∈ V ; θ(z) = +1}, V -= {z ∈ V ; θ(z) = -1}. We have that ∇h = +∇H, on V + , ∇h = -∇H, on V -.

We thus deduce that h = H on V + and h = -H on V -because h = H = 0 on ∂ V + \ ∂V = Z 0 . We know that H does not vanish in V + and V -, because H vanishes only on Z 0 . Hence H has constant sign on V + and on V -thanks to the maximum principle. These two signs are opposite, because if they were the same then the minimum (or maximum) of H would be 0 and would be inside the domain V , this contradicts the maximum principle. We can assume for example that H is non negative in V + and then h = |H| in V .

Proof of Lemma 6.38. By contradiction, assume that there exists p ∈ γ \ γ. Let t 0 ∈]0, 1[ such that γ(t 0 ) = p. Then we have ]0, 1[= γ-1 γ(]0, t 0 [) ∪ γ(]t 0 , 1[) since γ(]0, 1[) ⊂ γ(]0, 1[) and since γ(t 0 ) / ∈ γ. We then deduce that ]0, 1[= γ-1 (γ(]0, t 0 [)) ∪ γ-1 (γ(]t 0 , 1[)) .

But since γ and γ are homeomorphism onto their image we have that γ-1 (γ(]0, t 0 [)) and γ-1 (γ(]t 0 , 1[)) are two disjoint open sets. Thanks to the connectedness of ]0, 1[ we can deduce that 1) γ-1 (γ(]0, t 0 [)) =]0, 1[ and γ-1 (γ(]t 0 , 1[)) = ∅ or 2) γ-1 (γ(]0, t 0 [)) = ∅ and γ-1 (γ(]t 0 , 1[)) =]0, 1[. These two cases are similar. Let us assume that we are in case 1). We can then obtain that γ(]t 0 , 1[) ∩ γ(]0, 1[) = ∅.

This implies that γ(]t 0 , 1[) = ∅ or γ γ. The first assertion is impossible because γ is assumed to be a homeomorphism from [0, 1] onto its image and the second possibility is in contradiction with the hypothesis iii). Thus it holds that γ = γ.

Local behavior near a critical point of even order of h µ

This section is devoted to the proof of Theorem 6.4. It is very similar to the proof of Theorem 6.3. Here ω hµ (z 0 ) = 0, but since we assume that z 0 is a zero of even order of ω hµ there is no difficulty to find a holomorphic function g such that (∂ x h µ -i∂ y h µ ) 2 = g(z) 2 . Then the proof of Theorem 6.4 is a rather direct adaptation of the proof of Theorem 6.3 except that here because the function g vanishes at z 0 we can only show that the function θ defined as in the previous section is in BV loc (B R \ {z 0 }) for R sufficiently enough. This introduce a new technical difficulty. We drop the subscript µ in the rest of this section.

Lemma 6.39. Let h which satisfies (6.40), (6.41), (6.42). Let z 0 ∈ Ω be a zero of even order of ω h (z) = (∂ x h -i∂ y h) 2 (z) . Then there exist R > 0, a function θ : B R → {±1} and a harmonic function H : B R → R such that

∂ x h(z) -i∂ y h(z) = θ(z) (∂ x H(z) -i∂ y H(z)) , ∀ z ∈ B R (6.53)
Proof. Since z 0 is a zero of even order of ω h , we can find a neighborhood U of z 0 , n ∈ N and a holomorphic function f 1 : U → C such that f 1 (z 0 ) = 0 and (∂ x h -i∂ y h) 2 = (z -z 0 ) 2n f 1 (z). (6.54)

Since f 1 (z 0 ) = 0, we can choose a smaller neighborhood of z 0 still denoted by U such that in U there exists a holomorphic function denoted by ϕ 1 which satisfies ϕ 2 1 (z) = f 1 (z), and furthermore we can choose U = B(z 0 , R) for R small enough. We then have (∂ x h -i∂ y h) 2 = [(z -z 0 ) n ϕ(z)] 2 =: g(z) 2 .

(6.55)

We set F (z) := The function H is harmonic in B R and satisfies 2∂ z H = F (z) = g(z) = (z -z 0 ) n ϕ 1 (z).

Hence, thanks to (6.54) we deduce that there exists θ : U → {±1} such that Lemma 6.46. Under the same assumptions as in Lemma 6.44 with R sufficiently small there exists a finite number N ρ of curves Γ ρ j such that Γ ρ j ∩ A R ,ρ = ∅. We then have

supp(µ A R ,ρ ) = ∂ (A + R ,ρ ) \ ∂A R ,ρ = N ρ j=1 Γ ρ j .
Proof. As in Lemma 2 this is due to the coarea formula and the fact that the curves Γ ρ i are level curves of the harmonic function H.

The next result shows that, with R fixed if we take a larger annulus, then the number of curves in the decomposition of the support of µ is the same. This is due to the geometry of these curves since they are level curves of the harmonic function H. Lemma 6.47. Under the same assumptions as in Lemma 6.44 let ρ 1 < ρ 2 and R as before small enough. Then using the previous notations we have N ρ 2 = N ρ 1 and, up to re-order it holds Besides it holds that supp(µ A R,ρ 2 ) = supp(µ A R,ρ 1 ) ∩ A R,ρ 2 . We thus deduce that

Γ ρ 1 j ⊂ Γ
Nρ 1 j=1 Γ ρ 1 j ⊂ Nρ 2 j=1 Γ ρ 2 j .
We also recall that we have the existence of real numbers (c ρ 1 j ), j = 1, ..., N ρ 1 and (c ρ 2 j ), j = 1, ..., N ρ 2 such that Γ ρ 1 j is exactly one connected component of {H = c ρ 1 j } ∩ A R,ρ 1 and Γ ρ 2 j is exactly one connected component of {H = c ρ 2 j } ∩ A R,ρ 2 .

Assume that there exists c ρ 1 j 0 which is different from all the c ρ 2 i for i = 1, ..., N ρ 2 . Thanks to the maximum principle every connected component of level curves of the harmonic function H which lies in the ball B R meets the boundary of this ball if R is small enough. We thus obtain that Γ ρ 1 j 0 ∩ A R,ρ 2 = ∅ and then

Γ ρ 1 j 0 ∩ A R,ρ 2 ⊂ supp(µ A R,ρ 2 ).
As a consequence we obtain that Γ ρ 1 j 0 ∩A R,ρ 2 = Γ ρ 2 i 0 for some 1 ≤ i 0 ≤ N ρ 2 . This is a contradiction with our hypothesis on c ρ 1 j 0 . We then have

{c ρ 1 j } j = {c ρ 2 i } i .
With the same justification we prove that N ρ 1 = N ρ 2 . And then up to reorder we have Γ ρ 1 j ⊂ Γ ρ 2 j for j = 1, ..., N ρ 1 .

We are now in position to prove : Lemma 6.48. Let θ, H be such that (6.53) holds with h which satisfies (6.40), (6.41), (6.42).

As before we set B + R = {z ∈ B R ; θ(z) = +1}. Then there exists R > 0 small enough such that H 1 (∂ B + R \ ∂B R ) < +∞ and consequently θ is in BV (B R ).

Proof. By contradiction if H 1 (∂ B + R \ ∂B R ) = +∞ then for all sequence (ρ n ) of real numbers such that ρ n 0 we have and the union of these sets is increasing. We now use the previous Lemmas 6.44 and 6.47 to obtain that for R small enough there exists an integer N such that for all n ∈ N there are N simple connected rectifiable curves Γ ρn j and N real numbers c j such that

∂ A + R,ρn \ ∂A R,ρn = N j=1
Γ ρn j and Γ ρn j = {H = c j } ∩ A R,ρn . Furthermore we also have Γ ρn j ⊂ Γ ρm j if n > m. We then obtain

H 1 ∂ B + R ∩ A R,ρn ≤ H 1 N i=1 {H = c i } ∩ B R ≤ N i=1 H 1 ({H = c i } ∩ B R ) .
But for R small enough the level curves of H have a finite Hausdorff measure. Thus there exists M > 0 such that for all n ∈ N,

H 1 (∂ B + R ∩ A R,ρn ) ≤ M.
This is a contradiction with (6.59) and then H 1 (∂ B + R \ ∂B R ) < +∞. Now we prove that θ ∈ BV (B R ). We recall that in the proof of Lemma 6.40 we found that

∂ x θ -i∂ y θ = ∆h g = 1 g µ
in the sense of distributions where g = ∂ x H -i∂ y H. We then have

∂ x θ -i∂ y θ = 1 |g| 2 gµ = (∂ x H + i∂ y H)µ |∇H| 2 thus ∂ x θ = ∂xH |∇H| 2 µ and ∂ y θ = ∂yH |∇H| 2 µ. Now for all ϕ ∈ C 1 c (B R , R 2 ) with |ϕ| ≤ 1 B R θ div ϕ = -∂ x θ, ϕ 1 -∂ y θ, ϕ 2 = - B R ∂ x H |∇H| 2 ϕ 1 dµ + B R ∂ y H |∇H| 2 ϕ 2 dµ.
We now use Lemma 6.42 to say that

µ B R = -2∇H • ν + B R H 1 ∂ B + R \∂B R hence B R θ div ϕ = 2 ∂ B + R \∂B R ∇H • ν B + R ∂ x Hϕ 1 |∇H| 2 dH 1 -2 ∂ B + R \∂B R ∇H • ν B + R ∂ y Hϕ 2 |∇H| 2 dH 1 .
We thus deduce, using the fact that

| ∂xH |∇H| 2 | ≤ 1 and | ∂yH |∇H| 2 | ≤ 1, that | B R θ div ϕ| ≤ 4H 1 (∂ B + R \ ∂B R ) < +∞
for all ϕ ∈ C 1 c (B R , R 2 ) ; |ϕ| ≤ 1. This proves the claim.

From this point we have all the ingredients to pursue the proof of Theorem 6.4 as in the previous section.

Proof of Theorem 6.4. In particular |H| ∈ C ∞ (W k ) for all 1 ≤ k ≤ 2n + 3. We are now in position to state that Claim 6. The function θ is in BV loc (W k \ {z 0 }) for all 1 ≤ k ≤ 2n + 3. The proof of this fact is the same as the proofs of Lemma 6.22 and 6.40 in the previous sections. We have all the ingredients to repeat the arguments of Sections 3 and 4 in each sub-domains W k and obtain Claim 8. For all 1 ≤ k ≤ 2n + 3, there exist N k ∈ N, and N k simple connected rectifiable curves Γ k j , 1 ≤ j ≤ 2n + 3 such that

∂ W + k \ ∂W = N k j=1 Γ k j .
Furthermore there exist c k j real numbers such that Γ k j is exactly a connected component of {z ∈ U ; H(z) = c k j }.

Résumé

Cette thèse est dédiée à l'analyse de quelques problèmes variationnels motivés par le modèle de Ginzburg-Landau en supraconductivité. Dans la première partie on étudie l'existence de solutions pour les équations de Ginzburg-Landau sans champ magnétique et avec données au bord de type semi-rigides. Ces données consistent à prescrire le module de la fonction sur le bord du domaine ainsi que son degré topologique. Ici la méthode directe du calcul des variations ne peut pas s'appliquer car le degré n'est pas continu pour la convergence faible dans l'espace de Sobolev adapté. On dit que c'est un problème sans compacité : un phénomène de "bubbling" apparaît.

Dans le Chapitre 1 on étudie des conditions sous lesquelles la différence entre deux niveaux d'énergie est strictement optimale. Pour cela on adapte une technique due à Brezis-Coron. Ceci nous permet de redémontrer un résultat (précédemment obtenu par Berlaynd Rybalko et Dos Santos) d'existence de solutions stables pour les équations de Ginzburg-Landau dans des domaines multiplement connexes.

Dans le Chapitre 2 on considère les applications harmoniques à valeurs dans R 2 avec des conditions au bord de type degrés prescrits sur un anneau. On fait un lien avec la théorie des surfaces minimales dans R 3 grâce à la différentielle de Hopf. Ceci nous conduit à l'étude des surfaces minimales bordées par deux cercles dans des plans parallèles. On prouve l'existence de telles surfaces qui ne sont pas des caténoïdes grâce à un résultat de bifurcation.

On utilise alors les résultats obtenus pour déduire des théorèmes d'existence et de non existence de minimiseurs de l'énergie de Ginzburg-Landau à degrés prescrits dans un anneau. Dans ce troisième Chapitre on obtient des conclusions pour ε grand.

Le Chapitre 4 est dédié aux problèmes à degrés prescrits en dimension n ≥ 3. On y montre la non existence des minimiseurs de la n-énergie de Ginzburg-Landau dans un domaine difféomorphe à une boule. On étudie ensuite des points critiques de type min-max pour une énergie perturbée.

La deuxième partie est consacrée à l'analyse asymptotique des solutions des équations de Ginzburg-Landau lorsque ε tend vers zéro. Sandier et Serfaty ont étudié le comportement asymptotique des mesures de vorticité associées aux équations. Ils ont notamment trouvé des conditions critiques sur les mesures limites dans le cas des équations avec et sans champ magnétique. Nous nous intéressons alors à ces conditions dans le cas sans champ magnétique. Le problème de la régularité des mesures limites se ramène ainsi à l'étude de la régularité des fonctions stationnaires harmoniques dont le Laplacien est une mesure. Nous montrons que localement de telles mesures sont supportées par une union de lignes appartenant à l'ensemble des zéros d'une fonction harmonique.

Mots clés : Modèle de Ginzburg-Landau, Supraconductivité, Degrés prescrits, Perte de compacité, Bubbling, Applications harmoniques, Différentielle de Hopf, Surfaces minimales, Mesure de vorticité.

  differentials of solutions of (3.1) . . . . . . . . . . . . . . . . . . . . . . . . 3.4 The case c = 0 : holomorphic solutions . . . . . . . . . . . . . . . . . . . . . . . . 3.5 The case c = 0 : properties of radial solutions . . . . . . . . . . . . . . . . . . . . 3.5.1 Non existence of solutions with c = 0 in I p,q if p = q . . . . . . . . . . . . 3.5.2 The case c < 0 : geometry of the catenoid . . . . . . . . . . . . . . . . . . Chapitre 1

Figure 1 . 1 :

 11 Figure 1.1: Illustration de l'effet Meissner. Crédits : [eS11]

Figure 1

 1 Figure 1.2: Vortex (blanc) observés dans un supraconducteur de type II . Crédits : laboratoire de supraconductivité d'Oslo : http ://www.mn.uio.no/fysikk/english/

(1. 3 )

 3 Ici ν désigne la normale sortante à Ω, et ∂ ν représente la dérivée normale. De plus a ∧ b := 1 2i (a b -āb) pour (a, b), dans C, représente le déterminant de a, b vus comme vecteurs de R 2 .

Figure 1 . 3 :

 13 Figure 1.3: Une surface minimale bien connue : la caténoïde. Crédits : Matthias Weber, www.indiana.edu/ minimal/.

2

  dans l'espace I = {v ∈ H 1 (A, C); | tr |∂Ω v| = 1}. Les applications de l'espace I sont classifiées par leurs degrés sur le bord de A. Pour U ∈ {Ω, ω}, et u ∈ I on pose deg(u, ∂U ) = 1 2π ∂U u ∧ ∂ τ udτ ∈ Z (1.4) où τ désigne le vecteur unitaire tangent à ∂U qui vérifie que (ν, τ ) est une base orthonormée directe de R 2 , avec ν la normale sortante à U . On a alors I = p,q∈Z I p,q

. 10 )

 10 Théorème 1.10 (Lemme 3.5 dans [BMRS14]). Soit I = {v ∈ H 1 (D, C); | tr |S 1 v| = 1}. Les points critiques de E dans I d = {u ∈ I; deg(u, S 1 ) = d} sont exactement : a) les d-produits de Blaschke si d > 0, b) les conjugués des d-produits de Blaschke si d < 0, c) les constantes de module 1 si d = 0. d) De plus toutes ces solutions sont minimisantes dans les classes I d .

  Théorème 1.20 (Théorème 5.10, Chapitre 5). Pour d ∈ N * l'infimum m ε (d, Ω) n'est jamais atteint. Si on s'intéresse à la n-énergie de Dirichlet alors, dû à la rigidité des transformations conformes de l'espace en dimension n ≥ 3, le Théorème 1.10 est modifié, on obtient Théorème 1.21 (Théorème 5.10). Soit d ∈ Z * , l'infimum m(d, Ω) est atteint si et seulement si |d| = 1 est Ω est une boule euclidienne.

  Definition 2.2. ([BR10]) Let u ∈ H 1 (A, C) and V ∈ C ∞ (A,R) be a solution of the scalar boundary A, V = 1 on ∂Ω, V = 0 on ∂ω.

Figure 2 . 1 :

 21 Figure 2.1: Image of {re iφ ; µ < r < 1} by w in the case i).

Contents 3. 1

 1 Introduction and statement of the results . . . . . . . . . . . . . . . . 48 3.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.2.1 Notations and definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.2.2 Properties of solutions of (3.1) . . . . . . . . . . . . . . . . . . . . . . . 54 3.2.3 Minimizing sequences : Price lemma and insertion of bubbles . . . . . . 54 3.3 Hopf differentials of solutions of (3.1) . . . . . . . . . . . . . . . . . . 56 3.4 The case c = 0 : holomorphic solutions . . . . . . . . . . . . . . . . . . 59 3.5 The case c = 0 : properties of radial solutions . . . . . . . . . . . . . . 65 3.5.1 Non existence of solutions with c = 0 in I p,q if p = q . . . . . . . . . . . 65 3.5.2 The case c < 0 : geometry of the catenoid . . . . . . . . . . . . . . . . . 68 3.5.3 The case c > 0 : geometry of the helicoid . . . . . . . . . . . . . . . . . 75 3.6 Non rotationally symmetric minimal surface . . . . . . . . . . . . . . 75

  a, b). * a, b stands for the scalar product a, b = a 1 b 1 + a 2 b 2 = 1 2 (a b + āb). * D denotes the unit disc, S 1 denotes the unit circle. More generally we set D r := {z ∈ C; |z| < r} and C r := {z ∈ C; |z| = r}. * A r := {z ∈ C; r < |z| < 1}.

  ) thus we see how the Hopf quadratic differential appears in surface theory. Proposition 3.25. ([IKO12]) Let V ⊂ C be a domain and X = (u, h) : V → C × R = R 3 be the conformal representation of a surface (not necessarily minimal). Then * the function ∂ z u∂ z u admits a continuous branch of square root in V , * for each smooth closed curve Γ ⊂ V we have Re Γ -i ∂ z u∂ z udz = 0 * the real isothermal coordinate is given byh = Re z 0 z -2i ∂ z u∂ z udzwhere the line integral runs along any smooth curve γ ⊂ V beginning at a given point z 0 ∈ V and terminating at z.Since we are interested in harmonic C-valued functions we now explain how they are linked to minimal surface theory. Proposition 3.26. ([DHS10] p.72) Let X = (u 1 , u 2 , h) : V → R 3 be a conformal parametrization of a surface. Then this surface is minimal if and only if ∆X = 0 i.e. ∆u 1 = ∆u 2 = ∆h = 0. Now we can state a proposition which allows us to build a minimal surface from a C-valued harmonic function and its Hopf differential.

  .30) But we claim that deg(T, S 1 ) = deg(u, S 1 ) and deg(T, C ) = deg(u, C ). Indeed on ∂A we have T ∧ u = 0 because u ∧ ∂ r u = 0 on ∂A thanks to (3.1). Then on S 1 there exists a real function λ 1 such that ∂ r u(e iθ ) = λ 1 (θ)u(e iθ ).

  ) and the same argument as before implies that T = u on C or T = -u on C thus deg(T, C ) = deg(u, C ) and finally deg(u, S 1 ) = deg(u, C ). * If c < 0 then ∂ θ u does not vanish in A and we can consider the map R := ∂ θ u

  C); |u| = 1 a.e. on S 1 , deg(u, S 1 ) = p}, and E r (u) := Ar |∇u| 2 .

  p.82. We present a slightly different proof here. A direct computation shows that solutions of the problem (3.57) with µ = 0 are of the form w(r) = a tanh(pr) + b[1 -(pr) tanh(pr)], for a, b ∈ R. (3.61) For b = 0, the zeros of the function (3.61) occur at the solutions of coth(pr) = pr -c, where c = -a

Figure 3 . 1 :

 31 Figure 3.1: View of bifurcating solutions for p = 2 in the plane {z = 0}.

Figure 3 . 2 :

 32 Figure 3.2: Bifurcating solutions for p = 2.

Contents 4 . 1

 41 Introduction and main results . . . . . . . . . . . . . . . . . . . . . . . 86 4.2 Some known results and literature review . . . . . . . . . . . . . . . 90 4.2.1 Bounds for m ε (p, q) and cost to move degrees . . . . . . . . . . . . . . . 90 4.2.2 Some known Existence/Non Existence results : the case ε ∈]0, ∞[ . . . . 91 4.2.3 Some Existence/Non Existence results : the case ε = ∞ . . . . . . . . . 93 4.3 Existence Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.3.1 The key argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.3.2 Consequences of the key argument : existence of minimizers . . . . . . . 96 4.3.3 Comparison with the work of Berlyand&Golovaty [GB02] . . . . . . . . 98 4.3.4 Asymptotic behavior of minimizers as ε → +∞ . . . . . . . . . . . . . . 101 4.4 Non Existence Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.4.1 Strategy of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.4.2 Asymptotic analysis of v ε = w ε e -ıqθ . . . . . . . . . . . . . . . . . . . . 108 4.4.3 Reformulation of L ε (w ε ) and a first lower bound of L ε (w ε ) . . . . . . . 108 4.4.4 Lower bound of Lε (v ε ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 4.4.5 Last computations and conclusion . . . . . . . . . . . . . . . . . . . . . 115 4.5 Comments and perspectives . . . . . . . . . . . . . . . . . . . . . . . . 117

  Proposition 4.13. [Standard bubbling] Let ε ∈]0, ∞], η > 0, e ∈ {(1, 0), (0, 1)} and u ∈ I. There are v + , v -∈ I s.t. v + ∈ I deg(u)+e , v -∈ I deg(u)-e and

  |p | ≤ |p| and qq ≥ 0, |q | ≤ |q| . Lemma 4.21. Let P = (p, q) ∈ Z 2 , ε * ∈]0, ∞] and (ε n ) n be an increasing sequence s.t. ε n ↑ ε * or ε n = ε * for all n. Consider a sequence (u n ) n ⊂ I P s.t. E εn (u n ) ≤ m εn (P ) + o n (1).

  to the Hopf lemma. By Propositions 4.13&4.14 we have the existence of ũ ∈ I P s.t.m ε * (P ) ≤ E ε * (ũ) < E ε * (u) + π|P -P | = m ε * (P ) + π|P -P |.(4.10)By mimicking the argument which gives (4.9) we obtainm ε * (P ) = lim n m εn (P ) ≥ lim inf n E εn (u n ) ≥ m ε * (P ) + π|P -P |. (4.11)Clearly (4.11) is in contradiction with (4.10). Thus if p ≥ 0 then p ≤ p. Using the same argument we prove that if p ≥ 0 then p ≥ 0 and therefore p ∈ [0, p]. If p ≤ 0, we obtain, through the same method, that p ∈ [p, 0]. The same results hold for q instead of p. Hence we obtain that P ∈ A P .We now prove the last part of the proposition. Since we always have m ε * (P ) ≤ m ε * (P ) + π|P -P | (by Proposition 4.11.2), by using (4.11) we deduce that m ε * (P ) = m ε * (P ) + π|P -P |.

  Proposition 4.23. Let p > 0 and D be an annular type domain s.t.m ∞ (p, p) < m ∞ (p -1, p -1) + 2π (H) holds. Then minimizing sequences for m ∞ (p, p) are compact in H 1 and thus m ∞ (p, p) is attained. Proof. Let p > 0. Assume that m ∞ (p, p) < m ∞ (p -1, p -1) + 2π. Consider (u n ) n a minimizing sequence for m ∞ (p, p). Up to a subsequence we have the existence of u ∞ ∈ I s.t. u n u ∞ in H 1 (D, C). Let P := deg(u ∞ ). If P = (p, p) then we are done.Otherwise we have : P = (p, p). By Lemma 4.21 we have that u ∞ minimizes m ∞ (P ) and P ∈ A (p,p) . Thus, by Proposition 4.19 we have the existence of p ∈ [0, p -1] s.t. P = (p , p ) (here we used P = (p, p)).

  Proposition 4.24. Let p ∈ N * and let 0 < R p < 1 of Theorem 4.5. For an annular type domain D s.t. its conformal ratio (see Definition 4.2) satisfiesR p < R D < 1 we have m ∞ (p, p) < m ∞ (p -1, p -1) + 2π.Proposition 4.24 as two direct consequences : 1. If the hypothesis of Theorem 4.5 holds for a circular annular domain A then Proposition 4.22 holds. 2. A way to reformulate (in a weaker form) the hypothesis of Theorem 4.3 or Proposition 4.22 is to replace "m ∞ (p, p) < m ∞ (p -1, p -1) + 2π" by : • the conformal ratio of D satisfies R p < R D < 1 (0 < R p < 1 of Theorem 4.5) ; or equivalently • cap(D) > C p for C p = -2π ln R p .

Proposition 4. 26 .

 26 Let p ≥ 1 be an integer and let D be an annular type domain s.t. m ∞ (p, p) < m ∞ (p -1, p -1) + 2π. Thanks to that condition minimizers u ε of E ε in I p,p do exist for ε large and for ε = +∞ (cf. Proposition 4.22 and Proposition 4.23).

  ϕ(p) = exp ϕ(a) (dϕ(a).u p ) + h(|u p |), where h(|u p | = o(|u p |). Let λ : [η, 2η] → R + be a smooth non decreasing function, such that 1) λ(η) = λ (η) = 0 2) λ(2η) = 1, λ (2η) ϕ(a) (βO a .u p ) + h(|u p |) p ∈ B(a, η) ∩ Ω exp ϕ(a) [(O a (β(1 -λ(|u p |))Id) + λ(|u p |)S a ) u p ] + h(|u p |) p ∈ B(a, 2η) \ B(a, η) ϕ(p) p ∈ ∂Ω \ B(a, 2η).From the properties of the exponential map we have ϕ a (a) = exp ϕ(a) (0) = ϕ(a) and dϕ a (a) = βO a . From

  It holds that Φ(p) = exp Φ(a) dϕ(a).u T p + v N p ν exp Φ(a) (dϕ(a).u T p ) + h(|p -a|) with h(|p -a|) = o(|p -a| as p → a. By using Lemma 5.21 we can decompose dϕ(a) : T a ∂Ω → T Φ(a) S n-1 by writing dϕ(a) = O a S a I α 1 is continuous. We let M = {γ ∈ C 0 (Ω, I α ); γ = χ r on ∂Ω}, c = c(α, r, ε) = inf γ∈M {max a∈Ω E α ε [γ(a)]} (5.29) c 1 = c 1 (α, r, ε) = max a∈∂Ω E α ε [χ r (a)] .(5.30)Lemma 5.22. It holds that c 1 (α, r, ε) = n n 2 -1 |B n | + o r (1) × o ε (1) + o α (1), as r, α → 0 and ε → +∞. Proof. For a map v ∈ I we have lim α→0 E α ε (v) = E ε (v),by continuity of p → Ω |dv| p . Now the n-energy of χ r (a), for σ small enough, is almost equal to Φa(Bσ(a) |dM (1-r)Φ(a) (x)| n dx.

Proof.

  Since deg(γ(a), ∂Ω) = 1 we have from Lemma 5.8, E ε (γ(a)) ≥ n n 2 -1 |B n |, for all γ ∈ M . By contradiction, if we assume that inf{max a∈Ω E ε (γ(a))} = n n 2 -1 |B n | then there exists (γ k ) ⊂ M and (a

n 2 - 1

 21 |B n | + o r (1) × o ε (1) + o α (1) and c(α, r, ε) ≥ c 2 -o α (1) > n n 2 -1 |B n | -o α (1). We then choose r small enough such that o r (1) × o ε (1) < c 2 -n n 2 -1 |B n | 3 and α small enough such that o α (1) < c 2 -n n 2 -1 |B n | 3

  x∈ Ω |du m (x)| ≤ K. Then u m converges to u strongly in W 1,n and u is a solution of (5.1) with deg(u, ∂Ω) = 1 on the boundary.

Figure 6 . 1 :

 61 Figure 6.1: Near a regular point supp µ is a smooth curve.

Figure 6 . 2 :

 62 Figure 6.2: An example of the geometry of supp µ near a critical point of h µ .

d

  i ln |z -z i |, where M N = N i=1 |d i |. (6.28) Proposition 6.15. The points (z i ) 1≤i≤N ∈ R 2 form a stationary system of point vortices if and only if the function u(z) = 1 M N N i=1 d i ln |z -z i | is weakly stationary harmonic.

  lim δ→0 cap 1 (K ∩ E δ ) = 0, and ∀δ > 0 lim n→+∞ K\E δ |X n -X| = 0.(6.30)

.

  , r)| B(x,r) ν E d ∂E = ν E (x), and iii) |ν E (x)| = 1. Definition 6.25. Let E be a Lebesgue measurable set in R n and x ∈ R n . We say x ∈ ∂ E, the measure theoretic boundary or essential boundary of E if lim sup r→0 |B(x, r) ∩ E| r n > 0 and lim sup r→0 |B(x, r) \ E| r n > 0.(Here |A| denotes the n-Lebesgue measure of a set in R n ).The structure of the reduced boundary of a set of locally finite perimeter in R n is described by the following theorem :Theorem 6.26 ([EG92] p.205). Assume E has locally finite perimeter in R n . K k is a compact subset of a C 1 -hypersurface S k (k = 1, 2, ...). ii) Furthermore, ν E |S k is normal to S k (k = 1, ...) and iii) ∂E = H n-1 ∂ EWe have a relation between the reduced and the essential boundary. Proposition 6.27. i) ∂ E ⊂ ∂ E.

  Thus for all ϕ ∈ C ∞ c (B R , R) we have ∂ y θ∂ x H, ϕ = ∂ x θ∂ y H, ϕ ∂ y θ, ∂ x Hϕ = ∂ x θ, ∂ y Hϕ ∂ x θ, -∂ y Hϕ + ∂ y θ, ∂ x Hϕ = 0

  that ϕ has compact support in B R and the definition of ψ we find that ϕ ∈ C 1 c (B R , R 2 ), where we denotedν B + R H 1 -a.e. on ∂ B + R \ ∂B R .The last equality means that νB + R is orthogonal to (-∂ y H, ∂ x H) H 1 -a.e on ∂ B + R \ ∂B R and hence parallel to (∂ x H, ∂ y H). We also obtain that ν B + R is collinear to ∇h because ∇h = θ∇H in B R .

R dH 1

 1 since ∂ * B + R = ∂ * B - R and ν B + R = -ν B - R. Now because of the previous Lemma 6.29 we have that|∇H • ν B + R | = |∇H| = 0 in B R (recall that |∇H| = |∇h| in B R ). Hence we can deduce that the support of µ B R is ∂ * B + R \ ∂B Rand the lemma is proved.

  Definition 6.31. A curve Γ ⊂ R 2 is a Jordan curve if Γ = γ([a, b]) for some a, b ∈ R with a < b, and some continuous map γ, one-to-one on [a, b) and such that γ(a) = γ(b).

Figure 6 . 4 :

 64 Figure 6.4: Illustration of Lemma 6.37

ρ 2 j

 2 for j = 1, ..., N ρ 1 .Proof. Using the previous notations we have : supp(µ A R,ρ 1 ) =

lim n→+∞ H 1

 1 (∂ A + R,ρn \ ∂A R,ρn ) = +∞. (6.59) with A R,ρ = {z ∈ C; ρ < |z| < R} and A R,ρ + = A R,ρ ∩ B + R. This is due to the fact that∂ B + R \ ∂B R = n∈N ∂ B + R ∩ A R,ρn = ∂ A + R,ρn \ ∂A R,ρn

Figure 6 . 6 :

 66 Figure 6.6: Partition of W in disjoint open connected subsets.

  For 1 ≤ k ≤ 2n + 3 we setW + k = {z ∈ W k ; θ(z) = +1}.These sets are sets of locally finite perimeter in W k . As in the previous sections (see Lemma 6.29) we can obtain Claim 7. The generalized outer normal ν U + k is collinear to ∇h and ∇H, ∂U + k -almost everywhere in U k .

Table des

 des Minimiseurs de l'énergie de Ginzburg-Landau à degrés prescrits dans un anneau (en collaboration avec M.Dos Santos : soumis[DSR] 2015 ) . . . . 1.3.4 Points critiques de l'énergie de Ginzburg-Landau à degrés prescrits en dimension n ≥ 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Limites de mesures de vorticité associées aux équations de Ginzburg-Landau (soumis [Rod] 2015 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Perspectives de la première partie . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Perspectives de la deuxième partie . . . . . . . . . . . . . . . . . . . . . . Harmonic maps with prescribed degrees on the boundary and bifurcation of catenoids 3.1 Introduction and statement of the results . . . . . . . . . . . . . . . . . . . . . . 3.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Notations and definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Properties of solutions of (3.1) . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Minimizing sequences : Price lemma and insertion of bubbles . . . . . . . 3.3 Hopf

figures x 1 Introduction Générale 1.1 La supraconductivité et la théorie de Ginzburg-Landau . . . . . . . . . . . . . . . 1.1.1 La supraconductivité . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Le modèle de Ginzburg-Landau . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 L'énergie de Ginzburg-Landau sans champ magnétique avec donnée au bord de Dirichlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 L'énergie de Ginzburg-Landau avec données au bords semi-rigides . . . . . . . . 1.2.1 Un modèle physique intermédiaire . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Un problème à bord libre . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Un problème sans compacité . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Contributions aux problèmes à degrés prescrits . . . . . . . . . . . . . . . . . . . 1.3.1 Insertion de bulles au bord pour l'énergie de Ginzburg-Landau (en collaboration avec E.Sandier, publié : [RS14] ) . . . . . . . . . . . . . . . . . . 1.3.2 Applications harmoniques à degrés prescrits et surfaces minimales (en collaboration avec L.Hauswirth, soumis [HR] 2015 ) . . . . . . . . . . . . . 1.3.3 2 Insertion of bubbles at the boundary for the Ginzburg-Landau energy 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Proof of the fundamental lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Proof of Theorem 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Table des matières viii 3

  1.2.1 Un modèle physique intermédiaireDans[START_REF] Berlyand | Symmetry breaking in annular domains for a ginzburg-landau superconductivity model[END_REF], Berlyand et Voss ont proposé un modèle intermédiaire entre le modèle de Béthuel-Brézis-Hélein et le modèle complet avec champ magnétique. Dans ce modèle on considère toujours l'énergie de Ginzburg-Landau sans champ magnétique (1.2), mais on ne prescrit plus de donnée au bord de Dirichlet. On impose seulement |u| = 1, ainsi que le degré topologique de u sur le bord (noté deg(u, ∂Ω)). Notons que ces deux quantités ont un sens physique. En effet, on a vu que |u| 2 représente la densité de paire de Cooper, de plus deg(u, ∂Ω) mesure la vorticité des courants sur le bord du domaine (la vorticité est une quantité qui indique comment les courants "tourbillonnent"). Mathématiquement on cherche des points critiques de l'énergie E

ε dans I = {v ∈ H 1 (Ω, C); | tr |∂Ω v| = 1}. Ceci conduit aux équations d'Euler-Lagrange suivantes :

  On prouve que pour d ∈ N * , m ε (d, d, d) est atteint, puis par récurrence on montre que m ε (p, q, d) est atteint pour d ≥ max{p, q}. La preuve repose essentiellement sur deux ingrédients. Premièrement, on montre que si la limite d'une suite minimisante sort de sa classe initiale alors il y a une perte d'énergie qui est quantifiée et égale à un multiple entier de π, cet entier étant la différence des degrés initiaux et finaux (cf. Lemme du coût dans[START_REF]Ginzburg-Landau minimizers with prescribed degrees. Capacity of the domain and emergence of vortices[END_REF]). Ensuite on montre le lemme suivant :Lemme 1.6 (Proposition 20 dans[START_REF] Berlyand | Solutions with vortices of a semi-stiff boundary value problem for the Ginzburg-Landau equation[END_REF], Lemme 5 dans[START_REF] Santos | Local minimizers of the Ginzburg-Landau functional with prescribed degrees[END_REF]). Soit d ∈ N * et ε suffisamment petit. Si u ε ∈ I d p,q est une solution de l'equation de Ginzburg-Landau avec données au bord semirigides alors il existe v ∈ I d p+1,q telle que

  [START_REF]Mappings of least Dirichlet energy and their Hopf differentials[END_REF],[START_REF] Iwaniec | Doubly connected minimal surfaces and extremal harmonic mappings[END_REF],[START_REF] Iwaniec | n-harmonic mappings between annuli : the art of integrating free Lagrangians[END_REF], T.Iwaniec et al. ont considéré des applications (pas nécessairement harmoniques) dont la différentielle de Hopf s'écrit également H u (z) = c z 2 . Un de leur objectif était de minimiser l'énergie de Dirichlet dans une classe de limites d'homéomorphismes entre domaines doublement connexes. Dans ce problème on ne peut pas faire des variations standards (dans l'espace d'arrivée) pour obtenir les équations d'Euler-Lagrange associées à un minimiseur. En effet de telles variations ne préservent pas la classe d'homéomorphismes dans laquelle on cherche un minimiseur. En revanche les variations internes, ou variations du domaine, sont parfaitement adaptées. Ceci conduit à la définition : Définition 1.13. Soit Ω ⊂ R 2 un ouvert. On dit que u ∈ H 1 (Ω) est stationnaire harmonique si pour toute famille de difféomorphismes de Ω, notée φ t , telle que φ

  Ici •, • représente le produit scalaire de C identifié à R 2 . Physiquement j ε représente le courant superconducteur et µ ε est la vorticité de ce courant (elle quantifie la manière dont le courant tourne). En utilisant l'équation (1.14) on peut montrer que div(j ε ) = 0. Comme Ω est simplement connexe on peut écrire j ε = ∇ ⊥ h ε . D'après la définition de µ ε , la fonction h ε vérifie

		∆h ε = µ ε dans Ω ∂ ν h ε = j ε , τ sur ∂Ω	(1.17)
	et on peut déterminer h ε	10 de manière unique en imposant que sa moyenne soit nulle. Le premier
	résultat que nous énonçons affirme, en simplifiant, que pour ε petit il existe un nombre fini de
	vortex dans Ω et ce sont eux qui contribuent à la vorticité du courant :
	Théorème 1.22 (Partie A) du Théorème 13.2 dans [SS07]). Soit {u ε } ε>0 des solutions de
	(1.14) comme avant. Alors pour tout ε > 0 il existe une mesure ν ε = 2π i d ε i δ a ε i , où la somme
	est finie,

de vorticité 11 . On note n ε = i |d ε i |, et on a que ν ε vérifie les conclusions du théorème précédent. On remarque que νε nε est bornée et on s'intéresse à sa limite lorsque ε → 0. La connaissance de cette limite nous donne des informations sur ν ε pour ε petit et donc sur l'emplacement des vortex et leurs densités. Sandier et Serfaty ont établi dans [SS03a], [SS07] qu'une telle limite vérifie certaines conditions critiques.

  Soit {u ε } ε>0 comme avant. On suppose de plus que ∇u

	.19)
	pour p ∈]1, 2[.
	Soit {u ε } ε>0 comme avant, on cherche maintenant à comprendre l'emplacement des vortex. Pour
	cela on les note a ε i et on note d ε i leurs degrés. On s'intéresse alors à ν ε = 2π i d ε i δ a ε i , qui est
	appelée la mesure Théorème 1.23 (Théorème 3 dans [SS03a]).

  with σ small enough, and check that, from (2.19), (2.20),(2.21), (2.22),(2.23),(2.24),(2.25), it holds that

  1-p 1+ p .

	Hence if we have
	E(u

p ) > E(u 1 ) + 2π(p -1) (3.41) then u p can not be a minimizer of E in I p,p . Let us examine the condition (3.41) :

  1- 1+ . Besides from the definition of 2 (cf. (3.41)) it holds that if ≥ 2 then u 2 (z) = 1 1+ 2 (r 2 +

	2 r 2 )e i2θ satisfies that
	E(u 2 ) ≤ m(1, 1) + 2π.	(3.44)
	By contradiction if m(2, 2) is not attained for ≥ 2 . Then applying Lemma 3.18 and Lemma
	3.20 we can see that
	m(2, 2) = m(1, 1) + 2π.	(3.45)

  Proposition 4.22. Let D ⊂ R 2 be an annular type domain and let p ∈ N * s.t.

	m ∞ (p, p) < m ∞ (p -1, p -1) + 2π.	(H)
	Note that the minimizing property of (u ε n ) n combined with its non compactness property, implies
	that
	deg(u ε ) = (p, p).	(4.12)
	Indeed, if deg(u

Then, for sufficiently large ε, the minimizing sequences for m ε (p, p) are compact in H 1 (D) and thus m ε (p, p) is attained.

Proof. We argue by contradiction. We assume that -p ∈ N * and D are s.t. (H) holds, -there exists ε = ε k ↑ ∞ s.t. for all ε there is a minimizing sequence (u ε n ) n for m ε (p, p) satisfying :

(u ε n ) n is not compact for the strong topology of H 1 . For all ε = ε k , up to a subsequence (still denoted by u ε n ), there is

u ε ∈ I s.t. u ε n n→∞ u ε in H 1 (D,

C). By Lemma 4.21, we have that deg(u ε ) ∈ A (p,p) and that u ε minimizes m ε (deg(u ε )). ε ) = (p, p), then u ε ∈ I p,p . Moreover, by compact Sobolev embedding we have lim n

  the set {deg(u ε )} ⊂ A (p,p) is finite, we may consider a subsequence, still denoted by(ε k ) k , s.t. deg(u ε ) = P 1 ∈ A (p,p) \ {(p, p)}. Up to a subsequence in (ε k ) k , there exists u ∞ ∈ I s.t. u ε u ∞ .By Lemma 4.21 we have that P 2 := deg(u ∞ ) ∈ A P 1 ⊂ A (p,p) and u ∞ minimizes m ∞ (P 2 ). Therefore by Proposition 4.19 and the conformal invariance of the Dirichlet energy there is p 2 ∈ [0, p] s.t. P 2 = (p 2 , p 2 ). Moreover, since P 2 = (p 2 , p 2 ) ∈ A P 1 ⊂ A (p,p) \ {(p, p)} we have p 2 ∈ [0, p -1]. Hence it holds that (by Prop. 4.11.2)

  ) holds and a minimizer of E ε in I 2,2 exists if ε is large enough. On the other hand, the radius R 1 obtained in[START_REF] Golovaty | On uniqueness of vector-valued minimizers of the Ginzburg-Landau functional in annular domains[END_REF] is e Comparison of Hypotheses. As it is explained in Remark 2.14 of

						1) + 2π.	(4.16)
	Namely (4.16) implies (H).
	The study of (4.16) is easy to do (cf. [HR] proof of Theorem 5.4.) and gives :
			(4.16) holds if and only if R >	√	2 -1.
	Thus if R >	√	2 -1 then (H-1 16π 2	0.99 while	√	2 -1 0.41.
	-

  Lemma 4.27. Under the same hypothesis as in Proposition 4.26, we have that, up to a subsequence,u ε → u ∞ strongly in H 1 (D) and in C l loc (D), ∀l ∈ N (4.18)where u ∞ is a minimizer of E ∞ in I p,p .Proof. For ε large, if the domain D is s.t. m ∞ (p, p) < m ∞ (p -1, p -1) + 2π, denoting by u ε a minimizer of E ε in I p,p and by ũ∞ a minimizer of E ∞ in I p,p we have :

  the choice of λ we can check that ϕ a is C 1 . For η small enough since h(|u p |) = o(|u p |) we have that exp ϕ(a) (βO a u p ) + h(|u p |) is a diffeomorphism. Thus we only need to check that exp ϕ(a) [(O a (β(1 -λ(|u p |))Id) + λ(|u p |)S a ) u p ] is a diffeomorphism. To this end it suffices to show that β(1 -λ(|u p |))Id) + λ(|u p |)S a is a diffeomorphism. Firstobserve that this quantity is a positive definite symmetric matrix and hence a bijection. From its definition, to prove that it is a diffeomorphism we only need to prove that ∂ ∂|up| (u p → (β(1 -λ(|u p |))Id) + λ(|u p |)S a ) u p ) is bijective. Let v ∈ T a ∂Ω such that |v| = 1, for t small we compute

  ⊥ = (-∂ y , ∂ x ) and d j ∈ N. The points z i (t) are called vortices and d i are the degrees of vortices. The system is stationary if the vortices do not evolve in time, one then has

	N.	(6.23)
	with ∇ N
	j=1,j =i

  1 (6.46)for all ϕ ∈ C 1 c (R n , R n ).Since B + R is a set of finite perimeter in B R , we denote by ν B + Lemma 6.29. Let h as in Lemma 6.21. Let θ, H given by Lemma 6.21. LetB + R = {z ∈ B R ; θ(z) = +1}, thanks to Lemma 6.22, B +R is a set of finite perimeter in B R and we have : thegeneralized normal ν B + R is collinear to ∇h and ∇H, H 1 ∂ B + R \∂B R almost everywhere in B R .Proof. Let us recall that, because of (6.43) we have∂ x h -i∂ y h = θ(∂ x H -i∂ y H). (6.47)In the sense of distributions we have∂ x ∂ y h = ∂ y ∂ x h.Thus we obtain ∂ y (θ∂ x H) = ∂ x (θ∂ y H) and ∂ y θ∂ x H + θ∂ y ∂ x H = ∂ x θ∂ y H + θ∂ x ∂ y H. (6.48) Now since H is harmonic and hence C ∞ (B R ) it holds that ∂ 2 xy H = ∂ 2 yx H. Hence ∂ y θ∂ x H = ∂ x θ∂ y H.

	its measure theoretic (or genera-
	R
	lized) outer normal.

  E ∩ f -1 (t) is countably H 0 -rectifiable for dt-a.e. t ∈ R and E C k d E f x dH 1 (x) =

	H 0 (E ∩ f -1 (t))dt.	(6.50)
	R

Theorem 6.36 (Theorem 2.93 in [AFP00] p.101). Let f : R 2 → R be a Lipschitz function and let E be a countably H 1 -rectifiable subset of R 2 . Then the function t → H 0 (E ∩ f -1 (t)) is Lebesgue measurable in R,

En termes mathématiques u n'est pas une quantité invariante par changement de Jauge.

Cette liste n'est pas exhaustive.

Il est intéressant de noter que J.C.C Nitsche a employé un argument similaire, en utilisant une autre différentielle de Hopf, pour montrer que les seules surfaces minimales de type disque dans la boule avec bord libre dans la sphère sont les disques équatoriaux cf. Théorème 1 dans[START_REF] Nitsche | Stationary partitioning of convex bodies[END_REF]

Remerciements

(3.8)

The capacity of a domain is preserved under conformal transformations and for an annulus G = D R \ D r it holds cap(G) = 2π ln(R/r) . The capacity measures the "thickness" of the domain. Every annular domain A is conformally equivalent to an annulus A = {z ∈ C; < |z| < 1} with 2π ln(1/ ) = cap(A).

Many lemmas collected in the rest of this section were first proved for the G.L energy, or the G.L equation their proof can be found in [START_REF]Ginzburg-Landau minimizers with prescribed degrees. Capacity of the domain and emergence of vortices[END_REF], [BM], [START_REF] Santos | Local minimizers of the Ginzburg-Landau functional with prescribed degrees[END_REF]. Their adaptation to the Dirichlet energy is straightforward.

Properties of solutions of (3.1)

Proposition 3.14. Let u be a solution of (3.1) then u is in C ∞ (A).

Proof. Since we have ∆u = 0 in A, a classical result says that u ∈ C ∞ (A). It is the smoothness up to the boundary which is nontrivial and for that we refer to Lemma 4.4 in [BM].

An application of the maximum principle implies the following :

Proposition 3.15. Let u be a solution of (3.1) then |u| ≤ 1, in A.

Minimizing sequences : Price lemma and insertion of bubbles

Inspired by [START_REF] Kuwert | Minimizing the energy of maps from a surface into a 2-sphere with prescribed degree and boundary values[END_REF], we study the loss of degree in energy minimizing sequence. First we can obtain estimates of m(p, q). Lemma 3.16. ([DS09]) For u ∈ I r,s , (p, q) ∈ Z × Z and δ > 0 there exists v ∈ I p,q such that

In particular it holds that m(p, q) ≤ m(r, s) + π(|p -r| + |q -s|), and then (since m(0, 0) = 0) m(p, q) ≤ π(|p| + |q|).

(3.9)

The next lemma gives us some information about the "cost" for a weak limit to jump in another class.

Chapitre 5

Critical points of the n-Ginzburg-Landau energy with prescribed degrees

Abstract : Let Ω ⊂ R n be a smooth bounded simply connected domain, with n ≥ 2. We investigate here the problem of finding critical points of the n-Ginzburg-Landau energy

. Maps in I have a well-defined topological degree on ∂Ω. After proving that there are no minimizers of E ε with (non zero) prescribed degrees we develop a mountain pass approach for a perturbed energy E α ε = 1 n Ω |du| n+α + 1 4ε n Ω (1 -|u| 2 ) 2 , with α > 0 small. The idea of using a perturbed energy in the study of non-compact problem appeared in different contexts following the pioneer work of Sacks-Uhlenbeck concerning the minimal immersions of the 2-sphere.

Introduction

Let Ω ⊂ R n (n ≥ 2), be a smooth bounded simply connected domain. We set

with S a a symmetric endomorphism of T a ∂Ω and O a : T a ∂Ω → T Φ(a) S n-1 which preserves the inner product. We set

Here β is the same as in proposition 5.20. With λ :

Thanks to this choice of λ and µ we can see that Φ a is C 1 . Furthermore by using the properties of the exponential map we can see that dΦ a (a) = βR a where R a is an orthogonal matrix. For η (and thus δ 1 , δ 2 small enough) we have that

is also a diffeomorphism and this concludes the proof.

We then set χ r (a

where Φ a is as in Proposition 5.19 and M(1-r)Φ(a) : B n → B n is a modified Möbius map defined by

(5.28) 1) and I an interpolation between the Möbius map M (1-r)Φ(a) and the constant map Φ(a) whose energy goes to zero as σ goes to zero (we can take the same interpolation as in the proof of Lemma 5.9). From Proposition 5.19 the map a ∈ ∂Ω → χ r (a) ∈ Chapitre 6

Regularity properties of stationary harmonic functions whose Laplacian is a Radon measure Abstract : We study the regularity of Radon measures µ which satisfy that there exists a function h µ in H 1 , stationary harmonic such that ∆h µ = µ. Such conditions appear in physical contexts such as the study of a limiting vorticity measure associated to a family (u ε ) ε of solutions of the Ginzburg-Landau system without magnetic field. Under these conditions we prove that locally there exists a harmonic function H such that the support of the measure is contained in the set of zeros of H. Using the local structure of the set of zeros of harmonic functions we can thus obtain that locally the support of µ is a union of smooth simple curves.

Introduction and main results

Stationary harmonic functions arise in many physical problems such as the study of Ginzburg-Landau equations linked to superconductivity or the study of Euler equations in fluid mechanics. They are also related to limiting vorticities of stationary system of point vortices. Let Ω be a bounded open set in R 2 .

The function θH is continuous and ∇H(z 0 ) = 0. Besides the support of µ V is a union of smooth curves included in {z ∈ V ; H(z) = 0} which end at z 0 .

A key ingredient in the proof of the previous theorem is the local structure of the set of zeros of harmonic functions (see e.g. [START_REF] Hartman | On the local behavior of solutions of non-parabolic partial differential equations[END_REF] or [START_REF] Wen | Set of zeros of harmonic functions of two variables[END_REF]). Theorem 6.5 ( [START_REF] Wen | Set of zeros of harmonic functions of two variables[END_REF]). Let H be a harmonic function defined on an open set D ⊂ R 2 . We let Z 0 (H) := {z ∈ D; H(z) = 0}. Suppose z 0 ∈ D, H(z 0 ) = 0 and H is not identically zero. Then there exist a unique integer n = n(H, z 0 ) ≥ 1, a neighborhood U (z 0 ) of z 0 in D and n analytic curves

.n, where γ n+1 denotes γ 1 and ang(γ k , γ k+1 ) is the angle between γ k and γ k+1 at z 0 .

3) There exists an analytic diffeomorphism φ :

Remark : Note that in Theorem 6.4 it can happen that the support of µ is strictly contained in the set {z ∈ v; H(z) = 0}. In this case we can not have h µ = |H|. This is illustrated by the following example : we set h(re iϕ ) = θ(ϕ)r 2 cos(2ϕ), for r ∈ [0, 1], ϕ ∈ [0, 2π[ and

This function h satisfies (6.3), (6.4), (6.5). In particular one can check that ∆h = µ with supp

When z 0 ∈ supp(µ) is a zero of odd order of ω hµ we must use multivalued harmonic function.

Theorem 6.6. Let z 0 ∈ supp µ with (h µ , µ) which satisfy assumptions (6.3), (6.4), (6.5), and such that z 0 is a zero of odd order of ω hµ . We assume that h µ (z 0 ) = 0. Then there exist a neighborhood V of z 0 , a multivalued harmonic function

where int(A) denotes the interior of A and p * = 2p 2-p . We would like to mention that in [START_REF] Le | Regularity and nonexistence results for some free-interface problems related to Ginzburg-Landau vortices[END_REF], the author studied limiting vorticity measures associated to the Ginzburg-Landau system with magnetic field. This leads to conditions analog to (6.3), (6.4), (6.5). He investigated these conditions under the additional assumption that the measure µ is supported by a simple smooth curve. He then proved, among other things, that in that in this case µ has a fixed sign.

Connections to the Euler System

It turns out that conditions (6.4), (6.5) are also related to the Euler equations for incompressible flow in fluid mechanics. They can be written as follows :

where Ω is an open set of R 2 . In this system p is called the pressure and it is an unknown of the system. Here v We must be more specific to define the notion of solutions of the Euler system. Indeed we want to give a meaning to (6.19) for vector-fields which are only in L 2 (Ω). First note that thanks to the condition div(v) = 0 we can rewrite the stationary Euler system in the following form :

The divergence of a matrix is the sum of the divergence of the row. Let us denote by A, B := tr(A t B) the inner product between two matrices. Definition 6.10. Let Ω be an open set in R 2 . We say that v ∈ L 2 (Ω, R 2 ) is a weak solution of (6.21) if there exists p ∈ L 1 (Ω) such that

Proposition 6.11. Let h µ satisfy (6.3), (6.4), (6.5). We set v = ∇ ⊥ h µ . Then v is a weak solution of the stationary Euler system with vorticity equal to µ.

As before we set

We thus obtain that

Lemma 6.40. Let h which satisfies (6.40), (6.41), (6.42). Let R > 0 be small enough and θ : B R → {±1} such that (6.57) holds with H defined by (6.56). Then the function θ is in

Proof. In order to prove this result we can apply Lemma 6.22 of the previous section in any open subset W ⊂ B R such that g = ∂ x H -i∂ y H does not vanish in W .

Thus B + R and B - R are sets of locally finite perimeter in B R \ {z 0 }. Lemma 6.41. Let h as in Lemma 6.39. Let θ, H given by Lemma 6.39. Let B + R = {z ∈ B R ; θ(z) = +1}, thanks to Lemma 6.40 B + R is a set of locally finite perimeter in B R \ ({z 0 }). Furthermore the generalized normal ν B + R is collinear to ∇h and ∇H, H 1

The proof of this lemma is exactly the same as the one of Lemma 6.29. We can also copy the proof of Lemma 6.30 to obtain Lemma 6.42. Let h satisfy the hypothesis (6.40), (6.41), (6.42). Let θ, B R , B + R as in the previous lemma 6.29. Then the support of

We would like to apply Theorem 6.34 to the set B + R and continue the proof as in the previous section but we can not do that because B + R have only locally finite perimeter in B R \ {z 0 }. In fact we will show that this is just a technical issue and that B + R has indeed finite perimeter in B R but it requires some work. In a first time we work in an annular domain. Let 0 < ρ < R, we set

We first apply Theorem 6.34 to the set A + R,ρ .

Lemma 6.43. Let θ be such that (6.53) holds, θ ∈ BV loc (B R \ {z 0 }). Let A + R,ρ as before. There exist (possibly infinitely many) disjoint rectifiable Jordan curves γ ρ i such that

Proof. We have that θ ∈ BV (A R,ρ ). As in the proof of Lemma 6.35 one can show that A + R,ρ has finite perimeter in R 2 . We can then apply Theorem 6.34 to deduce the result. Lemma 6.44. Under the same assumptions as Lemma 6.43, there exist 0 < ρ < ρ < R < R and (possibly infinitely many) connected rectifiable simple curves Γ ρ j such that

(see Figure 6.5). Proof. The proof is the same as in Lemma 6.37, it uses the coarea formula (see 6.36).

Lemma 6.45. Under the same assumptions as in Lemma 6.43, for all j ∈ N, there exists

Proof. Again in order to prove this lemma we can follow line by line the proof of claim 1. The only difference is that here {z ∈ R 2 ; H(z) = c i }∩A R ,ρ is not necessarily connected if c i = 0.

Claim 3. There exist R > 0 small enough, a finite number N and N simple, connected, rectifiable curves Γ j such that

Furthermore for all 1 ≤ j ≤ N there exists c j such that Γ j is exactly a connected component of the level set {z ∈ C, H(z) = c j } ∩ B R .

Once we know that the function θ defined by (6.57) is in BV (B R ) for R small enough we can apply the same arguments as in the previous section to prove this claim, it results from an adaptation of Lemmas 6.30, 6.35, 6.37, and Claims 1, 2.

We can now conclude the proof of Theorem 6.4. The last claim proves that there exist a finite number of Γ j near Z 0 := {z ∈ B; H(z) = 0}. Thus there exists η > 0 such that dist(Z 0 , Γ i ) > η for all j ∈ N such that Γ j is not included in Z 0 . We then set V := B(z 0 , η 2 ). Because of the definition of η we can say that supp(µ V ) ⊂ {z ∈; H(z) = 0}.

We also set as usual V + = {z ∈ V ; θ(z) = +1} ,V -= {z ∈ V ; θ(z) = -1}. We have that ∇h = +∇H, on V + , ∇h = -∇H, on V -.

Note that in V the function θH is continuous since H = 0 at the discontinuity points of θ. Then θH is in H 1 (V ) since H is in H 1 . Computing ∇(θH) in the sense of distributions we obtain that ∇(θH) = θ∇H. Besides it comes

This proves that h -θH is constant in V , but evaluating this constant in z 0 we find that

This concludes the proof of Theorem 6.4.

Local behavior near a critical point of odd order of h µ

In this section we deal with the case where z 0 is a point in the support of µ and z 0 is a zero of odd order of ω hµ . This case is the most difficult. Indeed unlike the previous cases we can not find a holomorphic function g such that (∂ x h µ -i∂ y h µ ) 2 = g 2 . We must use multivalued holomorphic function to overcome this difficulty. We do not want to discuss here the notion of multivalued function. For us the prototype of multivalued function is z → z 1 2 . Such a multivalued function is single-valued up to a sign. Indeed given any complex number z different from 0 there exist exactly two complex numbers z 1 and z 2 such that z 2 i = z for i = 1, 2 and z 1 = -z 2 . In particular |z 1 2 | = |z| 1 2 is well defined. We drop the subscript µ during the rest of this section. Lemma 6.49. Let h which satisfies (6.40), (6.41), (6.42). Let z 0 ∈ Ω be a zero of odd order of ω h (z) = (∂ x h -i∂ y h) 2 (z). Then there exist W a neighborhood of z 0 , a function θ : W → {±1} and a function

Furthermore the function H 1 is such that : there exist an unique integer n ≥ 1, a small number r > 0 and a biholomorphism Φ : B(0, r) → W such that Φ(0) = z 0 and

Proof. Let z 0 be a zero of odd order of (∂ x h -i∂ y h) 2 . We can find a neighborhood U of z 0 , an integer n and a holomorphic function

Since f 1 (z 0 ) = 0, there exists a smaller neighborhood of z 0 , still denoted by U and a holomorphic function

We then set

, g is a multi-valued function which is single-valued up to a sign. As in the previous sections we want to take a primitive of the function g. However the fact that g is multivalued introduces a difficulty in this process. But we can show that we can choose a special form of a primitive of g. Claim 4. There exist a neighborhood U of z 0 and a single-valued holomorphic function ϕ 2 : U → C such that ϕ 2 (z) = 0 for all z ∈ U and

, for all z ∈ U (where g is defined by (6.62)) .

Proof. Let us assume that such a function ϕ 2 exists. We then have :

Since we want G (z) = g(z) = (z -z 0 ) n+ 1 2 ϕ 1 (z), the function ϕ 2 must satisfies the following complex ordinary differential equation :

In a neighborhood of z 0 we can expand ϕ 1 in power series

Thanks to an expansion in power series we have :

Thus we must have

We can check that if we set ϕ 2 (z) =

a primitive of g. Furthermore because ϕ ( z 0 ) = 0 we have a 0 = 0 and hence ϕ 2 (z 0 ) = 0. Thus ϕ 2 (z) = 0 in a neighborhood U of z 0 .

We then set

and

Note that H 1 is a multi-valued function which is single-valued up to a sign. The proof of the next claim is very similar to an analogous result for harmonic function (see e.g. [START_REF] Wen | Set of zeros of harmonic functions of two variables[END_REF]).

Claim 5. There exist a neighborhood W of z 0 , a number r > 0 and an analytic diffeomorphism Φ : B(0, r) → W such that Φ(0) = z 0 and

for all z ∈ B(0, r).

Proof. We have

, for all z ∈ U and ϕ 2 (z 0 ) = 0 (where U and ϕ 2 are given by Lemma 4). This last property allow us to find a neighborhood of z 0 , denoted by W , and a (single-valued) function ϕ 3 : U → C such that ϕ 3 (z) n+ 3 2 = ϕ 2 (z) for all z ∈ W . We thus obtain that

Note also that ϕ 3 (z 0 ) = 0. We let k(z) = (z -z 0 )ϕ 3 (z). We have that k is holomorphic, k(z 0 ) = 0 and k (z 0 ) = 0. We can thus apply an analytic version of the local inverse theorem to obtain that there exists a neighborhood of z 0 , still denoted by W and a number r > 0 such that k : W → B(0, r) is an analytic diffeomorphism (or biholomorphism). Now we set Φ = k -1 , we have that k

We then deduce that

for all z ∈ B(0, r) and the claim is proved.

One can check that

We thus deduce that there exists a function θ : W → {±1} such that

(6.66)

We then set

Note that the function θ does not play the same role as in the previous section. This is because the function H is not harmonic here. Furthermore H is only lipschitz and not smooth thus we can not use the same argument as in the previous section to prove that θ is in BV loc (W \ {z 0 }). Indeed to prove this we used the fact that

and we differentiated this expression in the sense of distributions, using the Leibniz rule. We can not do the same here since ∂ x H -i∂ y H is not a smooth function.

For this reason we work in W \ {z ∈ U ; H(z) = 0}. Thanks to Proposition 5 we know that this set is an union of 2n + 3 connected disjoint open sets (where n is defined in 5, see Figure 6.6).

Proof of Theorem 6.6. Let h which satisfies (6.40), (6.41), (6.42). Let W , θ, H be defined by (6.60) in Lemma 6.49. As before we set

with W k open and connected and W k ∩W j = ∅ if k = j. We also set W + k = {z ∈ W k ; θ(z) = +1}. We use the previous Claim 8 and we obtain that for all 1 ≤ k ≤ 2n + 3, since there are only a finite number of curves Γ k j such that ∂ W + k \ ∂W = N k j=1 Γ k j , with the Γ k j which are connected component of level curves of H then we can find η k such that

We then set η := min 1≤k≤2n+3 η k and V := B(z 0 , η). We have that θ is constant in each V ∩ W k since θ is constant in each B(z 0 , η k ) ∩ W k from the definition of η k .

We claim that V + = {z ∈ V ; θ(z) = +1} is a set of finite perimeter in V . Indeed we have that ∂V + \ ∂V ⊂ {z ∈ V ; H(z) = 0} (here we use the topological boundary ∂V + ) and H 1 ({z ∈ V ; H(z) = 0}) < +∞ from the last point of Lemma 6.49. Then applying Proposition 3.62 of [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF] we deduce that V + is a set of finite perimeter.

Note that in V the function θH is continuous since H = 0 at the discontinuity points of θ. Then θH is in H 1 (V ) since H is in H 1 . Computing ∇(θH) in the sense of distributions we obtain that ∇(θH) = θ∇H and it comes

this proves that h -θH is constant in V , but evaluating this constant in z 0 we find that h = θH in V.

Annexe A

On the stationary harmonic functions

This appendix is devoted to elementary results on stationary harmonic functions. These results are stated without proof in the introduction of Chapter 6. The original definition of stationary harmonic function is the following :

where E(h) = 1 2 Ω |∇h| 2 dx is the Dirichlet energy.

As shown by the following proposition we used an equivalent characterization.

Proposition A.2. A function h is stationary harmonic if and only if div T h = 0 in the sense of distributions, where

Proof. We first note that div(T h ) = 0 in the sense of distributions if and only if

where Dη denotes the differential of η (which is a 2 × 2 matrix) and A, B = tr( t AB) denotes the inner product of two matrices. Let

We can make the following change of variables y = x + tη(x) ⇔ x = y -tη(x) ⇒ x = y -tη(y) + o(t) (the last implication holds because η(x) = η(y) + o(1) when t goes to 0). We also have which is equivalent to div(T h ) = 0 with T h = ∇h t ∇h -1 2 |∇h| 2 I.

The equation (6.5) can also be interpreted in terms of holomorphic functions Proposition A.3. The condition div(T h ) = 0 is equivalent to ω h := |∂ x h| 2 -|∂ y h| 2 -2i∂ x h∂ y h is holomorphic in Ω.

Proof.

These are the Cauchy-Riemann equations for ω h written in the sense of distributions. We can rewrite them as ∂ z ω h = 0 where ∂ z = 1 2 (∂ x + i∂ y ). The operator ∂ z is elliptic and hence the elliptic regularity theory shows that ω h is smooth and then holomorphic because it satisfies the Cauchy-Riemann equations.

Proposition A.4. If h is harmonic in Ω then h is stationary harmonic in Ω.

Proof. Assume that ∆h = 0 in Ω. Recall that ∆v = 4∂ z ∂ z v and let us compute

Hence ∂ z ω h (z) = 0, that is ω h is holomorphic.

The converse of the previous proposition is not true. However if h is a stationary harmonic functions which statisfies the hypothesis (6.3), (6.4), (6.5) with µ ∈ L p , p > 1, then, using the same methods as in [START_REF] Sandier | Progress in Nonlinear Partial Differential Equations Birkhauser[END_REF] Chapter 13, one can show that h is harmonic.

Proposition A.5. If µ is in L p (Ω) for some p > 1 and satisfies (6.5),(6.4) then µ = 0.

Proof. Let µ be in L p (Ω) for some p > 1 and such that div(T µ ) = 0 and ∆h µ = µ. Let ρ n be a regularizing kernel, we set µ n := µ * ρ n , h n := h µ * ρ n and

One has µ n → µ in L p (Ω), and because ∇h µ is in L ∞ loc (Ω) one also has ∇h n → ∇h µ in L q loc (Ω), for all q ∈ [1, +∞[. Then µ n ∇h n → µ∇h µ , in L 1 loc (Ω) and T n → T µ , in L 1 loc (Ω).

The last equation implies that div(T n ) → div(T µ ) = 0 and µ n ∇h n → µ∇h µ in the sense of distributions. However div(T n ) = -∆h n ∇h n = µ n ∇h n thus µ∇h µ = lim n→+∞ div(T n ) = 0 in L 1 loc (Ω) and almost everywhere. From a property of Sobolev functions we have ∆h µ = 0 a.e. on the set F = {∇h µ = 0}, thus µ = 0 a.e. on F and µ = 0 on Ω \ F hence µ = 0 on Ω.