THE GINZBURG-LANDAU MODEL WITH A VARIABLE MAGNETIC FIELD

Keywords: Opérateur de Schrödinger magnétique, théorie spectrale, analyse semi-classique, fonctionnelle de Ginzburg-Landau avec champ magnétique variable Magnetic Schrödinger operator, spectral theory, semi-classical analysis, Ginzburg-Landau functional with variable magnetic field, pinning effect, superconducting

Je voudrais tout d'abord remercier Bernard Hellfer, mon directeur de thèse qui a dirigé ma thèse avec beaucoup de patience et qui a consacré du temps à mon travail en étant toujours disponible et en venant me chercher très souvent pour que l'on discute, ce qui m'a énormément encouragé pendant mes séjours à l'Université Paris-Sud. C'est un honneur pour moi de travailler avec lui. Son empreinte scientifique a laissé une influence énorme sur mes gouts et ma vision sur les mathématiques. Je le remercie sincèrement pour l'accueil chaleureux à Orsay dès mon arrivée en Janvier 2013.

J'exprime mes vifs remerciements à Ayman Kachmar, mon co-directeur de thèse qui est à l'origine de ce travail. Je lui suis infiniment reconnaissant, non seulement parce qu'il a accepté de me prendre en thèse, mais aussi parce qu'il a partagé ses idées avec moi et je lui en suis très reconnaissant. je le remercie d'avoir lu très serieusement beaucoup de versions préliminaires de mes travaux.

Je remercie l'Université Paris-Sud et l'Université Libanaise qui m'ont offert la possibilité de faire cette thèse en cotutelle dans le cadre d'un programme commun. Je désire en outre remercier l'Université Paris-Sud pour le soutien financier pour ma thèse qu'elle m'a accordé.

Table des matières

Dans un supraconducteur, deux effets différents permettent de faire léviter un aimant : l'effet Meissner, et le piégeage des vortex. L'effet Meissner est la propriété de repousser un champ magnétique appliqué en créant un contre champ opposé de même intensité (c.f. Figure 1.5),

• B 0 ∈ C ∞ (Ω) est le profil du champ magnétique appliqué, variable et satisfait : Dans cette fonctionnelle deux paramètres réels interviennent :

|B 0 | + |∇B 0 | > 0 in Ω ∇B 0 × n = 0 on Γ ∩ ∂Ω , ( 1
• Le paramètre κ > 0 est une caractéristique du matériau et il est appelé le 'paramètre de Ginzburg-Landau'. Le matériau est dit du type I si κ est suffisamment petit κ < 1 √ 2 et il est dit du type II lorsque κ est grand κ > 1 √ 2 . Mathématiquement, cela conduit à l'analyse des divers régimes asymptotiques κ -→ 0 ou κ -→ +∞. C'est ce dernier cas qui sera analysé dans notre thèse. Il est aussi appelé limite de London.

Introduction Générale

Invariance de jauge et équations

La fonctionnelle E κ,H,B 0 est invariante par changement de jauge : ψ → ψe i Φ A → A + ∇Φ ce qui signifie que pour n'importe quelle fonction Φ ∈ H 2 (Ω, R), on a E κ,H,B 0 (ψ, A) = E κ,H,B 0 (ψe i Φ , A + ∇Φ) .

Deux configurations équivalentes de jauge étant physiquement équivalentes, ceci nous conduit à ne chercher la solution du problème qu'à changement de jauge près. Il est alors mieux de restreindre la fonctionnelle à un sous-espace plus petit H 1 (Ω, C) × H 1 div (Ω), avec

H 1 div (Ω) = {A = (A 1 , A 2 ) ∈ H 1 (Ω) 2 : div A = 0 in Ω , A • ν = 0 on ∂Ω } ,
où ν est le vecteur normal extérieur de ∂Ω. En général, nous considérerons la fonctionnelle E κ,H,B 0 sur cet espace.

Nous définissons l'énergie de l'état fondamental de Ginzburg-Landau comme l'infimum de la fonctionnelle E κ,H,B 0 : E g (κ, H) = inf E κ,H,B 0 (ψ, A) : (ψ, A) ∈ H 1 (Ω; C) × H 1 div (Ω) .

(1.1.4)

Typiquement, Ω correspond au matériau supraconducteur 1 , c'est-à-dire au-dessous de sa température critique, entouré par un matériau normal c'est-à-dire au-dessus de sa température critique.

Puisque Ω est bornée, l'existence de minimiseurs est assez classique. Ainsi l'infimum est effectivement un minimum. La démonstration de l'existence de minimiseurs est rappelée par exemple dans le livre de Fournais et Helffer [START_REF] Fournais | Spectral Methods in Surface Superconducitivity[END_REF]Section 11.2].

Soit F : Ω → R 2 l'unique champ de vecteurs tel que, div F = 0 et curl F = B 0 dans Ω , ν • F = 0 sur ∂Ω.

(1.1.5) Si (ψ, A) ∈ H 1 (Ω, C) × H 1 div (Ω) est un point critique 2 de E κ,H,B 0 , donc, (ψ, A) est une solution faible du système constitué des équations suivantes, que nous appelons 'équations de Ginzburg- 1 Mentionnons ici que le paramètre de Ginzburg-Landau κ ne dépend que du matériau supraconducteur dans Ω 2 Nous disons que (ψ, A) est un point critique de Eκ,H,B 0 si pour chaque ( ψ, A) a support compact, nous avons :

d dt
Eκ,H,B 0 (ψ + t ψ, A + t A) |t=0 = 0.

1.1 Motivations et notations 5

Landau' (c.f. [44, Section 3.2]),

             -(∇ -iκHA) 2 ψ = κ 2 (1 -|ψ| 2 )ψ dans Ω -∇ ⊥ rot(A -F) = 1 κH Im(ψ (∇ -iκHA)ψ) dans Ω ν • (∇ -iκHA)ψ = 0 sur ∂Ω rot A = rot F sur ∂Ω .
(1. 1.6) Dans les équations ci-dessus, le champ vecteur ∇ ⊥ rot A est défini par

∇ ⊥ rot A = (-∂ x 2 (rot A), ∂ x 1 (rot A)).
Le vecteur ν désigne la normale unitaire en un point de ∂Ω pointant vers l'extérieur. L'analyse du système (

∞ (Ω, C) × C ∞ (Ω, R 2 ). (c.f. [14, Théorème F.2.1]).
Pour permettre à ce champ magnétique (représenté en noir) de passer à travers le vortex, le matériau développe des courants supraconducteurs (représentés en vert) :

j = i 2κH ψ(∇ -iκHA)ψ -ψ(∇ -iκHA)ψ
qui circulent autour de ces disques, sous forme de tourbillon qui justifie le nom 'vortex'. Autour de ces disques, la phase ϕ de ψ a un nombre d'enroulement non nul d = 1 2π ∂D ∂ϕ ∂τ d'éfinie sur le bord du disque, appelé le degré du vortex, où les vortex de degré 1 et -1 sont représentés.

Les champs critiques

Pour un κ donné, le comportement des minimiseurs et plus généralement des points critiques de la fonctionnelle est déterminé en fonction de la valeur du champ magnétique H.

Quand le champ magnétique est constant, il y a trois valeurs principales du H ou champs critiques H C 1 , H C 2 et H C 3 , pour lesquelles des transitions de phase se produisent.

En dessous du premier champ critique H C 1 , le supraconducteur est partout dans sa phase supraconductrice |ψ| = 0 et le champ magnétique ne pénètre pas (cela s'appelle l'effet Meissner ou état de Meissner (c.f. Figure 1.5). Mathématiquement, le champ magnétique induit rot A est asymptotiquement très petite.

À H C 1 , un premier vortex apparaît. Entre H C 1 et H C 2 les phases supraconductrices et normales (sous la forme de tourbillons) coexistent dans l'échantillon, et le champ magnétique pénètre à travers les vortex. C'est ce qu'on appelle l'état mixte (c.f. Figure 1.6). À H C 2 , quand κ est grand, une deuxième transition de phase se produit. |ψ| ≈ 0 à l'intérieur de l'échantillon, c'est-à-dire que la supraconductivité dans la plus grande partie de l'échantillon est perdue. Mathématiquement, le champ magnétique induit rot A et le profil du champ magnétique appliqué B 0 sont asymptotiquement égaux.

Entre H C 2 et H C 3 , la supraconductivité persiste près de la frontière, c'est ce qu'on appelle la supraconductivité de surface, et après H C 3 , la supraconductivité est complètement détruite et |ψ| = 0 dans tout l'échantillon. L'échantillon est alors complètement en phase normale. Nous allons donner maintenant le comportement asymptotique des trois champs critiques dans des plusieurs cas :

Le cas du champ magnétique constant :

H C 2 = κ B 0 et H C 3 ∼ κ Θ 0 B 0 .
Ici, Θ 0 est une constante universelle telle que, Θ 0 ∈ (1/2, 1).

Le cas du champ magnétique variable qui ne s'annule pas :

H C 2 = κ inf x∈Ω B 0 et H C 3 ∼ κ Θ 0 inf x∈∂Ω B 0 avec inf x∈Ω B 0 > Θ 0 inf x∈∂Ω B 0 .

Introduction Générale

Dans le cas où inf x∈Ω B 0 < Θ 0 inf x∈∂Ω B 0 , le phénomène de la supraconductivité de surface disparaît. Ceci nous donne que les comportements asymptotiques de H C 2 et H C 3 sont égaux.

H C 2 = H C 3 ∼ κ Θ 0 inf x∈∂Ω B 0 .
Le cas du champ magnétique variable qui s'annule le long d'une courbe simple et régulière Γ : Ici, λ 0 est introduite dans (4.1.31), λ 0 (R + , θ(x)) est le bas du spectre de l'opérateur qui est défini dans (4.1.33) et θ(x) désigne l'angle entre |∇B 0 (x)| et le vecteur normal -ν(x).

H C 2 = κ 2 λ 3 2 0 min x∈Ω∩Γ |∇B 0 (x)| et H C 3 ∼ κ 2 min x∈∂Ω∩Γ λ 0 (R + , θ(x))
Dans le cas où λ

H C 2 = H C 3 ∼ κ 2 min x∈∂Ω∩Γ λ 0 (R + , θ(x)) 3 2 |∇B 0 (x)|
.

Pour ce qui concerne le premier champ critique H C 1 , nous ronvoyons à Sandier et Serfaty [START_REF] Sandier | Vortices in the magnetic Ginzburg-Landau model[END_REF].

1.2 L'énergie de l'état fondamental de la fonctionnelle de Ginzburg-Landau avec un champ magnétique appliqué variable dans un domaine de R 2 9 1.2 L'énergie de l'état fondamental de la fonctionnelle de Ginzburg-Landau avec un champ magnétique appliqué variable dans un domaine de R 2

Problème posé

Ici on prend κ grand ce qui correspond à une hypothèse que le matériau est fortement de type II. Les propriétés supraconductrices sont décrites par les minimiseurs (ψ, A) de la fonctionnelle E κ,H,B 0 définie en (1.1.2). On s'intéresse juste à déterminer le comportement asymptotique du paramètre d'ordre ψ dans le régime où le paramètre de Ginzburg-Landau κ et le champ magnétique H sont grands et de même ordre, i.e.

∃ κ 0 ≥ 0 , ∀κ ≥ κ 0 , Λ min κ ≤ H ≤ Λ max κ , (1.2

Résultats principaux

Étant donné b ≥ 0. Nous définissons l'énergie de Ginzburg-Landau dans un ensemble ouvert

D ⊂ R 2 , G b,D (u) = D b|(∇ -iA 0 )u| 2 + 1 2 (1 -|u| 2 ) 2 dx , ∀u ∈ H 1 0 (D) .
Ici,

A 0 (x) = 1 2 (-x 2 , x 1 ) , ∀x = (x 1 , x 2 ) ∈ R 2 .
Compte tenu de la définition de l'énergie de l'état fondamental E g dans (1.1.4). Nous donnons une estimation asymptotique de E g , qui est valable quand H satisfait (1.2.1). Le comportement de l'énergie E g (κ, H) implique une fonction auxiliaire f qui est la limite de minimum de l'énergie de Ginzburg-Landau

G b,Q R avec Q R =] -R/2, R/2[×] -R/2, R/2[, f (b) = lim R→+∞ inf u∈H 1 0 (Q R ) G b,Q R (u) R 2 . (1.2.2)
La fonction f a les propriétés suivantes :

-Pour tous b ≥ 1, f (b) = 1 2 , et f (0) = 0.

f : [0, 1] -→ [0, 1 2 ] est une fonction croissante, continue et concave.

-

f (b) = b 2 ln 1 b (1 + o(1)) , lorsque b -→ 0 . (1.2.3) Introduction Générale
La fonction f a été introduite par Sandier-Serfaty dans [START_REF] Sandier | The decrease of bulk-superconductivity close to the second critical field in the Ginzburg-Landau model[END_REF], puis analysée en [START_REF] Aftalion | Lowest Landau level approach in superconductivity for the Abrikosov lattice close to H c 2[END_REF][START_REF] Fournais | The ground state energy of the three dimensional Ginzburg-Landau functional Part I : Bulk regime[END_REF]. Cette fonction joue un rôle important dans la description de la distribution de la supraconductivité dans la plupart des échantillons à deux et trois dimensions, voir [START_REF] Sandier | The decrease of bulk-superconductivity close to the second critical field in the Ginzburg-Landau model[END_REF], [START_REF] Aftalion | Lowest Landau level approach in superconductivity for the Abrikosov lattice close to H c 2[END_REF][START_REF] Fournais | Nucleation of bulk superconductivity close to critical magnetic field[END_REF][START_REF] Fournais | The ground state energy of the three dimensional Ginzburg-Landau functional Part I : Bulk regime[END_REF] et les articles récents [START_REF] Attar | The ground state energy of the two-dimensional Ginzburg-Landau functional with variable magnetic field[END_REF][START_REF] Helffer | The Ginzburg-Landau functional with vanishing magnetic field[END_REF].

Théorème 1.2.1. Supposons que B 0 vérifie (1.1.3) et H vérifie (1.2.1). Alors, l'énergie de l'état fondamental E g satisfait,

E g (κ, H) = κ 2 Ω f H κ |B 0 (x)| - 1 2 dx + o(κ 2 ) , (κ → +∞) . (1.2.4)
Remarque 1.2.2. Plus précisément, quand κ est grand, on peut montrer l'existence de C > 0 et τ 0 ∈ (1, 2) tels que ;

E g (κ, H) -κ 2 Ω f H κ |B 0 (x)| - 1 2 dx ≤ Cκ τ 0 . (1.2.5)
Notons que, le deuxième terme à droite dans (1.2.4) est petit par rapport au premier terme

κ 2 Ω f H κ |B 0 (x)| - 1 2
dx qui est donc le terme dominant.

Le théorème 1.2.1 a été prouvé par E. Sandier et S. Serfaty (cf. [START_REF] Sandier | The decrease of bulk-superconductivity close to the second critical field in the Ginzburg-Landau model[END_REF]Théorème 1.4]) lorsque le champ magnétique B 0 est constant (B 0 = 1). Cependant, ces auteurs ne contrôlent pas le reste avec la même précision dans [START_REF] Helffer | The Ginzburg-Landau functional with vanishing magnetic field[END_REF]. Pour être plus précis, l'énergie de l'état fondamental dans le cas B 0 = 1 qui était

κ 2 |Ω| f H κ - 1 2
, devient maintenant l'intégrale

κ 2 Ω f H κ |B 0 (x)| - 1 2
dx. L'approche utilisée dans la preuve du Théorème 1.2.1 est différente de celle du [START_REF] Sandier | The decrease of bulk-superconductivity close to the second critical field in the Ginzburg-Landau model[END_REF] et elle s'inspire plutôt de celle dans [START_REF] Fournais | Nucleation of bulk superconductivity close to critical magnetic field[END_REF] qui étudie le même problème lorsque Ω ⊂ R 3 et B 0 constant.

L'énoncé du Théorème 1.2.3 nous donne que l'énergie magnétique est petite, comparée à celle du terme dominant qui est d'ordre supérieur à O(κ τ 0 ). Théorème 1.2.3 (Estimation de l'énergie magnétique.). Avec les notations et l'hypothèses du Théorème 1.2.1, il existe deux constantes C > 0 et κ 0 > 0, telles que, si κ ≥ κ 0 , alors l'énergie magnétique satisfait,

(κH) 2 Ω | rot A -B 0 | 2 dx ≤ Cκ τ 0 .
(1.2.6)

Dans notre démonstration, la valeur de τ 0 dépend des propriétés de B 0 : nous trouvons que τ 0 = 7 4 lorsque B 0 ne s'annule pas en Ω et que τ 0 = 15 8 dans le cas général.

Nous allons maintenant donner une version locale du Théorème 1.2.1. Tout d'abord, si (ψ, A) ∈ H 1 (Ω; C) × H 1 div (Ω), nous introduisons la densité de l'énergie,

e(ψ, A) = |(∇ -iκHA)ψ| 2 -κ 2 |ψ| 2 + κ 2 2 |ψ| 4 .
1.2 L'énergie de l'état fondamental de la fonctionnelle de Ginzburg-Landau avec un champ magnétique appliqué variable dans un domaine de R 2

E(ψ, A; D) = κ 2 D f H κ |B 0 (x)| - 1 2 dx + o(κ 2 ) .
(1.2.9)

Dans le cas où B 0 = 1, Sandier et Serfaty (cf. [START_REF] Sandier | The decrease of bulk-superconductivity close to the second critical field in the Ginzburg-Landau model[END_REF]) ont donné un comportement asymptotique du E(ψ, A; D) quand (ψ, A) est un minimiseur de (1.1.2). Théorème 1.2.5 (Sandier-Serfaty). Il existe une constante κ 0 > 0 telle que, lorsque H vérifie (1.2.1), nous avons

E(ψ, A; D) = κ 2 |D| f H κ - 1 2 + o(κ 2 ) , (κ → +∞) . (1.2.10)
Le théorème suivant nous donne un comportement asymptotique du paramètre d'ordre ψ, quand (ψ, A) est un minimiseur global. Théorème 1.2.6 (Concentration du paramètre d'ordre.). Avec les notations et hypothèses du Théorème 1.2.1. Il existe deux constantes positives C, κ 0 et une constante τ 1 ∈ (-1, 0) telles que, si κ ≥ κ 0 , et D est un ouvert régulier tel que D ⊂ Ω, alors, 1. Si (ψ, A) ∈ H 1 (Ω; C) × H 1 div (Ω) est une solution de (1.1.6), alors,

1 2 D |ψ| 4 dx ≤ - D f H κ |B 0 (x)| - 1 2 dx + Cκ τ 1 .
(1.2.11)

2. Si (ψ, A) ∈ H 1 (Ω; C) × H 1 div (Ω) est un minimiseur de (1.1.2), alors,

D |ψ| 4 dx + 2 D f H κ |B 0 (x)| - 1 2 dx ≤ Cκ τ 1 .
(1.2.12)

La valeur de τ 1 trouvée dépend des propriétés de B 0 : nous trouvons que τ 1 = -1 4 lorsque B 0 ne s'annule pas en Ω et que τ 1 = - 1 8 dans le cas général. Notons que f (•) ∈ [0, 1 2 ], ce qui implique

- D 2 f H κ |B 0 (x)| -1 dx ≥ 0 .

Introduction Générale

Le théorème 1.2.6 a été prouvé par E. Sandier et S. Serfaty (cf. [START_REF] Sandier | The decrease of bulk-superconductivity close to the second critical field in the Ginzburg-Landau model[END_REF]) lorsque Ω ⊂ R 2 et B 0 est constant. Les techniques appliquées dans la démonstration sont étroitement liées à la définition de la fonction f (voir (1.2.2)), qui est la densité d'énergie de l'état fondamental associé à un champ appliqué constant b. Elle devient très grande à l'échelle naturelle du problème, mais reste petite à l'échelle de Ω et des variations de B 0 . À l'aide de cette fonction nous avons obtenu le comportement asymptotique de l'énergie dans le régime où H et κ sont grands et de même ordre.

La preuve des résultats ci-dessus consiste à subdiviser le domaine Ω en carrés de côté en excluant ceux qui rencontrent |B 0 | < , et à approximer l'énergie dans chaque carré en approximant B 0 par un B 0 constant, carré dans lesquel on utilisera les bornes supérieures et inférieurs triviales. Les valeurs de et sont également à optimiser pour que l'erreur soit la plus petite possible. Dans l'estimation du reste, nous nous référons aux estimations a priori des solutions des équations de Ginzburg-Landau (1.1.6) qui jouent un rôle essentiel dans le contrôle des erreurs résultant des diverses approximations.

De plus, nous utilisons la version locale de l'énergie pour démontrer que l'énergie magnétique est dans l'erreur, ce qui nous donne l'asymptotique de l'energie de l'état fondamental de Ginzburg-Landau dans un sous-domaine. Ceci nous indique que la supraconductivité est localisée dans la région où B 0 < κ H contrairement au cas lorsque le champ magnétique est constant. Il y a donc une différence importante entre nos résultats et ceux pour le champ magnétique constant. Quand le champ magnétique est d'intensité constante non nulle, (cf. [START_REF] Fournais | Spectral Methods in Surface Superconducitivity[END_REF]), il existe une constante universelle Θ 0 ∈ ( 1 2 , 1) telles que, si H = bκ et b > Θ -1 0 , alors ψ = 0 dans Ω. De plus, dans la même situation, lorsque H = bκ and 1 < b < Θ -1 0 , ψ est asymptotiquement petit partout sauf dans un petit voisinage de ∂Ω (cf. [START_REF] Pan | Surface superconductivity in applied magnetic fields above HC 2[END_REF]). Notre résultat est dans le même esprit que dans [START_REF] Pan | Schrödinger operators with non-degenerately vanishing magnetic fields in bounded domains[END_REF], où les auteurs ont établi que, sous l'hypothèse (1.1.3), pour κ assez grand, quand H = bκ 2 et b > b 0 , alors ψ = 0 in Ω. (b 0 est une constante).

Discussion des résultats principaux

Énergie et vorticité pour un modèle de Ginzburg-Landau avec un champ magnétique variable

Nous allons déterminer une formule asymptotique précise pour le minimum de l'énergie et montrer que les minimiseurs de l'énergie ont des vortex quand l'intensité du champ magnétique appliqué H varie entre deux échelles caractéristiques, et que le paramètre de Ginzburg-Landau κ tend vers l'infini, autrement dit, quand H satisfait,

C min κ 1 3 ≤ H κ lorsque κ -→ +∞ , (1.3
E g (κ, H) = κ 2 Ω f H κ |B 0 (x)| - 1 2 dx + o κH ln κ H . ( 1
E g (κ, H) = 1 2 |Ω| κH ln κ H -1 (1 + o(1)) , (κ → +∞) . (1.3.5)
La raison pour laquelle nous n'obtenons pas (1.3.3) sous la condition ci-dessus est probablement technique. La méthode consiste à construire des tests de configurations avec une condition au bord de Dirichlet. Nous ne pouvons pas construire des configurations périodiques comme dans [START_REF] Sandier | On the energy of type-II superconductors in the mixed phase[END_REF], parce que le champ magnétique B 0 est variable. L'approche utilisée dans la preuve du Théorème 1.2.5 est inspirée de celle dans [START_REF] Kachmar | The ground state energy of the three-dimensional Ginzburg-Landau model in the mixed phase[END_REF] qui étudie le même problème quand Ω ⊂ R 3 et B 0 est constante.

Rappelons que quand H vérifie (1.2.1), nous avons obtenu

E g (κ, H) = κ 2 Ω f H κ |B 0 (x)| - 1 2 dx + o (κH) , (κ -→ +∞) . (1.3.6)
Si nous supposons qu'il existe des constantes positives C min et C max avec H(κ) satisfaisant

C min κ 1 3 ≤ H(κ) ≤ C max κ , (1.3.7) alors (1.3.6) et (1.3.2) deviennent E g (κ, H) = κ 2 Ω f H κ |B 0 (x)| - 1 2 dx + o κH ln H κ + 1 . ( 1
E g (κ, H) = κ 2 Ω f H κ |B 0 (x)| dx + κ 3 H o(1) , (κ → +∞) .
(1.3.10) [START_REF] Sandier | On the energy of type-II superconductors in the mixed phase[END_REF] qui nous aident à obtenir des informations sur la distribution de la vorticité dans Ω. Partant de là, on peut associer à (ψ, A) une mesure de vorticité

E(ψ, A, D) ≥ κ 2 D f H κ |B 0 (x)| dx + o κH ln κ H , lorsque κ -→ +∞ . (1.3.12) 2. Si (ψ, A) est un minimiseur de (1.1.2) et H(κ) satisfait C 1 min κ 3 5 ≤ H κ lorsque κ -→ +∞ , (1.3
µ κ = 2π κH m i=1 d i δ a i ,
où les paires (a i , d i ) i sont les positions et les degrés des vortex de (ψ, A).

= m(κ) disques disjoints (D i (a i , r i )) m i=1 dans Ω tels que, lorsque κ -→ +∞ , 1. m i=1 r i ≤ (κH) 1 2 ln κ H -7 4 Ω 1 √ |B 0 (x)| dx (1 + o(1)) .

|ψ| ≥

µ κ = 2π κH m i=1 d i δ a i -→ B 0 (x) dx et |µ κ | = 2π κH m i=1 |d i |δ a i -→ |B 0 (x)| dx ,
au sens de la convergence faible

si R > 1 et 0 < b < 1 alors : e D (b, R) ≤ e N (b, R) + C R b 1 2 .
De plus, à l'aide de la fonction f introduite en (1.2.2) nous avons décrit le comportement asymptotique du e D (b, R), ce qui nous permet (après avoir subdiviser le domaine Ω en carrés de taille ) en utilisant l'inégalité ci dessus de minorer l'énergie dans chaque carré. Nous utilisons dans chaque carré des estimations qui nous permettent de montrer que le champ B 0 peut être considéré comme constant.

Dans la majoration de l'énergie, la méthode consiste à construire dans chaque carré des tests de configurations avec une condition aux limites de Dirichlet en utilisant le résultat de Proposition 3.2.4 qui contient la majoration de l'état fondamental e D . Notons que dans les deux estimations (majoration et minoration), nous approximons l'énergie en excluant les carrés qui rencontrent |B 0 | < .

Dans une autre partie nous montrons que l'énergie magnétique est dans l'erreur en utilisant les résultats sur l'énergie de l'état fondamental et nous déterminons une version locale de l'énergie. Ceci nous donne l'asymptotique de l'énergie de l'état fondamental dans un sousdomaine.

Dans la dernière partie, nous nous inspirons des méthodes de Sandier et Serfaty sur la distribution des vortex dans Ω. À nouvau nous resubdivisons chaque carré en M 2 carrés parmi lesquels on distingue entre bons et mauvais carrés, ceci nécessitant une optimisation fine des paramètres. Pour les détails nous renvoyons le lecteur à la Section 3.7. Nous démontrons que le nombre des mauvais carrés sont relativement petits, comparé à celui des bons carrés. En utilisant le résultat de la Proposition 3.7.3 qui a prouvé par Sandier et Serfaty, nous obtenons que les minimiseurs ont des vortex qui sont distribués dans Ω avec une densité non uniforme.

1.4 Énergie pour un modèle de Ginzburg-Landau avec un champ magnétique variable et un terme de "pinning"

Nous allons étudier un modèle de Ginzburg-Landau avec un champ magnétique variable et avec pinning. Le pinning traduit l'existence d'impuretés dans le matériau, il peut avoir des conséquences importantes sur la localisation des vortex (sujet que nous n'abordons pas ici) et aussi sur la température critique et l'apparition de la supraconductivité. Il est modélisé (selon la littérature physique) par un terme de poids a(x) qui correspond à l'inhomogénéité du matériau. L'énergie à étudier devient alors :

E κ,H,a,B 0 (ψ, A) = Ω |(∇ -iκHA)ψ| 2 + κ 2 2 (a(x, κ) -|ψ| 2 ) 2 dx + κ 2 H 2 Ω | curl A -B 0 | 2 dx ,
(1.4.1) où a(κ, x) peut dépendre de l'échelle κ. La fonction a(x, κ) est réelle, définie sur Ω × [κ 0 , +∞) pour un certain κ 0 > 0 et satisfait les hypothèses suivantes :

(H 1 ) ∀κ ≥ κ 0 , a(•, κ) ∈ C 1 (Ω) . (1.4.2) (H 2 ) sup x∈Ω, κ≥κ 0 |a(x, κ)| < +∞ . (1.4.3) (H 3 ) ∀κ ≥ κ 0 , sup x∈Ω |∇ x a(x, κ)| < +∞ . (1.4.4) (H 4 ) Il existe une constante positive C 1 , telle que, ∀κ ≥ κ 0 , L (∂{a(x, κ) > 0}) ≤ C 1 κ 1 2 , (1.4.5)
où L est la longueur de ∂{a(x, κ) > 0} dans Ω. Pour les détails nous renvoyons le lecteur à la section 4.3 (plus précisement à l'équation 4.3.1).

L'hypothèse H 3 nous donne un contrôle uniforme sur l'oscillation de a(•, κ) qui sera précisé ultérieurement par une hypothèse sur

L(κ) = sup x |∇ x a(x, κ)| . Si (ψ, A) ∈ H 1 (Ω, C) × H 1 div (Ω)
est un point critique de E κ,H,a,B 0 , donc, (ψ, A) est une solution faible du système constistué des équations suivantes, que nous appelons 'équations de Ginzburg-

Introduction Générale Landau',              -(∇ -iκHA) 2 ψ = κ 2 (a(x, κ) -|ψ| 2 )ψ dans Ω -∇ ⊥ rot(A -F) = 1 κH Im(ψ (∇ -iκHA)ψ) dans Ω ν • (∇ -iκHA)ψ = 0 sur ∂Ω rot A = rot F sur ∂Ω .
(1.4.6)

La fonctionelle E κ,H,a,B 0 a un point critique du type (0, A) avec A ∈ H 1 div (Ω) tel que rot A = B 0 . Un point critique ayant cette forme est appelé un état normal (ou une solution triviale). Il est donc naturel d'étudier si cet état normal est un minimiseur local de la fonctionnelle E κ,H,a,B 0 en présence d'un champ magnétique fort avec pinning. Le Hessien de E κ,H,a,B 0 en un état normal est donné par :

Hess (0,F) [φ, B] = Ω |(∇ -iκHF)φ| 2 -κ 2 a(x, κ)|φ| 2 dx + (κH) 2 Ω | rot B| 2 dx . (1.4.7)
Nous sommes conduits alors à étudier la positivité de la forme quadratique

Q Ω κHF,-κ 2 a (u) = Ω |(∇ -iκHF)u| 2 -κ 2 a(x, κ)|u| 2 dx (u ∈ H 1 (Ω)) . (1.4.8)
En observant que Q Ω κHF,-κ 2 a est semi-bornée inférieurement, i.e. qu'il existe une constante C telle que :

Ω |(∇ -iκHF)u| 2 -κ 2 a(x, κ)|u| 2 dx ≥ -C u 2 ,
nous considérons la réalisation auto-adjointe définie par le théorème de Friedrichs. C'est l'opérateur de Schrödinger magnétique P Ω κHF,-κ 2 a de domaine D(P Ω κHF,-κ 2 a ) :

P Ω κHF,-κ 2 a = -(∇ -iκHF) 2 , D(P Ω κHF,-κ 2 a ) = {u ∈ H 2 (Ω); ν • (∇ -iκHA)u | ∂Ω = 0} .
Notons µ 1 (κ, H) le bas du spectre de l'opérateur P Ω κHF,-κ 2 a . À l'aide du principe du min-max, µ 1 (κ, H) est défini par :

µ 1 (κ, H) = inf φ∈H 1 (Ω) φ =0 Q Ω κHF,-κ 2 a (φ) φ 2 L 2 (Ω)
.

(1.4.9)

Beaucoup d'articles sont consacrés à l'estimation de minimum de l'énergie de Ginzburg-Landau avec pinning, la plupart de ces papiers ont étudiée l'influence du pinning sur la localisation des vorticités. Dans le cas où B 0 = 0 dans (1.4.6), l'influence du pinning a été étudié par Lassoued et Mironescu dans [START_REF] Lassoued | Ginzburg-Landau type energy with discontinuous constraint[END_REF] et récemment par Michaël dans [START_REF] Michaël | The Ginzburg-Landau functional with a discontinuous and rapidly oscillating pinning term Part II : the non-zero degree case[END_REF]. Le pinning (i.e. la fonction a) dans [START_REF] Lassoued | Ginzburg-Landau type energy with discontinuous constraint[END_REF], est une fonction indépendante de κ, et est considéré dans [START_REF] Michaël | The Ginzburg-Landau functional with a discontinuous and rapidly oscillating pinning term Part II : the non-zero degree case[END_REF] comme une fonction périodique dépendant de κ. La version magnétique de la fonctionnelle dans [START_REF] Lassoued | Ginzburg-Landau type energy with discontinuous constraint[END_REF] a été étudiée dans [START_REF] Kachmar | The ground state energy of the three-dimensional Ginzburg-Landau model in the mixed phase[END_REF][START_REF] Kachmar | Magnetic vortices for a Ginzburg-Landau type energy with a discontinuous constraint[END_REF].

Dans [START_REF] Aftalion | Pinning phenomena in the Ginzburg-Landau model of superconductivity[END_REF], Aftalion, Sandier et Serfaty ont considéré une fonction a régulière dépendant de κ qui satisfait :

• L(κ)
κH.

• il existe une fonction a(x) continue, une constante positive a 0 et, pout tout κ ≥ 0, il existe deux fonctions σ(κ) = o ln ln

1 κ -1 2 et β(x, κ) ≥ 0 telles que, min B(x,σ(κ)) β(x, κ) = 0 , a(x, κ) = a(x) + β(x, κ) , et 0 < a 0 ≤ a(x) ≤ 1 .
Cette étude contient le cas quand a(x, κ) = a(x) (β = 0), mais elle contient aussi des cas avec des fonctions β(•, κ) dont l'oscillation en x peut croître avec κ.

La fonctionnelle E κ,H,a,B 0 dans (1.4.6) est proche des modèles de Bose-Einstein (voir [START_REF] Aftalion | Giant vortex and breakdown of strong pinning in a rotating Bose-Enstein condensate[END_REF][START_REF] Alama | Pinning effects and their breakdown for a Ginzburg-Landau model with normal inclusions[END_REF]).

Résultats principaux

Nous allons analyser comment le pinning apparaît dans l'asymptotique de l'énergie en présence d'un champ magnétique externe, variable et fort.

Nous nous concentrons sur le régime κ grand (i.e. κ → +∞) et nous étudions l'énergie de l'état fondamental de Ginzburg-Landau comme suit :

E g (κ, H, a, B 0 ) = inf E κ,H,a,B 0 (ψ, A) : (ψ, A) ∈ H 1 (Ω; C) × H 1 div (Ω) . ( 1
E g (κ, H, a, B 0 ) = κ 2 {a(x,κ)>0} a(x, κ) 2 f H κ |B 0 (x)| a(x, κ) dx+ κ 2 2 {a(x,κ)≤0} a(x, κ) 2 dx+o κ 2 .
(1.4.12)

Notons que quand Ω ∩ {a(x, κ) > 0} = ∅, nous obtenons directement du (1.4.1)

E κ,H,a,B 0 (ψ, A) ≥ κ 2 2 Ω a(x, κ) 2 dx = E κ,H,a,B 0 (0, F) .
(κH) 2 Ω | curl A -B 0 | 2 dx = o(κ 2 ) , lorsque κ -→ +∞ . (1.4.13)
Nous introduisons l'énergie locale de (ψ, A) dans un domaine D ⊂ Ω : • Estimation de l'énergie locale :

E 0 (ψ, A; a, D) = D |(∇ -iκHA)ψ| 2 dx + κ 2 2 D (a(x, κ) -|ψ| 2 ) 2 dx . ( 1
E 0 (ψ, A; a, D) = κ 2 D∩{a(x,κ)>0} a(x, κ) 2 f H κ |B 0 (x)| a(x, κ) dx + κ 2 2 D∩{a(x,κ)≤0} a(x, κ) 2 dx + o κ 2 . (1.4.15)
• Concentration du paramètre d'ordre :

D |ψ(x)| 4 dx = - D∩{a(x,κ)>0} a(x, κ) 2 2 f H κ |B 0 (x)| a(x, κ) -1 dx + o (1) . (1.4.16)
La formule (1.4.16) indique que ψ est localisé asymptotiquement dans la région où a > 0. Quand a(x, κ) = 1, le Théorème 1.4.3 a été prouvé dans [START_REF] Attar | The ground state energy of the two-dimensional Ginzburg-Landau functional with variable magnetic field[END_REF].

Nous allons maintenant étudier le troisième champ crtitique, i.e. le champ au-dessus duquel l'état normal (0, F) est le seul point critique de la fonctionnelle dans (1.4.6). Nous définissons les trois ensembles suivants :

N cp (κ) = {H > 0 : E κ,H,a,B 0 a un point critique non normal} ,
(1.4.17)

N (κ) = {H > 0 : E κ,H,a,B 0 a un minimiseur non normal} , (1.4.18) et N loc (κ) = {H > 0 : µ 1 (κ, H) < 0} . (1.4.19)
Ici, µ 1 (κ, H) a été défini dans (1.4.9). Nous renvoyons à [START_REF] Cancelier | Magnetic bottles with weak electric field[END_REF][START_REF] Kim | Estimate of the upper critical field and concentration for superconductor[END_REF][START_REF] Miqueu | Équation de Schrödinger avec champ magnétique qui s'annule[END_REF][START_REF] Pan | Schrödinger operators with non-degenerately vanishing magnetic fields in bounded domains[END_REF] pour des contributions précédentes.

21

Nous introduisons les champs critiques suivants (cf. e.g. [START_REF] Fournais | On the third critical field in Ginzburg-Landau theory[END_REF][START_REF] Lu | Estimates of the upper critical field for the Ginzburg-Landau equations of superconductivity[END_REF]) .

H cp C 3 (κ) = sup N cp (κ) , H cp C 3 (κ) = inf (R + \ N cp (κ)) , (1.4.20)
H C 3 (κ) = sup N (κ) , H C 3 (κ) = inf (R + \ N (κ)) , (1.4.21)
H loc C 3 (κ) = sup N loc (κ) , H loc C 3 (κ) = inf (R + \ N loc (κ)) . (1.4.22)
Au dessous du H C 3 , les états normaux perdront leur stabilité, et au dessus du H C 3 , l'état normal est (après une transformation de jauge) le seul point critique de la fonctionnelle dans (1.4.6).

Notre objectif est de determiner le comportement asymptotique de tous les champs critiques lorsque κ -→ +∞.

Il s'agit détudier des quantité spectrales liées à trois modèles qui dépendent de Γ (Γ étant vide ou non). Nous introduisons

Θ 0 = inf ξ∈R µ(ξ) ,
où µ est la valeur propre de l'opérateur

h N,ξ := - d 2 dt 2 + (t + ξ) 2 dans L 2 (R + ) ,
H C 3 (κ) = max sup x∈Ω a(x) |B 0 (x)| , sup x∈∂Ω a(x) Θ 0 |B 0 (x)| κ + O(κ 1
2 ) .

(1.4.23)

Nous introduisons,

λ 0 = inf τ ∈R λ(τ ) , (1.4.24)
où λ(τ ) est la valeur propre de l'opérateur autoadjoint :

M (τ ) = - d 2 dt 2 + 1 4 (t 2 + 2τ ) 2 in L 2 (R) . (1.4.25)
Nous considérons, pour tous θ ∈ (0, π) le bas du spectre λ(R 2 + , θ) de l'opérateur

P R 2 + A app,θ ,0 with A app,θ = - x 2 2 2 cos θ, x 2
H C 3 (κ) = max   sup x∈Γ∩Ω a(x) 3 2 λ 3 2 0 |∇B 0 (x)| , sup x∈Γ∩∂Ω a(x) 3 2 λ(R 2 + , θ(x)) 3 2 |∇B 0 (x)|   κ 2 + O κ 11 6
.

Introduction Générale

Ici, θ(x) désigne l'angle entre ∇B 0 (x) et le vecteur normal -ν(x).

Méthodes de démonstration

Notre but principal est de déterminer des estimations de l'énergie de l'état fondamental en présence d'un poids qui peut changer de signe et de donner une étude détaillée du troisième champ critique H C 3 quand a ne dépend pas du paramètre κ, qui peut être défini de plusieurs façons (voir (1.4.20)-(1.4.22)), mais qui correspond à la transition entre l'état où il y a supraconductivité de surface au bord du domaine, et l'état normal. Dans l'estimation de l'énergie, les techniques utilisées dans la démonstration sont inspirées de celles de [START_REF] Attar | The ground state energy of the two-dimensional Ginzburg-Landau functional with variable magnetic field[END_REF] et [START_REF] Attar | Energy and vorticity of the Ginzburg-Landau model with variable magnetic field[END_REF] (où le cas a(x, κ) = 1 a été traité). Au niveau technique, l'approche utilisée dans les preuves est différente de celles de [START_REF] Attar | The ground state energy of the two-dimensional Ginzburg-Landau functional with variable magnetic field[END_REF][START_REF] Fournais | The ground state energy of the three dimensional Ginzburg-Landau functional Part I : Bulk regime[END_REF][START_REF] Sandier | Vortices in the magnetic Ginzburg-Landau model[END_REF] puisque nous n'utilisons pas les estimations elliptiques uniformes. Ces estimations sont utilisées fréquemment dans les articles sur la fonctionnelle de Ginzburg-Landau (voir [START_REF] Fournais | Spectral Methods in Surface Superconducitivity[END_REF]) avec un pinning constant. Elles sont d'abord apparues dans [START_REF] Lu | Estimates of the upper critical field for the Ginzburg-Landau equations of superconductivity[END_REF] et ont été étendues au régime complet dans [START_REF] Fournais | Optimal uniform elliptic estimates for the Ginzburg-Landau system[END_REF]. Plus précisement nous donnons une généralisation de la formule (1.2.4), où l'intégrand doit être remplacer par

a 2 (x, κ) f H κ |B 0 (x)| a(x,κ)
dans la région où a(x, κ) > 0, et par a(x,κ) 2 2 ailleurs. De plus nous montrons que les vortex sont localisés dans la région où le pinning a est positif.

L'étude asymptotique du troisième champ critique H C 3 , nous conduit lorsque le champ magnétique appliqué B 0 est constant à l'analyse spectrale de l'opérateur de Gennes :

h N,ξ := - d 2 dt 2 + (t + ξ) 2 dans L 2 (R + ) ,
avec conditions aux limites de Neumann u (0) = 0. Dans le cas où le champ magnétique variable s'annule sur une courbe régulière, nous sommes conduit à faire l'analyse spectrale de l'opérateur de Montgomery :

M (τ ) = - d 2 dt 2 + 1 4 (t 2 + 2τ ) 2 dans L 2 (R) ,
et de l'opérateur de Pan et Kwek :

-(∇ -iA app,θ ) 2 dans L 2 (R + ) , où A app,θ = - x 2 2 2 cos θ, x 2

Chapitre 2

The ground state energy of the two dimensional Ginzburg-Landau functional with variable magnetic field

We consider the Ginzburg-Landau functional with a variable applied magnetic field in a bounded and smooth two dimensional domain. We determine an accurate asymptotic formula for the minimizing energy when the Ginzburg-Landau parameter and the magnetic field are large and of the same order. As a consequence, it is shown how bulk superconductivity decreases in average as the applied magnetic field increases.

Introduction

The functional and main results

We consider a bounded open simply connected set Ω ⊂ R 2 with smooth boundary. We suppose that Ω models a superconducting sample submitted to an applied external magnetic field. The energy of the sample is given by the Ginzburg-Landau functional,

E κ,H (ψ, A) = Ω |(∇ -iκHA)ψ| 2 -κ 2 |ψ| 2 + κ 2 2 |ψ| 4 dx + κ 2 H 2 Ω | curl A -B 0 | 2 dx .
(2.1.1)

Here κ and H are two positive parameters ; κ (the Ginzburg-Landau constant) is a material parameter and H measures the intensity of the applied magnetic field. The wave function (order parameter) ψ ∈ H 1 (Ω; C) describes the superconducting properties of the material. The induced magnetic field is curl A, where the potential A ∈ H 1 div (Ω), with H 1 div (Ω) is the space defined in (2.1.4) below. Finally, B 0 ∈ C ∞ (Ω) is the intensity of the external variable magnetic field and

The ground state energy of the two dimensional Ginzburg-Landau functional with variable magnetic field satisfies :

|B 0 | + |∇B 0 | > 0 in Ω . (2.1.2)
The assumption in (2.1.2) implies that for any open set ω relatively compact in Ω the set {x ∈ ω, B 0 (x) = 0} will be either empty, or consists of a union of smooth curves. Let F : Ω → R 2 be the unique vector field such that, div F = 0 and curl

F = B 0 in Ω , ν • F = 0 on ∂Ω. (2.1.3)
The vector ν is the unit interior normal vector of ∂Ω. The construction of F is recalled in the appendix. We define the space,

H 1 div (Ω) = {A = (A 1 , A 2 ) ∈ H 1 (Ω) 2 : div A = 0 in Ω , A • ν = 0 on ∂Ω }. (2.1.4) Critical points (ψ, A) ∈ H 1 (Ω; C) × H 1 div (Ω) of E κ,H are weak solutions of the Ginzburg-Landau equations,              -(∇ -iκHA) 2 ψ = κ 2 (1 -|ψ| 2 )ψ in Ω -∇ ⊥ curl(A -F) = 1 κH Im(ψ (∇ -iκHA)ψ) in Ω ν • (∇ -iκHA)ψ = 0 on ∂Ω curl A = curl F on ∂Ω . (2.1.5)
Here, curl

A = ∂ x 1 A 2 -∂ x 2 A 1 and ∇ ⊥ curl A = (∂ x 2 (curl A), -∂ x 1 (curl A)). If div A = 0, then ∇ ⊥ curl A = ∆A.
In this paper, we study the ground state energy defined as follows :

E g (κ, H) = inf E κ,H (ψ, A) : (ψ, A) ∈ H 1 (Ω; C) × H 1 div (Ω) . (2.1.6)
More precisely, we give an asymptotic estimate which is valid in the simultaneous limit κ → ∞ and H → ∞ in such a way that H κ remains asymptotically constant. The behavior of E g (κ, H) involves an auxiliary function g : [0, ∞) → [- 1 2 , 0] introduced in [START_REF] Sandier | The decrease of bulk-superconductivity close to the second critical field in the Ginzburg-Landau model[END_REF] whose definition will be recalled in (2.2.5) below. The function g is increasing, continuous, g(b) = 0 for all b ≥ 1 and g(0) = -1 2 .

Theorem 2.1.1. Let 0 < Λ min < Λ max . Under Assumption (2.1.2), there exist positive constants C, κ 0 and τ 0 ∈ (1, 2) such that if

κ 0 ≤ κ, Λ min ≤ H κ ≤ Λ max ,
then the ground state energy in (2.1.6) satisfies,

E g (κ, H) -κ 2 Ω g H κ |B 0 (x)| dx ≤ Cκ τ 0 . (2.1.7)
Theorem 2.1.1 was proved in [START_REF] Sandier | The decrease of bulk-superconductivity close to the second critical field in the Ginzburg-Landau model[END_REF] when the magnetic field is constant (B 0 (x) = 1). However, the estimate of the remainder is not explicitly given in [START_REF] Sandier | The decrease of bulk-superconductivity close to the second critical field in the Ginzburg-Landau model[END_REF].

The approach used in the proof of Theorem 2.1.1 is slightly different from the one in [START_REF] Sandier | The decrease of bulk-superconductivity close to the second critical field in the Ginzburg-Landau model[END_REF], and is closer to that in [START_REF] Fournais | Nucleation of bulk superconductivity close to critical magnetic field[END_REF] which studies the same problem when Ω ⊂ R 3 and B 0 constant. Corollary 2.1.2. Suppose that the assumptions of Theorem 2.1.1 are satisfied. Then the magnetic energy of the minimizer satisfies, for some positive constant C,

(κH) 2 Ω | curl A -B 0 | 2 dx ≤ Cκ τ 0 .
(2.1.8)

Remark 2.1.3. The value of τ 0 depends on the properties of B 0 : we find τ 0 = 7 4 when B 0 does not vanish in Ω and τ 0 = 15 8 in the general case.

Theorem 2.1.4. Suppose the assumptions of Theorem 2.1.1 are satisfied. There exist positive constants C, κ 0 and a negative constant τ 1 ∈ (-1, 0) such that, if κ ≥ κ 0 , and D is regular set such that D ⊂ Ω, then the following is true.

1. If (ψ, A) ∈ H 1 (Ω; C) × H 1 div (Ω) is a solution of (2.1.5), then, 1 2 D |ψ| 4 dx ≤ - D g H κ |B 0 (x)| dx + Cκ τ 1 .
(2.1.9)

2. If (ψ, A) ∈ H 1 (Ω; C) × H 1 div (Ω) is a minimizer of (2.1.1), then, D |ψ| 4 dx + 2 D g H κ |B 0 (x)| dx ≤ Cκ τ 1 .
(2.1.10)

Remark 2.1.5. The value of τ 1 depends on the properties of B 0 : we find τ 1 = -1 4 when B 0 does not vanish in Ω and τ 1 = -1 8 in the general case.

2.1.2 Discussion of main result :

If {x ∈ Ω : B 0 (x) = 0} = ∅ and H = bκ, b > 0, then g H κ |B 0 (x)| = 0 in D = x ∈ Ω : H κ |B 0 (x)| < 1
, and |D| = 0. Consequently, for κ sufficiently large, the restriction of ψ on D is not zero in L 2 (Ω). This is a significant difference between our result and the one for constant magnetic field. When the magnetic field is a non-zero constant, then (see [START_REF] Fournais | Spectral Methods in Surface Superconducitivity[END_REF]), there is a universal constant 0 ∈ ( 1 2 , 1) such that, if H = bκ and b > -1 0 , then ψ = 0 in Ω. Moreover, in the same situation, when H = bκ and 1 < b < -1 0 , then ψ is small every where except in a thin tubular neighborhood of ∂Ω (see [START_REF] Pan | Surface superconductivity in applied magnetic fields above HC 2[END_REF]). Our result goes in the same spirit as in [START_REF] Pan | Schrödinger operators with non-degenerately vanishing magnetic fields in bounded domains[END_REF], where the authors established under the Assumption (2.1.2) that when H = bκ 2 and b > b 0 , then ψ = 0 in Ω. (b 0 is a constant).

Notation.

Throughout the paper, we use the following notation :

• We write E for the functional E κ,H in (2.1.1).

• The letter C denotes a positive constant that is independent of the parameters κ and H, and whose value may change from a formula to another.

• If a(κ) and b(κ) are two positive functions, we write a(κ) b(κ) if a(κ)/b(κ) → 0 as κ → ∞.

• If a(κ) and b(κ) are two functions with b(κ) = 0, we write a(κ) ∼ b(κ) if a(κ)/b(κ) → 1 as κ → ∞.

• If a(κ) and b(κ) are two positive functions, we write a(κ) ≈ b(κ) if there exist positive constants c 1 , c 2 and κ 0 such that c 1 b(κ) ≤ a(κ) ≤ c 2 b(κ) for all κ ≥ κ 0 .

• If x ∈ R, we let [x] + = max(x, 0).

• Given R > 0 and x = (x 1 , x 2 ) ∈ R 2 , we denote by Q R (x) = (-R/2 + x 1 , R/2 + x 1 ) × (-R/2 + x 2 , R/2 + x 2
) the square of side length R centered at x.

• We will use the standard Sobolev spaces W s,p . For integer values of s these are given by

W n,p (Ω) := u ∈ L p (Ω) : D α u ∈ L p (Ω) for all |α| ≤ n .
• Finally we use the standard symbol H n (Ω) = W n,2 (Ω).

The limiting energy 2.2.1 Two-dimensional limiting energy

Given a constant b ≥ 0 and an open set D ⊂ R 2 , we define the following Ginzburg-Landau energy,

G σ b,D (u) = D b|(∇ -iσA 0 )u| 2 -|u| 2 + 1 2 |u| 4 dx , ∀ u ∈ H 1 0 (D). (2.2.1)
Here σ ∈ {-1, +1} and A 0 is the canonical magnetic potential,

A 0 (x) = 1 2 (-x 2 , x 1 ) , ∀ x = (x 1 , x 2 ) ∈ R 2 , (2.2.2) that satisfies : curl A 0 = 1 in R 2 .
We write

Q R = Q R (0) and let m 0 (b, R) = inf u∈H 1 0 (Q R ;C) G +1 b,Q R (u) . (2.2.3) Remark 2.2.1. As G +1 b,D (u) = G -1 b,D (u), it is immediate that, inf u∈H 1 0 (Q R ;C) G -1 b,Q R (u) = inf u∈H 1 0 (Q R ;C) G +1 b,Q R (u) . (2.2.4) 2.3 A priori estimates 27
The main part of the next theorem was obtained by Sandier-Serfaty [START_REF] Sandier | The decrease of bulk-superconductivity close to the second critical field in the Ginzburg-Landau model[END_REF] and Aftalion-Serfaty [START_REF] Aftalion | Lowest Landau level approach in superconductivity for the Abrikosov lattice close to H c 2[END_REF]Lemma 2.4]. However, the estimate in (2.2.7) is obtained by Fournais-Kachmar [START_REF] Fournais | The ground state energy of the three dimensional Ginzburg-Landau functional Part I : Bulk regime[END_REF]. Theorem 2.2.2. Let m 0 (b, R) be as defined in (2.2.3).

1. For all b ≥ 1 and R > 0, we have m 0 (b, R) = 0.

2. For any b ∈ [0, ∞), there exists a constant g(b) ≤ 0 such that,

g(b) = lim R→∞ m 0 (b, R) |Q R | and g(0) = - 1 2 .
(2.2.5)

3. The function [0, +∞) b → g(b)
is continuous, non-decreasing, concave and its range is the interval [-1 2 , 0].

4. There exists a constant α ∈ (0, 1 2 ) such that,

∀ b ∈ [0, 1] , α(b -1) 2 ≤ |g(b)| ≤ 1 2 (b -1) 2 . ( 2

.2.6)

5. There exist constants C and R 0 such that,

∀ R ≥ R 0 , ∀ b ∈ [0, 1] , g(b) ≤ m 0 (b, R) R 2 ≤ g(b) + C R . (2.2.7)

A priori estimates

The aim of this section is to give a priori estimates for solutions of the Ginzburg-Landau equations (2.1.5). These estimates play an essential role in controlling the errors resulting from various approximations. The starting point is the following L ∞ -bound resulting from the maximum principle. Actually, if (ψ, A) ∈ H 1 (Ω; C) × H 1 div (Ω) is a solution of (2.1.5), then

ψ L ∞ (Ω) ≤ 1 . (2.3.1)
The set of estimates below is proved in [15, Theorem 3.3 and Eq. 3.35] (see also [START_REF] Pan | Surface superconductivity in 3 dimensions[END_REF] for an earlier version).

Theorem 2.3.1. Let Ω ⊂ R 2 be bounded and smooth and B 0 ∈ C ∞ (Ω).

1. For all p ∈ (1, ∞) there exists

C p > 0 such that, if (ψ, A) ∈ H 1 (Ω, C) × H 1 div (Ω) is a solution of (2.1.5), then curl(A -F) W 1,p (Ω) ≤ C p 1 + κH + κ 2 κH ψ L ∞ (Ω) ψ L 2 (Ω) . (2.3.2)
2. For all α ∈ (0, 1) there exists

C α > 0 such that, if (ψ, A) ∈ H 1 (Ω, C) × H 1 div (Ω) is a solution of (2.1.5), then curl(A -F) C 0,α (Ω) ≤ C α 1 + κH + κ 2 κH ψ L ∞ (Ω) ψ L 2 (Ω) . (2.3.3) 3. For all p ∈ [2, ∞) there exists C > 0 such that, if κ > 0, H > 0 and (ψ, A) ∈ H 1 (Ω, C) × H 1 div (Ω) is a solution of (2.1.5), then (∇ -iκHA) 2 ψ p ≤ κ 2 ψ p , (2.3.4) (∇ -iκHA)ψ 2 ≤ κ ψ 2 , (2.3.5) curl(A -F) W 1,p (Ω) ≤ C κH ψ ∞ (∇ -iκHA)ψ p . ( 2
A -F W 2,p (Ω) ≤ C p 1 + κH + κ 2 κH ψ L ∞ (Ω) ψ L 2 (Ω) . (2.3.7)
The estimate is true for any p ∈ [2, ∞).

2. Using the Sobolev embedding Theorem we get, for all α ∈ (0, 1)

A -F C 1,α (Ω) ≤ C α 1 + κH + κ 2 κH ψ L ∞ (Ω) ψ L 2 (Ω) . (2.3.8)
3. Combining (2.3.5) and (2.3.6) (with p = 2) yields

curl(A -F) L 2 (Ω) ≤ C H ψ L ∞ (Ω) ψ L 2 (Ω) .
(2.3.9) Theorem 2.3.1 is needed in order to obtain the improved a priori estimates of the next theorem. Similar estimates are given in [START_REF] Pan | Surface superconductivity in 3 dimensions[END_REF].

Theorem 2.3.3. Suppose that 0 < Λ min ≤ Λ max . There exist constants κ 0 > 1, C 1 > 0 and for any α ∈ (0, 1), C α > 0 such that, if

κ ≥ κ 0 , Λ min ≤ H κ ≤ Λ max , (2.3.10) and (ψ, A) ∈ H 1 (Ω; C) × H 1 div (Ω) is a solution of (2.1.5), then (∇ -iκHA)ψ C(Ω) ≤ C 1 √ κH ψ L ∞ (Ω) , (2.3.11) A -F H 2 (Ω) ≤ C 1 curl(A -F) L 2 (Ω) + 1 √ κH ψ L 2 (Ω) ψ L ∞ (Ω) , (2.3
.12)

A -F C 0,α (Ω) ≤ C α curl(A -F) L 2 (Ω) + 1 √ κH ψ L 2 (Ω) ψ L ∞ (Ω) . ( 2
a H 2 (Ω) ≤ C curl a L 2 (Ω) + 1 √ κH ψ L 2 (Ω) ψ L ∞ (Ω) .
Proof of (2.3.13) : This is a consequence of the Sobolev embedding of H 2 (Ω) into C 0,α (Ω) for any α ∈ (0, 1) and (2.3.12).

Energy estimates in small squares

If (ψ, A) ∈ H 1 (Ω; C) × H 1 div (Ω)
, we introduce the energy density,

e(ψ, A) = |(∇ -iκHA)ψ| 2 -κ 2 |ψ| 2 + κ 2 2 |ψ| 4 .
We also introduce the local energy of (ψ, A) in a domain D ⊂ Ω :

E 0 (u, A; D) = D e(ψ, A) dx . (2.4.1)
Furthermore, we define the Ginzburg-Landau energy of (ψ, A) in a domain D ⊂ Ω as follows,

E(ψ, A; D) = E 0 (ψ, A; D) + (κH) 2 Ω | curl(A -F)| 2 dx . (2.4.2)
If D = Ω, we sometimes omit the dependence on the domain and write E 0 (ψ, A) for E 0 (ψ, A; Ω). We start with a lemma that will be useful in the proof of Proposition 2.4.2 below. Before we start to state the lemma, we define for all ( , x 0 ) such that Q (x 0 ) ⊂ Ω,

B Q (x 0 ) = sup x∈Q (x 0 ) |B 0 (x)| , (2.4.3)
where B 0 is introduced in (2.1.2). Later x 0 will be chosen in a lattice of R 2 .

Lemma 2.4.1. For any α ∈ (0, 1). there exist positive constants C and κ 0 such that if (2.3.10) holds, 0 < δ < 1, 0 < < 1, and (ψ, A) ∈ H 1 (Ω; C) × H 1 div (Ω) is a critical point of (2.1.1) (i.e. a solution of (2.1.5)), then, for any square

Q (x 0 ) relatively compact in Ω ∩ {|B 0 | > 0},
The ground state energy of the two dimensional Ginzburg-Landau functional with variable magnetic field there exists ϕ ∈ H 1 (Ω), such that,

E 0 (ψ, A; Q (x 0 )) ≥ (1 -δ)E 0 (e -iκHϕ ψ, σ B Q (x 0 ) A 0 (x -x 0 ), Q (x 0 )) -Cκ 2 δ -1 2α + δ -1 4 κ 2 + δ Q (x 0 ) |ψ| 2 dx , (2.4.4)
where σ denotes the sign of B 0 in Q (x 0 ).

Proof.

Construction of ϕ :

Let φ x 0 (x) = A(x 0 ) -F(x 0 ) • x,
where F is the magnetic potential introduced in (2.1.3). Using the estimate in (2.3.13), we get for all x ∈ Q (x 0 ) and α ∈ (0, 1) ,

|A(x) -∇φ x 0 -F(x)| = |(A -F)(x) -(A -F)(x 0 )| ≤ A -F C 0,α • |x -x 0 | α ≤ C √ λ κH α , (2.4.5)
where

λ = (κH) 2 curl(A -F) 2 L 2 (Ω) + 1 κH ψ 2 L 2 (Ω)
.

Using the bound ψ ∞ ≤ 1 and the estimate in (2.3.9), we get

λ ≤ Cκ 2 , (2.4.6)
which implies that

|A(x) -∇φ x 0 (x) -F(x)| ≤ C α H . (2.4.7)
We estimate the energy E 0 (ψ, A; Q (x 0 )) from below. We will need the function ϕ 0 introduced in Lemma 2.8.3 and satisfiying

|F(x) -σ B Q (x 0 ) A 0 (x -x 0 ) -∇ϕ 0 (x)| ≤ C 2 in Q (x 0 ). Let u = e -iκHϕ ψ , (2.4.8)
where ϕ = ϕ 0 + φ x 0 .

Lower bound :

We start with estimating the kinetic energy from below as follows. For any δ ∈ (0, 1), we write

|(∇ -iκHA)ψ| 2 = ∇ -iκH(σ B Q (x 0 ) A 0 (x -x 0 ) + ∇ϕ) ψ -iκH A -σ B Q (x 0 ) A 0 (x -x 0 ) -∇ϕ ψ 2 ≥ (1 -δ) ∇ -iκH(σ B Q (x 0 ) A 0 (x -x 0 ) + ∇ϕ) ψ 2 + (1 -δ -1 )(κH) 2 (A -∇φ x 0 -F)ψ + (F -σ B Q (x 0 ) A 0 (x -x 0 ) -∇ϕ 0 )ψ 2 .
Using the estimates in (2.4.7), (2.8.3) and the assumption in (2.3.10), we get, for any α ∈ (0, 1)

|(∇ -iκHA)ψ| 2 ≥ (1 -δ) ∇ -iκH(σ B Q (x 0 ) A 0 (x -x 0 ) + ∇ϕ) ψ 2 -Cκ 2 δ -1 2 α + δ -1 2 2 H 2 |ψ| 2 . (2.4.9)
Remembering the defintion of u in (2.4.8), then, we deduce the lower bound of E 0 ,

E 0 (ψ,A; Q (x 0 )) ≥ Q (x 0 ) (1 -δ)|(∇ -iκH(σ B Q (x 0 ) A 0 (x -x 0 ))u| 2 -κ 2 |u| 2 + κ 2 2 |u| 4 dx -Cκ 2 δ -1 2 2 κ + δ -1 2 α 2 Q (x 0 ) |ψ| 2 dx ≥ (1 -δ)E 0 (u, σ B Q (x 0 ) A 0 (x -x 0 ); Q (x 0 )) -Cκ 2 δ -1 4 κ 2 + δ -1 2α + δ Q (x 0 ) |ψ| 2 dx . (2.4.10)
This finishes the proof of Lemma 2.4.1.

Proposition 2.4.2. For all α ∈ (0, 1), there exist positive constants C, 0 and κ 0 such that, if (2.3.10) holds, κ ≥ κ 0 , ∈ (0, 1 2 ), ∈ (0, 0 ), 2 κ 2 > 1, (ψ, A) ∈ H 1 (Ω; C) × H 1 div (Ω) a critical point of (2.1.1), and

Q (x 0 ) ⊂ Ω ∩ {|B 0 | > }, then 1 |Q (x 0 )| E 0 (ψ, A; Q (x 0 )) ≥ g H κ B Q (x 0 ) κ 2 -C 3 κ 2 + 2α-1 + ( κ ) -1 + -1 κ 2 .
Here g(•) is the function introduced in (2.2.5), and B Q (x 0 ) is introduced in (2.4.3).

Proof.

Using the inequality ψ ∞ ≤ 1 and (2.4.4) to obtain,

E 0 (ψ, A; Q (x 0 )) ≥ (1 -δ)E 0 (u, σ B Q (x 0 ) A 0 (x -x 0 ); Q (x 0 )) -Cκ 2 δ -1 4 κ 2 + δ -1 2α + δ |Q (x 0 )| , (2.4.11)
where u is defined in (2.4.8).

Let b = H κ B Q (x 0 ) , R = κHB Q (x 0 ) .
(2.4.12)

Define the rescaled function,

v(x) = u R x + x 0 , ∀ x ∈ Q R . (2.4.13)
The ground state energy of the two dimensional Ginzburg-Landau functional with variable magnetic field Remember that σ denotes the sign of B 0 in Q (x 0 ). The change of variable y = R (x -x 0 ) gives :

E 0 (u, σ B Q (x 0 ) A 0 (x -x 0 ) ; Q (x 0 )) = Q R R ∇ y -iσ R A 0 (y) v 2 -κ 2 |v| 2 + κ 2 2 |v| 4 R dy = Q R |(∇ y -iσ A 0 )v| 2 - κ HB Q (x 0 ) |v| 2 + κ 2HB Q (x 0 ) |v| 4 dy = κ HB Q (x 0 ) Q R b |(∇ y -iσ A 0 )v| 2 -|v| 2 + 1 2 |v| 4 dy = 1 b G σ b ,Q R (v) . (2.4.14)
We still need to estimate from below the reduced energy

G σ b ,Q R (v). Since v is not in H 1 0 (Q R ), we introduce a cut-off function χ R ∈ C ∞ c (R 2 ) such that 0 ≤ χ R ≤ 1 in R 2 , supp χ R ⊂ Q R , χ R = 1 in Q R-1 , and |∇χ R | ≤ M in R 2 . (2.4.15) The constant M is universal. Let u R = χ R v . (2.4.16)
We have,

G σ b, Q R (v) = Q R b|(∇ -iσ A 0 )v| 2 -|v| 2 + 1 2 |v| 4 dx ≥ Q R b|χ R (∇ -iσ A 0 )v| 2 -|χ R v| 2 + 1 2 |v| 4 + (χ 2 R -1)|v| 2 dx ≥ G σ b ,Q R (χ R v) - Q R (1 -χ 2 R )|v| 2 dx -2 Q R (∇ -iσ A 0 )χ R v , ∇χ R v dy .
(2.4.17)

Having in mind (2.4.13) and (2.4.8), we get,

∇ y -iσ A 0 (y) v(y) = R ∇ x -iκHσ B Q (x 0 ) A 0 (x -x 0 ) u(x) .
Using the estimate in (2.3.11), (2.4.7) and (2.8.3) we get,

∇ y -iσ A 0 (y) v(y) ≤ R ∇ x -iκHσ B Q (x 0 ) (A + ∇ϕ) u(x) + κH R (A -σ B Q (x 0 ) A 0 (x -x 0 ) -∇ϕ)u(x) ≤ C 1 R κ + κ α + κ 2 2 . ( 2
Q R (∇ -iσ A 0 )χ R v , ∇χ R v dy ≤ C 1 R κ + κ α + κ 2 2 Q R \Q R-1 |∇χ R | dx ≤ C 1 κ + κ α+1 + κ 2 3 , (2.4.20)
and

Q R (1 -χ 2 R )|v| 2 dx ≤ |Q R \ Q R-1 | ≤ R . ( 2
G σ b ,Q R (v) ≥ G σ b ,Q R (u R ) -C 2 κ + κ α+1 + κ 2 3 ≥ G σ b ,Q R (u R ) -C 2 κ + κ 2 3 .
There are two cases :

Case 1 : σ = +1, when B 0 > 0, in Q (x 0 ). Case 2 : σ = -1, when B 0 < 0, in Q (x 0 ).
In Case 1, after recalling the definition of m 0 (b, R) introduced in (2.2.3), where b is introduced in (2.4.12) we get,

G +1 b ,Q R (v) ≥ m 0 (b, R) -C 2 κ + κ 2 3 . (2.4.22)
We get by collecting the estimates in (2.4.11)-(2.4.22) :

1 |Q (x 0 )| E 0 (ψ, A; Q (x 0 )) ≥ (1 -δ) b 2 m 0 (b, R) -C 2 κ + κ 2 3 -C δ -1 4 κ 2 + δ -1 2α + δ κ 2 ≥ (1 -δ) b 2 m 0 (b, R) -r(κ) , (2.4.23)
where

r(κ) = C 3 δ -1 4 κ 4 + δ -1 2α κ 2 + δκ 2 + 1 b 2 κ + κ 2 3 . ( 2
1 |Q (x 0 )| E 0 (ψ, A; Q (x 0 )) ≥ (1 -δ)R 2 b 2 g(b) -r(κ) , (2.4.25)
The ground state energy of the two dimensional Ginzburg-Landau functional with variable magnetic field with b defined in (2.4.12). By choosing δ = and using that Q (x 0 ) ⊂ {|B 0 | > }, we get, .4.26) This implies that,

r(κ) = O 3 κ 2 + 2α-1 + 1 ( κ) -1 + κ 2 . ( 2
1 |Q (x 0 )| E 0 (ψ, A; Q (x 0 )) ≥ g H κ B Q (x 0 ) κ 2 -C 3 κ 2 + 2α-1 + ( κ ) -1 + -1 κ 2 .
Similarly, in Case 2, according to Remark 2.2.1, we get that,

G -1 b ,Q R (v) ≥ m 0 (b, R) -C 2 κ + κ 2 3 ,
and the rest of the proof is as for Case 1.

2.5 Proof of Theorem 2.1.1 .

Proof. Let = (κ) and = (κ) be positive parameters such that κ -1 1 and κ -1 1 as κ → ∞. For some β ∈ (0, 1), µ ∈ (0, 1) to be determined later, we will choose = κ -β , = κ -µ .

(2.5.1)

Consider the lattice Γ := Z × Z and write for γ ∈ Γ ,

Q γ, = Q (γ). For any γ ∈ Γ such that Q γ, ⊂ Ω ∩ {|B 0 | > } let B γ, = inf x∈Q γ, |B 0 (x)| . (2.5.2) Let I , = γ : Q γ, ⊂ Ω ∩ {|B 0 | > } , N = Card I , , and
Ω , = int ∪ γ∈I , Q γ, .
It follows from (2.1.2) that :

N = |Ω| -2 + O( -2 ) + O( -1
) as → 0 and → 0 .

Let b = H κ B γ, , R = κHB γ, , (2.5.3)
and u R be a minimizer of the functional in (2.2.1), i.e.

m 0 (b, R) = Q R b|(∇ -iA 0 )u R | 2 -|u R | 2 + 1 2 |u R | 4 dx .
We will need the function ϕ γ introduced in Lemma 2.8.3 which satisfies

|F(x) -σ γ, B γ, A 0 (x -γ) -∇ϕ γ (x)| ≤ C 2 , in Q γ, ,
where

σ γ, is the sign of B 0 in Q γ, .
We define the function,

v(x) =            e -iκHϕγ u R R (x -γ) if x ∈ Q γ, ⊂ {B 0 > } e -iκHϕγ u R R (x -γ) if x ∈ Q γ, ⊂ {B 0 < -} 0 if x ∈ Ω \ Ω , . Since u R ∈ H 1 0 (Q R ), then v ∈ H 1 (Ω).
We compute the energy of the configuration (v, F). We get,

E(v, F) = Ω |(∇ -iκHF)v| 2 -κ 2 |v| 2 + κ 2 2 |v| 4 dx = γ∈I , E 0 (v, F; Q γ, ) . (2.5.4)
We estimate the term E 0 (v, F; Q γ, ) from above and we write :

E 0 (v, F; Q γ, ) = Q γ, |(∇ -iκHF)v| 2 -κ 2 |v| 2 + κ 2 2 |v| 4 dx = Q γ, ∇ -iκH σ γ, B γ, A 0 (x -γ) + ∇ϕ γ (x) v -iκH F -σ γ, B γ, A 0 (x -γ) -∇ϕ γ (x) v 2 -κ 2 |v| 2 + κ 2 2 |v| 4 dx ≤ Q γ, (1 + δ) ∇ -iκH σ γ, B γ, A 0 (x -γ) + ∇ϕ γ (x) v 2 -κ 2 |v| 2 + κ 2 2 |v| 4 dx + C(1 + δ -1 )(κH) 2 Q γ, F -σ γ, B γ, A 0 (x -γ) -∇ϕ γ (x) v 2 dx ≤ (1 + δ)E 0 (e -iκHϕγ v, σ γ, B γ, A 0 (x -γ); Q γ, ) + C(δκ 2 + δ -1 κ 4 4 ) Q γ, |v| 2 d x .
(2.5.5)

Having in mind that u R is a minimizer of the functional in (2.2.1), and using the estimate in

The ground state energy of the two dimensional Ginzburg-Landau functional with variable magnetic field

(2.3.1) we get :

Q γ, |v| 2 d x ≤ |Q γ, | .
Remark 2.2.1 and a change of variables give us,

Q γ, |(∇ -iκHσ γ, (B γ, A 0 (x -γ))e -iκHϕγ v| 2 -κ 2 |v| 2 + κ 2 2 |v| 4 dx = m 0 (b, R) b .
We insert this into (2.5.5) to obtain,

E 0 (v, F; Q γ, ) ≤ (1 + δ) m 0 (b, R) b + C(δκ 2 + δ -1 κ 4 4 ) 2 . (2.5.6)
We know from Theorem 2.2.2 that m 0 (b, R) ≤ g(b)R 2 + CR for all b ∈ [0, 1] and R sufficiently large, where b introduced in (2.5.3). We choose δ = in (2.5.6). That way we get,

E 0 (v, F; Q γ, ) ≤ g H κ B γ, 2 κ 2 + C 1 κ √ + + κ 2 3 2 κ 2 .
(2.5.7) Summing (2.5.7) over γ in I , , we recognize the lower Riemann sum of x → g H κ |B 0 (x)| . By monotonicity of g, g is Riemann-integrable and its integral is larger than any lower Riemann sum. Thus :

E(v, F) ≤ Ω , g H κ |B 0 (x)| dx κ 2 + C 1 κ √ + + κ 2 3 κ 2 .
(2.5.8)

Notice that using the regularity of ∂Ω and (2.1.2), there exists C > 0 such that :

|Ω \ Ω , | = O ( |∂Ω| + C ) , (2.5.9)
as and tend to 0. Thus, we get by using the properties of g in Theorem 2.2.2,

Ω , g H κ |B 0 (x)| dx ≤ Ω g H κ |B 0 (x)| dx + 1 2 |Ω \ Ω , |.
This implies that,

E(v, F) ≤ Ω g H κ |B 0 (x)| dx + C 1 κ √ + + + κ 2 3 κ 2 .
(2.5.10)

We choose in (2.5.1) .5.11) With this choice, we infer from (2.5.10),

β = 3 4 and µ = 1 8 . ( 2
E(v, F) ≤ Ω g H κ |B 0 (x)| dx + C 1 κ and 2 κ 2 = κ 3 8 > 1 . (2.5.13)
This finishes the proof of Proposition 2.5.1.

Remark 2.5.2. In the case when B 0 does not vanish in Ω, disappears and {x ∈ Ω; |B 0 (x)| > 0} = Ω. Consequently, the Ginzburg-Lundau energy of (v, F) in (2.4.2) satisfies :

E(v, F) ≤ Ω g H κ |B 0 (x)| dx + C 1 κ + + κ 2 3 κ 2 .
We take the same choice of β as in (2.5.11), then the ground state energy

E g (κ, H) in (2.1.6) satisfies, E g (κ, H) ≤ κ 2 Ω g H κ |B 0 (x)| dx + Cκ 7 4 .

Lower bound

We now establish a lower bound for the ground state energy E g (κ, H) in (2.1.6). The parameters and have the same form as in (2.5.1). Let B γ, = sup

x∈Q γ, |B 0 (x)| , (2.5.14)
and b γ, = H κ B γ, , R = κHB γ, , (2.5.15)
If (ψ, A) is a minimizer of (2.1.1), we have,

E g (κ, H) = E 0 (ψ, A; Ω , ) + E 0 (ψ, A; Ω \ Ω , ) + (κH) 2 Ω | curl A -F | 2 dx , (2.5.16)
where, for any D ⊂ Ω, the energy E 0 (ψ, A; D) is introduced in (2.4.1). Since the magnetic energy term is positive, we may write,

E g (κ, H) ≥ E 0 (ψ, A; Ω , ) + E 0 (ψ, A; Ω \ Ω , ) .
(2.5.17) Thus, we get by using (2.3.1), (2.3.11), and (2.5.9) :

|E 0 (ψ, A; Ω \ Ω , )| ≤ Ω\Ω , |(∇ -iκHA)ψ| 2 + κ 2 |ψ| 2 + κ 2 2 |ψ| 4 dx ≤ |Ω \ Ω , | C 1 κ 2 ψ 2 L ∞ (Ω) + κ 2 ψ 2 L ∞ (Ω) + κ 2 2 ψ 4 L ∞ (Ω) ≤ C 2 ( + )κ 2 . (2.5.18)
To estimate E 0 (ψ, A; Ω , ), we notice that,

E 0 (ψ, A; Ω , ) = γ∈I , E 0 (ψ, A; Q γ, ) .
The ground state energy of the two dimensional Ginzburg-Landau functional with variable magnetic field Using Proposition 2.4.2 with α = 2 3 and (2.5.18) with β = 3 4 and µ = 1 8 in (2.5.1), we get,

E 0 (ψ, A; Ω , ) ≥ γ∈I , g H κ B Q (x 0 ) 2 κ 2 -C 3 κ 2 + 2α-1 + ( κ ) -1 + -1 κ 2 ≥ κ 2 γ∈I , g H κ B Q (x 0 ) 2 -C 1 κ 15 8
,

and E 0 (ψ, A; Ω \ Ω , ) ≥ -C 2 κ 15 8 . (2.5.19)
As for the upper bound, we can use the monotonicity of g and recognize that the sum above is an upper Riemann sum of g. In that way, we get,

E 0 (ψ, A; Ω , ) ≥ κ 2 Ω , g H κ |B 0 (x)| dx -C 1 κ 15 8 .
Notice that Ω , ⊂ Ω and that g ≤ 0, we deduce that,

E 0 (ψ, A; Ω , ) ≥ κ 2 Ω g H κ |B 0 (x)| dx -C 1 κ 15 8 . (2.5.20)
Finally, putting (2.5. [START_REF] Fournais | On the transition to the normal phase for superconductors surrounded by normal conductors[END_REF]) and (2.5.20) into (2.5.17), we obtain

E g (κ, H) ≥ κ 2 Ω g H κ |B 0 (x)| dx -Cκ 15 8 .
(2.5.21)

Remark 2.5.3. When B 0 does not vanish, the local energy in Q (x 0 ) in Proposition 2.4.2 becomes :

1 |Q (x 0 )| E 0 (ψ, A; Q (x 0 )) ≥ g H κ B Q (x 0 ) κ 2 -C 3 κ 2 + 2α-1 + ( κ) -1 κ 2 .
Similarly, we choose α = 2 3 and = κ -34 , we get

E 0 (ψ, A; Ω \ Ω , ) ≥ -C 2 κ 7 4 , (2.5.22) and E 0 (ψ, A; Ω , ) ≥ κ 2 Ω g H κ |B 0 (x)| dx -C 1 κ 7 4 . (2.5.23)
As a consequence of (2.5.22) and (2.5.23), (2.5.21) becomes

E g (κ, H) ≥ κ 2 Ω g H κ |B 0 (x)| dx -Cκ 2.5.3 Proof of Corollary 2.1.2 If (ψ, A
) is a minimizer of (2.1.1), we have,

E(ψ, A; Ω) = E 0 (ψ, A; Ω) + (κH) 2 Ω | curl(A -F)| 2 dx. (2.5.25) Theorem 2.1.1 tells us that E(ψ, A; Ω) ≤ κ 2 Ω g H κ |B 0 (x)| dx + Cκ τ 0 .
This implies that

E 0 (ψ, A; Ω) + (κH) 2 Ω | curl(A -F)| 2 dx ≤ κ 2 Ω g H κ |B 0 (x)| dx + C 1 κ τ 0 . ( 2
E 0 (ψ, A; Ω) ≥ κ 2 Ω g H κ |B 0 (x)| dx -C 2 κ τ 0 . (2.5.27)
Putting (2.5.27) into (2.5.26), we get

-C 2 κ τ 0 + κ 2 Ω g H κ |B 0 (x)| dx + (κH) 2 Ω | curl(A -F)| 2 dx ≤ κ 2 Ω g H κ |B 0 (x)| dx + C 1 κ τ 0 . (2.5.28)
By simplification, we obtain

(κH) 2 Ω | curl(A -F)| 2 dx ≤ C κ τ 0 .
(2.5.29)

Local Energy Estimates

The object of this section is to give an estimates to the Ginzburg-Landau energy (2.4.2) in the open set D ⊂ Ω.

Main statements

Theorem 2.6.1. There exist positive constants κ 0 such that if (2.3.10) is true and D ⊂ Ω is an open set, then the local energy of the minimizer satisfies,

E(ψ, A; D) -κ 2 D g H κ |B 0 (x)| dx = o(κ 2 ) . (2.6.1) For all ( , x 0 ) such that Q (x 0 ) ⊂ Ω ∩ {|B 0 | > }, we define B Q (x 0 ) = inf x∈Q (x 0 ) |B 0 (x)| , (2.6.2)
where B 0 is introduced in (2.1.2).

Proposition 2.6.2. For all α ∈ (0, 1), there exist positive constants C, 0 and κ 0 such that if

(2.3.10) is true, κ ≥ κ 0 , ∈ (0, 1 2 ), ∈ (0, 0 ), 2 κ 2 > 1, (ψ, A) ∈ H 1 (Ω; C) × H 1 div (Ω) is a minimizer of (2.1.1), and Q (x 0 ) ⊂ Ω ∩ {|B 0 | > }, then, 1 |Q (x 0 )| E 0 (ψ, A; Q (x 0 )) ≤ g H κ B Q (x 0 ) κ 2 + C 3 κ 2 + 2α-1 + ( κ √ ) -1 κ 2 .
Here g(•) is the function introduced in (2.2.5) and E 0 is the functional in (2.4.1).

Proof. As explained earlier in the proof of Lemma 2.4.1 in (2.4.5), we may suppose after performing a gauge transformation that the magnetic potential A satisfies,

|A(x) -F(x)| ≤ C α H , ∀ x ∈ Q (x 0 ) . (2.6.3) Let b = H κ B Q (x 0 ) , R = κHB Q (x 0 ) , (2.6.4
)

and u R ∈ H 1 0 (Q R ) be the minimizer of the functional G +1 b,Q R introduced in (2.2.1). Let χ R ∈ C ∞ c (R 2 ) be a cut-off function such that, 0 ≤ χ R ≤ 1 in R 2 , supp χ R ⊂ Q R+1 , χ R = 1 in Q R ,
and |∇χ R | ≤ C for some universal constant C. Let η R (x) = 1 -χ R R (x -x 0 ) for all x ∈ R 2 and = 1 + 1 R .
This implies that,

η R (x) = 0 in Q (x 0 ) (2.6.5) 0 ≤ η R (x) ≤ 1 in Q (x 0 ) \ Q (x 0 ) (2.6.6) η R (x) = 1 in Ω \ Q (x 0 ) . (2.6.7)
Consider the function w(x) defined as follows,

w(x) = η R (x)ψ(x) in Ω \ Q (x 0 ), and, if x ∈ Q (x 0 ), w(x) =        e iκHϕ u R R (x -x 0 ) if Q (x 0 ) ⊂ {B 0 > } ∩ Ω e iκHϕ u R R (x -x 0 ) if Q (x 0 ) ⊂ {B 0 < -} ∩ Ω .
Notice that by construction, w = ψ in Ω \ Q (x 0 ). We will prove that, for any δ ∈ (0, 1) and α ∈ (0, 1),

E(w, A; Ω) ≤ E(ψ, A; Ω \ Q (x 0 )) + (1 + δ) bR m 0 (b, R) + r 0 (κ) 2 , (2.6.8)
and for some constant C, r 0 (κ) is given as follows,

r 0 (κ) = C δ + δ -1 4 κ 2 + δ -1 2α + 1 κ √ κ 2 .
(2.6.9)

Proof of (2.6.8) : With E 0 defined in (2.4.1), we write,

E 0 (w, A; Ω) = E 1 + E 2 , (2.6.10)
where

E 1 = E 0 (w, A; Ω \ Q (x 0 )) , E 2 = E 0 (w, A; Q (x 0 )) .
(2.6.11)

We estimate E 1 and E 2 from above. Starting with E 1 and using (2.6.7), we get,

E 1 = Ω\Q (x 0 ) |(∇ -iκHA)η R ψ| 2 -κ 2 |η R ψ| 2 + κ 2 2 |η R ψ| 4 dx = Ω\Q (x 0 ) η 2 R |(∇ -iκHA)ψ| 2 + |∇η R ψ| 2 + 2R η R (∇ -iκHA)ψ , ∇η R ψ -κ 2 η 2 R |ψ| 2 + κ 2 2 η 4 R |ψ| 4 dx = E 0 (ψ, A; Ω \ Q (x 0 )) + R(ψ, A) , (2.6.12)
where

R(ψ, A) = Q (x 0 )\Q (x 0 ) (η 2 R -1) |(∇ -iκHA)ψ| 2 -κ 2 |ψ| 2 + |ψ∇η R | 2 + κ 2 2 (η 4 R -1)|ψ| 4 + 2 η R (∇ -iκHA)ψ, ψ∇η R dx . Noticing that Q (x 0 ) \ Q (x 0 ) ≤ √ κHB Q (x 0 )
and using (2.6.6) together with the estimates in

(2.3.1), (2.3.10), (2.3.11) and |∇η R | ≤ C R , we get, |R(ψ, A)| ≤ C κ √ .
(2.6.13) Inserting (2.6.13) in (2.6.12), we get the following estimate,

E 1 ≤ E 0 (ψ, A; Ω \ Q (x 0 )) + C κ √ . (2.6.14)
We estimate the term E 2 in (2.6.11). We will need the function ϕ 0 introduced in Lemma 2.8.3 and satisfying

|F(x) -σ B Q (x 0 ) A 0 (x -x 0 ) -∇ϕ 0 (x)| ≤ C 2 in Q (x 0 ),

where σ denotes the

The ground state energy of the two dimensional Ginzburg-Landau functional with variable magnetic field sign of B 0 . We start with the kinetic energy term and write for any δ ∈ (0, 1) :

E 2 = Q (x 0 ) ∇ -iκH(σ B Q (x 0 ) A 0 (x -x 0 ) + ∇ϕ(x)) w -iκH A -σ B Q (x 0 ) A 0 (x -x 0 ) + ∇ϕ(x) 2 + -κ 2 |w| 2 + κ 2 2 |w| 4 dx ≤ Q (x 0 ) (1 + δ) ∇ -iκH(σ B Q (x 0 ) A 0 (x -x 0 ) + ∇ϕ(x)) w 2 -κ 2 |w| 2 + κ 2 2 |w| 4 dx + (1 + δ -1 )(κH) 2 Q (x 0 ) (A -∇φ x 0 -F)w + (F -σ B Q (x 0 ) A 0 (x -x 0 ) -∇ϕ 0 (x))w 2 dx .
(2.6.15)

Using the estimate in (2.6.3) together with (2.3.10) and (2.3.1), we deduce the upper bound,

E 2 ≤ (1 + δ)E 0 (e -iκHϕ w, σ B Q (x 0 ) A 0 (x -x 0 ); Q (x 0 )) + C(δ -1 2α + δ -1 4 κ 2 + δ)κ 2 2 , (2.6.16)
where α ∈ (0, 1). There are two cases :

Case 1 : If B 0 > in Q (x 0 ), then σ = +1 and w(x) =    e iκHϕ u R R (x -x 0 ) in Q (x 0 ) η R (x)ψ(x) in Ω \ Q (x 0 ) .
The change of variable y = R (x -x 0 ) and (2.4.12) gives us :

E 0 (e -iκHϕ w, σ B Q (x 0 ) A 0 (x -x 0 ); Q (x 0 )) = Q R R ∇ y -i R A 0 (y) u R 2 -κ 2 |u R | 2 + κ 2 2 |u R | 4 R dy = Q R |(∇ y -iA 0 (y))u R | 2 - κ HB Q (x 0 ) |u R | 2 + κ 2HB Q (x 0 ) |u R | 4 dy = κ HB Q (x 0 ) Q R b |(∇ y -iA 0 (y))u R | 2 -|u R | 2 + 1 2 |u R | 4 dy = 1 b G +1 b ,Q R (u R ) , (2.6.17)
where

G +1 b,Q R is the functional from (2.2.1). Case 2 : If B 0 < -in Q (x 0 ), then σ = -1 and w(x) =    e iκHϕ u R R (x -x 0 ) in Q (x 0 ) η R (x)ψ(x) in Ω \ Q (x 0 ) .
Similarly, like in case 1, we have,

E 0 (e -iκHϕ w, σ B Q (x 0 ) A 0 (x -x 0 ); Q (x 0 )) = 1 b G -1 b ,Q R (u R ) = 1 b G +1 b ,Q R (u R ) .
In both cases we see that,

E 0 (e -iκHϕ w, σ B Q (x 0 ) A 0 (x -x 0 ); Q (x 0 )) = 1 b G +1 b ,Q R (u R ) = m 0 (b, R) b .
(2.6.18)

Inserting (2.6.18) into (2.6.16), we get,

E 2 ≤ (1 + δ) 1 b m 0 (b, R) + C(δ + δ -1 4 κ 2 + δ -1 2α )κ 2 2 . ( 2
E 0 (w, A) ≤ E 0 (ψ, A; Ω \ Q (x 0 )) + (1 + δ) 1 b m 0 (b, R) + C(δ + δ -1 4 κ 2 + δ -1 2α κ 2 + ( κ √ ) -1 ) 2 κ 2 .
(2.6.20) This proves (2.6.8). Now, we show how (2.6.8) proves Proposition 2.6.2. By definition of the minimizer (ψ, A), we have,

E(ψ, A) ≤ E(w, A; Ω) . Since E(ψ, A; Ω) = E(ψ, A; Ω \ Q (x 0 )) + E 0 (ψ, A; Q (x 0 )), the estimate (2.6.8) gives us, E 0 (ψ, A; Q (x 0 )) ≤ (1 + δ) b m 0 (b, R) + r 0 (κ) ,
where r 0 (κ) is defined in (2.6.9). Dividing both sides by |Q (x 0 )| = 2 , we get,

1 |Q (x 0 )| E 0 (ψ, A, Q (x 0 )) ≤ (1 + δ) b 2 m 0 (b, R) + C δ + δ -1 4 κ 2 + 1 κ √ + δ -1 2α κ 2 . (2.6.21)
The inequality in (2.2.7) tell us that m 0 (b, R) ≤ R 2 g(b) + CR for all b ∈ [0, 1] and R sufficiently large. We substitute this into (2.6.21) and we select δ = , so that

r 0 (κ) = κ 2 O ( κ √ ) -1 + 3 κ 2 + 2α-1 .
Using (2.4.12) we get,

1 |Q (x 0 )| E(ψ, A, Q (x 0 )) ≤ (1 + δ)R 2 b 2 g(b) + CR b 2 + κ 2 O ( κ √ ) -1 + 3 κ 2 + 2α-1 ≤ g H κ B Q (x 0 ) κ 2 + C ( κ √ ) -1 + 3 κ 2 + 2α-1 κ 2 .
This establishes the result of Proposition 2.6.2.

The ground state energy of the two dimensional Ginzburg-Landau functional with variable magnetic field 2.6.2 Proof of Theorem 2.6.1, upper bound

The parameters and have the same form as in (2.5.1) and we take the same choice of β and µ as in (2.5.11). Consider the lattice Γ := Z × Z and write, for

γ ∈ Γ , Q γ, = Q (γ). For any γ ∈ Γ such that Q (γ) ⊂ Ω ∩ {|B 0 | > }, let : I , (D) = {γ : Q γ, ⊂ D ∩ {|B 0 | > }} , N = Card I , (D),
and

D , = int ∪ γ∈I , (D) Q γ, .
Notice that, by (2.1.2),

N = |D| -2 + O( -2 ) + O( -1
) as → 0 and → 0 .

If (ψ, A) is a minimizer of (2.1.1), we have, E(ψ, A; D) = E 0 (ψ, A; D , ) + E 0 (ψ, A; D \ D , ) + (κH) 2 Ω | curl A -F | 2 dx . (2.6.22)
Using Corollary 2.1.2, we may write,

E(ψ, A; D) ≤ E 0 (ψ, A; D , ) + E 0 (ψ, A; D \ D , ) + Cκ τ 0 .
(2.6.23)

Here τ 0 ∈ (1, 2). Notice that |D \ D , | = O ( |∂D , | + ) . (2.6.24)
We get by using (2.3.1) and (2.3.11) :

|E 0 (ψ, A; D \ D , )| ≤ |D \ D , | C 1 κ 2 ψ 2 L ∞ (D) + κ 2 ψ 2 L ∞ (D) + κ 2 2 ψ 4 L ∞ (D) ≤ C 2 ( + )κ 2 .
(2.6.25)

To estimate E 0 (ψ, A; D , ), we notice that,

E 0 (ψ, A; D , ) = γ∈I , (D) E 0 (ψ, A; Q γ, ) .
Using Proposition 2.6.2 and the estimates in (2.6.25) with β = 3 4 , α = 2 3 and µ = 1 8 , we get,

E 0 (ψ, A; D) ≤ γ∈I , (D) g H κ B Q (x 0 ) κ 2 2 + C 3 κ 2 + 2α-1 + ( κ √ ) -1 + κ 2 + C 1 κ τ 0 ≤ κ 2 γ∈I , (D) g H κ B Q (x 0 ) 2 + C 2 κ τ 0 , where B Q (x 0 ) = sup x∈Q (x 0 ) B 0 (x) .
Recognizing the lower Riemann sum of x → g H κ B 0 (x) , and using the monotonicity of g we get :

E 0 (ψ, A; D) ≤ κ 2 D , g H κ B 0 (x) dx + C 2 κ τ 0 . (2.6.26)
Thus, we get by using (2.6.24) and the property of g in Theorem 2.2.2,

κ 2 D , g H κ B 0 (x) dx ≤ κ 2 D g H κ B 0 (x) dx + C 3 κ τ 0 .
This finishes the proof of the upper bound.

Lower bound

We keep the same notation as in the derivation of the upper bound. We start with (2.6.22) and write,

E(ψ, A; D) ≥ E 0 (ψ, A; D , ) + E 0 (ψ, A; D \ D , ) . (2.6.27)
Similarly, as we did for the Lower bound 2.5.2, we get,

E(ψ, A; D) ≥ κ 2 D g H κ B 0 (x) dx -Cκ τ 0 . (2.6.28)
This finish the proof of Theorem 2.6.1.

2.7

Proof of Theorem 2.1.4 2.7.1 Proof of (2.1.9)

Let (ψ, A) be a solution of (2.1.5) and τ 1 = τ 0 -2. Then ψ satisfies,

-(∇ -iκHA) 2 ψ = κ 2 (1 -|ψ| 2 )ψ in Ω . (2.7.1)
We multiply both sides of the equation in (2.7.1) by ψ then we integrate over D. An integration by parts gives us,

D |(∇ -iκHA)ψ| 2 -κ 2 |ψ| 2 + κ 2 |ψ| 4 dx - ∂D ν • (∇ -iκHA)ψ ψ dσ(x) = 0 . ( 2
If (ψ, A
) is a minimizer of (2.1.1), then (2.7.3) is still true. We apply in this case Theorem 2.6.1 to write an upper bound of E 0 (ψ, A; D). Consequently, we deduce that,

1 2 D |ψ| 4 dx ≥ - D g H κ B 0 (x) dx -Cκ τ 1 .
(2.7.5)

Combining the upper bound in (2.7.5) with the lower bound in (2.7.4) finishes the proof of Theorem 2.1.4.

Useful gauge transformation

2.8.1 L p -regularity for the curl-div system

We consider the two dimensional case. We denote, for k ∈ N, by W k,p div (Ω) the space

W k,p div (Ω) = {A ∈ W k,p (Ω), divA = 0 and A • ν = 0 on ∂Ω}.
Then we have the following L p regularity for the curl-div system.

Proposition 2.8.1. Let 1 ≤ p < ∞. If A ∈ W 1,p div (Ω) satisfies curl A ∈ W k,p (Ω), for some k ≥ 0, then A ∈ W k+1,p div (Ω).
Proof. If A belongs to W 1,p div (Ω) and curl A ∈ L p (Ω), then there exists ψ ∈ W 2,p (Ω) such that A = (-∂ x 2 ψ, ∂ x 1 ψ), -∆ψ = curl A, with ψ = 0 on ∂Ω. This is simply the Dirichlet L p problem for the Laplacian (See [START_REF] Fournais | Optimal uniform elliptic estimates for the Ginzburg-Landau system[END_REF], Section A.1). The result we need for proving the proposition is then that if -∆ψ is in addition in W k,p (Ω) then ψ ∈ W k+2,p (Ω). This is simply an L p regularity result for the Dirichlet problem for the Laplacian which is described in ( [START_REF] Fournais | Optimal uniform elliptic estimates for the Ginzburg-Landau system[END_REF], Section F.4).

Construction of ϕ

x 0 . Lemma 2.8.2. If B 0 ∈ L 2 (Ω), then there exists a unique F ∈ H 1 div (Ω) such that, curl F = B 0 . (2.8.1)
Proof. The proof is standard, see [START_REF] Girault | Finite Elements Methods for Navier-Stokes Equations[END_REF]. Let

F = ∂ x 2 f -∂ x 1 f , where f ∈ H 2 (Ω) H 1 0 (Ω) is the unique solution of -∆f = B 0 in Ω . (2.8.2)
2.8 Useful gauge transformation

47

Then we deduce from the Dirichlet condition satisfied by f that τ • ∇f = 0 on ∂Ω which is equivalent to ν • F = 0 on ∂Ω . This finishes the proof of Lemma 2.8.2.

We continue with a lemma that will be useful in estimating the Ginzburg-Landau functional.

Lemma 2.8.3. There exists a positive constant C such that, if ∈ (0, 1) and x 0 ∈ Ω are such that Q (x 0 ) ⊂ Ω, then for any x 0 ∈ Q (x 0 ), there exists a function ϕ 0 ∈ H 1 (Ω) such that the magnetic potential F satisfies,

|F(x) -∇ϕ 0 (x) -B 0 ( x 0 )A 0 (x -x 0 )| ≤ C 2 , x ∈ Q (x 0 ) , (2.8.3)
where B 0 is the function introduced in (2.1.2) and A 0 is the magnetic potential introduced in (2.2.2).

Proof. We use Taylor formula near x 0 to order 2 and get :

F(x) = F( x 0 ) + M (x -x 0 ) + O(|x -x 0 | 2 ) , ∀x ∈ Q (x 0 ) , (2.8.4)
where

M = DF( x 0 ) =     ∂F 1 ∂x 1 | x 0 ∂F 1 ∂x 2 | x 0 ∂F 2 ∂x 1 | x 0 ∂F 2 ∂x 2 | x 0     .
We can write M as the sum of two matrices, M = M s + M as , where

M s = M +M t 2 is symmetric and M as = M -M t 2 is antisymmetric. Notice that curl F( x 0 ) = ∂F 2 ∂x 1 | x 0 - ∂F 1 ∂x 2 | x 0 = B 0 ( x 0 ). Consequently, M as = 0 -B 0 /2 B 0 /2 0 .
Substitution into M gives as that,

M (x -x 0 ) = ∇φ 0 (x) + B 0 ( x 0 )A 0 (x -x 0 ) ,
where A 0 (x) = 1 2 (-x 2 , x 1 ) and the function φ 0 is defined by

φ 0 (x) = 1 2 M + M t 2 (x -x 0 ), (x -x 0 ) . Let ϕ 0 (x) = φ 0 (x) + (F( x 0 ) + M (x 0 -x 0 )) • x .
Substitution into (2.8.4) gives as that,

F = B 0 ( x 0 )A 0 (x -x 0 ) + ∇ϕ 0 (x) + O |x -x 0 | 2 . Notice that, if x ∈ Q (x 0 ) , then |x -x 0 | ≤ √ 2.
This finishes the proof of Lemma 2.8.3.

The ground state energy of the two dimensional Ginzburg-Landau functional with variable magnetic field Remark 2.8.4. We will apply this lemma by considering x 0 such that B 0 (

x 0 ) = sup Q (x 0 ) B 0 (x) or B 0 ( x 0 ) = inf Q (x 0 ) B 0 (x) .
Chapitre 3

Energy and vorticity of the Ginzburg-Landau model with variable magnetic field

We consider the Ginzburg-Landau functional with a variable applied magnetic field in a bounded and smooth two dimensional domain. The applied magnetic field varies smoothly and is allowed to vanish non-degenerately along a curve. Assuming that the strength of the applied magnetic field varies between two characteristic scales, and the Ginzburg-Landau parameter tends to +∞, we determine an accurate asymptotic formula for the minimizing energy and show that the energy minimizers have vortices. The new aspect in the presence of a variable magnetic field is that the density of vortices in the sample is not uniform.

Introduction

We consider a bounded, open and simply connected set Ω ⊂ R 2 with smooth boundary. We suppose that Ω models a superconducting sample subject to an applied external magnetic field. The energy of the sample is given by the Ginzburg-Landau functional,

E κ,H (ψ, A) = Ω |(∇ -iκHA)ψ| 2 + κ 2 2 (1 -|ψ| 2 ) 2 dx+κ 2 H 2 Ω | curl A-B 0 | 2 dx . (3.1.1)
Here κ and H are two positive parameters, to simplify we will consider that H = H(κ). The wave function (order parameter) ψ ∈ H 1 (Ω; C) and the magnetic potential A ∈ H 1 div (Ω). The space H 1 div (Ω) is defined in (3.1.4) below. Finally, the function B 0 ∈ C ∞ (Ω) gives the intensity of the external variable magnetic field. Let Γ = {x ∈ Ω, B 0 (x) = 0}, then, we assume that B 0 satisfies :

|B 0 | + |∇B 0 | > 0 in Ω ∇B 0 × n = 0 on Γ ∩ ∂Ω . (3.1.2)
The assumption in (3.1.2) implies that for any open set ω relatively compact in Ω the set Γ ∩ ω will be either empty, or consists of a union of smooth curves. Here, the definition of the functional (3.1.1) is taken as in [START_REF] Fournais | Spectral Methods in Surface Superconducitivity[END_REF]. In [START_REF] Dombrowski | Semiclassical analysis with vanishing magnetic fields[END_REF], the scaling for the intensity of the external magnetic field (denoted by h) is different. We choose the scaling from [START_REF] Fournais | Spectral Methods in Surface Superconducitivity[END_REF] for convenience when estimating the ground state energy of the functional.

Let F : Ω → R 2 be the unique vector field such that, div F = 0 and curl F = B 0 in Ω , ν • F = 0 on ∂Ω.

(3.1.

3)

The vector ν is the unit interior normal vector of ∂Ω. We define the space,

H 1 div (Ω) = {A = (A 1 , A 2 ) ∈ H 1 (Ω) 2 : div A = 0 in Ω , A • ν = 0 on ∂Ω }. (3.1.4) Critical points (ψ, A) ∈ H 1 (Ω; C) × H 1 div (Ω) of E κ,H are weak solutions of the Ginzburg-Landau equations,              -(∇ -iκHA) 2 ψ = κ 2 (1 -|ψ| 2 )ψ in Ω -∇ ⊥ curl(A -F) = 1 κH Im(ψ (∇ -iκHA)ψ) in Ω ν • (∇ -iκHA)ψ = 0 on ∂Ω curl A = curl F on ∂Ω . (3.1.5)
Here, curl

A = ∂ x 1 A 2 -∂ x 2 A 1 and ∇ ⊥ curl A = (∂ x 2 (curl A), -∂ x 1 (curl A)).
For a solution (ψ, A) of (3.1.5), the function ψ describes the superconducting properties of the material and (κH curl A) is the induced magnetic field. The number κ is a parameter describing the properties of the material, and the number H measures the variation of the intensity of the applied magnetic field. We focus on the regime of large values of κ, κ → +∞.

In this paper, we study the ground state energy defined as follows :

E g (κ, H) = inf E κ,H (ψ, A) : (ψ, A) ∈ H 1 (Ω; C) × H 1 div (Ω) . (3.1.6)
More precisely, we give an asymptotic estimate valid when H(κ) satisfies :

C min κ 1 3 ≤ H(κ) κ as κ -→ +∞ , (3.1.7)
where C min is a positive constant. The behavior of E g (κ, H) involves a function f : [0, 1] -→ [0, 1 2 ] introduced in (3.2.10) below. The function f is the limit of a simplified Ginzburg-Landau type functional. It has been defined by Sandier-Serfaty in [START_REF] Sandier | The decrease of bulk-superconductivity close to the second critical field in the Ginzburg-Landau model[END_REF], and then analyzed in [START_REF] Aftalion | Lowest Landau level approach in superconductivity for the Abrikosov lattice close to H c 2[END_REF][START_REF] Fournais | The ground state energy of the three dimensional Ginzburg-Landau functional Part I : Bulk regime[END_REF]. This function plays an important role in describing the distribution of superconductivity in the bulk of 2D and 3D samples, see [START_REF] Sandier | The decrease of bulk-superconductivity close to the second critical field in the Ginzburg-Landau model[END_REF], [START_REF] Aftalion | Lowest Landau level approach in superconductivity for the Abrikosov lattice close to H c 2[END_REF][START_REF] Fournais | Nucleation of bulk superconductivity close to critical magnetic field[END_REF][START_REF] Fournais | The ground state energy of the three dimensional Ginzburg-Landau functional Part I : Bulk regime[END_REF] and the recent papers [START_REF] Attar | The ground state energy of the two-dimensional Ginzburg-Landau functional with variable magnetic field[END_REF][START_REF] Helffer | The Ginzburg-Landau functional with vanishing magnetic field[END_REF].

In Section 3.2, we will state various properties of the function f . Note for the moment that the function f is increasing, continuous and f (b) = 1 2 , for all b ≥ 1. Under the assumption that B 0 (x) satisfies (3.1.2) and that the function H = H(κ) satisfies

C 1 κ ≤ H ≤ C 2 κ , (3.1.8)
where C 1 and C 2 are positive constants, we obtained 1 in [START_REF] Attar | The ground state energy of the two-dimensional Ginzburg-Landau functional with variable magnetic field[END_REF] that

E g (κ, H) = κ 2 Ω f H κ |B 0 (x)| dx + o (κH) , as κ -→ +∞ . (3.1.9)
In this paper, we generalize this result to the case when H(κ) satisfies (3.1.7).

Theorem

(b) = b 2 ln 1 b (1 + ŝ(b)) , as b -→ 0 , with ŝ(b) = o(1).
As a consequence of the behaviour of f above, (3.1.10) becomes

E g (κ, H) = 1 2 κH Ω |B 0 (x)| ln κ H|B 0 (x)| dx (1 + o(1)) . (3.1.11)
When the magnetic field is constant (i.e B 0 is a constant function), (3.1.11) is proved in [START_REF] Sandier | On the energy of type-II superconductors in the mixed phase[END_REF] under the relaxed condition ln κ κ H κ .

(3.1.12)

The reason why we do not obtain (3.1.11) under the relaxed condition (3.1.12) is probably technical. The method is to construct test configurations with a Dirichlet boundary condition. We can not construct periodic configurations as in [START_REF] Sandier | On the energy of type-II superconductors in the mixed phase[END_REF] because the magnetic field B 0 is variable. The approach used in the proof of Theorem 3.1.1 is close to that in [START_REF] Kachmar | The ground state energy of the three-dimensional Ginzburg-Landau model in the mixed phase[END_REF] which studies the same problem when Ω ⊂ R 3 and B 0 is constant. If we assume that there exist positive constants C min and C 1 and H(κ) satisfies

C min κ 1 3 ≤ H(κ) ≤ C 1 κ , (3.1.13)
1 After a change of notation

52

Energy and vorticity of the Ginzburg-Landau model with variable magnetic field then (3.1.9) and (3.1.10) can be rewritten in a unique statement :

E g (κ, H) = κ 2 Ω f H κ |B 0 (x)| dx + o κH ln H κ + 1 . ( 3
(κH) 2 Ω | curl A -B 0 | 2 dx = o κH ln κ H , as κ -→ +∞ . (3.1.15) If (ψ, A) ∈ H 1 (Ω; C) × H 1 div (Ω), we introduce the energy density, e(ψ, A) = |(∇ -iκHA)ψ| 2 + κ 2 2 (1 -|ψ| 2 ) 2 .
We also introduce the local energy of (ψ, A) in a domain D ⊂ Ω :

E 0 (ψ, A; D) = D e(ψ, A) dx . (3.1.16)
Furthermore, we define the Ginzburg-Landau energy of (ψ, A) in a domain D ⊂ Ω as follows,

E(ψ, A; D) = E 0 (u, A; D) + (κH) 2 Ω | curl(A -F)| 2 dx . ( 3
E(ψ, A, D) ≥ κ 2 D f H κ |B 0 (x)| dx + o κH ln κ H , as κ -→ +∞ . (3.1.18) 2. If H(κ) satisfies C 1 min κ
where C 1 min is a positive constant, then

E(ψ, A, D) ≤ κ 2 D f H κ |B 0 (x)| dx + o κH ln κ H , as κ -→ +∞ . (3.1.20)
As a consequence of the proof of Theorem 3.1.1, the methods used in [START_REF] Sandier | On the energy of type-II superconductors in the mixed phase[END_REF] allow us to obtain information regarding the distribution of vortices in Ω. When the magnetic field is constant (i.e B 0 is a constant), it is proved in [START_REF] Sandier | On the energy of type-II superconductors in the mixed phase[END_REF] that ψ has vortices whose density tends to be uniform. In Section 3.7 we will prove that, if (ψ, A) is a minimizer of (3.1.1) and B 0 (x) is a variable magnetic field, then, ψ has vortices that are distributed everywhere in Ω but with a non uniform density.

The next theorem was proved by E. Sandier and S. Serfaty in [START_REF] Sandier | On the energy of type-II superconductors in the mixed phase[END_REF] when the magnetic field is constant (B 0 (x) = 1). Theorem 3.1.6. Suppose that Assumption (3.1.2) holds and that H(κ) satisfies (3.1.7). Let (ψ, A) be a minimizer of (3.1.1). Then there exists m = m(κ) disjoint disks

(D i (a i , r i )) m i=1 in Ω such that, as κ -→ +∞ , 1. m i=1 r i ≤ (κH) 1 2 ln κ H -7 4 Ω 1 √ |B 0 (x)| dx (1 + o(1)) . 2. |ψ| ≥ 1 2 on ∪ i ∂D i . 3. If d i = deg ψ |ψ| , ∂D i is the winding number of ψ |ψ| on ∂D i , then as κ -→ +∞ µ κ = 2π κH m i=1 d i δ a i -→ B 0 (x) dx and |µ κ | = 2π κH m i=1 |d i |δ a i -→ |B 0 (x)| dx ,
in the weak sense of measures2 , where dx is the Lebesgue measure on R 2 restricted to Ω .

The measure µ describes the distribution of vortices see Fig. 3.1, and it is called the vorticity measure, the function o(1) is bounded independently of the choice of the minimizer (ψ, A).

Notation.

Throughout the paper, we use the following notation : • Given R > 0 and

• If a(κ)
x = (x 1 , x 2 ) ∈ R 2 , Q R (x) = (-R/2+x 1 , R/2+x 1 )×(-R/2+x 2 , R/2+x 2 )
denotes the square of side length R centered at x and we write Q R = Q R (0).

A reference problem

In this section, we will introduce the function f appearing in Theorem 3.1.1, and recall its main properties as proved previously in [START_REF] Sandier | The decrease of bulk-superconductivity close to the second critical field in the Ginzburg-Landau model[END_REF] and [START_REF] Aftalion | Lowest Landau level approach in superconductivity for the Abrikosov lattice close to H c 2[END_REF][START_REF] Fournais | The ground state energy of the three dimensional Ginzburg-Landau functional Part I : Bulk regime[END_REF]. In addition, we will give a new estimate on f in Proposition 3.2.4 below.

Consider two constants b ∈ (0, 1) and R > 0. If u ∈ H 1 (Q R ), we define the following Ginzburg-Landau energy,

F σ b,Q R (u) = Q R b|(∇ -iσA 0 )u| 2 + 1 2 1 -|u| 2 2 dx , (3.2.1)
where σ ∈ {-1, +1} and

A 0 (x) = 1 2 (-x 2 , x 1 ) , ∀ x = (x 1 , x 2 ) ∈ R 2 . (3.2.2)
Notice that the magnetic potential A 0 satisfies :

curl A 0 = 1 in R 2 .
We introduce the two ground state energies

e N (b, R) = inf F +1 b,Q R (u) : u ∈ H 1 (Q R ; C) (3.2.3) e D (b, R) = inf F +1 b,Q R (u) : u ∈ H 1 0 (Q R ; C) . (3.2.4)
The minimization of the functional F +1 b,Q R over 'magnetic periodic' functions appears naturally

E R = u ∈ H 1 loc (R 2 ; C) : u(x 1 + R, x 2 ) = e iR x 2 2 u(x 1 , x 2 ), u(x 1 , x 2 + R) = e -iR x 1 2 u(x 1 , x 2 ) , (3.2
.5) together with the ground state energy

e p (b, R) = inf F +1 b,Q R (u) : u ∈ E R . (3.2.6)
Since F +1 b,Q R is bounded from below, there exists for each e # (b, R) with # ∈ {N, D, p}, a ground state (minimizer). Note also that by comparison of the three domains of minimization it is clear that e N (b, R) ≤ e p (b, R) ≤ e D (b, R) .

(3.2.7)

In the three cases, if u is such a ground state, u satisfies the Ginzburg-Landau equation

b(∇ -iA 0 ) 2 u = (1 -|u| 2 )u ,
and it results from a standard application of the maximum principle that

|u| ≤ 1 . (3.2.8) As F +1 b,Q R (u) = F -1 b,Q R (u), it is also immediate that, inf u∈H 1 (Q R ;C) F +1 b,Q R (u) = inf u∈H 1 (Q R ;C) F -1 b,Q R (u) . (3.2.9)
In the next theorem we will define the limiting function f , which describes the ground state energy of both two and three dimensional superconductors subject to high magnetic fields (see [START_REF] Fournais | The ground state energy of the three dimensional Ginzburg-Landau functional Part I : Bulk regime[END_REF]). Theorem 3.2.1. Let e p (b, R) be as introduced in (3.2.6).

For any

b ∈ [0, ∞), there exists a constant f (b) ≥ 0 such that f (b) = lim R-→∞ e p (b, R) |Q R | = lim R-→∞ e D (b, R) |Q R | . (3.2.10) 2. For all b ≥ 1, f (b) = 1 2 . 3. The function [0, ∞) b -→ f (b) is continuous, non-decreasing and its range is the interval [0, 1/2]. 4. As b -→ 0 + , f (b) satisfies f (b) = b 2 ln 1 b (1 + ŝ(b)) , (3.2.11)
where the function ŝ : (0, +∞) -→ (-∞, +∞) satisfies

lim b-→0 ŝ(b) -→ 0 .
Energy and vorticity of the Ginzburg-Landau model with variable magnetic field 5. There exist universal constants C and R 0 such that

∀R ≥ R 0 , ∀b ∈ [0, 1], f (b) - e p (b, R) R 2 ≤ C R .
(3.2.12)

6. There exist positive constants b 0 , R 0 and a function err : (0, 1) × (0, +∞) -→ (0, +∞) , (3.2.13)

such that ∀ ≥ 0 , ∃η ≥ 0 if |b| + 1 R < η then |err(b, R)| < , (3.2.14)
and

∀b ∈ (0, b 0 ) , ∀R ∈ (R 0 , +∞) , e N (b, R) R 2 ≥ f (b)(1 -err(b, R)) . (3.2.15)
The limiting function f was defined in ( [START_REF] Aftalion | Lowest Landau level approach in superconductivity for the Abrikosov lattice close to H c 2[END_REF], [START_REF] Sandier | The decrease of bulk-superconductivity close to the second critical field in the Ginzburg-Landau model[END_REF], [START_REF] Kachmar | The ground state energy of the three-dimensional Ginzburg-Landau model in the mixed phase[END_REF]). then,

e N (b, R) ≤ e D (b, R) -CR b 1 2 . (3.2.17)
Proof. Without loss of generality, we can suppose σ = +1. Let u ∈ H 1 (Q R ) be a minimizer of the functional in (3.2.1), i.e. such that :

e N (b, R) = F +1 b,Q R (u) = Q R b|(∇ -iA 0 )u| 2 + 1 2 1 -|u| 2 2 dx . (3.2.18) We introduce a cut-off function χ R,b ∈ C ∞ c (R 2 ) such that 0 ≤ χ R,b ≤ 1 in R 2 , suppχ R,b ⊂ Q R , χ R,b = 1 in Q R-b 1 2 . (3.2.19)
In addition, the function χ R,b can be chosen such that for some universal constants C and C , we have,

|∇χ R,b | ≤ Cb -1 2 and |∆χ R,b | ≤ C b -1 , ∀R ≥ 1 and ∀b ∈ (0, 1) . (3.2.20) 3.2 A reference problem 57 Let u R,b (x) = χ R,b (x)u(x). Then u R,b ∈ H 1 0 (Q R ) and consequently e D (b, R) ≤ F +1 b,Q R (u R,b ) . (3.2.21) We rewrite F +1 b,Q R (u R,b ) as follows, F +1 b,Q R (u R,b ) = Q R b|(∇ -iA 0 )χ R,b u| 2 + 1 2 1 -|χ R,b u| 2 2 dx = Q R b|(∇ -iA 0 )χ R,b u| 2 + 1 2 1 -2|u| 2 + |χ R,b u| 4 + 2(|u| 2 -|χ R,b u| 2 ) dx ≤ Q R b|(∇ -iA 0 )χ R,b u| 2 + 1 2 1 -|u| 2 2 dx + Q R \Q R-b 1 2 1 -|χ R,b | 2 |u| 2 dx . (3.2.22)
We estimate from above the term

Q R |(∇ -iA 0 )χ R,b u| 2 dx as follows : Q R |(∇ -iA 0 )χ R,b u| 2 dx = (∇ -iA 0 )χ R,b u, (∇ -iA 0 )χ R,b u = ∇χ R,b u + χ R,b (∇ -iA 0 )u, ∇χ R,b u + χ R,b (∇ -iA 0 )u = ∇χ R,b u, ∇χ R,b u + χ R,b (∇ -iA 0 )u, χ R,b (∇ -iA 0 )u + ∇χ R,b u, χ R,b (∇ -iA 0 )u + χ R,b (∇ -iA 0 )u, ∇χ R,b u .
An integration by parts yields,

∇χ R,b u, χ R,b (∇ -iA 0 )u = -∇χ R,b u, ∇χ R,b u -χ R,b ∆χ R,b u, u -χ R,b (∇ -iA 0 )u, ∇χ R,b u , (3.2.23) which implies that Q R |(∇ -iA 0 )χ R,b u| 2 dx = χ R,b (∇ -iA 0 )u, χ R,b (∇ -iA 0 )u -χ R,b ∆χ R,b u, u . (3.2.24)
Putting (3.2.24) into (3.2.22), we get

F +1 b,Q R (u R,b ) ≤ Q R b|χ R,b (∇ -iA 0 )u| 2 + 1 2 1 -|u| 2 2 dx + Q R \Q R-b 1 2 1 -|χ R,b | 2 |u| 2 dx + b Q R \Q R-b 1 2 |∆χ R,b | |u| 2 dx .

19), it

Energy and vorticity of the Ginzburg-Landau model with variable magnetic field is easy to check that,

F +1 b,Q R (u R,b ) ≤ F +1 b,Q R (u) + CR b 1 2 .
Using (3.2.21) and (3.2.18), we get

e D (b, R) ≤ e N (b, R) + CR b 1 2 . Corollary 3.2.3. With f (b) introduced in (3.2.10), it holds, f (b) = lim R-→+∞ e N (b, R) R 2 . (3.2.26)
Proof. We have from (3.2.7) and (3.2.17) that, for any b ∈ (0, 1) ,

e D (b, R) -CR b 1 2 ≤ e N (b, R) ≤ e D (b, R) .
Having in mind (3.2.10), we divide all sides of this inequality by R 2 and then take the limit as

R -→ +∞ . That gives us f (b) = lim R-→∞ e N (b, R) R 2 .
Proposition 3.2.4 (Fournais). There exists a positive constant C, such that if (3.2.16) is satisfied, then

e D (b, R) R 2 ≤ f (b) + C √ b R , (3.2.27) e D (b, R) R 2 ≥ f (b) . (3.2.28)
Proof. We have already seen that

f (b) = lim R→+∞ e D (b, R) R 2 . (3.2.29) Let us first prove (3.2.28). Let n ∈ N * and R > 0. Let u ∈ H 1 0 (Q R ) be a minimizer of F +1 b,Q R (i.e. e D (b, R) = F +1 b,Q R (u)). We extend u to a function u ∈ H 1 0 (Q nR ) by 'magnetic periodicity' as follows u(x 1 + R, x 2 ) = e iR x 2 2 u(x 1 , x 2 ) , u(x 1 , x 2 + R) = e -iR x 1 2 u(x 1 , x 2 ) . Let J n = {j ∈ Z, 1 ≤ j ≤ n 2 }. Notice that, the square Q nR is formed exactly of n 2 squares (Q R (x j 0 )) j∈J n .
We define in each Q R (x j 0 ) the following function

u j = u | Q R (x j 0 )
.

Observe that u j is a minimizer of

F +1 b,Q R (x j 0 ) in H 1 0 (Q R (x j 0 )
) and if we extend u j by 0 outside of

3.2 A reference problem 59 Q R (x j 0 )
, keeping the same notation u j for this extension, we have, u = i∈J n u j . Using magnetic translation invariance, it is easy to check that

F +1 b,Q nR ( u) = j∈J n F +1 b,Q R (u j ) = n 2 e D (b, R) .
Consequently, we get e D (b, nR) ≤ n 2 e D (b, R) .

We now divide both sides of this inequality by n 2 R 2 then we take the limit as n -→ ∞. Having in mind (3.2.10), this gives (3.2.28).

We prove (3.2.27). If n ∈ N * and j = (j 1 , j 2 ) ∈ Z 2 , we denote by

K j = I j 1 × I j 2 ,
where ∀m ∈ Z,

I m = 2m + 1 -n 2 - 1 2 , 2m + 1 -n 2 + 1 2 .
For all R > 0, we set Q R,j = {Rx : x ∈ K j } .

Let J n = {j = (j 1 , j 2 ) ∈ Z 2 : 0 ≤ j 1 , j 2 ≤ n -1} and Q nR = -nR 2 , nR 2 × -nR 2 , nR 2 . Then the family (Q R,j ) is a covering of Q nR , formed exactly of n 2 squares. Let u = u nR ∈ H 1 0 (Q nR ) be a minimizer of of F +1 b,Q nR i.e. F +1 b,Q nR (u) = e D (b, nR).
We have the obvious decomposition,

Q nR |u(x)| 4 dx = i∈J n Q R,j |u(x)| 4 dx . (3.2.30) Let χ = χ R,b 1 2 (x -x j 0 ), where χ R,b 1 2 is the cut-off function introduced in (3.2.19). The function u satisfies -b(∇ -iA 0 ) 2 u = (1 -|u| 2 )u in Q nR . It results from an integration by parts that e D (b, nR) = F +1 b,Q nR (u) = - 1 2 Q nR (|u(x)| 4 -1) dx . (3.2.31)
We may write,

Q R,j |(∇ -iA 0 )χu| 2 dx = (∇ -iA 0 )χu, (∇ -iA 0 )χu = ∇χ u, ∇χ u + χ (∇ -iA 0 )u, χ (∇ -iA 0 )u + 2 ∇χ u, χ (∇ -iA 0 )u = ∇χ u, ∇χ u + (∇ -iA 0 )(χ 2 u), (∇ -iA 0 )u .
Energy and vorticity of the Ginzburg-Landau model with variable magnetic field An integration by parts gives us

Q R,j |(∇ -iA 0 )χu| 2 dx = Q R,j |∇χ| 2 |u| 2 dx -χ 2 u, (∇ -iA 0 ) 2 u . (3.2.32)
Using (3.2.32), we may express the energy F +1 b,Q R,j (χu) as follows :

F +1 b,Q R,j (χu) = Q R,j b|(∇ -iA 0 )χu| 2 -|χu| 2 dx + 1 2 Q R,j |χu| 4 + 1 dx = -χ 2 u, (b(∇ -iA 0 ) 2 + 1)u + b Q R,j |∇χ| 2 |u| 2 dx + 1 2 Q R,j (χ 4 |u| 4 + 1)dx .
Using the equation (b(∇ -iA 0 ) 2 + 1)u = |u| 2 u and the inequality χ 4 ≤ χ 2 , we get

F +1 b,Q R,j (χu) ≤ b Q R,j |∇χ| 2 |u| 2 dx - 1 2 Q R,j χ 2 |u| 4 -1 dx ≤ b Q R,j |∇χ| 2 |u| 2 dx + 1 2 Q R,j 1 -χ 2 dx - 1 2 Q R,j |u| 4 -1 dx ≤ - 1 2 Q R,j |u| 4 -1 dx + Cb 1 2 R .
Since each χu has support in a square of side length R, we get

F +1 b,Q R,j (χu) ≥ e D (b, R) . (3.2.33)
We sum over the n 2 squares (Q R,j ) j∈J n (that cover Q nR ), and get

n 2 e D (b, R) ≤ - 1 2 Q nR (|u| 4 -1) dx + Cb 1 2 Rn 2 .
Using (3.2.31), we obtain

n 2 e D (b, R) ≤ e D (b, nR) + Cn 2 Rb 1 2 .
Dividing by n 2 R 2 , we obtain

e D (b, R) R 2 ≤ e D (b, nR) (nR) 2 + CR -1 b 1 2 .
We take the limit n → +∞ and get (3.2.27).

Upper bound of the energy

The aim of this section is to give an upper bound on the ground state energy E g (κ, H) introduced in (3.1.6).

In the sequel, for some choice of ρ ∈ (0, 1) to be determined later (see (3.3.11)), we consider triples ( , x 0 , x 0 ) such that Q (x 0 ) ⊂ {|B 0 | > ρ} ∩ Ω and x 0 ∈ Q (x 0 ). In this situation, we say that this triple is ρ-admissible, that the pair ( , x 0 ) is ρ-admissible and the corresponding square

Q (x 0 ) is a ρ-admissiblle. Let R = κH|B 0 ( x 0 )| (3.3.1)
We introduce the function :

w ,x 0 , x 0 (x) =    e iκHϕ x 0 , x 0 u R R (x -x 0 ) if x ∈ Q (x 0 ) ⊂ {B 0 > ρ} ∩ Ω e iκHϕ x 0 , x 0 u R R (x -x 0 ) if x ∈ Q (x 0 ) ⊂ {B 0 < -ρ} ∩ Ω , (3.3.2)
where u R ∈ H 1 0 (Ω) is a minimizer of the functional in (3.2.1) and ϕ x 0 , x 0 is the function introduced in [5, Lemma A.3] that satisfies

|F(x) -σ |B 0 ( x 0 )|A 0 (x -x 0 ) -∇ϕ x 0 , x 0 (x)| ≤ C 2 , (x ∈ Q (x 0 )) , (3.3.3)
where

B 0 ( x 0 ) = curl F( x 0 ), A 0 is the magnetic potential introduced in (3.2.2) and σ is the sign of B 0 (x) in Q (x 0 ). Proposition 3.3.1. Under Assumption (3.1.
2), there exist positive constants C and κ 0 such that if κ ≥ κ 0 , ∈ (0, 1), δ ∈ (0, 1), ρ > 0, 2 κHρ > 1, and ( , x 0 , x 0 ) is a ρ-admissible triple, then,

1 |Q (x 0 )| E 0 (w ,x 0 , x 0 , F, Q (x 0 )) ≤ (1 + δ)κ 2 f H κ |B 0 ( x 0 )| + C 1 H + δ -1 4 κH κH . (3.3.4) Proof. Let b = H κ |B 0 ( x 0 )| . (3.3.5)
We estimate E 0 (w ,x 0 , x 0 , F, Q (x 0 )) from above. We write for any δ ∈ (0, 1)

E 0 (w ,x 0 , x 0 , F, Q (x 0 )) = Q (x 0 ) |(∇ -iκHF)w ,x 0 , x 0 | 2 + κ 2 2 (1 -|w ,x 0 , x 0 | 2 ) 2 dx ≤ (1 + δ) Q (x 0 ) |(∇ -iκH(σ |B 0 ( x 0 )|A 0 (x -x 0 ) + ∇ϕ x 0 , x 0 (x)))w ,x 0 , x 0 | 2 + κ 2 2 (1 -|w ,x 0 , x 0 | 2 ) 2 dx + C(κH) 2 δ -1 Q (x 0 ) F -(σ |B 0 ( x 0 )|A 0 (x -x 0 ) + ∇ϕ x 0 , x 0 (x))w ,x 0 , x 0 2 dx ≤ (1 + δ)E 0 w ,x 0 , x 0 , σ |B 0 ( x 0 )|A 0 (x -x 0 ) + ∇ϕ x 0 , x 0 (x), Q (x 0 ) + Cδ -1 6 (κH) 2 . (3.3.6)
Using (3.2.9), the definition of w ,x 0 , x 0 and the change of variable y = R (x -x 0 ), we obtain

E 0 w ,x 0 , x 0 , σ |B 0 ( x 0 )|A 0 (x -x 0 ) + ∇ϕ x 0 , x 0 (x), Q (x 0 ) = Q R R ∇ y -i R A 0 (y) u R (y) 2 + κ 2 2 1 -|u R (y)| 2 2 2 R 2 dy = 1 b F +1 b,Q R (u R ) . (3.3.7) Since u R ∈ H 1 0 (Q R ) is a minimizer of F +1 b,Q R , then F +1 b,Q R (u R ) = e D (b, R) . (3.3.8) Proposition 3.2.4 tells us that e D (b, R) R 2 ≤ f (b) + C √ b R for all b ∈]0,
E 0 (w ,x 0 , x 0 , σ |B 0 ( x 0 )|A 0 (x -x 0 ) + ∇ϕ x 0 , x 0 (x), Q (x 0 )) ≤ R 2 f (b) b + C R √ b , (3.3.9)
with b defined in (3.3.5). We get by collecting the estimates in (3.3.6)-(3.3.7) that,

E 0 (w ,x 0 , x 0 , F, Q (x 0 )) ≤ (1 + δ)R 2 f (b) b + C 1 R √ b + C 2 δ -1 6 (κH) 2 . ( 3
1 |Q (x 0 )| E 0 (w ,x 0 , x 0 , F, Q (x 0 )) ≤ (1 + δ)κ 2 f H κ |B 0 ( x 0 )| + C κ + δ -1 4 (κH) 2 ,
which finishes the proof of Proposition 3.3.1.

Remark 3.3.2. We select , δ and ρ as follow :

= (κH) -1 4 , ρ = (κH) -1 3 . ( 3
E g (κ, H) ≤ κ 2 Ω f H κ |B 0 (x)| dx + o κH ln κ H , as κ -→ +∞ . (3.3.13)
Proof. Let ∈ (0, 1), δ and ρ be the parameters depending on κ and chosen as in Remark 3.3.2.

As we did in the previous paper [5, Proposition 5.1], we consider the lattice Γ := Z × Z and write, for γ, γ ∈ Γ , Q γ, = Q (γ) and w ,x 0 , x 0 = w ,γ, γ .

For any γ ∈ Γ such that Q γ, is ρ-admissible square, let

B γ, = inf x∈Q γ, |B 0 (x)| (3.3.14)
and

I ,ρ = γ; Q γ, ⊂ Ω ∩ {|B 0 | > ρ} , N = card I ,ρ . (3.3.15)
Then as κ → +∞, we have :

N = |Ω| -2 + O( -1 ) + O(ρ -2 ). (3.3.16)
Note that (3.3.16) results from both conditions in Assumption 3.1.2. For all x ∈ Ω, we define,

s(x) = γ∈J ,ρ w ,γ, γ (x) , (3.3.17)
where w ,γ, γ has been extended by 0 outside of Q γ, . Remember the functional E κ,H in (3.1.1).

We compute the energy of the test configuration (s, F). Since curl F = B, we get,

E κ,H (s, F, Ω) = γ∈J ,ρ E 0 (w ,γ, γ , F, Q γ, ) . (3.3.18)
Recall that for any γ ∈ Q γ, , B 0 ( γ) satisfies (3.3.3). Then, we select γ ∈ Q γ, such that

|B 0 ( γ)| = B γ, .
Using Proposition 3.3.1 and noticing that |Q γ, | = 2 , we get for any δ ∈ (0, 1) We have, for any ρ 0 > 0

γ∈J ,ρ E 0 (w ,γ, γ , F, Q γ, ) ≤ κ 2 (1 + δ) γ∈J ,ρ f H κ B γ, 2 + r(κ, H, ) , (3.3
κ 2 Ω f H κ |B 0 (x)| dx ≥ κ 2 Ω∩{|B 0 |>ρ 0 } f H κ |B 0 (x)| dx ≥ κ 2 f H κ ρ 0 |Ω ∩ {|B 0 | > ρ 0 }| .
In view of (3.2.11), for all positive constant C there exists ρ 0 > 0 such that if H ≤ Cκ and ρ 0 C 1 is sufficiently small for some C 1 > 0, then as κ -→ +∞

κ 2 Ω f H κ |B 0 (x)| dx ≥ C 2 κHρ 0 2 ln κ Hρ 0 (1 + o(1)) ,
where C 2 is a positive constant.

In particular, when (3.1.7) is satisfied, we see that,

r(κ, H, ) κ 2 Ω f H κ |B 0 (x)| dx . (3.3.22)

A priori estimates of minimizers

The aim of this section is to give a priori estimates on the solutions of the Ginzburg-Landau equations (3.1.5). These estimates play an essential role in controlling the error resulting from various approximations. The starting point is the following L ∞ -bound resulting from the maximum principle.

If (ψ, A) ∈ H 1 (Ω; C) × H 1 div (R 2
) is a solution of (3.1.5), then

ψ L ∞ (Ω) ≤ 1 . (3.4.1)
Next we prove an estimate on the induced magnetic potential.

Proposition 3.4.1. Suppose that the magnetic field H is a function of κ and satisfies (3.1.7).

Let α ∈ (0, 1). There exist positive constants κ 0 and C such that, if κ ≥ κ 0 and (ψ, A) is a

A -F H 2 (Ω) ≤ C H Ω f H κ |B 0 (x)| dx 1 2 , A -F C 0,α (Ω) ≤ C H Ω f H κ |B 0 (x)| dx 1 2
.

Here F is the magnetic potential introduced in (3.1.3).

Proof. The estimate in C 0,α -norm is a consequence of the continuous Sobolev embedding of

H 2 (Ω) in C 0,α (Ω). It is easy to show that curl(A -F) L 2 (Ω) ≤ 1 κH E g (κ, H) 1 2 , (3.4.2) and (∇ -iκHA)ψ L 2 (Ω) ≤ E g (κ, H) 1 2 . (3.4.3)
Notice that under Assumption (3.1.7), it follows from Theorem 3.3.3 and Remark 3.

3.4 that curl(A -F) L 2 (Ω) ≤ 1 H Ω f H κ |B 0 (x)| dx 1 2 , (3.4.4)
(∇ -iκHA)ψ L 2 (Ω) ≤ κ Ω f H κ |B 0 (x)| dx 1 2
.

(3.4.5)

Let a = A -F. We will prove that

a H 2 (Ω) ≤ C H Ω f H κ |B 0 (x)| dx 1 2
.

Since diva = 0 and a • ν = 0 on ∂Ω, we get by regularity of the curl-div system see [14,

Appendix A.5] a H 2 (Ω) ≤ C curl a H 1 (Ω) . (3.4.6)
The second equation in (3.1.5) reads as follows :

-∇ ⊥ curl a = 1 κH Im(ψ(∇ -iκHA)ψ) .
The estimates in (3.4.1) and the bound in (3.4.6), give us

a H 2 (Ω) ≤ C curl a L 2 (Ω) + 1 κH (∇ -iκHA)ψ L 2 (Ω) .
Inserting the estimates in (3.4.4) and (3.4.5) into this upper bound finishes the proof of the proposition.

Energy and vorticity of the Ginzburg-Landau model with variable magnetic field Construction of a gauge transformation :

Let φ x 0 (x) = (A(x 0 ) -F(x 0 )) • x,
where F is the magnetic potential introduced in (3.1.3) and ( , x 0 ) a ρ-admissible pair. Choosing α ∈ (0, 1) and using the estimate of A -F C 0,α (Ω) given in Proposition 3.4.1, we get for all x ∈ Q (x 0 ),

|A(x) -∇φ x 0 -F(x)| = |(A -F)(x) -(A -F)(x 0 )| ≤ A -F C 0,α (Ω) |x -x 0 | α ≤ C α λ α , (3.5.1)
where

λ = 1 H Ω f H κ |B 0 (x)| dx 1 2 . ( 3.5.2)
Using (3.2.11), it is clear that, under condition (3.1.7)

λ 2 = O 1 κH ln κ H , (3.5.3)
as κ -→ +∞.

Proposition 3.5.1. For all α ∈ (0, 1), there exist positive constants C and κ 0 such that if

κ ≥ κ 0 , ∈ (0, 1 2 ), δ ∈ (0, 1), ρ > 0, 2 κHρ > 1, (ψ, A) ∈ H 1 (Ω; C) × H 1 div (Ω)
is a minimizer of (3.1.1), and ( , x 0 , x 0 ) a ρ-admissible triple, then,

1 |Q (x 0 )| E 0 (ψ, A; Q (x 0 )) ≥ (1 -δ)κ 2 f H κ |B 0 ( x 0 )| -C κ + δ -1 (κH) 2 4 + δ -1 κ 2 H 2 λ 2 2α . Proof. Let x 0 ∈ Q (x 0 ). Recall the function ϕ x 0 , x 0 satisfiying (3.3.3). For all x ∈ Q (x 0 ), let u(x) = e -iκHϕ ψ(x) , (3.5.4)
where ϕ = ϕ x 0 , x 0 + φ x 0 and φ x 0 is introduced in (3.5.1).

Estimate of E 0 in Q (x 0 ) : As we did in [5, Lemma 4.1], we have, for any δ ∈ (0, 1) and α ∈ (0, 1)

E 0 (ψ, A; Q (x 0 )) ≥ (1 -δ)E 0 (u, σ |B 0 ( x 0 )|A 0 (x -x 0 ); Q (x 0 )) -Cδ -1 (κH) 2 4 + λ 2 2α Q (x 0 )
Q R v ,x 0 , x 0 (x) =    u R x + x 0 if x ∈ Q R ⊂ {{B 0 > ρ} ∩ Ω} u R x + x 0 if x ∈ Q R ⊂ {{B 0 < -ρ} ∩ Ω} .
(3.5.7)

Using (3.2.9), and the change of variable y = R (x -x 0 ), we get

E 0 (u, σ |B 0 ( x 0 )|A 0 (x -x 0 ); Q (x 0 )) = 1 b F +1 b,Q R (v ,x 0 , x 0 ) . (3.5.8)
Here

F +1 b,Q R is introduced in (3.2.1). Since v ,x 0 , x 0 ∈ H 1 (Q R ), we have F +1 b,Q R (v ,x 0 , x 0 ) ≥ e N (b, R) .
(3.5.9)

By collecting (3.2.17)-(3.2.28) and the lower bound in (3.5.9), we get,

F +1 b,Q R (v ,x 0 , x 0 ) ≥ R 2 f (b) -CRb 1 2 . (3.5.10)
As a consequence, (3.5.8) gives us

E 0 (u, σ |B 0 ( x 0 )|A 0 (x -x 0 ); Q (x 0 )) ≥ R 2 f (b) b -C R √ b . ( 3
E 0 (ψ, A; Q (x 0 )) ≥ (1 -δ)R 2 f (b) b -C R √ b -C δ -1 (κH) 2 4 + λ 2 2α 2 .
Having in mind (3.5.6), we get for any α ∈ (0, 1)

E 0 (ψ, A; Q (x 0 )) ≥ (1 -δ)κ 2 f H κ |B 0 ( x 0 )| -C κ + δ -1 (κH) 2 4 + δ -1 κ 2 H 2 λ 2 2α 2 .
(3.5.12) This finishes the proof of Proposition 3.5.1.

68

Energy and vorticity of the Ginzburg-Landau model with variable magnetic field 1 .

δ -1 (κH) 2 4 κH ln κ H = 1 ln κ H 1-α 4 1 , δ -1 κH ln C 0 κ H 2α κH ln κ H = ln C 0 κ H ln κ H 1-α 4 (κH) α 2 1 , (κH)
Remark 3.5.3. As a byproduct of the proof, we get also a useful estimate. Using the bound |ψ| ≤ 1, it results from (3.5.5) :

(

1 -δ) |Q (x 0 )| E 0 (ψ, σ |B 0 ( x 0 )|A 0 (x -x 0 ) + ∇ϕ, Q (x 0 )) ≤ 1 |Q (x 0 )| E 0 (ψ, A, Q (x 0 )) + Cδ -1 (κH) 2 ( 4 + λ 2 2α
) .

E 0 (ψ, σ |B 0 ( x 0 )|A 0 (x -x 0 ) + ∇ϕ, Q (x 0 )) ≤ E 0 (ψ, A, Q (x 0 )) + 2 κH ln κ H r(κ) , (3.5
E(ψ, A; D) ≥ κ 2 D f H κ |B 0 (x)| dx + o κH ln κ H , as κ -→ +∞ .
Proof. The proof is similar to the one of Theorem 3. To estimate E 0 (ψ, A; D ,ρ ), we notice that,

3.3. Let D ,ρ = int ∪ γ∈I ,ρ Q γ, ( 3
E 0 (ψ, A; D ,ρ ) = γ∈I ,ρ E 0 (ψ, A; Q γ, ) .
Recall that for any γ ∈ Q γ, we have B 0 ( γ) satisfies (3.3.3). Then, we select γ such that

|B 0 ( γ)| = B γ, .
Using (3.5.12), similarly as we did in the upper bound we recognize the upper Riemann sum, and get

E 0 (ψ, A; D ,ρ ) ≥ κ 2 (1 -δ) D ,ρ f H κ |B 0 (x)| dx -C κ + δ -1 (κH) 2 4 + δ -1 κ 2 H 2 λ 2 2α .
(3.5.19) Notice that using the regularity of ∂D and (3.1.2), there exists C > 0 such that ∀ ∈ (0, 1), ∀ρ ∈ (0, 1), |D \ D ,ρ | ≤ C( + ρ) .

(3.5.20)

We get by using property (3) of f in Theorem 3.2.1, Assumption (3.1.7) and for some choice of ρ to be determined later

D ,ρ f H κ |B 0 (x)| dx ≥ D f H κ |B 0 (x)| dx - D\D ,ρ f H κ |B 0 (x)| dx ≥ D f H κ |B 0 (x)| dx -C H κ |D \ D ,ρ | . (3.5.21)
This implies that

E g (κ, H) ≥ κ 2 (1 -δ) D f H κ |B 0 (x)| dx -r (κ, H, ) , (3.5.22)
where r (κ, H,

) = O κH + κHρ + κ + δ -1 (κH) 2 4 + δ -1 κ 2 H 2 λ 2 2α .
Having in mind (3.5.3), then, the remainder term becomes

r (κ, H, ) = O κH + κHρ + κ + δ -1 (κH) 2 4 + δ -1 κH ln C 0 κ H 2α .
The choice of the parameters δ in (3.5.13) and ρ, in (3.3.11) implies all error terms to be of lower order compared with κH ln κ H . This finishes the proof of Theorem 3.5.4.

Remark 3.5.

E(ψ, A; Ω) = E 0 (ψ, A; Ω) + (κH) 2 Ω | curl A -B 0 | 2 dx .
Using the estimate of E(ψ, A; Ω) in Theorem 3.1.1, we get, as κ -→ +∞

E 0 (ψ, A; Ω) + (κH) 2 Ω | curl A -B 0 | 2 dx ≤ κ 2 Ω f H κ |B 0 (x)| dx + o κH ln κ H . (3
E 0 (ψ, A; Ω) ≥ κ 2 Ω f H κ |B 0 (x)| dx + o κH ln κ H .
Therefore, (3.5.25) becomes

κ 2 Ω f H κ |B 0 (x)| dx + o κH ln κ H + (κH) 2 Ω | curl A -B 0 | 2 dx ≤ κ 2 Ω f H κ |B 0 (x)| dx + o κH ln κ H . (3.5.26)
By simplification, we get (3.1.15).

Proof of Theorem 3.1.5 : upper bound

One aim of this section is to derive a sharp estimate of E 0 (ψ, A; R .

Q (x 0 )), when (ψ, A) ∈ H 1 (Ω; C) × H 1 div (Ω)
Proposition 3.6.1. For α ∈ (0, 1), there exist positive constants C and

κ 0 such that if κ ≥ κ 0 , ∈ (0, 1 2 ), δ ∈ (0, 1), ρ > 0, 2 κHρ ≥ 1, (ψ, A) ∈ H 1 (Ω; C) × H 1 div (Ω)
is a minimizer of (3.1.1), and ( , x 0 , x 0 ) a ρ-admissible triple, then, For any α ∈ (0, 1), there exist positive constants C α and κ 0 such that if κ ≥ κ 0 , H(κ) satisfies (3.1.7), is chosen as in (3.3.11), δ as in (3.5.13), 2 κHρ ≥ 1, (ψ, A) is a minimizer of (3.1.1), and ( , x 0 , x 0 ) a ρ-admissible triple, then

1 |Q (x 0 )| E 0 (ψ, A; Q (x 0 )) ≤ (1 + δ)κ 2 f H κ |B 0 ( x 0 )| + C κ + δ -1 4 κ 2 H 2 + δ -1 κ 2 H 2 λ 2 2α , (3.6
1 |Q (x 0 )| E 0 (ψ, σ |B 0 ( x 0 )|A 0 (x -x 0 ) + ∇(ϕ x 0 , x 0 + φ x 0 ), Q (x 0 )) -κ 2 f H κ |B 0 ( x 0 )| ≤ C α κH ln κ H α 4 , (3.6.2)
where A 0 is the magnetic potential introduced in (3.2.2), σ denotes the sign of B 0 , φ x 0 is defined in (3.5.1) and ϕ x 0 , x 0 is the function satisfying (3.3.3).

Proof.

Lower bound : We refer to (3.5.11) and (3.5.6). We obtain

1 |Q (x 0 )| E 0 (ψ, σ |B 0 ( x 0 )|A 0 (x -x 0 ) + ∇(ϕ x 0 , x 0 + φ x 0 ), Q (x 0 )) ≥ κ 2 f H|B 0 ( x 0 )| κ -C κ , (3.6.
3) where C is a positive constant. If we select as in (3.3.11), we get

1 |Q (x 0 )| E 0 (ψ, σ |B 0 ( x 0 )|A 0 (x -x 0 ) + ∇(ϕ x 0 , x 0 + φ x 0 ), Q (x 0 )) ≥ κ 2 f H|B 0 ( x 0 )| κ -C (κ 5 H) 1 4 .
(3.6.4) Assumption (3.1.7) permits to verify that the remainder (κ 5 H)

1 4 = O(κH(ln κ H ) α 4
). Upper bound : Collecting (3.5.14) and (3.6.1), we get for any α ∈ (0, 1), the existence of

C > 0 such that 1 |Q (x 0 )| E 0 (ψ, σ |B 0 ( x 0 )|A 0 (x -x 0 ) + ∇(ϕ x 0 , x 0 + φ x 0 ), Q (x 0 )) ≤ κ 2 f H κ |B 0 ( x 0 )| + C κ + δ -1 4 κ 2 H 2 + δ -1 κ 2 H 2 λ 2 2α , (3.6.5)
where λ is introduced in (3.5.2). Using (3.5.3) and selecting as in (3.3.11) and δ as in (3.5.13), we get the existence of a constant

C α such that 1 |Q (x 0 )| E 0 (ψ, σ |B 0 ( x 0 )|A 0 (x -x 0 ) + ∇(ϕ x 0 , x 0 + φ x 0 ), Q (x 0 )) ≤ κ 2 f H κ |B 0 ( x 0 )| + C α κH ln κ H α 4 . (3.6.6)
This achieves the proof of the lemma.

The next lemma will be useful in the proof of Theorem 3.1.5.

Lemma 3.6.4. For any C 1 > 0, there exist positive constants C and κ 0 such that if ∈ (0, 1),

κ 0 ≤ κ and (ψ, A) ∈ H 1 (Ω; C) × H 1 div (Ω) is a solution of (3.1.5), then V (Γ,C 1 ) |(∇ -iκHA)ψ| 2 dx ≤ Cκ 2 1 + 1 κ 3 2 , (3.6.7) where V (Γ, C 1 ) = x ∈ Ω : dist(x, Γ) ≤ C 1 and d(x, ∂Ω) ≥ C 1 .
Proof. Using (3.4.5) and the fact that the range of f is the interval [0, 1/2] , we get

(∇ -iκHA)ψ L 2 (Ω) ≤ Cκ . (3.6.8)
Hence the improvment given by the lemma is when 1 C κ -2 ≤ ≤ 0 . Let C 2 > C 1 and for small enough we define the following sets D 1 = V (Γ, C 1 ) and

D 2 = V (Γ, C 2 ) . We can construct a cut-off function χ ∈ C ∞ c (Ω) such that 0 ≤ χ ≤ 1 in R 2 , suppχ ⊂ D 2 ⊂⊂ Ω , χ = 1 in D 1 and |∇χ | ≤ C in R 2 , (3.6.9)
where C is a positive constant independent of . The minimizer ψ satisfies

κ 2 ψ(1 -|ψ| 2 ) = -(∇ -iκHA) 2 ψ in Ω .
(3.6.10)

We multiply the above equation by χ ψ, it results from an integration by parts that

κ 2 D 2 χ (1 -|ψ| 2 )|ψ| 2 dx = D 2 (∇ -iκHA)ψ χ (∇ -iκHA)ψ + ψ∇χ dx = D 2 χ |(∇ -iκHA)ψ| 2 dx + D 2
∇χ ψ (∇ -iκHA)ψ dx .

(3.6.11)

Using Hölder inequality, we have

D 2 ∇χ ψ (∇ -iκHA)ψ dx ≤ (∇ -iκHA)ψ L 2 (Ω,C 2 ) |∇χ | ψ L 2 (D 2 ) .
(3.6.12)

Notice that |D 2 | ≤ C . Using (3.6.9) and the bound ψ ∞ ≤ 1, we obtain

|∇χ | ψ L 2 (D 2 ) ≤ C -1 2 .
(3.6.13) Putting (3.6.8) and (3.6.13) into (3.6.12), we get

D 2 ∇χ ψ(∇ -iκHA)ψ dx ≤ C κ -1 2 , ( 3
D 2 χ |(∇ -iκHA)ψ| 2 dx ≤ κ 2 D 2 χ (1 -|ψ| 2 )|ψ| 2 dx + C κ -1 2 .
(3.6.15)

The lemma easily follows from the control of the area of D 2 and from observing that χ = 1 on D 1 .

Remark 3.6.5. We get a similar estimate by replacing in the lemma Γ by the boundary ∂D of a regular open set D compactly contained in Ω.

End of the proof of Theorem 3.1.5. The proof of (3.1.18) is already obtained in Theorem 3.5.4. Hence it remains only to give the proof of (3.1.20). We keep the same notation as in (3.3.14), (3.3.15) and (3.5.16). If (ψ, A) is a minimizer of (3.1.1), we start with (3.1.17) and write,

E(ψ, A; D) = E 0 (ψ, A; D ,ρ ) + E 0 (ψ, A; D \ D ,ρ ) + (κH) 2 Ω | curl A -F | 2 dx . (3.6.16)
To estimate E 0 (ψ, A; D ,ρ ), we notice that,

E 0 (ψ, A; D ,ρ ) = γ∈I ,ρ E 0 (ψ, A; Q γ, ) .
Remark 3.6.2 tells us that the error terms in (3.6.1) are of order κH ln κ H . Therfore, using (3.6.1), we get

E 0 (ψ, A; D ,ρ ) ≤ κ 2 γ∈I ,ρ f H κ |B 0 ( γ)| 2 + o κH ln κ H , as κ -→ +∞ . We select γ ∈ Q (γ) such that |B 0 ( γ)| = B γ,
, where B γ, is defined in (3.3.14). By monotonicity of f , f is Riemann-integrable and its integral is larger than any lower Riemann sum. Thus

E 0 (ψ, A; D ,ρ ) ≤ κ 2 D ,ρ f H κ |B 0 (x)| dx + o κH ln κ H , as κ -→ +∞ . (3.6.17)
Moreover, recalling that f is a positive function and D ,ρ ⊂ D, (3.6.17) becomes

E 0 (ψ, A; D ,ρ ) ≤ κ 2 D f H κ |B 0 (x)| dx + o κH ln κ H , as κ -→ +∞ . (3.6.18)
For estimating E 0 (ψ, A; D \ D ,ρ ), we use Lemma 3.6.4, Remark 3.6.5 and we keep the same choice of and ρ as in (3.3.11), which implies ρ , we obtain that

D\D ,ρ |(∇ -iκHA)ψ| 2 dx ≤ C( κ 2 + κ -1 2 ) . (3.6.19)
Adding the second term in the energy leads to

E 0 (ψ, A; D \ D ,ρ ) ≤ C( κ 2 + κ -1 2 ) . (3.6.20)
The second term in the right hand side is controlled by the first one if

κ 3 2 1 .
This is effectively satisfied with our choice of and the condition on H(κ).

In order to obtain the term κ 2 in (3.6.20) comparatively small with κH ln κ H , we need a stronger condition than (3.1.7) on H(κ). In fact, we have

κ 2 κH ln κ H = κ 3 H 5 1

Vortices

This section is devoted to the proof of Theorem 3.1.6. We keep the choice of given in (3.3.11) :

= (κH) -1 4 ,
but we select ρ and α as follows :

ρ = ln κ H -1 2 , α = 1 2 . ( 3
E 0 (ψ, A, Q (x 0 )) = Q (x 0 ) |(∇ -iκHA)ψ| 2 + κ 2 2 (1 -|ψ| 2 ) 2 dx .

Division of the square Q (x 0 )

Let H = H(κ) be a function satisfying (3.1.7). For reasons that will become clear in Proposition 3.7.3, we need to divide

Q (x 0 ) into N = M 2 disjoint open squares (Q j δ(κ) ) j∈J such that Q (x 0 ) = ∪ j∈J Q j δ(κ) , with M = 2 7 8 (κH) 1 4 ln κ H -7 8 , (3.7.2)
where for t ∈ R, [t] denotes the integer part of t.

The side length of theses squares is consequently For all > 0 there exists κ 0 > 0 such that ∀κ ≥ κ 0 , H satisfying (3.1.7), ρ introduced in (3.7.1) and any ρ-admissible triple ( , x 0 , x 0 )

δ(κ) = M ∼ 2 -
|b(κ, H, x 0 )| + 1 R(κ, H, x 0 ) < .
In fact, we have as

κ -→ +∞ R(κ, H, x 0 ) ≥ 2 -7 8 ln κ H 7 8 ρ 1 2 ≥ 1 C ln κ H 5 8
1 .

Since B 0 ∈ C ∞ (Ω), we have also where β = |B 0 ( x 0 )|. We define the function : 8 , sup

0 < b(κ, H, x 0 ) ≤ H κ β 0 1 , (3.7
h(κ, H) = max    ln κ H - 3
β 0 ≥β≥(ln κ H ) -1 2 err( b(κ, H, β), R(κ, H, β))    , (3.7.8)
where err(b, R) is defined in Proposition 3.2.13. Notice that h satisfies h(κ, H) = o(1) , as κ -→ +∞ .

(3.7.9)

Next, we will use a method introduced by E. Sandier and S. Serfaty in [START_REF] Sandier | On the energy of type-II superconductors in the mixed phase[END_REF]. We distinguish in the family indexed by J two types of squares respectively called the 'nice squares' (Q j δ(κ) ) which are indexed in J n and the 'bad squares' (Q j δ(κ) ) indexed in J b . The set J n is the set of indices j ∈ J such that

E 0 (ψ, σ |B 0 ( x 0 )|A 0 (x -x 0 ) + ∇ϕ, Q j δ(κ) ) ≤ δ(κ) 2 κ 2 f H κ |B 0 ( x 0 )| (1 + h(κ, H) 1 2
). (3.7.10)

The set J b is the set of indices j ∈ J such that

E 0 (ψ, σ |B 0 ( x 0 )|A 0 (x -x 0 ) + ∇ϕ, Q j δ(κ) ) > δ(κ) 2 κ 2 f H κ |B 0 ( x 0 )| (1 + h(κ, H) 1 2
). (3.7.11)

Hence we have J = J n ∪ J b . We denote by N g the cardinal of J n and by N b the cardinal of J b .

Lemma 3.7.1. There exist positive constants C and κ 0 such that if κ ≥ κ 0 , then

N b ≤ C h(κ, H) 1 2 1 -h(κ, H) 1 2
N n , (3.7.12)

where h is introduced in (3.7.8).

Proof. Recall that A 0 is the magnetic potential introduced in (3.2.2), φ x 0 is defined in (3.5.1) and that, for x 0 ∈ Q (x 0 ), ϕ x 0 , x 0 is the function satisfying (3.3.3).

Having in mind the definition of b and R in (3.7.4) and their properties, and using (3.2.15), we

E 0 (ψ, σ |B 0 ( x 0 )|A 0 (x -x 0 ) + ∇ϕ, Q j δ(κ) ) ≥ e N (b, R) b ≥ R 2 b f (b) (1 -err(b, R)) , (3.7.13)
where ϕ = φ x 0 + ϕ x 0 , x 0 , e N is introduced in (3.2.3), b = b(κ, H, x 0 ) and R = R(κ, H, x 0 ). As a consequence of (3.7.8), (3.7.13) becomes

E 0 (ψ, σ |B 0 ( x 0 )|A 0 (x -x 0 ) + ∇ϕ, Q j δ(κ) ) ≥ κ 2 δ(κ) 2 f H κ |B 0 ( x 0 )| (1 -h(κ, H)) . (3.7.14) Notice that |Q (x 0 )| = j∈J Q j δ(κ) = (N n + N b ) δ(κ) 2 . (3.7.15)
Thus, we may write

j∈J E 0 ψ, σ |B 0 ( x 0 )|A 0 (x -x 0 ) + ∇ϕ, Q j δ(κ) = E 0 (ψ, σ |B 0 ( x 0 )|A 0 (x -x 0 ) + ∇ϕ, Q (x 0 )) .
(3.7.16) When (3.1.7) is satisfied, we have uniformly

2 κHρ = (κH) 1 2 ln κ H -1 2 1 , as κ -→ +∞ .
Hence the assumptions of Proposition 3.6.3 are satisfied. Putting (3.7.16) into (3.6.2), using (3.7.1) and (3.7.15), we get

j∈J E 0 ψ, σ |B 0 ( x 0 )|A 0 (x -x 0 ) + ∇ϕ, Q j δ(κ) -κ 2 δ(κ) 2 f H κ |B 0 ( x 0 )| ≤ C (N b + N n ) δ(κ) 2 κH ln κ H 1 8 . (3.7.17)
Using the monotonicity of f and remembering that ( , x 0 , x 0 ) is a ρ-admissible triple, we get,

f H κ |B 0 ( x 0 )| ≥ f H κ ρ . ( 3
E 0 ψ, σ |B 0 ( x 0 )|A 0 (x -x 0 ) + ∇ϕ, Q j δ(κ) -κ 2 δ(κ) 2 f H κ |B 0 ( x 0 )| ≤ C (N b + N n ) κ 2 δ(κ) 2 f H κ |B 0 ( x 0 )| h(κ,
N b κ 2 δ(κ) 2 f H κ |B 0 ( x 0 )| h(κ, H) 1 2 ≤ j∈J b E 0 ψ, σ |B 0 ( x 0 )|A 0 (x -x 0 ) + ∇ϕ, Q j δ(κ) -κ 2 δ(κ) 2 f H κ |B 0 ( x 0 )| ≤ j∈J E 0 ψ, σ |B 0 ( x 0 )|A 0 (x -x 0 ) + ∇ϕ, Q j δ(κ) -κ 2 δ(κ) 2 f H κ |B 0 ( x 0 )| + j∈J n κ 2 δ(κ) 2 f H κ |B 0 ( x 0 )| h(κ, H) ≤ C (N b + N n )κ 2 δ(κ) 2 f H κ |B 0 ( x 0 )| h(κ, H) + N n κ 2 δ(κ) 2 f H κ |B 0 ( x 0 )| h(κ, H).
We divide both sides by More precisely, we mean that N b = N n e(κ, H, , x 0 , x 0 ), with e(κ, H, , x 0 , x 0 ) is uniformly o(1) for any κ ≥ κ 0 , any ρ-admissible triple ( , x 0 , x 0 ), any H satisfying (3.1.7). Now we recall an important result of Sandier-Serfaty [START_REF] Sandier | On the energy of type-II superconductors in the mixed phase[END_REF]. Define the energy of (u,

κ 2 δ(κ) 2 f H κ |B 0 ( x 0 )| h(κ, H)
A) ∈ H 1 (D; C)× H 1 div (D; R) in a domain D ⊂ R 2 as follows J D (u, A) = D |(∇ -iA)u| 2 + κ 2 2 (1 -|u| 2 ) 2 + | curl A -h ex | 2 dx . (3.7.25)
The next proposition is essentially proved 3 in [46, Proposition 5.1].

Proposition 3.7.3. Let ĥ : (0, +∞) -→ (0, +∞) such that lim t-→+∞ ĥ(t) = 0, there exist two functions s 1 , s 2 : (0, +∞) -→ (0, +∞) satisfying , as κ -→ ∞ .

lim t-→+∞ s 1 (t) = 0 , lim t-→+∞ s 2 (t) = 0 . ( 3
(3.7.27)

If (u, A) ∈ C 1 (K; C) × C 1 (K; R 2 ) verifies J K (u, A) ≤ h ex γ(κ) 2 ln κ √ h ex (1 + ĥ(κ)) , (3.7.28)
then, there exist disjoint disks (D(a i , r i )) m i=1 such that :

1.

r i ≤ h -1 2 ex 2. |u| > 1 2 on ∪ i ∂D(a i , r i ) 3. If d i = deg u |u| , ∂D(a i , r i ) , then, as κ -→ +∞ 2π m i=1 d i ≥ h ex γ(κ) 2 (1 -s 1 (κ)) and 2π m i=1 |d i | ≤ h ex γ(κ) 2 (1 + s 2 (κ)) . (3.7.29)
We will present a proof of Proposition 3.7.3 in Appendix A.

The next lemma will give us that δ(κ), the side length of the square Q j δ(κ) , satisfies (3.7.27) and will be useful in Proposition 3.7.5. Lemma 3.7.4. Under the assumptions of the previous subsection we have

δ(κ) 2 = 1 ε 1 (κ, , x 0 , x 0 , H) 1 κH |B 0 ( x 0 )| ln κ H |B 0 ( x 0 )| , as κ -→ +∞ (3.7.30) δ(κ) 2 = ε 2 (κ, , x 0 , x 0 , H) 1 κH |B 0 ( x 0 )| ln κ H |B 0 ( x 0 )| 2 , as κ -→ +∞ , (3.7.31)
where ε 1 (κ, , x 0 , x 0 , H) and ε 2 (κ, , x 0 , x 0 , H) are uniformly o(1) as κ -→ +∞. 3 We replaced ε by 1 κ . We can indeed verify that only the upper bound of JK (u, A) is needed with no additional condition on f (ε) and that the o(1) are actually uniformity under uniform assumptions. Note also that we do not use in this proposition that (u, A) is a critical point of JΩ.

Proof.

Proof of (3.7.30). We know that for all ρ-admissible triple ( , x 0 , x 0 )

1 κH|B 0 ( x 0 )| ln κ H|B 0 ( x 0 )| ≤ 1 κHρ ln κ Hρ ≤ 3 2 1 κH ln κ H 3 2 , Also, we know that 1 κH ln κ H 3 2 2 -7 4 1 κH ln κ H 7 4 ∼ δ(κ) 2 .
Proof of (3.7.31). On the other hand, we have

1 κH|B 0 ( x 0 )| ln κ H|B 0 ( x 0 )| 2 ≥ 1 κHβ 0 ln κ Hβ 0 2 ≥ C 1 κH ln κ H 2 ,
where C is a positive constant and β 0 is introduced in (3.7.6).

It is clear that

1 κH ln κ H 2 2 -7 4 1 κH ln κ H 7 4 ∼ δ(κ) 2 .
We can prove the following result regarding the vortices of the minimizers in the 'nice squares'. We start with the admissible squares contained in Ω ∩ {B 0 > 0}. Proposition 3.7.5. Under Assumptions (3.1.2) and (3.1.7) there exists s 1 , s 2 : (0, +∞) -→ (0, +∞) two functions satisfying (3.7.26) and such that, for any ( , x 0 , x 0 ) such that Q j δ(κ) ⊂ Ω ∩ {B 0 > ρ} and x 0 ∈ Q j δ(κ) for which Q j δ(κ) is a nice square, and any minimizer (ψ, A) ∈ H 1 (Ω, C) × H 1 div (Ω, R 2 ) of (3.1.1), there exist disjoint disks (D(a i,j , r i,j ))

m j i=1 in Q j δ(κ) such that • m j i=1 r i,j ≤ (κHB 0 ( x 0 )) -1 2 • |ψ| > 1 2 on ∪ j ∂D(a i,j , r i,j ) • If d i,j
is the winding number of ψ |ψ| restricted to ∂D(a i,j , r i,j ), then Proof.

(3.7.32) 2π m j i=1 d i,j ≥ δ(κ) 2 κHB 0 ( x 0 )(1 -s 1 (κ)) , as κ -→ +∞ , ( 3
We will apply Proposition 3.7.3 with

K = Q j δ(κ) , γ(κ) = δ(κ), h ex = κHB 0 ( x 0 ), u = e -iκHϕ ψ and A(x) = κH B 0 ( x 0 ) A 0 (x -x 0 ) , (3.7
.35) where A 0 is the magnetic potential introduced in (3.2.2) and ϕ = φ x 0 + ϕ x 0 , x 0 , with φ x 0 defined in (3.5.1) and ϕ x 0 , x 0 in (3.3.3).

Vortices

81

Let us verify that the conditions of the proposition are satisfied for this choice. First, we start by proving (3.7.28). Since curl A 0 = 1, then, curl A = κH B 0 ( x 0 ) = h ex , and consequently

Q j δ(κ) | curl A -h ex | 2 dx = 0 . (3.7.36)
This implies that, for any j ∈ J n

J K (u, A) = Q j δ(κ) |(∇ -iκH(B 0 ( x 0 )A 0 (x -x 0 ) + ∇ϕ))ψ| 2 + κ 2 2 1 -|ψ| 2 2 dx = E 0 (ψ, B 0 ( x 0 )A 0 (x -x 0 ) + ∇ϕ, Q j δ(κ) ) . (3.7.37) Since Q j δ(κ) is a nice square, then, J K (u, A) ≤ δ(κ) 2 κ 2 f H κ B 0 ( x 0 ) (1 + h(κ, H) 1 2 ) , as κ -→ +∞ . (3.7.38)
As a consequence of (3.2.11), (3.7.38) becomes

J K (u, A) ≤ 1 2 δ(κ) 2 κHB 0 ( x 0 ) ln κ HB 0 ( x 0 ) (1 + ĥ(κ, H)) , as κ -→ +∞ , (3.7.39)
where ĥ(κ, H) = h(κ, H)

1 2 + ŝ H κ + ŝ H κ h(κ, H) 1 2
.

Notice that the function ĥ(κ, H) is uniformly o( .

Thanks to Lemma 3.7.4, we get that (3.7.27) is satisfied and in this way we achieve the proof of Proposition 3.7.5.

In light of Lemma 3.7.1, we deduce from Proposition 3.7.5 the distribution of vortices in a ρ-admissible square Q . Proposition 3.7.6. Suppose that Assumptions (3.1.2) and (3.1.7) are true. There exists two functions s 1 , s 2 : (0; +∞) -→ (0; +∞) satisfying (3.7.26) and the following is true. Let (ψ, A) ∈ H 1 (Ω, C) × H 1 div (Ω, R 2 ) be a minimizer of (3.1.1) and ( , x 0 ) such that Q (x 0 ) ⊂ Ω ∩ {B 0 > ρ}.

Energy and vorticity of the Ginzburg-Landau model with variable magnetic field

There exist a family of disjoint disks (indexed by

K = K ,x 0 ) (D( a k , r k )) k∈K in Q (x 0 ) such that • k∈K r k ≤ (κHB 0 ( x 0 )) -1 2 δ(κ) 2 (1 + o(1)) , as κ -→ +∞ (3.7.40) • |ψ| > 1 2 on ∪ k ∂D( a k , r k ) (3.7.41) • If d k is the winding number of ψ |ψ| restricted to ∂D( a k , r k ), then as κ -→ +∞ 2π k∈K d k ≥ 2 κH B 0 ( x 0 ) (1 -s 1 (κ)) and 2π k∈K | d k | ≤ 2 κH B 0 ( x 0 ) (1 + s 2 (κ)) . (3.7.42)
Here, the function o( 1) is bounded independently of the choice of x 0 and the minimizer (ψ, A).

Proof. Recall that Q (x 0 ) is decomposed into N n 'nice squares' (Q j δ(k) ) j∈J n and N b 'bad squares' (Q j δ(k) ) j∈J b .
In every nice square Q j δ(κ) , Proposition 3.7.5 tells us that there exist disjoint disks (D(a i,j , r i,j ))

m j i=1
such that (3.7.32), (3.7.33) and (3.7.34) hold. Let (D( a k , r k )) k∈K = (D(a i,j , r i,j )) i,j be the family of disjoint disks in

∪ j∈J n Q j δ(κ) . Clearly k∈K r k = j∈J n m j i=1 r i,j .
This implies that k∈K

r k ≤ (κHB 0 ( x 0 )) -1 2 N n .
Having in mind (3.7.15) and (3.7.24) , we have, as κ -→ +∞ ,

N n δ(κ) 2 2 -→ 1 . ( 3
d k = 2π j∈J n m j i=1 d i,j ≥ N n δ(κ) 2 κHB 0 ( x 0 ) (1 -s 1 (κ)) ≥ 2 κHB 0 ( x 0 ) (1 -s 1 (κ)) , (3.7.44)
and

2π k∈K | d k | = 2π j∈J n m j i=1 |d i,j | ≤ N n δ(κ) 2 κHB 0 ( x 0 ) (1 + s 2 (κ)) ≤ 2 κHB 0 ( x 0 ) (1 + s 2 (κ)) . (3.7.45)
This finishes the proof of Proposition 3.7.6.

Proof of Theorem

3.1.6 Let (ψ, A) ∈ H 1 (Ω, C) × H 1 div (Ω, R 2
) be a minimizer of (3.1.1) and Γ := Z × Z a lattice of R 2 . For all γ ∈ Γ , we consider the family of squares Q (γ) and γ ∈ Q (γ). Consider an open set S ⊂ Ω ∩ {B 0 > 0} such that the boundary of S is smooth. Let

J = {γ ; Q (γ) ⊂ S ∩ {B 0 > ρ} } , ( 3
such that Q (γ) ⊂ Ω ∩ {B 0 > ρ} and γ ∈ Q (γ) B 0 ( γ) 2 (1 -s 1 (κ)) ≤ 2π κH k∈K ,γ d k,γ ≤ B 0 ( γ) 2 (1 + s 2 (κ)) .

Introduction

We consider a bounded, open and simply connected set Ω ⊂ R 2 with smooth boundary. We suppose that Ω models an inhomogeneous superconducting sample submitted to an applied external magnetic field. The energy of the sample is given by the so called pinned Ginzburg-Landau functional,

E κ,H,a,B 0 (ψ, A) = Ω |(∇ -iκHA)ψ| 2 + κ 2 2 (a(x, κ) -|ψ| 2 ) 2 dx + κ 2 H 2 Ω | curl A -B 0 | 2 dx .
(4.1.1) Here κ and H are two positive parameters such that κ describes the properties of the material, and H measures the variation of the intensity of the applied magnetic field. The modulus |ψ| 2 of the wave function (order parameter) ψ ∈ H 1 (Ω; C) measures the density of the superconducting electron Cooper pairs. The magnetic potential A belongs to H 1 div (Ω) where

H 1 div (Ω) = {A = (A 1 , A 2 ) ∈ H 1 (Ω) 2 : div A = 0 in Ω , A • ν = 0 on ∂Ω } , (4.1.2)
with ν being the unit interior normal vector of ∂Ω.

The function κH curl A gives the induced magnetic field.

When ψ ≡ 0 and (ψ, A) is a minimizer or a critical point of the functional, we call this pair normal state. In our case it is easy to see normal minimizers (if any) are necessarily in the form (0, A) with A in H 1 div (Ω) such that curl A = B 0 . This solution is unique and denoted by F. A natural question will be to determine under which condition this normal solution is a minimizer.

The function B 0 ∈ C ∞ (Ω) is the intensity of the external magnetic field which is variable in our problem. Let Γ = {x ∈ Ω :

B 0 (x) = 0} . (4.1.3)
We assume that either Γ is empty or that B 0 satisfies :

|B 0 | + |∇B 0 | > 0 in Ω ∇B 0 × n = 0 on Γ ∩ ∂Ω . (4.1.4)
The assumption in (4.1.4) implies that for any open set ω relatively compact in Ω, Γ ∩ ω is either empty, or consists of a union of smooth curves.

The energy E κ,H,a,B 0 considered here is slightly different from the classical Ginzburg-Landau energy in the sense that there is a varying term denoted by a(x, κ) penalizing the variations of the order parameter ψ and called the pinning term. This term arises also naturally in the microscopic derivation of the Ginzburg-Landau theory from BCS theory (see [START_REF] Frank | Derivation of Ginzburg-Landau theory for a one-dimensional system with contact interaction[END_REF]) without any a priori assumption on the sign of a. Assumption 4.1.1. The function a(x, κ) is real, defined on Ω × [κ 0 , +∞), and satisfies for some κ 0 > 0 the following assumptions :

(A 1 ) ∀κ ≥ κ 0 , a(•, κ) ∈ C 1 (Ω) . (4.1.5) (A 2 ) sup x∈Ω, κ≥κ 0 |a(x, κ)| < +∞ . (4.1.6) (A 3 ) ∀κ ≥ κ 0 , sup x∈Ω |∇ x a(x, κ)| < +∞ . (4.1.7) (A 4 ) There exists a positive constant C 1 , such that, ∀κ ≥ κ 0 , L (∂{a(x, κ) > 0}) ≤ C 1 κ 1 2 , (4.1.8)
where L is the "length" of ∂{a(x, κ) > 0} in Ω in a sense that will be explained in (4.3.1). The assumption in (A 3 ) gives a uniform control for any κ of the oscillation of a(., κ) which will be made precise later by an assumption on L(κ). Notice that the normal state (0, F) is a critical point of the functional in (4.1.1). It is standard, starting from a minimizing sequence, to prove the existence of minimizers in H 1 (Ω; C)×H 1 div (Ω) of the functional E κ,H,a,B 0 . A minimizer (ψ, A) of (4.1.1) is a weak solution of the Ginzburg-Landau equations,

L(κ) = sup

             -(∇ -iκHA) 2 ψ = κ 2 (a(x, κ) -|ψ| 2 ) ψ in Ω (a) -∇ ⊥ curl(A -F) = 1 κH Im(ψ (∇ -iκHA)ψ) in Ω (b) ν • (∇ -iκHA)ψ = 0 on ∂Ω (c) curl A = curl F on ∂Ω (d) .
(4.1.12)

Here, curl

A = ∂ x 1 A 2 -∂ x 2 A 1 and ∇ ⊥ curl A = (∂ x 2 (curl A), -∂ x 1 (curl A)).
Let us introduce the magnetic Schrödinger operator in an open set Ω in R 2 :

P Ω A,V = -(∇ -iA) 2 + V (x) , (4.1.13)
where A ∈ H 1 div ( Ω) and V is a continuous function bounded from below. The form domain of P Ω A,V is

V( Ω) = {u ∈ L 2 ( Ω) , (∇ -iA)u ∈ L 2 ( Ω) , (V + C) 1 2 u ∈ L 2 ( Ω)} ,
and its operator domain is given by

D(P Ω A,V ) := {u ∈ V( Ω) , P Ω A,V u ∈ L 2 ( Ω), ν • (∇ -iA)u = 0 on ∂ Ω} .
Then, (4.1.12) a,c reads

P Ω A,V ψ = -κ 2 |ψ| 2 ψ , with A = κHA, ψ ∈ D(P Ω A,V ) and V = -κ 2 a .
There are many papers on the Ginzburg-Landau functional with a pinning term, most of them study the influence of the pinning term on the location of vortices, i.e. the zeros of the minimizing order parameter. For the functional without a magnetic field (i.e. B 0 = 0 in (4.1.1)), the influence of the pinning term is studied in [START_REF] Lassoued | Ginzburg-Landau type energy with discontinuous constraint[END_REF] and more recently in [START_REF] Michaël | The Ginzburg-Landau functional with a discontinuous and rapidly oscillating pinning term Part II : the non-zero degree case[END_REF] and the references therein. The pinning term (i.e. the function a) in [START_REF] Lassoued | Ginzburg-Landau type energy with discontinuous constraint[END_REF] is a step function independent of κ ; more complicated κ-dependent periodic step functions are considered in [START_REF] Michaël | The Ginzburg-Landau functional with a discontinuous and rapidly oscillating pinning term Part II : the non-zero degree case[END_REF]. The magnetic version of the functional in [START_REF] Lassoued | Ginzburg-Landau type energy with discontinuous constraint[END_REF] is studied in [START_REF] Kachmar | The ground state energy of the three-dimensional Ginzburg-Landau model in the mixed phase[END_REF][START_REF] Kachmar | Magnetic vortices for a Ginzburg-Landau type energy with a discontinuous constraint[END_REF].

In [START_REF] Aftalion | Pinning phenomena in the Ginzburg-Landau model of superconductivity[END_REF], Aftalion, Sandier and Serfaty considered a smooth and κ-dependent pinning term a satisfying :

(H 1 ) L(κ) κH.
90 Pinning with a variable magnetic field of the two dimensional Ginzburg-Landau model (H 2 ) There exist a continuous function a(x), a positive constant a 0 and, for all κ ≥ 0, there exist two functions σ(κ) = o ln ln β(x, κ) = 0 , a(x, κ) = a(x) + β(x, κ) , and 0 < a 0 ≤ a(x) ≤ 1 .

The study contains the case when a(x, κ) = a(x) (β = 0) but also cases with a κ-control of the x-oscillation of β(•, κ) which could increase with κ. In the scales of this paper, the results in [START_REF] Aftalion | Pinning phenomena in the Ginzburg-Landau model of superconductivity[END_REF] are valid when the parameter H is of order | ln κ| κ as κ -→ +∞. Extending the discussion, the functional in (4.1.1) is close to models of Bose-Einstein condensates (see e.g. [START_REF] Aftalion | Giant vortex and breakdown of strong pinning in a rotating Bose-Enstein condensate[END_REF][START_REF] Alama | Pinning effects and their breakdown for a Ginzburg-Landau model with normal inclusions[END_REF]).

In this paper, we will analyze how the pinning term appears in the asymptotics of the energy in the presence of a strong external variable magnetic field (see Theorem 4.1.2 below). Also, we discuss the influence of the pinning on the asymptotic expression of the third critical field H C 3 (see Theorems 4.1.6 and 4.1.7).

We focus on the regime of large values of κ, κ → +∞ and we study the ground state energy defined as follows,

E g (κ, H, a, B 0 ) = inf E κ,H,a,B 0 (ψ, A) : (ψ, A) ∈ H 1 (Ω; C) × H 1 div (Ω) . (4.1.14)
More precisely, we give an asymptotic estimate which is valid in the simultaneous limit κ -→ +∞ and H(κ) -→ +∞ with the constraint that H(κ) κ remains asymptotically of uniform size, that is satisfying

λ min ≤ H(κ) κ ≤ λ max (κ ≥ κ 0 ) , ( 4
L(κ) = O(κ 1 2 ) as κ → +∞ . (4.1.16)
The ground state energy in (4.1.14) satisfies

E g (κ, H, a, B 0 ) = κ 2 {a(x,κ)>0} a(x, κ) 2 f H κ |B 0 (x)| a(x, κ) dx + κ 2 2 {a(x,κ)≤0} a(x, κ) 2 dx + o κ 2 , as κ -→ +∞ . (4.1.17)
When Ω ∩ {a(x, κ) > 0} = ∅, we obtain directly from (4.1.14)

E κ,H,a,B 0 (ψ, A) ≥ κ 2 2 Ω a(x, κ) 2 dx = E κ,H,a,B 0 (0, F) .
Hence the minimizer of E κ,H,a,B 0 is the normal state. In physical terms, this case corresponds to the case when we are above the critical temperature. We will describe later cases when the remainder term in (4.1.17) is indeed small compared with the leading order term (see Section 4.6).

The assumptions in Theorem 4.1.2 contain the case when the function a is constant and equals 1, which was proved in [START_REF] Attar | The ground state energy of the two-dimensional Ginzburg-Landau functional with variable magnetic field[END_REF] under Assumption (4.1.15).

Along the proof of Theorem 4.1.2, we obtain an estimate of the 'magnetic energy' as follows :

Corollary 4.1.3. Under the assumptions of Theorem 4.1.2, we have

(κH) 2 Ω | curl A -B 0 | 2 dx = o(κ 2 ) , as κ -→ +∞ . (4.1.18)
If D is a domain in Ω, we introduce the local energy in D of (ψ, A) ∈ H 1 (Ω; C) × H 1 div (Ω) by :

E 0 (ψ, A; a, D) = D |(∇ -iκHA)ψ| 2 dx + κ 2 2 D (a(x, κ) -|ψ| 2 ) 2 dx . (4.1.19)
The next theorem gives an estimate of the local energy E 0 (ψ, A; a, D).

|ψ(x)| 4 dx = - D∩{a(x,κ)>0} a(x, κ) 2 2 f H κ |B 0 (x)| a(x, κ) -1 dx + o (1) , as κ -→ +∞ . (4.1.21)
Formula (4.1.21) indicates that ψ is asymptotically localized in the region where a > 0. When a(x, κ) = 1, Theorem 4.1.5 was proved in [START_REF] Attar | The ground state energy of the two-dimensional Ginzburg-Landau functional with variable magnetic field[END_REF].

The techniques that we are going to use here are inspired from those of [START_REF] Attar | The ground state energy of the two-dimensional Ginzburg-Landau functional with variable magnetic field[END_REF] and [START_REF] Attar | Energy and vorticity of the Ginzburg-Landau model with variable magnetic field[END_REF] (where the case a = 1 was treated). At a technical level, our proof is slightly different than the proofs in [START_REF] Attar | The ground state energy of the two-dimensional Ginzburg-Landau functional with variable magnetic field[END_REF][START_REF] Fournais | The ground state energy of the three dimensional Ginzburg-Landau functional Part I : Bulk regime[END_REF][START_REF] Dombrowski | Semiclassical analysis with vanishing magnetic fields[END_REF] since we do not use the uniform elliptic estimates. These important estimates are frequently used in the papers about the Ginzburg-Landau functional (see [START_REF] Fournais | Spectral Methods in Surface Superconducitivity[END_REF]) with a constant pinning term. They appeared first in [START_REF] Lu | Estimates of the upper critical field for the Ginzburg-Landau equations of superconductivity[END_REF] and were then extended to the full regime in [START_REF] Fournais | Optimal uniform elliptic estimates for the Ginzburg-Landau system[END_REF].

Compared with other papers studying the pinned functional, one novelty here is that the pinning term has no definite sign, another one being the consideration of a variable (and a potentially vanishing) applied magnetic field.

The rest of this paper is devoted to the study of third critical field, i.e. the field above which the normal state (0, F) is the only critical point of the functional in (4.1.1), in the case when the pining term a is independent of κ (i.e. a(x, κ) = a(x)). We define the set : Notice that the above set is bounded (see Theorem 4.8.5). We also introduce the two sets :

N cp (κ) = {H > 0 : E κ,
N (κ) = {H > 0 : E κ,H,a,B 0 has a non-normal minimizer} . (4.1.23) N loc (κ) = {H > 0 : µ 1 (κ, H) < 0} . (4.1.24)
Here, µ 1 (κ, H) is the ground state energy of the semi-bounded quadratic form

Q Ω κHF,-κ 2 a (φ) = Ω |(∇ -iκHF)φ| 2 -κ 2 a(x, κ)|φ| 2 dx , (4.1.25)
i.e.

µ 1 (κ, H) = inf φ∈H 1 (Ω) φ =0 Q Ω κHF,-κ 2 a (φ) φ 2 L 2 (Ω) . (4.1.26)
Note that µ 1 (κ, H) is the lowest eigenvalue of P Ω κHF,-κ 2 a . Here, we refer to [START_REF] Cancelier | Magnetic bottles with weak electric field[END_REF][START_REF] Kim | Estimate of the upper critical field and concentration for superconductor[END_REF][START_REF] Miqueu | Équation de Schrödinger avec champ magnétique qui s'annule[END_REF][START_REF] Pan | Schrödinger operators with non-degenerately vanishing magnetic fields in bounded domains[END_REF] for previous contributions. We introduce the following critical fields (cf. e.g. [START_REF] Fournais | On the third critical field in Ginzburg-Landau theory[END_REF][START_REF] Lu | Estimates of the upper critical field for the Ginzburg-Landau equations of superconductivity[END_REF]) .

H cp C 3 (κ) = sup N cp (κ) , H cp C 3 (κ) = inf (R + \ N cp (κ)) , (4.1.27) H C 3 (κ) = sup N (κ) , H C 3 (κ) = inf (R + \ N (κ)) , (4.1.28) H loc C 3 (κ) = sup N loc (κ) , H loc C 3 (κ) = inf (R + \ N loc (κ)) . (4.1.29)
Below H C 3 , normal states will loose their stability and above H C 3 , the normal state is (up to a gauge transformation) the only critical point of the functional in (4.1.1).

Our aim is to determine the asymptotics of all the critical fields as κ -→ +∞. This involves spectral quantities related to three models depending on Γ being empty or not. Let us introduce

Θ 0 = inf ξ∈R µ(ξ) ,
where µ is the lowest eigen value of the operator

h N,ξ := - d 2 dt 2 + (t + ξ) 2 in L 2 (R + ) ,
subject to the Neumann boundary condition u (0) = 0.

Theorem 4.1.6. Suppose that Γ = {x ∈ Ω : B 0 (x) = 0} = ∅ and that a ∈ C 1 (Ω) satisfies {a > 0} = ∅. Then, as κ -→ +∞, all the six critical fields satisfy an asymptotic expansion in the form :

H C 3 (κ) = max sup x∈Ω a(x) |B 0 (x)| , sup x∈∂Ω a(x) Θ 0 |B 0 (x)| κ + O(κ 1 2 ) . (4.1.30)
We introduce

λ 0 = inf τ ∈R λ(τ ) , (4.1.31)
where λ(τ ) is the lowest eigenvalue of the selfadjoint realization of the differential operator

M (τ ) = - d 2 dt 2 + 1 4 (t 2 + 2τ ) 2 in L 2 (R) . (4.1.32)
We consider, for any θ ∈ (0, π) the bottom of the spectrum λ(R 2 + , θ) of the operator

P R 2 + A app,θ ,0 with A app,θ = - x 2 2 2 cos θ, x 2
H C 3 (κ) = max   sup x∈Γ∩Ω a(x) 3 2 λ 3 2 0 |∇B 0 (x)| , sup x∈Γ∩∂Ω a(x) 3 2 λ(R 2 + , θ(x)) 3 2 |∇B 0 (x)|   κ 2 + O κ 11 6
.

Here θ(x) denotes the angle between ∇B 0 (x) and the inward normal vector -ν(x).

Organization of the paper

The rest of the paper is split into eleven sections. Section 4.2 analyzes the model problem with a constant magnetic field and a constant pinning term. Section 4.

Notation.

Throughout the paper, we use the following notation : • Let a + ( x 0 , κ) = [a( x 0 , κ)] + and a -( x 0 , κ) = [a( x 0 , κ)] -where, for any x ∈ R, [x] + = max(x, 0) and [x] -= max(-x, 0).

• If b 1 (κ)
• Given R > 0 and x = (x 1 , x 2 ) ∈ R 2 , Q R (x) = (-R/2+x 1 , R/2+x 1 )×(-R/2+x 2 , R/2+x 2 )
denotes the square of side length R centered at x = (x 1 , x 2 ) and we write

Q R = Q R (0).

A reference problem

The reference problem is obtained by freezing the pinning term and the magnetic field. This approximation will appear to be reasonable in squares avoiding the boundary and the zero set Γ of the magnetic field B 0 .

A useful function

Consider R > 0, b > 0, ζ ∈ {-1, +1} and α ∈ R . We define the following Ginzburg-Landau energy with constant magnetic field on

H 1 (Q R ) by u → F ζ,α b,Q R (u) = Q R b|(∇ -iζA 0 )u| 2 + 1 2 α -|u| 2 2 dx , (4.2.1)
where

A 0 (x) = 1 2 (-x 2 , x 1 ) , ∀ x = (x 1 , x 2 ) ∈ R 2 . (4.2.2)
We have two cases according to the sign of α : Case 1. α > 0 :

We notice that

F ζ,α b,Q R (u) = α 2 F ζ,1 b,Q R ( u) , (4.2.3) where b = b α and u = u √ α . ( 4

.2.4)

We introduce the two ground state energies

e N (b, R, α) = inf F +1,α b,Q R (u) : u ∈ H 1 (Q R ; C) (4.2.5) e D (b, R, α) = inf F +1,α b,Q R (u) : u ∈ H 1 0 (Q R ; C) . (4.2.6) As F +1,α b,Q R (u) = F -1,α b,Q R (u), it is immediate that, inf F +1,α b,Q R (u) = inf F -1,α b,Q R (u) . ( 4
e N (b, R, α) = α 2 e N b α , R, 1 = α 2 e N b α , R , (4.2.8) and e D (b, R, α) = α 2 e D b α , R, 1 = α 2 e D b α , R . (4.2.9)
As a consequence of (4.2.3) and ( 4.

2.4), u is a minimizer of F ζ,1 b,Q R if and only if u is a minimizer of F ζ,α b,Q R . In particular any minimizer of F ζ,α b,Q R satisfies |u| ≤ √ α . (4.2.10) Recall from [18, Theorem 2.1] that, f (b) = lim R-→∞ e D (b, R) R 2 . ( 4
∀R ≥ R M , ∀ b > 0, ∀ α > 0 such that 0 < b α ≤ M , we have e N (b, R, α) ≥ e D (b, R, α) -C M α 2 R b α 1 2
(4.2.12)

α 2 f b α ≤ e D (b, R, α) R 2 ≤ α 2 f b α + C M α 3 2 √ b R . (4.2.13)
Case 2. α ≤ 0 : When α ≤ 0, we write α = -α 0 , α 0 ≥ 0 and (4.2.1) becomes

F ζ,α b,Q R (u) = Q R b|(∇ -iζA 0 )u| 2 + 1 2 α 0 + |u| 2 2 dx . (4.2.14) It is clear that, F ζ,α b,Q R (u) ≥ 1 2 α 2 0 R 2 and F ζ,α b,Q R (0) = 1 2 α 2 0 R 2 .
As a consequence, we have

1 2 α 2 0 R 2 ≤ e D (b, R, α) ≤ F ζ,α b,Q R (0) = 1 2 α 2 0 R 2 .
When α = 0, it is easy to show that

F ζ,α b,Q R (u) = 0 .
96 Pinning with a variable magnetic field of the two dimensional Ginzburg-Landau model

Notice that the only minimizer of F ζ,α b,Q R is u = 0 . Thus, for any α ≤ 0 , we obtain

e D (b, R, α) R 2 = 1 2 α 2 . (4.2.15)

Upper bound of the energy

The aim of this section is to give an upper bound of the ground state energy E g (κ, H, a, B 0 ) introduced in (4. 1.14) under Assumption (4.1.15). For this we cover Ω by (the closure of) disjoint open squares (Q (γ)) γ whose centers γ belong to a square lattice Γ = Z × Z.

We will get an upper bound by matching together approximate minimizers, in each square Q (γ) contained in Ω, obtained by freezing the pinning term and the magnetic field at a suitable point γ. The size of the square will be chosen as a function of κ. We start with estimates in a given square Q (x 0 ) and will take later x 0 = γ . About Assumption (A 4 ). We first explain what was meant in Assumption (A 4 ). By L(∂{a > 0}) ≤ C 1 κ 1 2 we mean the existence of C 2 > 0 and κ 0 such that :

∀κ ≥ κ 0 , ∀ ≤ C 2 κ -1 2 , card {γ ∈ Γ ∩ Ω with Q (γ) ∩ ∂{a > 0} ∩ Ω = ∅} ≤ C 1 κ 1 2 -1 . (4.3.1)
Using Assumption (4.1.9), for any x 0 ∈ Q (x 0 ) and κ ≥ κ 0 , we observe that,

|a(x, κ) -a( x 0 , κ)| ≤ sup x |∇ x a(x, κ)| |x -x 0 | ≤ √ 2 L(κ) , ∀x ∈ Q (x 0 ) . (4.3.2)
Definition 4.3.1 (ρ-admissible). Let ρ ∈ (0, 1). We say that triple

( , x 0 , x 0 ) is ρ-admissible if Q (x 0 ) ⊂ {|B 0 | > ρ} ∩ Ω and x 0 ∈ Q (x 0 ).
In this case, we also say that the pair ( , x 0 ) is ρ-admissible and the corresponding square Q (x 0 ) is ρ admissible.

We recall from [6, Section 3] the definition of the test function,

w ,x 0 , x 0 (x) =    e iκHϕ x 0 , x 0 u R R (x -x 0 ) if x ∈ Q (x 0 ) ⊂ {B 0 > ρ} ∩ Ω e iκHϕ x 0 , x 0 u R R (x -x 0 ) if x ∈ Q (x 0 ) ⊂ {B 0 < -ρ} ∩ Ω , (4.3.3) where u R ∈ H 1 0 (Ω) is a minimizer of F +1,1 b,Q R satisfying by (4.2.10) | u R | ≤ 1 and ϕ x 0 , x 0 is the function introduced in [5, Lemma A.3] that satisfies |F(x) -B 0 ( x 0 )A 0 (x -x 0 ) -∇ϕ x 0 , x 0 (x)| ≤ C 2 , ∀x ∈ Q (x 0 ) . (4.3.4)
Here B 0 = curl F and A 0 is the magnetic potential introduced in (4.2.2).

Upper bound of the energy

97

Let us introduce the function : such that if κ ≥ κ 0 , ∈ (0, 1), δ ∈ (0, 1), ρ > 0, 2 κHρ > 1 and ( , x 0 , x 0 ) is a ρ-admissible triple, then,

w ,x 0 , x 0 (x) = a + ( x 0 , κ) w ,x 0 , x 0 (x) , ∀x ∈ Q ( x 0 ) . ( 4
1 |Q (x 0 )| E 0 (w ,x 0 , x 0 , F; a, Q (x 0 )) ≤ (1 + δ)κ 2 a + ( x 0 , κ) 2 f H |B 0 ( x 0 )| κ a + ( x 0 , κ) + 1 2 a -( x 0 , κ) 2 + C 1 κ + δ -1 2 L(κ) 2 + δ -1 κ 2 4 κ 2 . (4.3.7) Proof. Let R = κH|B 0 ( x 0 )| and b = H |B 0 ( x 0 )| κ . (4.3.8)
First we estimate κ 2 2 Q (x 0 ) (a(x, κ) -|w ,x 0 , x 0 | 2 ) 2 dx from above. Using (4.3.2), we get the existence of a constant C > 0 such that for any δ ∈ (0, 1) and any κ ≥ κ 0 ,

κ 2 2 Q (x 0 ) a(x, κ) -|w ,x 0 , x 0 | 2 2 dx ≤ (1 + δ) κ 2 2 Q (x 0 ) a( x 0 , κ) -|w ,x 0 , x 0 | 2 2 dx + (1 + δ -1 ) κ 2 2 Q (x 0 ) (a( x 0 , κ) -a(x, κ)) 2 dx ≤ (1 + δ) κ 2 2 Q (x 0 ) a( x 0 , κ) -|w ,x 0 , x 0 | 2 2 dx + Cδ -1 κ 2 4 L(κ) 2 . (4.3.9)
The estimate of Q (x 0 ) |(∇-iκHF)w ,x 0 , x 0 | 2 dx from above is the same as in [6, Proposition 3.1]. We have

Q (x 0 ) |(∇ -iκHF)w ,x 0 , x 0 | 2 dx ≤ (1 + δ) Q (x 0 ) ∇ -iκH(B 0 ( x 0 )A 0 (x -x 0 ) + ∇ϕ x 0 , x 0 ) w ,x 0 , x 0 2 dx + Cδ -1 κ 4 6 | w ,x 0 , x 0 | 2 . (4.3.10)
98 Pinning with a variable magnetic field of the two dimensional Ginzburg-Landau model From (4.1.10), by collecting (4.3.9), (4.3.10) and (4.3.6), we find that,

E 0 (w ,x 0 , x 0 , F; a, Q (x 0 )) ≤ (1 + δ)E 0 w ,x 0 , x 0 , B 0 ( x 0 )A 0 (x -x 0 ) + ∇ϕ x 0 , x 0 ; a( x 0 , κ), Q (x 0 ) + Cδ -1 (κ 2 4 L(κ) 2 + κ 4 6 a + ( x 0 , κ)) . (4.3.11)
As we did in [START_REF] Attar | Energy and vorticity of the Ginzburg-Landau model with variable magnetic field[END_REF], we use the change of variable y = R (x -x 0 ) and obtain

E 0 w ,x 0 , x 0 , B 0 ( x 0 )A 0 (x -x 0 ) + ∇ϕ x 0 , x 0 ; a( x 0 , κ), Q (x 0 ) = Q R a + ( x 0 , κ) R ∇ -i R ζ A 0 (y) u R (y) 2 + κ 2 2 a( x 0 , κ) -a + ( x 0 , κ) | u R (y)| 2 2 2 R 2 dy.
Here, we denote by ζ the sign of B 0 (x 0 ). We distinguish between two cases : Case 1 : When a( x 0 , κ) > 0, we get

E 0 w ,x 0 , x 0 , B 0 ( x 0 )A 0 (x -x 0 ) + ∇ϕ x 0 , x 0 ; a( x 0 , κ), Q (x 0 ) = a( x 0 , κ) 2 b F ζ ,1 b/a( x 0 ,κ),Q R ( u R ) .
From (4.2.7) and (4.2.8), we obtain,

E 0 w ,x 0 , x 0 , B 0 ( x 0 )A 0 (x -x 0 ) + ∇ϕ x 0 , x 0 ; a( x 0 , κ), Q (x 0 ) = 1 b e D (b, R, a( x 0 , κ)) . (4.3.12)
As a consequence of the upper bound in (4.2.13), the ground state energy e D (b, R, a( x 0 , κ)) in (4.3.12) is bounded for all b > 0 and R ≥ 1 by :

e D (b, R, a( x 0 , κ)) ≤ a( x 0 , κ) 2 R 2 f b a( x 0 , κ) + C M a( x 0 , κ) 3 2 R √ b . (4.3.13)
With the choice of R in (4.3.8), we have effectively R ≥ 1 which follows from the assumption R ≥ √ κHρ > 1. We get from (4.3.12) and (4.3.13) the estimate

E 0 (w ,x 0 , x 0 , ζ |B 0 ( x 0 )|A 0 (x -x 0 ) + ∇ϕ x 0 , x 0 ; a( x 0 , κ), Q (x 0 )) ≤ a( x 0 , κ) 2 R 2 b f b a( x 0 , κ) + C M a( x 0 , κ) 3 2 R √ b , (
E 0 (w ,x 0 , x 0 , F; a( x 0 , κ), Q (x 0 )) ≤ (1 + δ) a( x 0 , κ) 2 R 2 b f b a( x 0 , κ) + C M a 3 2 R √ b + Cδ -1 (κ

99

Here, we have used the fact that a( x 0 , κ) ≤ sup x∈Ω, κ≥κ 0 a(x, κ) = a .

Case 2 : When a( x 0 , κ) ≤ 0 , we have,

E 0 (w ,x 0 , x 0 , F; a( x 0 , κ), Q (x 0 )) = κ 2 2 Q (x 0 ) a(x, κ) 2 dx .
From (4.3.2), we get the existence of a constant C > 0 such that for any δ ∈ (0, 1),

E 0 (w ,x 0 , x 0 , F; a( x 0 , κ), Q (x 0 )) ≤ (1 + δ) κ 2 2 a( x 0 , κ) 2 2 + C δ -1 κ 2 4 L(κ) 2 . (4.3.16)
The results of cases 1-2, we obtain,

E 0 (w ,x 0 , x 0 , F; a( x 0 , κ), Q (x 0 )) ≤ (1 + δ)κ 2 a + ( x 0 , κ) 2 f H |B 0 ( x 0 )| κ a + ( x 0 , κ) + 1 2 a -( x 0 , κ) 2 2 + C κ a 3 2 + δ -1 κ 2 2 L(κ) 2 + δ -1 κ
κ 2 , δ -1 κ 2 2 L(κ) 2 ≤ κ 23 12 κ 2 , δ -1 κ 4 4 = κ 21 12 κ 2 , 2 κHρ = κ 3 24
1 . , then, the ground state energy E g (κ, H, a, B 0 ) in (4.1.14) satisfies

E g (κ, H, a, B 0 ) ≤ κ 2 {a(x,κ)>0} a(x, κ) 2 f H |B 0 (x)| κ a(x, κ) dx + κ 2 2 {a(x,κ)≤0} a(x, κ) 2 dx + o(κ 2 ) , as κ -→ ∞ . (4.3.20)
Proof. Let ∈ (0, 1), δ and ρ be chosen as in (4.3.18) and (4.3.19). We consider the lattice 100 Pinning with a variable magnetic field of the two dimensional Ginzburg-Landau model Γ := Z × Z and write, for γ ∈ Γ , Q γ, = Q (γ). In the next decomposition we keep the ρ-admissible boxes Q (γ) in Ω which in addition are either contained in {a > 0} or in {a ≤ 0}. Hence we introduce Under Assumption (4.1.8), we have,

I + ,ρ = γ; Q γ, ⊂ Ω ∩ {|B 0 | > ρ ; a > 0} , I - ,ρ = γ; Q γ, ⊂ Ω ∩ {|B 0 | > ρ ; a ≤ 0} , ( 4
N + + N -= |Ω| -2 + O(κ 1 2 -1 + -1 + ρ -2 ) , as κ → +∞ . (4.3.23) In (4.3.23), κ 1
2 -1 appears when treating the boundary of the set {a(x, κ) > 0} (using Assumption (A 4 ) as explained in (4.3.1)), -1 appears in the treatment of the boundary and ρ -2 appears when treating the neighborhood of Γ. In each ρ-admissible Q (γ), we consider some γ (to be chosen later) such that ( , γ, γ) be a ρ-admissible triple. We consider w ,γ, γ and extend it by 0 outside of Q γ, , keeping the same notation for this extension. Then we define

s(x) = γ∈I + ,ρ ∪I - ,ρ w ,γ, γ (x) . (4.3.24)
We compute the Ginzburg-Landau energy of the test configuration (s, F) in Ω. Since curl F = B 0 , we get,

E κ,H,a,B 0 (s, F, Ω) = γ∈I + ,ρ ∪I - ,ρ E 0 (w ,γ, γ , F; a( γ, κ), Q γ, ) . (4.3.25)
Notice that for any γ ∈ Q γ, , a( γ, κ) satisfies (4.3.2) with x = γ and x 0 = γ , and B 0 ( γ) satisfies (4.3.4). We recall that f is a continuous, non-decreasing function (see [START_REF] Attar | Energy and vorticity of the Ginzburg-Landau model with variable magnetic field[END_REF]Theorem 2.1]) and that B 0 and a(•, κ) are in

C 1 . Then, in each box Q γ, , we select γ ∈ Q γ, such that |a( γ, κ)| 2 f H B 0 ( γ) κ a( γ, κ) = inf γ∈Q γ, |a( γ, κ)| 2 f H B 0 ( γ) κ a( γ, κ) (if γ ∈ I + ,ρ ) and |a( γ, κ)| 2 = inf γ∈Q γ, |a( γ, κ)| 2 (if γ ∈ I - ,ρ ) .
Using Proposition 4.3.2 and noticing that |Q γ, | = 2 , we get the existence of C > 0 such that, 4.4 A priori estimates of minimizers 101 for any δ ∈ (0, 1)

γ∈I + ,ρ ∪I - ,ρ E 0 (w ,γ, γ , F; a( γ, κ), Q γ, ) ≤ κ 2 (1 + δ) γ∈I + ,ρ inf γ∈Q γ, [a( γ, κ)] 2 + f H B 0 ( γ) κ a( γ, κ) 2 + κ 2 (1 + δ) γ∈I - ,ρ inf γ∈Q γ, [a( γ, κ)] 2 - 2 2 + C γ∈I + ,ρ ∪I - ,ρ κ + δ -1 κ 2 2 L(κ) 2 + δ -1 κ 4 4 2 . (4.3.26)
We recognize the lower Riemann sum of the function

x -→ [a(x, κ)] 2 + f H B 0 (x) κ a(x,κ) in (∪ γ∈I + ,ρ Q γ, )
and the function

x -→ [a(x, κ)] 2 -in (∪ γ∈I - ,ρ Q γ, ) . Notice that {∪ γ∈I ,ρ Q γ, } ⊂ Ω.
E κ,H,a,B 0 (s, F, Ω) ≤ κ 2 {a(x,κ)>0} a(x, κ) 2 f H κ |B 0 (x)| a(x, κ) dx + κ 2 2 {a(x,κ)≤0} a(x, κ) 2 dx + C κ 23 12 .
(4.3.27) Since (ψ, A) is a minimizer of the functional E κ,H,a,B 0 in (4.1.1), we get

E g (κ, H, a, B 0 ) ≤ E κ,H,a,B 0 (s, F, Ω) .
This finishes the proof of Theorem 4.3.4.

A priori estimates of minimizers

The aim of this section is to give a priori estimates for the solutions of the Ginzburg-Landau equations (4.1.12). In the case when a(x, κ) = 1 the starting point is an L ∞ estimate of ψ. This estimate can be easly extended in the general case considered in this paper when (4.1.12) a and (4.1.12) c hold. Let us introduce :

a(κ) = sup x∈Ω a(x, κ) . (4.4.1)
Proposition 4.4.1. Let κ > 0 ; if (ψ, A) is a critical point (see (4.1.12)), then, |ψ(x)| 2 ≤ max {a(κ), 0} , ∀x ∈ Ω . (4.4.2)
Proof. We distinguish between two cases : Case 1 : a(κ) ≤ 0 .

Multiplying the equation for ψ in (4.1.12) a by ψ and integrating over Ω, we get

Ω |(∇ -iκHA)ψ| 2 dx = κ 2 Ω (a(x, κ) -|ψ| 2 )|ψ| 2 dx . (4.4.3)
Since (a(x, κ) -|ψ| 2 ) ≤ -|ψ| 2 , we obtain that |ψ| 2 = 0 almost everywhere. Case 2 : a(κ) > 0 .

102 Pinning with a variable magnetic field of the two dimensional Ginzburg-Landau model

We will show that ψ ∈ C 0 (Ω). In fact, (ψ, A) satisfies (4.1.12) a , ψ ∈ L p (Ω) for all 2 ≤ p < +∞ and A ∈ H 1 div (Ω) → L p (Ω). Thus, ψ ∈ W 2,q (Ω) for all q < 2. As a consequence of the continuous Sobolev embedding of W j+m,q (Ω) into C j (Ω) for any q > 2 m , we obtain that ψ ∈ C 0 (Ω). Define for any κ > 0 the following open set :

Ω + = x ∈ Ω : |ψ(x)| > a(κ) , (4.4.4)
and the following functions on

Ω + φ = ψ |ψ| , ψ = |ψ| -a(κ) + φ .
It is clear that ∇ |ψ| -a(κ)

+ = 1 Ω + ∇ |ψ| -a(κ) = 1 Ω + ∇|ψ| .
Notice that ψ ∈ H 1 (Ω), so applying [14, Proposition 3.1.2], we get the property that ∇ |ψ| -a(κ)

+ ∈ L 2 (Ω), which implies that |ψ| -a(κ) + ∈ H 1 (Ω).
We introduce an increasing cut-off function χ ∈ C ∞ (R) such that,

χ(t) = 0 for t ≤ 1 4 a(κ) 1 for t ≥ 3 4 a(κ) , (4.4.5)
and define

φ = χ(|ψ|) ψ |ψ| . (4.4.6)
Since χ(|ψ|) ψ |ψ| is smooth with bounded derivatives and ψ ∈ H 1 (Ω), the chain rule gives that φ ∈ H 1 (Ω) . Furthermore,

(∇ -iκHA) ψ = 1 Ω + φ ∇|ψ| + |ψ| -a(κ) + (∇ -iκHA) φ. (4.4.7)
Using (4.4.5) and (4.4.6), we get

1 Ω + (∇ -iκHA)ψ = 1 Ω + (∇ -iκHA)(|ψ| φ) = 1 Ω + { φ ∇|ψ| + |ψ|(∇ -iκHA) φ} . (4.4.8)
We have on Ω + that |φ| = | φ| = 1 . Therefore

φ∇φ + φ∇φ = φ∇φ + φ∇φ = ∇|φ| 2 = 0 .
So, Re(1 Ω + φ∇φ) = 0 . This implies by using (4.4.7) and (4.4.8) that

Re (∇ -iκHA) ψ • (∇ -iκHA)ψ = 1 Ω + |∇|ψ|| 2 + |ψ| -a(κ) |ψ||(∇ -iκHA) φ| 2 .
4.4 A priori estimates of minimizers 103 Multiplying (4.1.12) a by ψ and using (4.1.12) c , it results from an integration by parts over Ω that

0 = Re Ω (∇ -iκHA) ψ(∇ -iκHA)ψ + ψ(|ψ| 2 -a)ψ dx ≥ Re Ω (∇ -iκHA) ψ(∇ -iκHA)ψ + ψ |ψ| 2 -a(κ) ψ dx ≥ Ω + |∇|ψ|| 2 + (|ψ| -a(κ)) |ψ||(∇ -iκHA) φ| 2 + |ψ| + a(κ) |ψ| -a(κ) 2 |ψ| dx .
Since the integrand is non-negative in Ω + , we easily conclude that Ω + has measure zero, and consequently, we get that |ψ| ∈ L ∞ (Ω) .

Since Ω + has measure zero and ψ ∈ C 0 (Ω), we get

|ψ(x)| 2 ≤ a(κ) , ∀x ∈ Ω . Corollary 4.4.2. Let κ > 0 ; If (ψ, A) ∈ H 1 (Ω; C) × H 1 div (Ω) is a critical point, we have, |ψ(x)| 2 ≤ max {a, 0} , ∀x ∈ Ω , (4.4.9)
where a = sup κ a(κ) was introduced in (4.1.10).

The following estimates play an essential role in controlling the errors resulting from various approximations (see Section 4.5). These estimates are simpler than the delicate elliptic estimates in [START_REF] Fournais | Optimal uniform elliptic estimates for the Ginzburg-Landau system[END_REF] and [START_REF] Lu | Estimates of the upper critical field for the Ginzburg-Landau equations of superconductivity[END_REF].

Proposition 4.4.3. Suppose that (4.1.15) holds. Let β ∈ (0, 1). There exist positive constants κ 0 and C such that, if κ ≥ κ 0 and (ψ, A) is a minimizer of (4.1.1), then

curl(A -F) L 2 (Ω) ≤ C H . (4.4.10) A -F H 2 (Ω) ≤ C H , (4.4.11)
A -F C 0,β (Ω) ≤ C H . (4.4.12)
Here we recall that F is the magnetic potential defined by

curl F = B 0 , F ∈ H 1 div (Ω) . ( 4
(A -F) L 2 (Ω) ≤ 1 κH E g (κ, H, a, B 0 ) 1 2 ≤ 1 κH κ 2 {a(x,κ)>0} a(x, κ) 2 f H κ |B 0 (x)| a(x, κ) dx + κ 2 2 {a(x,κ)≤0} a(x, κ) 2 dx 1 2 . ( 4.4.14)
Using (4.1.6) and the bound f (b) ≤ 1 2 , we get,

curl(A -F) L 2 (Ω) ≤ C H . (4.4.15)
As in [6, Proposition 4.1], we prove that

A -F H 2 (Ω) ≤ C H . (4.4.16)
Now, the estimate in C 0,β -norm is a consequence of the continuous Sobolev embedding of H 2 (Ω) in C 0,β (Ω).

Lower bounds for the global and local energies

In this section, we suppose that D is an open set with smooth boundary such that D ⊂ Ω (or D = Ω). We will give a lower bound of the ground state energy E g (κ, H, a, B 0 ) introduced in (4.1.14).

Proposition 4.5.1. Under Assumptions (4.1.4)-(4.1.7), there exist for all β ∈ (0, 1) positive constants C and κ 0 such that if κ ≥ κ 0 , ∈ (0, 1 2 ), δ ∈ (0, 1), ρ > 0, 2 κHρ > 1, (ψ, A) is a minimizer of (4.1.1), h ∈ C 1 (Ω), h ∞ ≤ 1 and ( , x 0 , x 0 ) is a ρ-admissible triple, then,

1 |Q (x 0 )| E 0 (hψ, A; a, Q (x 0 )) ≥ (1 -δ)κ 2 a + ( x 0 , κ) 2 f H κ |B 0 ( x 0 )| a + ( x 0 , κ) + 1 2 a -( x 0 , κ) 2 -Cκ 2 δ -1 2 L(κ) 2 + δ -1 κ 2 4 + δ -1 2β + (κ ) -1 + L(κ) , (4.5.1)
where L(κ) is introduced in (4.1.9).

Proof. We distinguish between two cases according to the sign of a( x 0 , κ).

We begin with the case when a( x 0 , κ) ≤ 0 . We have,

E 0 (hψ, A; a, Q (x 0 )) = Q (x 0 ) |(∇ -iκHA)hψ| 2 dx + κ 2 2 Q (x 0 ) (a(x, κ) -|hψ| 2 ) 2 dx ≥ κ 2 2 Q (x 0 ) a(x, κ) 2 dx -κ 2 Q (x 0 ) a(x, κ)|hψ| 2 dx .
Using (4.3.2), (4.4.9) and the assumptions on h, the simple decomposition a(x, κ) = a( x 0 , κ) + 4.5 Lower bounds for the global and local energies 105 (a(x, κ) -a( x 0 , κ)) yields for any δ ∈ (0, 1)

κ 2 2 Q (x 0 ) a(x, κ) 2 dx ≥ (1 -δ) κ 2 2 Q (x 0 ) a( x 0 , κ) 2 dx + (1 -δ -1 ) κ 2 2 Q (x 0 ) (a(x, κ) -a( x 0 , κ)) 2 dx ≥ (1 -δ) κ 2 2 a( x 0 , κ) 2 |Q (x 0 )| -Cδ -1 κ 2 2 L(κ) 2 |Q (x 0 )| , (4.5.2)
and

-κ 2 Q (x 0 ) a(x, κ)|hψ| 2 dx ≥ -κ 2 Q (x 0 ) a( x 0 , κ)|hψ| 2 dx -C L(κ) κ 2 |Q (x 0 )| ≥ -C L(κ) κ 2 |Q (x 0 )| . (4.5.3)
Collecting (4.5.2) and (4.5.3), we get,

1 |Q (x 0 )| E 0 (hψ, A; a, Q (x 0 )) ≥ (1 -δ) κ 2 2 a( x 0 , κ) 2 -Cδ -1 κ 2 2 L(κ) 2 -C L(κ) κ 2 . (4.5.4)
Now, we treat the case when a( x 0 , κ) > 0 . Let

φ x 0 (x) = (A(x 0 ) -F(x 0 )) • x,
where F is the magnetic potential introduced in (4.4.13). Using the estimate of A -F C 0,β (Ω) given in Proposition 4.4.3, we get for any β ∈ (0, 1) the existence of a constant C such that for all x ∈ Q (x 0 ),

|A(x) -∇φ x 0 -F(x)| ≤ C β H . (4.5.5)
Let x 0 ∈ Q (x 0 ) and ϕ = ϕ x 0 , x 0 + φ x 0 with ϕ x 0 , x 0 satisfying (4.3.4). We define the function in

Q (x 0 ), u(x) = e -iκHϕ hψ(x) . (4.5.6)
Similarly to (4.3.9), we have, for any δ ∈ (0, 1),

κ 2 2 Q (x 0 ) a(x, κ) -|hψ| 2 2 dx ≥ (1 -δ) κ 2 2 Q (x 0 ) a( x 0 , κ) -|hψ| 2 2 dx -Cδ -1 κ 2 4 L(κ) 2 . (4.5.7)
Using the same techniques as in [5, Lemma 4.1], we get, for any β ∈ (0, 1),

Q (x 0 ) |(∇-iκHA)hψ| 2 dx ≥ (1-δ) Q (x 0 ) |(∇-iκH(ζ |B 0 ( x 0 )|A 0 (x-x 0 )+∇ϕ(x)))hψ| 2 dx -Cδ -1 (κH) 2 4 + 2β H 2 Q (x 0 ) |hψ| 2 dx . (4.5.8)
106 Pinning with a variable magnetic field of the two dimensional Ginzburg-Landau model Thus, by collecting (4.5.7) and (4.5.8), using (4.1.7), (4.4.9) and h L ∞ (Ω) ≤ 1, we get

E 0 (hψ, A; a( x 0 , κ), Q (x 0 )) ≥ (1 -δ)E 0 (e -iκHϕ hψ(x), ζ |B 0 ( x 0 )|A 0 (x -x 0 ); a( x 0 , κ), Q (x 0 )) -Cδ -1 κ 2 4 L(κ) 2 -C 1 δ -1 κ 2 H 2 4 + 2β H 2 2 . (4.5.9)
Let R and b be as in (4.3.8). Let us introduce the function v ,x 0 , x 0 in Q R as follows :

v ,x 0 , x 0 (x) =    u R x + x 0 if x ∈ Q R ⊂ {B 0 > ρ} ∩ Ω u R x + x 0 if x ∈ Q R ⊂ {B 0 < -ρ} ∩ Ω , (4.5.10)
where u is defined in (4.5.6).

Similarly to (4.3.12), we use the change of variable y = R (x -x 0 ) and get

E 0 (e -iκHϕ hψ(x), ζ κH|B 0 ( x 0 )|A 0 (x -x 0 ); a( x 0 , κ), Q (x 0 )) = 1 b F +1,a( x 0 ,κ) b,Q R (v ,x 0 , x 0 ) , (4.5.11)
where

F +1,a( x 0 ,κ), b,Q R is introduced in (4.2.1). Since v ,x 0 , x 0 ∈ H 1 (Q R )
then, using (4.2.12) and (4.2.13), we get

1 b F +1,a( x 0 ,κ) b,Q R (v ,x 0 , x 0 ) ≥ 1 b e N (b, R, a( x 0 , κ)) ≥ 1 b e D (b, R, a( x 0 , κ)) -C M a( x 0 , κ) 3 2 R √ b ≥ a( x 0 , κ) 2 R 2 b f b a( x 0 , κ) -C M R √ b . (4.5.12)
Inserting (4.5.12) into (4.5.11), we get

E 0 (e -iκHϕ hψ(x), ζ κH|B 0 ( x 0 )|A 0 (x -x 0 ); a( x 0 , κ), Q (x 0 )) ≥ a( x 0 , κ) 2 R 2 b f b a( x 0 , κ) -C M R √ b . (4.5.13)
Having in mind (4.3.8) and (4.5.13), we get from (4.5.9),

1 |Q (x 0 )| E 0 (hψ, A; a( x 0 , κ), Q (x 0 )) ≥ (1 -δ)κ 2 a( x 0 , κ) 2 f H κ |B 0 ( x 0 )| a( x 0 , κ) -Cδ -1 κ 2 2 L(κ) 2 -C 1 δ -1 κ 2 H 2 4 + 2β H 2 -C 2 κ . (4.5.14)
The estimates in (4.5.4) and (4. This choice and Assumption (4.1.15) permit to have the assumptions in Proposition 4.5.1 satisfied and make the error terms in its statement of order o(κ 2 ). We have as κ -→ ∞ ,

δ -1 κ 4 4 = κ 21 12 κ 2 , δ -1 κ 2 2β = κ 29 24 κ 2 , δ -1 κ 2 2 L(κ) 2 = κ 23 12 κ 2 , κ = κ 19 12 κ 2 , L(κ) κ 2 = κ 23 12 κ 2 , 2 κHρ = κ 3 24
1 .

The next theorem presents a lower bound of the local energy in a relatively compact smooth domain D in Ω. We deduce the lower bound of the global energy by replacing D by Ω.

Theorem 4.5.3. Under Assumptions (4.1.4)-(4.1.8), if (4.1.15) holds, L(κ) ≤ C κ 1 2 with C > 0, h ∈ C 1 (Ω), h ∞ ≤ 1, (ψ, A) is a minimizer of (4.1.1) and D an open set in Ω, then as κ -→ +∞, E(hψ, A; a, B 0 , D) ≥ E 0 (hψ, A; a, D) ≥ κ 2 D∩{a(x,κ)>0} a(x, κ) 2 f H κ |B 0 (x)| a(x, κ) dx + κ 2 2 D∩{a(x,κ)≤0} a(x, κ) 2 dx + o κ 2 . (4.5.16)
Proof. The proof is similar to the one in Theorem 4.3.4 and we keep the same notation. Let

D + ,ρ = int ∪ γ∈I + ,ρ Q γ, and
D - ,ρ = int ∪ γ∈I - ,ρ Q γ, ,
where γ ∈ I + ,ρ and γ ∈ I - ,ρ are introduced in (4.3.21). Thanks to Proposition 4.5.1, we can easily prove the existence of positive constant C such that for any δ ∈ (0, 1) and β ∈ (0, 1),

E 0 (hψ, A; a, D) ≥ κ 2 (1 -δ) D + ,ρ ∩{a(x,κ)>0} a(x, κ) 2 f H κ |B 0 (x)| a(x, κ) dx + 1 2 D - ,ρ ∩{a(x,κ)≤0} a(x, κ) 2 dx -C r(κ, , δ, ρ, L(κ), β) ,
where

r(κ, , δ, ρ, L(κ), β) = κ 2 + κ 2 ρ + κ + δ -1 κ 2 2 L(κ) 2 + δ -1 κ 4 4 + δ -1 κ 2 2β + L(κ) κ 2 . (4.5.17)
Notice that using the regularity of ∂D, (4.1.4) and (4.1.8) (see (4.3.1)), we get the existence of 108 Pinning with a variable magnetic field of the two dimensional Ginzburg-Landau model constants C 1 > 0 and C 2 > 0 such that,

∀ ≤ C 2 κ -1 2 , ∀ρ ∈ (0, 1) , |D \ D + ,ρ | + |D \ D - ,ρ | ≤ C 1 (κ 1 2 + ρ) . (4.5.18)
This implies by using (4.1.7) and the upper bound f ≤ 1 2 ,

D + ∩{a(x,κ)>0} a(x, κ) 2 f H κ |B 0 (x)| a(x, κ) dx ≥ D + ,ρ ∩{a(x,κ)>0} a(x, κ) 2 f H κ |B 0 (x)| a(x, κ) dx - 1 2 a |D \ D ,ρ | (4.5.19)
and

1 2 D -∩{a(x,κ)≤0} a(x, κ) 2 dx ≥ 1 2 D ,ρ ∩{a(x,κ)≤0} a(x, κ) 2 dx - 1 2 a |D \ D - ,ρ | , (4.5.20)
where a is introduced in (4.1.10). Collecting (4.5.19) and (4.5.20), using Assumptions (4.1.6) and (4.5.18), we find that,

E 0 (hψ, A; a, D) ≥ κ 2 (1 -δ) D∩{a(x,κ)>0} a(x, κ) 2 f H κ |B 0 (x)| a(x, κ) dx + 1 2 D∩{a(x,κ)≤0} a(x, κ) 2 dx -C r(κ, , δ, ρ, L(κ), β) , (4.5.21)
where r(κ, , δ, ρ, L(κ), β) satisfies (4.5.17). Under Assumption (4.1.15), the choice of the parameters ρ, , L(κ) in (4.3.18), δ in (4.3.19) and β in (4.5.15), implies that all error terms are of lower order compared to κ 2 . As a consequence of (4.1.15), the inequality (4.5.21) becomes as κ -→ +∞

E 0 (hψ, A; a, D) ≥ κ 2 D∩{a(x,κ)>0} a(x, κ) 2 f H κ |B 0 (x)| a(x, κ) dx + 1 2 D∩{a(x,κ)≤0} a(x, κ) 2 dx + o(κ 2 ) . (4.5.22)
Moreover, we know that E(hψ, A; a, B 0 , D) ≥ E 0 (hψ, A; a, D) .

This achieves the proof of Theorem 4.5.3.

As we now show, Theorem 4.5.3 permits to achieve the proof of two statements presented in the introduction :

Proof of Corollary 4.1.3. If (ψ, A) is a minimizer of (4.1.1), we have E g (κ, H) = E 0 (ψ, A; a, Ω) + (κH) 2 Ω | curl A -F | 2 dx , (4.5
E 0 (ψ, A; a, Ω) = κ 2 {a(x,κ)>0} a(x, κ) 2 f H κ |B 0 (x)| a(x, κ) dx + κ 2 2 {a(x,κ)≤0} a(x, κ) 2 dx + o κ 2 .
E 0 (ψ, A; a, D c ) ≥ κ 2 D c ∩{a(x,κ)>0} a(x, κ) 2 f H κ |B 0 (x)| a(x, κ) dx + 1 2 D c ∩{a(x,κ)≤0} a(x, κ) 2 dx + o(κ 2 ) . (4.5.25)
We can decompose E 0 (ψ, A; a, D) as follow :

E 0 (ψ, A; a, D) = E 0 (ψ, A; a, Ω) -E 0 (ψ, A; a, D c ) .
Using (4.5.24) and (4.5.25), we get

E 0 (ψ, A; a, D) ≤ κ 2 D∩{a(x,κ)>0} a(x, κ) 2 f H κ |B 0 (x)| a(x, κ) dx + 1 2 D∩{a(x,κ)≤0} a(x, κ) 2 dx + o(κ 2 ) . (4.5.26)

Study of examples

In this section, we will describe situations where the remainder term in (4.1.17) is indeed small as κ → +∞ compared with the leading order term

E L g (κ, H, a, B 0 ) := κ 2 {a(x,κ)>0} a(x, κ) 2 f σ |B 0 (x)| a(x, κ) dx + 1 2 {a(x,κ)≤0} a(x, κ) 2 dx , (4.6.1) where
, σ = H κ . (4.6.2)
110 Pinning with a variable magnetic field of the two dimensional Ginzburg-Landau model Note that 0 < λ min ≤ σ ≤ λ max , so that σ will be considered as an independent parameter in [λ min , λ max ]. We will also explore, case by case how one can verify Assumption (A 4 ) as formulated precisely in (4.3.1).

Proposition 4.6.1. Suppose (4.1.4) and (4.1.15) hold. Let a(x, κ) = a(x) where a(x) ∈ C 1 (Ω) is a function independent of κ and satisfies,

     {x ∈ Ω : a(x) > 0} = ∅ , or {x ∈ Ω : a(x) < 0} = ∅ . (4.6.3)
There exist positive constants C and κ 0 such that,

∀κ ≥ κ 0 , E L g (κ, H, a, B 0 ) ≥ C κ 2 .
Proof. Since a(x, κ) = a(x), the energy E L g becomes :

E L g (κ, H, a, B 0 ) := κ 2 {a(x)>0} a(x) 2 f σ |B 0 (x)| a(x) dx + 1 2 {a(x)≤0} a(x) 2 dx .
Each term being positive, it is clear that the leading term is positive if {x ∈ Ω : a(x) < 0} = ∅.

If {x ∈ Ω : a(x) < 0} = ∅ and {x ∈ Ω : a(x) > 0} = ∅, there exist ρ 0 > 0, a 0 > 0 and a disk D(x 0 , r 0 ) such that

D(x 0 , r 0 ) ⊂ {a(x) > a 0 } ∩ {|B 0 | > ρ 0 } .
Using the monotonicity of f and the bound of a(x) in (4.1.6), we may write

{a(x)>0} a(x) 2 f H κ |B 0 (x)| a(x) dx ≥ D(x 0 ,r 0 ) a(x) 2 f σ |B 0 (x)| a(x) dx ≥ π r 2 0 a 2 0 f ρ 0 a σ , (4.6.4)
where a is introduced in (4.1.10).

In particular, when (4.1.15) is satisfied, there exists κ 0 > 0 such that

∀κ ≥ κ 0 , {a(x)>0} a(x) 2 f H κ |B 0 (x)| a(x) dx ≥ π r 2 0 a 2 0 f ρ 0 a λ min . (4.6.5)
Proposition 4.6.2 (Verification of (A 4 )). Suppose that the function a satisfies (see Fig. 4.1), where Γ defined as follows :

|a| + |∇a| > 0 in Ω , ∇a × n = 0 on Γ ∩ ∂Ω , (4.6.6)
Γ = {x ∈ Ω : a(x) = 0} . (4.6.7)
Then Assumption (A 4 ) is satisfied.

Proof. From (4.6.6), we observe that,

card {γ ∈ Γ ∩ Ω with Q (γ) ∩ ∂{a > 0} = ∅} = card {γ ∈ Γ ∩ Ω with Q (γ) ∩ Γ = ∅} .
Let ∈ (0, 1), we introduce the domain

D = {x ∈ Ω : dist(x, Γ) ≤ } .
Now we give a rough upper bound for the area of D .

By assumption Γ consists of a finite number of connected curves, which are either closed in Ω or join two points of ∂Ω. Let us consider the first case, we denote by Γ (1) such a curve. We can parametrize this curve using the standard tubular coordinates (s, t), where s measures the arc-length in Γ (1) and t measures the distance to Γ (1) (see [START_REF] Fournais | Spectral Methods in Surface Superconducitivity[END_REF]Appendix F] for the detailed construction of these coordinates).

In the neighborhood of Γ (1) , we choose one point γ 0 on Γ (1) corresponding to (0, 0). Let N ∈ N and L the length of Γ (1) . We consider for i = 0, ..., N , s i = i N L (modulo LZ) and γ i = (s i , 0). Notice that, there exists a positive constant C such that,

| dist(γ i , γ i+1 )| = (1 + i )|s i -s i+1 | , - C N ≤ i < 0 .
112 Pinning with a variable magnetic field of the two dimensional Ginzburg-Landau model Thus,

x ∈ Ω : dist x, Γ (1) ≤ L N ≤ i Q L N ((s i , 0)) .
Coming back to our problem, we select N = L and we note that

L N + 1 ≤ ≤ L N ,
which implies that,

|D | ≤ L 2 N 1 + O 1 N ≤ L 1 + O 1 N = L(1 + O( )) .
Hence we have shown that, lim sup

→0 |D | ≤ L .
In a similar fashion, we prove that lim inf

→0 |D | ≥ L .
and, as a consequence, we end up with the following conclusion :

lim →0 |D | = L . (4.6.8)
Coming back to Assumption (A 4 ), we now observe that all the Q (γ) touching Γ are inside D √ 2 , hence we get, by comparison of the area

2 card {γ ∈ Γ ∩ Ω with Q (γ) ∩ Γ = ∅} ≤ C ,
and consequently, there exist positive constants C 1 , C 2 and κ 0 such that

∀κ ≥ κ 0 , ∀ ≤ C 2 κ -1 2 , card {γ ∈ Γ ∩ Ω with Q (γ) ∩ ∂{a > 0} = ∅} ≤ C 1 -1 ,
which is a stronger form of (A 4 ).

4.6.1

The case with a κ-dependent oscillation. There exists a positive constant M 0 such that if M ≥ M 0 , then,

113 in R 2 with Γ T 1 ,T 2 = T 1 Z × T 2 Z. There exists a positive constant M 0 such that if M ≥ M 0 , then, D ϕ(M x) dx = |D| T 1 T 2 T 1 0 T 2 0 ϕ(t 1 , t 2 )dt 1 dt 2 + O(M -1
D φ(M x, x) dx = D φ(x) dx + O(M -1 ) ,
where, x) where α(•) ∈ C 1 (Ω) is a Γ T 1 ,T 2 -periodic function 1 . Then the leading order term E L g defined in (4.6.1) satisfies,

φ(x) = 1 T 1 T 2 T 1 0 T 2 0 φ((t 1 , t 2 ), x) dt 1 dt 2 .
E L g (κ, H, a, B 0 ) = κ 2 Ω φ + (x) dx + κ 2 |Ω| φ -+ o(κ 2 ) , as κ → +∞ .
Here,

φ + (x) = 1 T 1 T 2 T 1 0 T 2 0 α + (t 1 , t 2 ) 2 f σ |B 0 (x)| α + (t 1 , t 2 ) dt 1 dt 2 ,
and

φ -= 1 T 1 T 2 T 1 0 T 2 0 α -(t 1 , t 2 ) 2 dt 1 dt 2 .

Proof.

We first estimate the second term in (4.6.1). We apply Lemma 4.6.3 with D = Ω, M = κ and ϕ = α 2 -, we obtain,

Ω a -(x, κ) 2 dx = |Ω| T 1 T 2 T 1 0 T 2 0 α -(t 1 , t 2 ) 2 dt 1 dt 2 + O(κ -1 2 ) , 1 see Fig. 4.2
114 Pinning with a variable magnetic field of the two dimensional Ginzburg-Landau model and consequently,

κ 2 {a(x)≤0} a(x, κ) 2 dx = κ 2 |Ω| T 1 T 2 T 1 0 T 2 0 α -(t 1 , t 2 ) 2 dt 1 dt 2 + O(κ 3
2 ) . Now, we estimate the first term in (4.6.1). We first prove that f is a Lipschitz function in [b 0 , 1] with b 0 ∈ (0, 1). We consider this restriction because when b → 0 + (see [6, Theorem 2.1]), f satisfies,

f (b) = b 2 ln 1 b (1 + o(1)
) , (4.6.12) and f is not a Lipschitz function at 0. We recall the definition of f

f (b) = lim R-→∞ e D (b, R) R 2 (∀b ∈ [0, 1]) , where e D (b, R) = inf u F +1,+1 b,Q R (u) := inf u Q R b|(∇ -iA 0 )u| 2 + 1 2 1 -|u| 2 2 dx .
From the definition, we can conclude that f is concave and hence locally Lipschitz in (0, +∞) (see [START_REF] Giaquinta | Mathematical Analysis, Foundations and Advanced Techniques for Functions of Several Variables[END_REF]Theorem 2.35]). For completion we write below a proof making explicit the Lipschitz constant. For b > 0, let u b ,R ∈ H 1 0 (Q R ) be a minimizer of F +1,+1 b ,Q R . Then for all b ∈ (0, 1), we have, Dividing by R 2 and taking the limit as R → +∞, we obtain

e D (b, R) ≤ F +1,+1 b,Q R (u b ,R ) ≤ e D (b , R) + (∇ -iA 0 )u b ,R 2
L 2 (Q R ) |b -b | .
f (b) ≤ f (b ) + | f (b )| b |b -b | .
Using the asymptotic behavior of f in (4.6.12) as b → 0 + , we finally obtain the existence of C such that

f (b) ≤ f (b ) + C log 1 b 0 |b -b | , ∀b, b with 1 > b > b 0 and 1 > b > b 0 .
Exchanging b and b , we have proved the Lemma 4.6.6. f is locally Lipschitz in (0, +∞). More precisely, there exists C such that for To continue, we consider

R 2 × Ω ρ (t, x) → φ(t, x) = α + (t) 2 f σ |B 0 (x)| α + (t) ,
where, Ω ρ := Ω ∩ {|B 0 | > ρ}.

The periodicity condition in (4.6.9) is clear. Let us verify the Lipschitz property. Let

b 0 = λ min α 0 ρ ,
where, λ min is introduced in (4.1.15) and α 0 = sup α + (t).

Let > 0, I + = {t ∈ R : α + (t) ≥ } and I -= {t ∈ R : α + (t) ≤ }, we distinguish between two cases : Case 1 : (α + (t) ≥ ). We observe that for (x, t) ∈ Ω ρ × I + , we have

b 0 ≤ σ |B 0 (x)| α + (t) ≤ σ |B 0 (x)| .
Thus, for any t ∈ I + and for any x, x ∈ Ω ρ , we get

α + (t) 2 f σ |B 0 (x)| α + (t) -α + (t) 2 f σ |B 0 (x )| α + (t) = α + (t) 2 | f (b) -f b | ≤ C log 1 ρ |B 0 (x)| -|B 0 (x )| . (4.6.15)
Therefore, using also the Lipschitz property for x → |B 0 (x)|, we get that Ω ρ x → φ(t, x) is uniformly Lipschitz for t ∈ I + . Case 2 : (α + (t) ≤ ). We observe that for (x, t) ∈ Ω ρ × I -,

σ |B 0 (x)| α + (t) ≥ σ |B 0 (x)| .
We note that f (b) = 1 2 , ∀b ≥ 1 (see [START_REF] Fournais | The ground state energy of the three dimensional Ginzburg-Landau functional Part I : Bulk regime[END_REF]Theorem 2.1]). For this reason we choose = λ min 2 ρ , which implies that for (x, t) ∈ Ω ρ × I -,

σ |B 0 (x)| α + (t) ≥ 2 and f σ |B 0 (x)| α + (t) = 1 2 .
Thus, for any t ∈ I -and for any x, x ∈ Ω ρ , we get

α + (t) 2 f σ |B 0 (x)| α + (t) -α + (t) 2 f σ |B 0 (x )| α + (t) = α + (t) 2 2 - α + (t) 2 2 = 0 . ( 4
Ωρ a + (x, κ) 2 f σ |B 0 (x)| a + (x, κ) dx = Ωρ φ(x) dx + O ρ (κ -1 2 ) , (4.6.17)
where φ is introduced in (4. 6.11).

Coming back to the integral over Ω, we get, for any ρ ∈ (0, ρ 0 ) and for any κ ≥ κ 0 with ρ 0 small enough and κ 0 large enough, 2 ) . x and γ = κ

Ω a + (x, κ) 2 f σ |B 0 (x)| a + (x, κ) dx = Ω φ(x) dx + O(ρ) + O ρ (κ - 1
1 2 γ yields, card {γ ∈ Γ ∩ Ω with Q (γ) ∩ ∂{x ∈ Ω : a(x, κ) > 0} = ∅} = card {γ ∈ Γ κ 1 2 ∩ κ 1 2 Ω with Q κ 1 2 (γ ) ∩ Γ = ∅} , where, Γ = {y ∈ R 2 | α(y) = 0} .
Let ∈ (0, 1), we introduce the domain

D ,M = {y ∈ M • Ω : dist(y, Γ) ≤ } .
Thanks to (4.6.8) and the periodicity assumption, we get the existence of positive constants C, M 0 and 0 such that, for any

∈ (0, 0 ), M ≥ M 0 | D ,M | ≤ C M .
In the sequel, we choose M = κ 1 2 and = M √ 2 . We note that, there exist constants c > 0 and κ 0 > 0 such that, ∀κ ≥ κ 0 , ∀ ≤ c κ -1 2 , 0 < ≤ 0 .

We now observe that all the Q

κ 1 2 (γ) touching Γ are inside D κ 1 2 √ 2 ,κ 1
2 , hence we get, by comparison of the areas

κ 2 card {γ ∈ Γ κ 1 2 ∩ κ 1 2 Ω with Q κ 1 2 (γ ) ∩ Γ κ = ∅} ≤ C √ 2 κ .
There exist positive constants C 1 and C 2 , such that,

∀κ ≥ κ 0 , ∀ ≤ C 2 κ -1 2 , card {γ ∈ Γ ∩ Ω with Q (γ) ∩ ∂{x ∈ Ω : a(x, κ) > 0} = ∅} ≤ C 1 -1 .
4.6.1.3 Second example.

This example was considered by Aftalion, Sandier and Serfaty (see (H 2 )).

∀κ ≥ κ 0 , E L g (κ, H, a, B 0 ) ≥ τ 1 κ 2 .
118 Pinning with a variable magnetic field of the two dimensional Ginzburg-Landau model

Proof. We can write,

κ 2 {a(x,κ)>0} a(x, κ) 2 f H κ |B 0 (x)| a(x, κ) dx ≥ κ 2 {a(x)>0} a(x, κ) 2 f H κ |B 0 (x)| a(x, κ) dx ≥ κ 2 {a(x)>0} a(x) 2 f H κ |B 0 (x)| a dx .
(4.6.20)

Here we have used that f is increasing, the nonnegativity of β to get a(x, κ) ≥ a(x), Assumption (A 2 ) to estimate f from below, and {a(x) > 0} ⊂ {a(x, κ) > 0}.

Proceding like in (4.6.4), there exist τ 1 > 0 and κ 0 > 0 such that, This example is similar to the previous example, but here we suppose that

∀κ ≥ κ 0 , κ 2 {a(x,κ)>0} a(x, κ) 2 f H κ |B 0 (x)| a(x, κ) dx ≥ τ 1 κ 2 . ( 4
β(x, κ) = α(κ 1 2 x) , where α(•) is a Γ T 1 ,T 2 -periodic positive function in R 2 .
Proposition 4.6.9. Suppose that (4.1.4) and (4.1.15) hold. Let a(x, κ) = a(x) + α(κ

1 2 x), where α(•) is a Γ T 1 ,T 2 -periodic positive bounded function in R 2 , a(•) ∈ C 1 (Ω) and {a < 0} ∩ Ω = ∅.
Then the leading order term E L g defined in (4.6.1) satisfies,

E L g (κ, H, a, B 0 ) = κ 2 Ω φ(x) dx + o(κ 2 ) , as κ → +∞ .
Here,

φ(x) = 1 T 1 T 2 T 1 0 T 2 0 (a(x) + α(t 1 , t 2 )) 2 f σ |B 0 (x)| a(x) + α(t 1 , t 2 ) dt 1 dt 2 .
The proof of Proposition 4.6.9 is similar to that of Proposition 4.6.5.

4.6.2 Upper bound of the main term.

It is easy to show that E L g is less than Cκ 2 for some C > 0. Indeed, using the bound of a in (4.1.6) and the bound f (b) ≤ 1 2 , we have,

κ 2 {a(x,κ)>0} a(x, κ) 2 f H κ |B 0 (x)| a(x, κ) dx ≤ Cκ 2 ,
and

κ 2 2 {a(x,κ)≤0} a(x, κ) 2 dx ≤ Cκ 2 .

Proof of Theorem 4.1.5

The technique that will be used in this proof has been introduced by Helffer-Kachmar in [START_REF] Helffer | The Ginzburg-Landau functional with vanishing magnetic field[END_REF] for the case a(x, κ) = 1. The proof is decomposed into three steps :

Step 1 : Case D = Ω . Let (ψ, A) be a solution of (4.1.12). Thanks to (4.4.3), we have,

Ω |(∇ -iκHA)ψ| 2 dx = κ 2 Ω (a(x, κ) -|ψ| 2 )|ψ| 2 dx = κ 2 2 Ω (a(x, κ) 2 -|ψ(x)| 4 ) dx - κ 2 2 Ω (a(x, κ) -|ψ| 2 ) 2 dx .
Having in mind the definition of E 0 (ψ, A; a, Ω), we get, Step 2 : Upper bound.

κ 2 2 Ω (a(x, κ) 2 -|ψ(x)| 4 ) dx = E 0 (ψ, A; a, Ω) . ( 4
Let D ⊂ Ω be a regular domain and, for ∈ (0, 1),

D = {x ∈ D : dist(x, ∂D) ≥ } . (4.7.4) We introduce a cut-off function χ ∈ C ∞ c (R 2 ) such that 0 ≤ χ ≤ 1 in R 2 , suppχ ⊂ D , χ = 1 in D and |∇χ | ≤ C in R 2 , (4.7.5)
where C is a positive constant. We multiply both sides of (4.1.12) a by χ 2 ψ. It results from an integration by parts that

D |(∇ -iκHA)χ ψ| 2 -κ 2 a χ 2 |ψ| 2 + κ 2 χ 2 |ψ| 4 dx = D |∇χ | 2 |ψ| 2 dx = O( -1 ) . (4.7.6)
Here, we have used the fact that

|∇ χ | 2 = O( -2 ), |D | = O(
) and the bound of ψ in (4.4.9). We notice that χ 4 ≤ χ 2 ≤ 1. We add to both sides the term κ 2 2 D a 2 dx to obtain,

D |(∇ -iκHA)χ ψ| 2 + κ 2 2 a 2 -κ 2 a |χ ψ| 2 + κ 2 |χ ψ| 4 dx ≤ C -1 + κ 2 2 D a 2 dx .
This implies that

E 0 (χ ψ, A; a, D) ≤ κ 2 2 D (a 2 -χ 4 |ψ| 4 ) dx + C -1 .
Using (4.7.5), we get

D |ψ| 4 dx = D χ 4 |ψ| 4 dx + D (1 -χ 4 )|ψ| 4 dx ≤ D χ 4 |ψ| 4 dx + C , (4.7.7)
and consequently,

E 0 (χ ψ, A; a, D) ≤ κ 2 2 D (a 2 -|ψ| 4 ) dx + C( -1 + ) . ( 4
D c |ψ| 4 dx ≤ - D c ∩{a(x,κ)>0} a(x, κ) 2 2 f H κ |B 0 (x)| a(x, κ) -1 dx + o (1) . (4.7.12)
Step 3 : Lower bound. We can decompose D |ψ| 4 dx as follows :

Extension of the Giorgi-Phillips Theorem

In this section we extend a result of Giorgi-Phillips [START_REF] Giorgi | The breakdown of superconductivity due to strong fields for the Ginzburg-Landau model[END_REF], in the two cases when the external magnetic field B 0 is variable (i.e. Γ = ∅) and when the external magnetic field B 0 is constant (i.e. Γ = ∅), with a pinning term. We recall that the normal solution (0, F) is a trivial solution of the Ginzburg-Landau system (4.1.12). We will show that this solution is a global minimizer, when κ and H are sufficiently large. We first establish a priori estimates for a critical point (ψ, A) of the G-L-functional.

4.8.1

Estimates of A and of (∇ -iκHF)ψ .

We need the following estimates on A and on (∇ -iκHF)ψ which are independent of the assumption of Γ.

Theorem 4.8.1. There exist positive constants C 1 , C 2 and C 3 such that, if (4.1.6) hold, κ > 0, H > 0 and (ψ, A) is a solution of (4.1.12), then,

(∇ -iκHA)ψ L 2 (Ω) ≤ C 1 κ ψ L 2 (Ω) , (4.8.1) curl(A -F) L 2 (Ω) ≤ C 2 H ψ L 2 (Ω) ψ L 4 (Ω) , (4.8.2)
(∇ -iκHF)ψ L 2 (Ω) ≤ C 3 κ ψ L 2 (Ω) . (4.8.3)
Proof. We first prove (4.8.1). In the case when a ≤ 0 with a introduced in (4.1.10), we get using (4.4.9) that ψ = 0 and hence (4.8.1) is proved.

In the case when a > 0, thanks to (4.4.9), we have,

0 ≤ (a -|ψ| 2 ) ≤ a . (4.8.4)
We recall that if (ψ, A) is a solution of (4.1.12) then, (see (4.4.3))

Ω |(∇ -iκHA)ψ| 2 dx = κ 2 Ω (a(x, κ) -|ψ| 2 )|ψ| 2 dx .
Using (4.1.6) and (4.8.4), we obtain (4.8.1). Now, we prove (4.8.2). We obtain from the equation in (4.1.12) b the following estimate (see [START_REF] Fournais | Spectral Methods in Surface Superconducitivity[END_REF]Equation (11.9b)]) :

κH Ω | curl(A -F)| 2 dx ≤ (∇ -iκHA)ψ L 2 (Ω) (A -F)ψ L 2 (Ω) .
Using (4.8.1) and applying Hölder's inequality, we get

κH Ω | curl(A -F)| 2 dx ≤ C κ ψ L 2 (Ω) ψ L 4 (Ω) A -F L 4 (Ω) .
We get by regularity of the curl-div system (see [14, A.7]),

A -F H 1 (Ω) ≤ C curl(A -F) L 2 (Ω) , (4.8.5)
where C is a positive constant. By the Sobolev embedding theorem, we get,

A -F L 4 (Ω) ≤ C Sob A -F H 1 (Ω) ≤ C curl(A -F) L 2 (Ω) . (4.8.6)
Consequently,

κH Ω | curl(A -F)| 2 dx ≤ κ ψ L 2 (Ω) ψ L 4 (Ω) curl(A -F) L 2 (Ω) ,
which leads to (4.8.2).

(A -F)ψ 2 L 2 (Ω) ≤ A -F 2 L 4 (Ω) ψ 2 L 4 (Ω) ≤ C H 2 ψ 4 L 4 (Ω) ψ 2 L 2 (Ω) , (4.8.7)
Using (4.8.1), (4.8.7) and the bound of ψ above, Young's inequality gives,

(∇ -iκHF)ψ 2 L 2 (Ω) ≤ 2 (∇ -iκHA)ψ 2 L 2 (Ω) + 2 (κH) 2 (A -F)ψ 2 L 2 (Ω) ≤ 2 C κ 2 ψ 2 L 2 (Ω) . (4.8.8)
4.8.2 The case Γ = ∅.

For ξ ∈ R, we consider the Neumann realization h N,ξ in L 2 (R + ) associated with the operator

-d 2 dt 2 + (t + ξ) 2 , i.e. h N,ξ := - d 2 dt 2 + (t + ξ) 2 , D(h N,ξ ) = {u ∈ B 2 (R + ) : u (0) = 0} , (4.8.9)
where, B 2 (R + ) = {u ∈ L 2 (R + ) : t p u (q) ∈ L 2 (R + ), ∀p, q ∈ N s.t. p + q ≤ 2} .

M. Dauge and B. Helffer [START_REF] Dauge | Eigenvalues variation, I, Neumann problem for Sturm-Liouville operators[END_REF] (see also Fournais-Helffer [14, Proposition 4.2.2]) have proved that the lowest eigenvalue µ of h N,ξ admits a minimum Θ 0 , which is attained at a unique point ξ 0 < 0, and satisfies :

Θ 0 = inf ξ µ(ξ) = µ(ξ 0 ) < 1 . ( 4
µ N (BF; Ω) = inf ψ∈H 1 (Ω) ψ =0 P Ω BF,0 ψ, ψ ψ 2 L 2 (Ω)
.

(4.8.13)

In [START_REF] Fournais | Spectral Methods in Surface Superconducitivity[END_REF], it is proved that In [START_REF] Pan | Schrödinger operators with non-degenerately vanishing magnetic fields in bounded domains[END_REF], it is proved that In the next theorem, we give a simple proof of the result which says that (0, F) is the unique minimizer of the functional when H is sufficiently large and when B 0 is variable. This result was obtained in [START_REF] Giorgi | The breakdown of superconductivity due to strong fields for the Ginzburg-Landau model[END_REF] for the case with constant magnetic field and with a constant pinning term.

Theorem 4.8.5. Let Ω ⊂ R 2 be a smooth, bounded, simply-connected open set, the pinning term a satisfying (4.1.6), and the magnetic field B 0 satisfying (4.1.4). Then, there exist positive constants C and κ 0 , such that, if

H ≥ Cκ 2 , κ ≥ κ 0 .
Then (0, F) is the unique solution to (4.1.12).

Proof. Similarly to the proof of Theorem 4.8.3, we assume that we have a non normal critical point (ψ, A) for E κ,H,a,B 0 . Therefore, we get from (4.8.3) that,

µ N (BF; Ω) ≤ C κ 2 (B = κH) .
Thanks to Theorem 4.8.4, we get a contradiction, if H ≥ Cκ 2 and C is sufficiently large.

4.9 Asymptotics of µ 1 (κ, H) : the case with non vanishing magnetic field

The aim of this section is to give an estimate for the lowest eigenvalue µ 1 (κ, H) of the operator P Ω κHF,-κ 2 a (see (4.1.26)) in the case when Γ = ∅ with a κ-independent pinning (i.e. a(x, κ) = a(x)). Recall that the set Γ is introduced in (4. 1.3).

Without loss of generality we suppose that B 0 > 0 in Ω. Our results will be formulated 126 Pinning with a variable magnetic field of the two dimensional Ginzburg-Landau model by introducing :

Λ 1 (B 0 , a, σ) = min inf x∈Ω {σ B 0 (x) -a(x)} , inf x∈∂Ω {Θ 0 σ B 0 (x) -a(x)} , (4.9.1)
where σ is a positive constant.

In the case when the pinning term is constant (i.e. a(x) = a 0 ), (4.9.1) becomes as follows :

Λ 1 (B 0 , a, σ) = σ min inf x∈Ω {B 0 (x)} , Θ 0 inf x∈∂Ω {B 0 (x)} -a 0 .
This case was treated by Pan and Kwek [START_REF] Lu | Eigenvalue problems of Ginzburg-Landau operator in bounded domains[END_REF].

Let Q Ω B F,-B
σ a be the quadratic form of P Ω BF,-B σ a , i.e.

Q Ω B F,-B σ a (ψ) = Ω |(∇ -iBF)ψ| 2 - B σ a(x)|ψ| 2 dx . ( 4
∈ I, B ≥ B 0 , ψ ∈ H 1 (Ω) \ {0} and a ∈ C 1 (Ω), then, Q Ω BF,-B σ a (ψ) ψ 2 L 2 (Ω) ≥ B σ Λ 1 (B 0 , a, σ) -C B 3 4 . (4.9.3)
Proof. The proof is a consequence of the following inequality that we take from [14, Prop. 9.2.1],

∀ ψ ∈ H 1 (Ω) , Ω |(∇ -iBF)ψ| 2 dx ≥ Ω U (x) -CB 3/4 |ψ| 2 dx ,
where

U (x) = B B 0 (x) if dist(x, ∂Ω) ≥ B -3/8 , Θ 0 B B 0 (x) if dist(x, ∂Ω) < B -3/8 , (4.9.4)
B ≥ B0 , B0 and C are two constants independent of B.

Clearly, there exist two constants C > 0 and B 0 > 0 such that, for all σ ∈ I, we have,

U (x) - B σ a(x) ≥ B σ Λ 1 (B 0 , a, σ) -C B 3/4 .
Coming back to our initial parameters κ and H, we obtain : Theorem 4.9.2. Let Ω ⊂ R 2 be an open bounded set with smooth boundary and Γ = ∅. Suppose that (4.1.15) holds and a ∈ C 1 (Ω), then,

µ 1 (κ, H) ≥ κ 2 Λ 1 B 0 , a, H κ + O(κ 3
2 ) , as κ → +∞ .

Here, Λ 1 is introduced in (4.9.1). Let us verify that the conditions of the proposition are satisfied for this choice.

It is trivial that σ ∈ I. Now, as κ → +∞, we have,

B = σ κ 2 → +∞ .
This implies that, as κ → +∞,

µ 1 (κ, H) ≥ κ 2 Λ 1 B 0 , a, H κ + O(κ 3
2 ) .

This finishes the proof of the theorem.

Upper bound

Proposition 4.9.3 (Upper bound in the bulk). Suppose that Ω ⊂ R 2 is an open bounded set with smooth boundary ∂Ω, λ max > 0 and Γ = ∅. For any x 0 ∈ Ω, there exist positive constants C and B 0 such that, if σ ∈ (0, λ max ], B ≥ B 0 and a ∈ C 1 (Ω), then,

µ B,σ ≤ B σ {σ B 0 (x 0 ) -a(x 0 )} + C B 1 2 .
(4.9.5)

Here,

µ B,σ = inf ψ∈H 1 (Ω)\{0} Q Ω BF,-B σ a (ψ) ψ 2 L 2 (Ω) , (4.9.6)
where Q Ω BF,-B σ a is introduced in (4.9.2).

Proof. Thanks to (4.9.2), we have,

Q Ω BF,-B σ a (u) = Ω |(∇ -iBF)u(x)| 2 dx - B σ Ω a(x)|u(x)| 2 dx .
The upper bound of the first term in the right hand side above is based on the construction of Gaussian quasi-mode (see [14, Subsection 2.4.2] for the case with constant pinning) centered at

x 0 ∈ Ω, ϕ 1 (x) = e i B φ 0 χ B + 1 2 (x -x 0 ) u BB 0 (x 0 ) (x -x 0 ) .
Here, χ is a cut-off function equal to 1 in a neighborhood of 0 such that supp χ ⊂ D(0, 1), the function φ 0 satisfies (4.3.4) and the function u defined as follows :

u(x) = π -1 4 √ 2 e -|x| 2 2 .
We note that supp ϕ 1 ⊂ Ω for B large enough. As in [14, (2.35)], we get the existence of a 128 Pinning with a variable magnetic field of the two dimensional Ginzburg-Landau model positive constant B 0 such that, for any B ≥ B 0 ,

Ω |(∇ -iBF)ϕ 1 (x)| 2 dx Ω |ϕ 1 (x)| 2 dx ≤ B B 0 (x 0 ) + O(B 1
2 ) . (4.9.7)

To derive the upper bound for the second term, we use Taylor's formula for the function a near

x 0 , |a(x) -a(x 0 )| ≤ C B -1 2 , x ∈ D x 0 , B -1 2 . ( 4
.9.8)

Using (4.9.8), since supp ϕ 1 ⊂ D x 0 , B -1 2 , we get,

- Ω a(x)|ϕ 1 (x)| 2 dx ≤ -a(x 0 ) Ω |ϕ 1 (x)| 2 dx + C B -1 2 Ω |ϕ 1 (x)| 2 dx , (4.9.9)
and consequently

- B σ Ω a(x)|ϕ 1 (x)| 2 dx Ω |ϕ 1 (x)| 2 dx ≤ - B σ a(x 0 ) + C B 1 2 . ( 4
.9.10)

Collecting (4.9.7) and (4.9.10), we finish the proof of Proposition 4.9.6.

Remark 4.9.4. When

inf x∈Ω {σ B 0 (x) -a(x)} < inf x∈∂Ω {Θ 0 σ B 0 (x) -a(x)} ,
we notice that, if the infimum of σ B 0 (x) -a(x) was attained on ∂Ω, (i.e. there exists x 0 ∈ ∂Ω such that inf x∈Ω {σ B 0 (x) -a(x)} = σ B 0 (x 0 ) -a(x 0 )), we would have,

σ B 0 (x 0 ) -a(x 0 ) < Θ 0 σ B 0 (x 0 ) -a(x 0 ) ,
which is impossible, since Θ 0 < 1. Hence, we can choose x 0 ∈ Ω, such that,

σ B 0 (x 0 ) -a(x 0 ) = inf x∈Ω {σ B 0 (x) -a(x)} ,
and we apply Proposition 4.9.3 with B = κH and σ = H κ .

Thus, we get the existence of a positive constant κ 0 such that, if, κ ≥ κ 0 and κ 0 κ -1 < H < λ max κ , (4.9.11) then,

µ 1 (κ, H) ≤ κ 2 inf x∈Ω H κ B 0 (x) -a(x) + O(κ) , as κ → +∞ . ( 4
.9.12) Proposition 4.9.5 (Upper bound near the boundary). Suppose that Ω ⊂ R 2 is an open bounded set with a smooth boundary, λ max > 0 and Γ = ∅. For any x 0 ∈ ∂Ω and for any σ ∈ (0, λ max ],

then µ 1 (κ, H) < 0 .

Remark 4.9.9. The conclusion in Lemma 4.9.8 is valid in both cases where Γ = ∅ and Γ = ∅.

Proof of Lemma 4.9.8.

Let > 0. Choose x 0 ∈ Ω such that a(x 0 ) > 0. We introduce a cut-off function χ ∈ C ∞ c (R 2 ) satisfying : 0 ≤ χ ≤ 1 in R 2 , suppχ ⊂ B(x 0 , ) , χ = 1 in B (x 0 , /2) and |∇χ | ≤ C . (4.9.15)
The min-max principle yields,

µ (1) (κ, H) χ 2 L 2 (Ω) ≤ Ω |(∇ -iκHF)χ | 2 dx -κ 2 Ω a(x)|χ (x)| 2 dx .
Using the assumptions on χ and the fact that

F ∈ C ∞ (Ω), a trivial estimate is, Ω |(∇ -iκHF)χ | 2 dx = B(x 0 , ) |∇χ (x)| 2 dx + κ 2 H 2 B(x 0 , ) |F χ (x)| 2 dx ≤ C (1 + (κ H ) 2 ) . (4.9.16)
We write by Taylor's formula applied to the function a near x 0 ,

-κ 2 Ω a(x)|χ (x)| 2 dx ≤ -a(x 0 ) κ 2 2 + C κ 2 3 . ( 4
.9.17)

Collecting (4.9.16) and (4.9.17), we obtain,

µ (1) (κ, H) χ 2 L 2 (Ω) ≤ -a(x 0 ) κ 2 2 + C(κ 2 3 + 1 + (κ H ) 2 ) .
We select = κ -1 2 and note that κH < C max . We find that,

µ (1) (κ, H) χ 2 L 2 (Ω) ≤ -a(x 0 ) κ + C κ 1 2 + 1 + C 2 max κ -1 .
Since χ = 0 and a(x 0 ) > 0, we deduce that, for κ sufficiently large, µ

(κ, H) < 0 .

0 ≥ 0 such that if κ ≥ κ 0 , H ≤ κ max sup x∈Ω a(x) B 0 (x) , sup x∈∂Ω a(x) Θ 0 B 0 (x) -C κ 1 2 , (4.10.1) then, µ 1 (κ, H) < 0 . Moreover, κ max sup x∈Ω a(x) B 0 (x) , sup x∈∂Ω a(x) Θ 0 B 0 (x) -C κ 1 2 ≤ H loc C 3 .
Proof. To apply the previous results, we take

λ max = max sup x∈Ω a(x) B 0 (x) , sup x∈∂Ω a(x) Θ 0 B 0 (x) + 1 .
We have two cases :

Case 1. If sup x∈Ω a(x) B 0 (x) > sup x∈∂Ω a(x) Θ 0 B 0 (x)
.

then, there exists x 0 ∈ Ω (the supremum of a(x) B 0 (x) can not be attained on the boundary, since a(x)

Θ 0 B 0 (x) > a(x) B 0 (x) ), such that, sup x∈Ω a(x) B 0 (x) = a(x 0 ) B 0 (x 0 ) .
If (4.10.1) is satisfied for some C > 0, then,

H κ ≤ a(x 0 ) B 0 (x 0 ) -C κ -1 2 ,
that we can write in the form,

κ 2 H κ B 0 (x 0 ) -a(x 0 ) ≤ -C M κ 3 2 ,
where M > 0 is a constant independent of C. Suppose that κH ≥ B 0 where B 0 is selected sufficiently large such that we can apply Remark 4.9.4. (Thanks to Lemma 4.9.8, µ 1 (κ, H) < 0 when κH < B 0 ). Remark 4.9.4 tells us that there exist positive constants C 1 and κ 0 such that, for κ ≥ κ 0 , By compactness, there exists x 0 ∈ ∂Ω, such that,

µ 1 (κ, H) ≤ κ 2 inf x∈Ω H κ B 0 (x) -a(x) + C 1 κ ≤ κ 2 H κ B 0 (x 0 ) -a(x 0 ) + C 1 κ 3 2 (4.10.2) ≤ (C 1 -C M ) κ 3 2 . ( 4
sup x∈∂Ω a(x) Θ 0 B 0 (x) = a(x 0 ) Θ 0 B 0 (x 0 ) If (4.10.1) is satisfied for some C > 0, then, κ 2 H κ Θ 0 B 0 (x 0 ) -a(x 0 ) ≤ -C M κ 3 2 .
Thanks to Remark 4.9.6, we get the existence of positive constants C 2 and κ 0 such that, for κ ≥ κ 0 , This finishes the proof of the proposition.

µ 1 (κ, H) ≤ κ 2 inf x∈∂Ω H κ Θ 0 B 0 (x) -a(x) + C 2 κ ≤ κ 2 H κ Θ 0 B 0 (x 0 ) -a(x 0 ) + C 2 κ 3
Proposition 4.10.2. Suppose that {a > 0} = ∅, λ max > 0 and Γ = ∅. There exist constants C > 0 and κ 0 > 0 such that if

κ ≥ κ 0 , λ max κ ≥ H > κ max sup x∈Ω a(x) B 0 (x) , sup x∈∂Ω a(x) Θ 0 B 0 (x) + C κ 1 2 , (4.10.6) then, µ 1 (κ, H) > 0 .
Moreover,

H loc C 3 ≤ κ max sup x∈Ω a(x) B 0 (x) , sup x∈∂Ω a(x) Θ 0 B 0 (x) + C κ 1 2 .
Proof. To apply the previous results, we take

λ min = 1 2 max sup x∈Ω a(x) B 0 (x) , sup x∈∂Ω a(x) Θ 0 B 0 (x)
.

If (4.10.6) holds for some C > 0, then, for any x ∈ Ω, we have,

H κ B 0 (x) -a(x) ≥ C κ -1 2 , (4.10.7)
and, for any x ∈ ∂Ω, we have,

H κ Θ 0 B 0 (x ) -a(x ) ≥ C κ -1 2 . (4.10.8)
Having in mind the definition of Λ 1 in (4.9.1), the estimates in (4.10.7) and in (4.10.8) give us that for κ large enough,

Λ 1 B 0 , a, H κ ≥ C κ -1 2 .
Thanks to Theorem 4.9.2, we get the existence of positive constants C and κ 0 such that, for κ ≥ κ 0 , In this subsection we give a lower bound of the critical field H cp C 3 (see (4.1.27)) and we give an upper bound of the critical field H cp C 3 in the case when the magnetic field B 0 is constant with a pining term. We start with a proposition which measures the effect of the localization at the boundary when H is sufficiently large. Proposition 4.10.3. Suppose that Γ = ∅ and (4.10.6) holds. There exists a positive constant C, such that if (ψ, A) is a solution of (4.1.12), then the following estimate holds : Next, we define λ = κ -3 4 , and χ κ as follows :

µ 1 (κ, H) ≥ κ 2 Λ 1 B 0 , a, H κ -C κ 3 2 ≥ (C -C ) κ 3 2 . ( 4
ψ 2 L 2 (Ω) ≤ C κ -3 8 ψ 2 L 4 (Ω) . ( 4
χ κ (x) = χ dist(x, ∂Ω) λ , ∀x ∈ Ω . ( 4
|(∇ -iκHA)χ κ ψ| 2 dx ≥ κ H Ω curl F |χ κ ψ| 2 dx -κ H curl(A -F) L 2 (Ω) χ κ ψ 2 L 4 (Ω) . Noticing that curl F = B 0 (x) and curl(A -F) L 2 (Ω) ≤ c H ψ L 2 (Ω) , we have, Ω |(∇ -iκHA)χ κ ψ| 2 dx ≥ κ H Ω B 0 (x) |χ κ ψ| 2 dx -c κ ψ L 2 (Ω) χ κ ψ 2 L 4 (Ω) .
Implementing a Cauchy-Schwarz inequality, we get

Ω |(∇ -iκHA)χ κ ψ| 2 dx ≥ κ H Ω B 0 (x) |χ κ ψ| 2 dx -c 2 ψ 2 L 2 (Ω) -κ 2 χ κ ψ 4 L 4 (Ω) . ( 4
Ω κ H B 0 (x) -κ 2 a(x) |χ κ ψ| 2 dx ≤ c 2 Ω |ψ| 2 dx+ Ω |∇χ κ | 2 |ψ| 2 dx-κ 2 Ω χ 2 κ -χ 4 κ |ψ| 4 dx .
As a consequence of (4.10.11), the inequality above becomes,

C κ 3 2 |χ κ ψ(x)| 2 dx ≤ c 2 Ω |ψ| 2 dx + Ω |∇χ κ | 2 |ψ| 2 dx -κ 2 Ω χ 2 κ -χ 4 κ |ψ| 4 dx . Notice that -κ 2 Ω χ 2 κ -χ 4 κ |ψ| 4 dx ≤ 0 . Decomposing the integral Ω |ψ| 2 dx = Ω |χ κ ψ| 2 dx + Ω (1 -χ 2
κ )|ψ| 2 dx, using (4.10.11) and choosing C such that C ≥ 2c 2 , we get, Recall that λ = κ -3 4 , we observe that,

1 2 C κ 3 2 |χ κ ψ(x)| 2 dx ≤ c 2 + χ 2 L ∞ (R) λ -2 {x∈Ω: dist(x,Γ)≤λ} |ψ| 2 dx .
|χ κ ψ(x)| 2 dx ≤ 4 χ 2 L ∞ (R) {x∈Ω: dist(x,Γ)≤λ} |ψ| 2 dx ,
and consequently, we get,

|ψ(x)| 2 dx ≤ 4 χ 2 L ∞ (R) + 1 {x∈Ω: dist(x,Γ)≤λ} |ψ| 2 dx . By choosing C = max 2 c 2 , 4 χ 2 L ∞ (R) + 1 , we obtain, ψ 2 L 2 (Ω) ≤ C κ -3 8 ψ 2 L 4 (Ω) .
Theorem 4.10.4. Supose that Γ = ∅ and {a > 0} = ∅. There exists C > 0 and κ 0 such that, if H satisfies

H > κ max sup x∈Ω a(x) B 0 (x) , sup x∈∂Ω a(x) Θ 0 B 0 (x) + C κ 1 2 , (4.10.16)
then (0, F) is the unique solution to (4.1.12). Moreover,

H cp C 3 ≤ κ max sup x∈Ω a(x) B 0 (x) , sup x∈∂Ω a(x) Θ 0 B 0 (x) + C κ 1 2 .
Proof. We first observe that it results from Giorgi-Phillips like Theorem 4.8.3 that it remains only to prove the theorem under the stronger Assumption (4.10.6). Suppose now that (ψ, A) is a solution of (4.1.12) with ψ = 0, we observe that,

0 < κ 2 ψ 4 L 4 (Ω) = - Ω |(∇ -iκHA)ψ| 2 -κ 2 a(x)|ψ| 2 dx := . ( 4
.10.17)

We can write,

-≥ (1 - √ κ -1 ) Ω |(∇ -iκHF)ψ| 2 dx -κ 2 Ω a(x)|ψ| 2 dx - (κH) 2 √ κ -1 Ω |(A -F)ψ| 2 dx ≥ µ 1 (κ, H) ψ 2 L 2 (Ω) - √ κ -1 (∇ -iκHF)ψ 2 L 2 (Ω) - (κH) 2 √ κ -1 (A -F)ψ 2 L 2 (Ω) .
(4.10.18)

We reffer to (4.8.3) and (4.8.7), we have, Having in mind that ψ = 0 and > 0 (see (4.10.17)), we deduce for κ sufficiently large µ 1 (κ, H) < 0, which is in contradiction with Proposition 4.10.2. Therefore, we conclude that ψ = 0, which is what we needed to prove. Proposition 4.10.5. Supose that Γ = ∅ and {a > 0} = ∅. There exists C > 0 and κ 0 such that, if H satisfies

-≥ µ 1 (κ, H) ψ 2 L 2 (Ω) -C √ κ ψ 2 L 2 (Ω) . ( 4
H ≤ κ max sup x∈Ω a(x) B 0 (x) , sup x∈∂Ω a(x) Θ 0 B 0 (x) -C κ 1 2 , (4.10.22)
then there exists a solution (ψ, A) of (4.1.12) with ψ L 2 (Ω) = 0. Moreover,

κ max sup x∈Ω a(x) B 0 (x) , sup x∈∂Ω a(x) Θ 0 B 0 (x) -C κ 1 2 ≤ H cp C 3 .
Proof. We use (tψ * , F), with t sufficiently small and ψ * an eigenfunction associated with µ 1 (κ, H), as a test configuration for the functional (4.1.1), i.e.

Ω |(∇ -iκHF)ψ * | 2 -κ 2 a(x)|ψ * | 2 dx = µ 1 (κ, H) ψ * 2 L 2 (Ω) .
Proposition 4.10.1 tells us that there exists a constant C such that, under Assumption (4.10.22), µ 1 (κ, H) < 0 . Therefore,

C 1 (κ, H) := Ω |(∇ -iκHF)ψ * | 2 -κ 2 a(x)|ψ * | 2 dx < 0 .
We can write,

E κ,H,a,B 0 (tψ * , F) = t 2 Ω |(∇ -iκHF)ψ * | 2 -κ 2 a(x)|ψ * | 2 dx + t 4 κ 2 2 Ω |ψ * | 4 dx + κ 2 2 Ω a(x) dx = t 2 C 1 (κ, H) + t 2 κ 2 2 Ω |ψ * | 4 dx + E κ,H,a,B 0 (0, F) .
We choose t such that,

C 1 (κ, H) + t 2 κ 2 2 Ω |ψ * | 4 dx < 0 .
Thus, we get E κ,H,a,B 0 (tψ * , F) < E κ,H,a,B 0 (0, F) .

Hence a minimizer, which is a solution of (4.1.12), will be non-trivial. First, we will prove the following inclusion,

N loc (κ) ⊂ N (κ) .
We see that if H / ∈ N (κ), then (0, F) is a local minimizer of E κ,H,a,B 0 . Thus, the Hessian of the functional E κ,H,a,B 0 at the normal state (0, F) should be positive. For every ( φ, A) ∈ H 1 (Ω) × H 1 div (Ω) we have,

E κ,H,a,B 0 (t φ, F + t A) = t 2 Q Ω κHF,-κ 2 a ( φ) + (κH) 2 Ω | curl A| 2 dx + O(t 3 ) .
This implies that the Hessian of the functional E κ,H,a,B 0 at the normal state (0, F) can be written as follows :

Hess (0,F) [ φ, A] = Q Ω κHF,-κ 2 a ( φ) + (κH) 2 Ω | curl A| 2 dx .
Since Hess (0,F) [ φ, A] ≥ 0, we get that µ 1 (κH) ≥ 0 , and consequently H / ∈ N loc (κ). Hence we obtain the above inclusion. On the other hand, if (ψ, A) is a minimizer of the functional in (4.1.1) with ψ = 0, then (ψ, A) is a solution of (4.1.12), and we have the following inclusion, N (κ) ⊂ N cp (κ) , and consequently,

N loc (κ) ⊂ N (κ) ⊂ N cp (κ) . ( 4
H loc C 3 (κ) ≤ H C 3 (κ) ≤ H cp C 3 (κ) , (4.10.24)
Using (4.10.23), we observe that,

R + \ N cp (κ) ⊂ R + \ N (κ) ⊂ R + \ N loc (κ) .
From the definition of all the critical fields, we conclude that,

H loc C 3 (κ) ≤ H C 3 (κ) ≤ H cp C 3 (κ) . ( 4
a(x) B 0 (x) , sup x∈∂Ω a(x) Θ 0 B 0 (x) -C κ 1 2 ≤ H loc C 3 ≤ H cp C 3 ≤ κ max sup x∈Ω a(x) B 0 (x) , sup x∈∂Ω a(x) Θ 0 B 0 (x) + C κ 1 2 . (4.10.26)
As a consequence, we have proved Theorem 4.1.6 for the six critical fields.

Remark 4.10.6. As in [START_REF] Fournais | Spectral Methods in Surface Superconducitivity[END_REF], it would be interesting to show that all the critical fields coincide when κ is large enough. . After a change of notation, we deduce an estimate for µ 1 (κ, H).

Lower bound

In the absence of a pinning term, that is when a = 1, Pan and Kwek [START_REF] Pan | Schrödinger operators with non-degenerately vanishing magnetic fields in bounded domains[END_REF] gave the lower bound for the lowest eigenvalue µ(BF) of P Ω BF,0 when B → +∞. In this subsection, we determine a lower bound for µ 1 when κ → +∞ and the pinning term is present. We first recall the definition of λ 0 in (4.1.31), the definition of Γ in (4.1.3) and for any θ ∈ (0, π) we recall that λ(R 2 + , θ) is the bottom of the spectrum of the operator P R 2

+

A app,θ ,0 , with

A app,θ = - x 2 2 2 cos θ, x 2
1 2 sin θ .

We then define for any σ > 0,

Λ 1 (B 0 , a, σ) = min inf x∈Γ∩Ω λ 0 σ |∇B 0 (x)| 2 3 -a(x) , inf x∈Γ∩∂Ω λ(R 2 + , θ(x)) σ |∇B 0 (x)| 2
Here, for x ∈ Γ ∩ ∂Ω, θ(x) denotes the angle between ∇B 0 (x) and the inward normal vector -ν(x).

We start with a proposition that states a lower bound of µ 1 (κ, H) in the case when Γ = ∅.

Proposition 4.11.1. Let I be a closed interval in (0, ∞). There exist two positive constants

B 0 > 0 and C > 0 such that if B ≥ B 0 , σ ∈ I, ψ ∈ H 1 (Ω) \ {0} and a ∈ C 1 (Ω), then, Q Ω BF,-( B σ ) 2 3 a (ψ) ψ 2 L 2 (Ω) ≥ B σ 2 3 Λ 1 (B 0 , a, σ) -CB -1 18
.

(4.11.3)

Proof. Let = B -7/29 . We define the following sets,

D 1 = {x ∈ Ω : dist(x, Γ) < 2 } , D 2 = {x ∈ Ω : dist(x, Γ) > } .
Let h j be a partition of unity satisfying

2 j=1 h 2 j = 1 , 2 j=1 |∇h j | 2 ≤ C -2 = CB 14/29 and supp h j ⊂ D j (j ∈ {1, 2}) .
There holds the following decomposition, We cover the curve Γ by a family of disks

Q Ω BF,-( B σ ) 2 3 a (ψ) = Q D 1 BF,-( B σ ) 2 3 a (h 1 ψ) + Q D 2 BF,-( B σ ) 2 3 a (h 2 ψ) - 2 j=1 Ω |∇h j | 2 |ψ| 2 dx ≥ Q D 1 BF,-( B σ ) 2 3 a (h 1 ψ) + Q D 2 BF,-( B σ ) 2
D(ω j , ) ⊂ {x ∈ R 2 : dist(x, Γ) ≤ 2 } and D 1 ⊂ j D(ω j , ) (ω j ∈ Γ) .
Consider a partition of unity satisfying

j χ 2 j = 1 , j |∇χ j | 2 ≤ C -2 and supp χ j ⊂ D(ω j , ) .
Moreover, we can add the property that :

either suppχ j ∩ Γ ∩ ∂Ω = ∅ , either ω j ∈ Γ ∩ ∂Ω .
We may write,

Q D 1 BF,-( B σ ) 2 3 a (h 1 ψ) = int Q D 1 BF,-( B σ ) 2 3 a (χ j h 1 ψ)+ bnd Q D 1 BF,-( B σ ) 2 3 a (χ j h 1 ψ)- j D 1 |∇χ j | 2 |h 1 ψ| 2 dx , (4
.11.5) where 'int' is in reference to the j's such that ω j ∈ Γ ∩ Ω and 'bnd' is in reference to the j's such 140 Pinning with a variable magnetic field of the two dimensional Ginzburg-Landau model that ω j ∈ Γ ∩ ∂Ω. For the last term on the right side of (4.11.5), we get using the assumption on χ j :

D 1 |∇χ j | 2 |h 1 ψ| 2 dx ≤ C -2 D 1 |h 1 ψ| 2 dx = C B 14/29 D 1 |h 1 ψ| 2 dx .
(4.11.6)

We have to find a lower bound for

Q D 1 BF,-( B σ ) 2 3 a
(h 1 ψ) for each j such that ω j ∈ Γ ∩ Ω and for each j such that ω j ∈ Γ ∩ ∂Ω. Thanks to [START_REF] Miqueu | Équation de Schrödinger avec champ magnétique qui s'annule[END_REF], we have,

Ω |(∇ -iBF)χ j h 1 ψ| 2 dx ≥ B 2 3 Ω (λ 0 |∇B 0 (ω j )| 2 3 -CB -1/18 |χ j h 1 ψ| 2 dx .
Using Taylor's formula, we can write in every disk D(w j , ),

|a(x) -a(w j )| ≤ C = CB -7/29 ≤ CB -1/18 . (4.11.7)
In that way, we get, In a similar fashion, the analysis in [START_REF] Miqueu | Équation de Schrödinger avec champ magnétique qui s'annule[END_REF] and (4.11.7) yields, bnd

int Q D 1 BF,-( B σ ) 2 3 a (χ j h 1 ψ) ≥ int B σ 2 3 λ 0 σ |∇B 0 (ω j )| 2 3 -a(ω j ) -CB -1/18 |χ j h 1 ψ| 2 dx ≥ B σ 2 3 inf x∈Γ∩Ω λ 0 σ |∇B 0 (x)|
Q D 1 BF,-( B σ ) 2 3 a (χ j h 1 ψ) ≥ bnd B σ 2 3 λ(R 2 + , θ(ω j )) σ |∇B 0 (ω j )| 2 3 -a(ω j ) -CB -1/18 |χ j h 1 ψ| 2 dx ≥ B σ 2 3 inf x∈Γ∩∂Ω λ(R 2 + , θ(x)) σ |∇B 0 (x)| 2 3 -a(x) -CB -1/18 bnd |χ j h 1 ψ| 2 dx .
(4.11.9)

We insert (4.11.8), (4.11.9) and (4.11.6) into (4.11.5) to obtain, We observe that there exists a gauge function ϕ j satisfying (see [5, Equation (A.3)]), F(x) -(B 0 (ω j )A 0 (x -ω j ) + ∇ϕ j ) ≤ C 2 1 in D(ω j , 1 ) .

Q D 1 BF,-( B σ ) 2 3 a (h 1 ψ) ≥ B σ 2 3 Λ 1 (B 0 , a, σ) -CB -1/
Using Cauchy-Schwarz inequality, we may write, Ω |(∇ -iBF)χ j h 2 ψ| 2 dx ≥ 1 2 Ω |(∇ -iB B 0 (ω j )A 0 (x -ω j ))e -iBϕ j χ j h 2 ψ| 2 dx

-C B 2 4
1 Ω |χ j h 2 ψ| 2 dx .

We are reduced to the analysis of the Neumann realization of the Schrödinger operator with a constant magnetic field equal to B B 0 (ω j ) in our case.

Notice that by the assumption on h 2 , there exist constants M > 0 and B 0 > 0 such that, for all j, |B 0 (ω j )| ≥ M in the support of h 2 . Thus, ∀j, B|B 0 (ω j )| ≥ M B = M B 22/29 1 .

Moreover, the magnetic potentials A 0 (x) and A 0 (x -ω j ) are gauge equivalent since A 0 (x -ω j ) = A 0 (x) -A 0 (ω j ) = A 0 (x) -∇(A 0 (ω j ) • x) .

Thanks to Theorem 4.8.2, there exists a constant B 0 such that, for any B ≥ B 0 , we write by the min-max principle, Proof. To apply the results of Proposition 4.11.3, we take B = κH and σ = H κ 2 . We see for κ sufficiently large that σ ∈ (0, λ max ) and B large. In this subsection we will prove Theorem 4.1.7 for H loc C 3 and H loc C 3 . We first recall some useful results from [START_REF] Pan | Schrödinger operators with non-degenerately vanishing magnetic fields in bounded domains[END_REF] about the relation between the eigenvalues λ 0 and λ(R 2 + , θ), introduced in (4. 1.31) and in (4.1.33). Theorem 4.12.1.

(i) λ(R 2 + , 0) = λ 0 .

(ii) If 0 < θ < π, then λ(R 2 + , θ) < λ 0 .

The next proposition gives the region where µ 1 (κ, H) < 0 that allows us to obtain an information about H loc C 3 (see (4.1.29)) in the case when the magnetic field B 0 is constant with a pining term. Thanks to the assumption in (4.1.4), we have, for all x ∈ Γ ∩ ∂Ω, 0 < θ(x) < π. Theorem 4.12.1 then tells us that,

∀ x ∈ Γ ∩ ∂Ω , a (x) 3 2
λ(R 2 + , θ(x)) where M > 0 is a constant independent of C.

Suppose that κH ≥ B 0 where B 0 is selected sufficiently large such that we can apply Theorem 4.11.4. (Thanks to Lemma 4.9.8, µ 1 (κ, H) < 0 when κH < B 0 ). By Theorem 4.11.4, there exist positive constants C 1 and κ 0 such that, for κ ≥ κ 0 , Proof. Having in mind the definition of Λ 1 in (4.11.2), under the assumption in (4.12.4), we get for κ large enough, Λ 1 B 0 , a, H κ 2 ≥ C M κ -1 6 , (4.12.5)

µ 1 (κ, H) ≤ κ 2 inf x∈Γ∩Ω λ 0 H κ 2 |∇B 0 (x)|
where M > 0 is a constant independent of the constant C.

Thanks to Theorem 4.11.2, we get the existence of positive constants C and κ 0 such that, for κ ≥ κ 0 ,

µ 1 (κ, H) ≥ (C M -C ) κ 11 6
To finish the proof, we choose C sufficiently large such that C M > C .

4.12.2 Analysis of H cp C 3 .

Proposition 4.12.4 below is an adaptation of an analogous result obtained in [START_REF] Helffer | The Ginzburg-Landau functional with vanishing magnetic field[END_REF] for the functional in (4.1.1) with a constant pinning term. Proposition 4.12.4 is valid when Γ = ∅. Proposition 4.12.4 says that, if (ψ, A) is a critical point of the functional in (4.1.1) and H is of order κ 2 , then |ψ| is concentrated near the set Γ.

Proposition 4.12.4. Let ε > 0. There exist two positive constants C and κ 0 such that, if κ ≥ κ 0 , H ≥ ε κ 2 , (4.12.6) and (ψ, A) is a solution of (4.1.12), then Here, we have used the fact that h 4 ≤ h 2 since 0 ≤ h ≤ 1.

ψ 2 L 2 (Ω) ≤ C κ -
By assumption (4.1.4), |∇B 0 | does not vanish on Γ, hence

|B 0 (x)| ≥ 1 M κ -1 2
in {dist(x, Γ) ≥ λ} , (4.12.8)

for some constant M > 0. Thus, by using (4.1.10) and (4.12.6), we get, ε M κ , (4.12.9) then (0, F) is the unique solution to (4.1.12). Moreover, Proof. In light of the result in Theorem 4.8.5, we may assume the extra condition that H ≤ λ max κ 2 for a sufficiently large constant λ max .

H cp C 3 ≤ max   sup x∈Γ∩Ω a(x)
We take the constant C in (4.12.9) as in Proposition 4.12.3. In that way, under the assumption in (4.12.9), we have µ 1 (κ, H) < 0 . (4.12.10) Suppose now that (ψ, A) is a solution of (4.1.12) with ψ = 0. Similarly, as in the proof of Theorem 4.10.4, we have,

-≥ µ 1 (κ, H) ψ 2 L 2 (Ω) -C √ κ ψ 2 L 2 (Ω) , (4
.12.11) where = κ 2 ψ 4 L 4 (Ω) is introduced in (4.10.17). To apply the result of Proposition 4.12.4, we take (4.12.12) Putting (4.12.12) into (4.12.11), we obtain,

-≥ µ 1 (κ, H) ψ 2 L 2 (Ω) -C κ -1
We conclude that, for κ ≥ κ 0 and κ 0 a sufficiently large constant, µ 1 (κ, H) < 0, which is in contradiction with (4.12.10). Therefore, we conclude that ψ = 0.

Following the argument given in Proposition 4.10.5, we get : Proposition 4.12.6. Supose that Γ = ∅ and {a > 0} = ∅. There exists C > 0 and κ 0 such that, if κ ≥ κ 0 and H satisfies We can rewrite J K (u, A) as follows :

H ≤ max   sup x∈Γ∩Ω a(x)
J K (u, A) = 1 2 K |∇|u|| 2 + |u| 2 |∇ϕ -A| 2 + 1 2ε 2 |1 -|u| 2 | 2 + |h -h ex | 2 dx . (A.3.6)
The simple inequality (a -b) 2 ≥ 0 yields,

|∇|u|| 2 + 1 2ε 2 |1 -|u| 2 | 2 ≥ |∇|u|| √ 2|1 -|u| 2 | ε .
Notice that |u| = t in γ t . Therefore, using the co-area formula, we find that, Proof. The proofs of (P1)-(P3) follow by a straightforward computation. We will prove (P4). For all i, R i + r i ≥ 0 and R i -r i ≥ 0. Thus, Multiplying the above inequality by |d i | and summing over i yields the result in (P4).

K |∇|u|| 2 dx + K 1 2ε 2 |1
R 2 i -r 2 i = (R i + r i )(R i -r i ) ≤   k j=1 (R j +

A.4 Construction of vortex disks for S 1 valued functions

The next proposition establishes the existence of disjoint disks as in Theorem A.1.1.

Proposition A.4.1. Let V ⊂ R 2 be an open set and ω ⊂ R 2 be a compact set. Assume that v : V \ ω -→ S 1 , A : V -→ R 2 be C 1 and r(ω) < Step 4.3 : The auxiliary field and the degree.

Here, we will explore a relationship between the degree of u/|u| and the auxiliary field ĥ. Let (ω i ) i be the collection of connected components of Ω tε . If ω i K, let d ω i = deg(e iϕ , ∂ω i ). Now, suppose that ω i K. We have, using the definition of the degree, (A. where we have essentially used Stoke's theorem and (A.5.10).

Since curl(∇ϕ) = 0, then applying the operator curl on (A.5.10) yields, -∆ ĥ = -curl(∇ ⊥ ĥ) = curl(∇ϕ) -curl A = -ĥ . (A.5.16)

This gives that, -∆ ĥ + ĥ = 0 , in K \ Ω tε . (A.5.17)

Let K ⊂ K be an open subset of K such that ∂K ∩ Ω tε = ∅. Suppose that the boundary of K is piecewise smooth. We integrate (A.5.17 Step 4.4 : A lower bound of the total degree.

For every β > 0, we introduce the following subset of the square K, K β = {x ∈ K : dist(x, ∂K) > β} .

Notice that, for all 0 < β < /2, K t occupies a square of side-length -2β. Later, we will choose α = α(ε) ∈ (0, 1) such that 1 √ h ex α 4 (ε → 0 + ) . To see this, notice that

T c ⊂ π 1 i B i π 2 i B i ,
where π 1 : (x 1 , x 2 ) → x 1 and π 2 : (x 1 , x 2 ) → x 2 .

Consequently, we see that,

|T c | ≤ π 1 i B i + π 2 i B i ≤ 4 i r i < α 2 .
A In particular, we have that ∂K β 0 ∩ Ω t = ∅ , and

B i ⊂K β 0 d B i = i, ω i ⊂K β 0 d ω i ,
where (ω i ) i are the connected components of Ω t .

We define the sub-collection of (B i ) k i=1 to be (B i ) B i ⊂K β 0 ). Now, we can apply (A.5.19) with K = K β 0 and get, 2π Now, we have the following simple inequalities,

B i ⊂K β 0 d B i ≥ K β 0 ĥ - ∂K β 0 ∂ ĥ ∂n ≥ K β 0 h ex - K β 0 ĥ -h ex -
K β 0 h ex ≥ h ex ( -α) 2 , K β 0 ĥ -h ex ≤ K β 0 | ĥ -h ex | 2 1 2
and

∂K β 0 ∂ ĥ ∂n ≤ 2 1 2   ∂K β 0 ∂ ĥ ∂n 2   1 2
.

The last two inequalities are obtained via Cauchy-Schwarz. Now, we put these inequalities into (A.

  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii 1 Introduction Générale 1.1 Motivations et notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Supraconducteurs et champs magnétiques . . . . . . . . . . . . . . . . . . 1.1.2 Présentation du modèle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 Invariance de jauge et équations . . . . . . . . . . . . . . . . . . . . . . . 1.1.5 Les vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.6 Les champs critiques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 L'énergie de l'état fondamental de la fonctionnelle de Ginzburg-Landau avec un champ magnétique appliqué variable dans un domaine de R 2 . . . . . . . . . . . 1.2.1 Problème posé . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Résultats principaux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Méthodes de démonstration. . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Discussion des résultats principaux . . . . . . . . . . . . . . . . . . . . . . 1.3 Énergie et vorticité pour un modèle de Ginzburg-Landau avec un champ magnétique variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Énoncé des résultats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Méthodes de démonstration. . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Énergie pour un modèle de Ginzburg-Landau avec un champ magnétique variable et un terme de "pinning" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

 1 . . . 1.4 Vortex de degré +1 et -1, les flèches représentent le courant. . . . . . . . . . . . 1.5 Etat Meissner. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Etat mixte. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Vortices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Schematic representation of Ω with pinning term independent of κ and with variable magnetic field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Schematic representation of a domain with a κ-dependent oscillation pinning and with vanishing magnetic field along Γ. . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Schematic representation of some domain with pinning term dependent of κ and with vanishing magnetic field along Γ. . . . . . . . . . . . . . . . . . . . . . . . . Supraconducteurs et champs magnétiques Dans les années 50, les physiciens Vitali Lazarevitch Ginzburg et Lev Davidovitch Landau ont proposé la théorie éponyme de la supraconductivité. La supraconductivité est un phénomène caractérisé par l'absence de résistance électrique et l'expulsion du champ magnétique. Elle permettrait de transporter de l'électricité sans perte d'énergie. A très basses températures, le matériau devient supraconducteur. Lorsqu'il est soumis à un champ magnétique de faible intensité, le champ magnétique est repoussé, comme indique la figure ci-dessous.

Figure 1 . 1 -

 11 Figure 1.1 -Champ magnétique repoussé par un superconducteur.

  .1.3) où, Γ = {x ∈ Ω : B 0 (x) = 0} .

Figure 1 . 2 -

 12 Figure 1.2 -Représentation schématique d'un champ magnétique qui peut s'annuler sur des courbes régulières.

Figure 1 . 4 -

 14 Figure 1.4 -Vortex de degré +1 et -1, les flèches représentent le courant.

Figure 1 . 5 -

 15 Figure 1.5 -Etat Meissner.Figure 1.6 -Etat mixte.

Figure 1 . 6 -

 16 Figure 1.5 -Etat Meissner.Figure 1.6 -Etat mixte.

3 2

 3 |∇B 0 (x)| > min x∈∂Ω∩Γ λ 0 (R + , θ(x)) |∇B 0 (x)| .

3 2

 3 |∇B 0 (x)| < min x∈∂Ω∩Γ λ 0 (R + , θ(x)) |∇B 0 (x)| ,la supraconductivité de surface disparaît. Alors, nous ne distinguons pas entre H C 2 et H C 3 , de plus

11 Nous 2 Ω| 8 ) 1 . 2 . 4 (

 1128124 introduisons aussi l'énergie locale de ψ dans un domaine D ⊂ Ω :E 0 (ψ, A; D) = D e(ψ, A) dx .(1.2.7)De plus, nous definissons l'énergie locale de Ginzburg-Landau de (ψ, A) dans un domaine D ⊂ Ω comme suit,E(ψ, A; D) = E 0 (ψ, A; D) + (κH) rot A -B 0 | 2 dx . (1.2.Théorème Estimation de l'énergie locale.). Supposons que D ⊂ Ω soit un ensemble ouvert. Alors, il existe une constante κ 0 > 0 telle que, lorsque H vérifie (1.2.1) et (ψ, A) un minimiseur de (1.1.2), nous avons

  Si Γ = ∅ et H = bκ, b > 0, alors, f H κ |B 0 (x)| -1 2 = 0 en D = x ∈ Ω : H κ |B 0 (x)| < 1 , et |D| = 0 .

1. 3

 3 Énergie et vorticité pour un modèle de Ginzburg-Landau avec un champ magnétique variable 13 Par conséquent, pour κ assez grand, nous obtenons le résultat suivant : D |ψ| 4 dx > D dans L 2 (Ω) , où D est une constante positive dépendant de D.

.4. 14 )

 14 Le théorème suivant, nous donne une estimation de l'énergie locale E 0 (ψ, A; a, D) et aussi un comportement asymptotique de la norme L 4 dans D du paramètre d'ordre ψ, quand (ψ, A) est un minimiseur global. Théorème 1.4.3. Avec les notations et les hypothèses du Théorème 1.4.1, si (ψ, A) est un minimiseur de (1.4.1) et D est un ensemble régulier tel que D ⊂ Ω, alors, lorsque κ → +∞,

Théorème 1 . 4 . 4 .

 144 avec la condition de Neumann au bord u (0) = 0. Supposons que B 0 > 0 dans Ω et que a ∈ C 1 (Ω) vérifie {a > 0} = ∅. Alors, lorsque κ -→ +∞, tous les champs critiques satisfont :

Théorème 1 . 4 . 5 .

 145 Supposons que Γ = {x : B 0 (x) = 0} = ∅, que (1.1.3) est satisfaite et que a ∈ C 1 (Ω) vérifie {a > 0} = ∅. Lorsque κ -→ +∞, les six champs critiques dans (1.4.20)-(1.4.22) satisfont :

Remark 3 . 1 . 2 .

 312 Notice that when H(κ) satisfies (3.1.8) we have o(κH) = o κH ln H κ + 1 .

.1. 14 )

 14 Remark 3.1.3. When the set Γ = {x ∈ Ω, B 0 (x) = 0} consists of a finite number of smooth curves and the intensity of the magnetic field H satisfies κ H ≤ O(κ 2 ), then the energy E g (κ, H) in (3.1.1) is estimated in [27]. Theorem 3.1.1 admits the following corollary which is useful in the proof of Theorem 3.1.5 below. The content of Corollary 3.1.4 gives us that the magnetic energy is small compared with the leading term in (3.1.14). Corollary 3.1.4. Suppose that the assumptions of Theorem 3.1.1 hold. Then, the magnetic energy of the minimizer satisfies

  .1.17) If D = Ω, we sometimes omit the dependence on the domain and write E 0 (ψ, A) for E 0 (ψ, A; Ω). The next theorem gives a local version of Theorem 3.1.1. Theorem 3.1.5. Under Assumption (3.1.2), if (ψ, A) is a minimizer of (3.1.1) and D is regular set such that D ⊂ Ω, then the following is true. 1. If H(κ) satisfies (3.1.7), then,

  and b(κ) are two positive functions, we write a(κ) b(κ) if a(κ)/b(κ) → 0 as κ → ∞. • If a(κ) and b(κ) are two functions with b(κ) = 0, we write a(κ) ∼ b(κ) if a(κ)/b(κ) → 1 as κ → ∞. • If a(κ) and b(κ) are two positive functions, we write a(κ) ≈ b(κ) if there exist positive constants c 1 , c 2 and κ 0 such that c 1 b(κ) ≤ a(κ) ≤ c 2 b(κ) for all κ ≥ κ 0 .

Figure 3 .

 3 Figure 3.1 -Vortices.

3. 2

 2 A reference problem 55 in the proof. Let us introduce the following space

( 3 .

 3 2.25) By using the bound |u| ≤ 1, (3.2.20) and the assumption on the support of χ R,b in (3.2.

.3. 10 )

 10 Remembering the definition of R and b in (3.3.1) and (3.3.5) respectively, we get

3. 4 A

 4 priori estimates of minimizers 65 minimizer of (3.1.1), then

3. 5

 5 Proof of Theorem 3.1.5 : Lower boundIn this section, we suppose that D is an open set with smooth boundary such that D ⊂ Ω. We will give a lower bound of the energy E(ψ, A; D) introduced in (3.1.17), when (ψ, A) is a minimizer of the functional in (3.1.1).

|ψ| 2

 2 dx.(3.5.5)

  .5.11) with b and R introduced in (3.5.6). Inserting (3.5.11) into (3.5.5) and using the bound of ψ in (3.4.1), we get

Remark 3 . 5 . 2 .

 352 For any α ∈ (0, 1), we keep the same choice of , ρ as in(3.3.11) and choose δ as follows : Assumption (3.1.7) permit us to have the assumptions of Proposition 3.5.1 satisfied and make the error terms in its statement of order o κH ln κ H . We have as κ -→ +∞ ,

  (3.5.14) Using(3.5.3) and choosing , ρ as in(3.3.11) and δ as in (3.5.13), we get a function r : (0, +∞) -→ (0, +∞) satisfying lim t-→+∞ r(t) = 0 and

. 15 )

 15 for any x 0 in Q (x 0 ) . The next theorem presents the lower bound of the local energy in the domain D such that D ⊂ Ω and we deduce the lower bound of the global energy by replacing D with Ω. Theorem 3.5.4. Under Assumption (3.1.2), if H(κ) satisfies (3.1.7), (ψ, A) ∈ H 1 (Ω, C) × H 1 div (Ω) is a minimizer of (3.1.1) and D ⊂ Ω is open, then,

2 Ω|

 2 I ,ρ was introduced in(3.3.15). If (ψ, A) is a minimizer of (3.1.1), we haveE(ψ, A; D) = E 0 (ψ, A; D ,ρ ) + E 0 (ψ, A; D \ D ,ρ ) + (κH) curl A -B 0 | 2 dx ,where E 0 (ψ, A; D) is introduced in (3.1.16). Since the magnetic energy term and the energy in D \ D ,ρ are positive, we may write, E(ψ, A; D) ≥ E 0 (ψ, A; D ,ρ ) .(3.5.18) 3.5 Proof of Theorem 3.1.5 : Lower bound 69

.5. 25 )

 25 Remark 3.5.5 tells us that

R by b 1 2

 2 is a minimizer of (3.1.1). The proof of the next proposition is similar to that in [5, Proposition 6.2], by replacing 1

. 1 ) 3 . 6

 136 where λ is introduced in (3.5.2). Remark 3.6.2. Under Assumption (3.1.7), with the choices of , ρ in (3.3.11) and δ in (3.5.13), we get that the error terms in (3.6.1) are of order κH ln κ H Proof of Theorem 3.1.5 : upper bound 71 Proposition 3.6.3.

.6. 14 ) 3 . 6

 1436 Proof of Theorem 3.1.5 : upper bound 73 and consequently

3 ) 7 8 2

 372 Let us introduced for all ρ-admissible triple ( , x 0 , x 0 ) the functions b and R byR(κ, H, x 0 ) = 2 -|B 0 ( x 0 )| 1 and b(κ, H, x 0 ) = H κ |B 0 ( x 0 )| . (3.7.4)Notice that b(κ, H, x 0 ) and 1 R(κ, H, x 0 ) are uniformly o(1) as κ -→ +∞, in the following sense :

  , H, β) = b(κ, H, x 0 ) and R(κ, H, β) = R(κ, H, x 0 ) ,(3.7.7)

3. 7

 7 Vortices 77 get from (3.5.8) and (3.5.9) the following inequality

1 2

 1 to get (3.7.12). Remark 3.7.2. Using (3.7.9) and (3.1.7), we obtain uniformlyN b N n , as κ -→ +∞ . (3.7.24)

3. 7 79 3 . 7 . 3

 7373 Vortices The results of Sandier-Serfaty.

  j | ≤ δ(κ) 2 κHB 0 ( x 0 )(1 + s 2 (κ)) , as κ -→ +∞ . (3.7.34)

x

  |∇ x a(x, κ)| , (4.1.9) a = sup x∈Ω, κ≥κ 0 a(x, κ) (4.1.10)

Theorem 4 . 1 . 2 .

 412 .1.15) where λ min , λ max are positive constants such that λ min < λ max . The behavior of E g (κ, H, a, B 0 ) involves a function f : [0, +∞) -→ [0,1 2 ] introduced in [6, Theorem 2.1]. The function f is increasing, continuous and f (b) =1 2 , for all b ≥ 1. Suppose that Assumption 4.1.1 and (4.1.15) hold, and

Theorem 4 . 1 . 4 . 2 D∩{a 2

 41422 Under the assumptions of Theorem 4.1.2, if (ψ, A) is a minimizer of (4.1.1) and D is regular set such that D ⊂ Ω, then E 0 (ψ, A; a, D) = κ D∩{a(x,κ)≤0} a(x, κ) 2 dx + o κ 2 , as κ -→ +∞ . (4.1.20) Theorem 4.1.4 will be useful in the proof of the next theorem which gives the asymptotic behavior of the order parameter ψ, when (ψ, A) is a global minimizer. Theorem 4.1.5. Under the assumptions of Theorem 4.1.2, if (ψ, A) is a minimizer of (4.1.1) and D is a regular set such that D ⊂ Ω, then D

  H,a,B 0 has a non-normal critical point} . (4.1.22)

Theorem 4 . 1 . 7 .

 417 Suppose that Γ = {x : B 0 (x) = 0} = ∅, that (4.1.4) holds and that a ∈ C 1 (Ω) satisfies {a > 0} = ∅. As κ -→ +∞, the six critical fields in (4.1.27)-(4.1.29) satisfy the asymptotic expansion :

and b 2

 2 (κ) are two positive functions on [κ 0 , +∞), we write b1 (κ) b 2 (κ) if b 1 (κ)/b 2 (κ) → 0 as κ → ∞. • If b 1 (κ) and b 2 (κ) are two functions with b 2 (κ) = 0, we write b 1 (κ) ∼ b 2 (κ) if b 1 (κ)/b 2 (κ) → 1 as κ → ∞.• If b 1 (κ) and b 2 (κ) are two positive functions, we write b 1 (κ) ≈ b 2 (κ) if there exist positive constants c 1 , c 2 and κ 0 such that c 1 b 2 (κ) ≤ b 1 (κ) ≤ c 2 b 2 (κ) for all κ ≥ κ 0 .

Theorem 4 . 3 . 4 .

 434 Under Assumptions (4.1.4)-(4.1.8), if (4.1.15) holds and L(κ) ≤ C κ 1 2

  .3.21) and N + = card I + ,ρ , N -= card I - ,ρ . (4.3.22)

5 . 14 ) 1 . 4 . 5

 514145 achieve the proof of Proposition 4.5.Application 4.5.2. We keep the same choice of , ρ, L(κ) and δ as in (4.3.18), (4.3.19) and choose : Lower bounds for the global and local energies 107

( 4 .

 4 5.24) Putting (4.5.24) and (4.1.17) into (4.5.23), we finish the proof of Corollary 4.1.3. Proof of Theorem 4.1.4. Noticing that (4.5.22) is valid when h = 1 and D replaced by D c := Ω \ D for any open domain D ⊂ Ω with smooth boundary, then we get :

Figure 4 . 1 -

 41 Figure 4.1 -Schematic representation of Ω with pinning term independent of κ and with variable magnetic field.

5 .

 5 Suppose that (4.1.4) and (4.1.15) hold. Let a(x, κ) = α(κ 1 2

2 L 2 ( 2 L 2

 2222 Now, we estimate (∇ -iA 0 )u b ,R Q R ) from above. Coming back to the definition, we get the existence of a positive constant C, such that for any b ∈ [b 0 , 1] and for any b ∈ [b 0 , 1], (∇ -iA 0 )u b ,R (Q R ) ≤ e D (b , R) b . This implies that, e D (b, R) ≤ e D (b , R) + e D (b , R) b |b -b | .

4. 6

 6 Study of examples 115 any b 0 > 0, | f (b) -f (b )| ≤ C log 1 b 0 |b -b | , ∀b, b with 1 > b > b 0 and 1 > b > b 0 . (4.6.13) In addition, we have | f (b) -f (b )| ≤ 2 |b -b | , ∀b, b with b >

  have used the fact that φ is a bounded function in Ω. Let us show that the remainder term s(κ) in the right hand side in (4.6.18) is o(1). The remainder term has the form s 1 (κ)+s 2 (κ) with s 1 (κ) = O(ρ) and s 2 (κ) = O ρ (κ -1 2). Let us show that it is o(1). Given ε > 0, there exists ρ ε > 0 such that |s 1 (κ)| ≤ ε 2 , for all κ ≥ κ 0 . Then, ρ = ρ ε being chosen, we can find κ ε ≥ κ 0 such that, for any κ ≥ κ , |s 2 (κ)| ≤ ε 2 .

Figure 4 . 2 -

 42 Figure 4.2 -Schematic representation of a domain with a κ-dependent oscillation pinning and with vanishing magnetic field along Γ.

Proposition 4 . 6 . 8 .

 468 Suppose that (4.1.4) and (4.1.15) hold. Let a(x, κ) = a(x) + β(x, κ), where β(x, κ) is a nonnegative function and {a > 0} ∩ Ω = ∅, (see Fig.4.3). There exist positive constants τ 1 and κ 0 such that,

.6. 21 )Figure 4 . 3 -

 2143 Figure 4.3 -Schematic representation of some domain with pinning term dependent of κ and with vanishing magnetic field along Γ.

D |ψ| 4

 4 dx = Ω |ψ| 4 dx -D c |ψ| 4 dx Thanks to Remark 4.7.1, using the asymptotics in (4.7.3) obtained in Step 1 when D = Ω and the upper bound in Step 2 , we get D |ψ| 4 dx ≤ -D∩{a(x,κ)>0} a(x, κ) 2 2 f H κ |B 0 (x)| a(x, κ) -1 dx + o (1) . (4.7.13)

  µ N (BF; Ω) the lowest eigenvalue of the Schrödinger operator P Ω BF,0 (see (4.1.13)) with Neumann condition in L 2 (Ω) :

Theorem 4 . 8 . 2 . 2 +A

 4822 Suppose that Ω ⊂ R 2 is an open bounded set with smooth boundary and Γ = ∅.recall that λ(R 2 + , θ) is the bottom of the spectrum of the operator P R app,θ ,0 , with A app,θ = -

Theorem 4 . 8 . 4 . 3 = α 1 (

 48431 Suppose that (4.1.4) holds and Γ = ∅. Then lim B-→+∞ µ N (BF, Ω) B 2

.9. 2 )

 2 Proposition 4.9.1. Let Ω ⊂ R 2 be an open bounded set with smooth boundary, I a closed interval in (0, +∞) and Γ = ∅. There exist positive constant C and B 0 such that if σ

4. 9

 9 Asymptotics of µ 1 (κ, H) : the case with non vanishing magnetic field 127 Proof. We apply Proposition 4.9.1 with B = κH , σ = H κ and I = [λ min , λ max ] .

4. 10 3 .

 103 Proof of Theorem 4.1.6 4.10.1 Analysis of H loc C In this subsection we give a lower bound of the critical field H loc C 3 (see (4.1.29)) and we give an upper bound of the critical field H loc C 3 in the case when the magnetic field B 0 is constant with a 4.10 Proof of Theorem 4.1.6 131 pining term. Proposition 4.10.1. Suppose that {a > 0} = ∅ and Γ = ∅. There exist constants C > 0 and κ

2 ( 4 5 )

 245 By choosing C such that C M > C 2 , we get, µ 1 (κ, H) < 0 .

4. 10

 10 Proof of Theorem 4.1.6 135

4. 10 . 3

 103 End of the proof of Theorem 4.1.6

3 a(h 2 29 Ω

 3229 ψ) -CB 14/|ψ| 2 dx . (4.11.4)

2 3 -

 3 a(x) -CB -1/18 int |χ j h 1 ψ| 2 dx . (4.11.8)

4. 11 Consider a partition of unity satisfying j χ 2 j = 1 , j |∇χ j | 2 ≤ C - 2 1 1 Ω

 1121221 Asymptotics of µ 1 (κ, H) : the case with vanishing magnetic field 141 and supp χ j ⊂ D(ω j , 1 ) .There holds the decomposition formula,Ω |(∇ -iBF)h 2 ψ| 2 dx = j Ω |(∇ -iBF)χ j h 2 ψ| 2 dxj Ω |∇χ j | 2 |h 2 ψ| 2 dx ≥ j Ω |(∇ -iBF)χ j h 2 ψ| 2 dx -C -2|h 2 ψ| 2 dx ,(4.11.11)

Theorem 4 .

 4 11.4 is valid when κH ≥ κ 0 and κ 0 is sufficiently large.

4. 12

 12 Proof of Theorem 4.1.7 4.12.1 Analysis of H loc C 3 .

Proposition 4 . 12 . 2 .0 2 λ(R 2 +

 412222 Suppose that {a > 0} = ∅ and Γ = ∅. There exist constants C > 0 and κ 0 ≥ 0 such that ifκ ≥ κ 0 , H ≤ max |∇B 0 (x)| , sup x∈Γ∩∂Ω a(x)3 , θ(x))

3 2 3 . 3 2

 333 |∇B 0 (x)|   κ 2 -C κProof. We have two cases : Case 1. Here, we suppose that, |∇B 0 (x)| .

00

  |∇B 0 (x)| . Thus, there exists x 0 ∈ Ω ∩ Γ such that (the supremum of a(x) |∇B 0 (x 0 )| . If (4.12.1) is satisfied for some C > 0, then,

2 3 -a(x) + C 1 κ 11 6 ≤ κ 2 λ 0 H κ 2 3 -a(x 0 ) + C 1 κ 11 6 ≤ (C 1 -C M ) κ 11 6 . ( 4 . 12 . 3 )0 6 . 2 +

 23623616412362 |∇B 0 (x 0 )| 2 By choosing C such that C M > C 1 , we get, µ 1 (κ, H) < 0 . Case 2. Here, we suppose that sup x∈Γ∩∂Ω |∇B 0 (x)|.The assumption in (4.12.1) and the upper bound in Theorem 4.11.4 give us, for all κ ≥ κ 0 , κH ≥ B 0 and B 0 a sufficiently large constant,µ 1 (κ, H) ≤ (C 1 -C M ) κ 11where M > 0 is a constant independent of C. By choosing C such that C M > C 1 , we get,µ 1 (κ, H) < 0 .This finishes the proof of the proposition.The next proposition gives us a lower bound ofH loc C 3 (see (4.1.29)). This is obtained by localizing the region where µ 1 (κ, H) > 0 holds. Proposition 4.12.3. Suppose that {a > 0} = ∅, λ max > 0 and Γ = ∅. There exist constants C > 0 and κ 0 > 0 such that if κ ≥ κ 0 , λ max κ ≥ , θ(x))

3 2 6 ,

 36 |∇B 0 (x)|   κ 2 + C κ 11 (4.12.4) then, µ 1 (κ, H) > 0 .

5 2 -κ 2 a Ω |hψ| 2 dx ≤ c 2 Ω( 1 -M κ 5 2 -κ 2 a -c 2 ΩFor κ large enough, ε M κ 5 2 -κ 2 a -c 2

 52215252 |ψ| 2 dx + Ω |∇h| 2 |ψ| 2 dx . h 2 )|ψ|2 dx and using the assumption on h, we have,ε |hψ(x)| 2 dx ≤ (c 2 + C κ) Ω\Ω λ |ψ| 2 dx .Thanks to the assumption on the support of h, we get further,Ω |ψ(x)| 2 dx ≤ 2 M ε C κ -3 2 + 1 Ω\Ω λ |ψ| 2 dx .

4. 12 7 149 2 Ω\Ω λ |ψ| 4 dx 1 2 ≤ C κ -1 4 Ω |ψ| 4 dx 1 2 .

 1272242 Proof of Theorem 4.1.Recall that λ = κ -1 2 . The Cauchy Schwarz inequality yields,Ω\Ω λ |ψ(x)| 2 dx ≤ |Ω \ Ω λ | 1/This finishes the proof of the proposition. Now, we can give an upper bound of the critical field H cp C 3 in the case when Γ = ∅ and with a pining term. Theorem 4.12.5. Supose that Γ = ∅ and {a > 0} = ∅. There exists C > 0 and κ 0 such that, if H satisfies

ψ 2 L 2

 22 (Ω) ≤ C κ -1 4 ψ 2 L 4 (Ω) = Cκ -5 4 √ .

3 . 6 .

 36 then there exists a solution (ψ, A) of (4.1.12) with ψ L 2 (Ω)End of the proof of Theorem 4.1.7All the critical fields are contained in the interval [H loc C 3 , H cp C 3 ](the proof of this statement is exactly as the one given for (4.10.24) and (4.10.25)). By Proposition 4.12.2 and Theorem 4.12.5, we get the existence of positive constants C and κ 0 , such that for κ ≥ κ 0 , (4.12.14) As a consequence, we have proved that the asymptotics in Theorem 4.1.7 is valid for for the six critical fields in (4.1.27), (4.1.28) and (4.1.29). Proof. Knowing that |(∇ -iA)u| 2 = |∇|u|| 2 + |u| 2 |∇ϕ -A| 2 .

  ) over K \ Ω tε . Using Integration by Parts, we obtain, K \Ωt ε -∆ ĥ + ĥ dx = -, ω i ⊂K } ∂ω i ∂ ĥ ∂n .(A.5.18)A.5 Proof of Theorem A.1.1167Here, we used the assumption that ∂K ∩ Ω t = ∅ to write, ∂Ωt ∂ ĥ ∂n = {i, ω i ⊂K } ∂ω i ∂ ĥ ∂n .Using (A.5.18), (A.5.15) and the assumption ∂K ∩ Ω tε = ∅, we geti ⊂K } ∂ω i ∂ ĥ ∂n + {i, ω i ⊂K } ω i ĥ = {i, ω i ⊂K } 2π d ω i . (A.5.19)

(A. 5 . 20 ) 2 √B

 5202 Since α satisfies (A.5.20) and i r i = 1 hex , then the (Lebesuge) measure of the set,T = {β ∈ (0, α] : ∂K β ∩ i

22 )

 22 Since the region K β 0 occupies a square of side length -2β 0 , then,|K β 0 | = ( -2β 0 ) 2 and |∂K β 0 | = 4( -2 β 0 ) .Since β 0 ∈ (0, α] and α < 4 , then,( -α) 2 ≤ |K β 0 | ≤ 2 and 4( -2α) ≤ |∂K β 0 | ≤ 4 .

  Table des matières . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Liste des Figures . . .

Table des figures

 des 1.1 Champ magnétique repoussé par un superconducteur. . . . . . . . . . . . . . . . 1.2 Représentation schématique d'un champ magnétique qui peut s'annuler sur des courbes régulières. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Représentation schématique d'un supraconducteur avec tourbillons (vortex).

  Théorème 1.2.7 (Sandier-Serfaty). Supposons que Ω ⊂ R 2 , B 0 = 1, H vérifie (1.2.1) et que D est un ouvert régulier tel que D ⊂ Ω. Alors,

	1. Si (ψ, A) ∈ H 1 (Ω; C) × H 1 div (Ω) est une solution de (1.1.6), alors,
	1 2 D	|ψ| 4 dx ≤ -f H κ	-	1 2	|D| + o(1) , (κ → +∞) .	(1.2.13)
	2. Si (ψ, A) ∈ H 1 (Ω; C) × H 1 div (Ω) est un minimiseur de (1.1.2), alors,

D |ψ| 4 dx + 2|D| f H κ -1 2 ≤ o(1) , (κ → +∞) . (1.2.14)

1.2.3 Méthodes de démonstration.

  1.3 Énergie et vorticité pour un modèle de Ginzburg-Landau avec un champ magnétique variable 15 Le théorème suivant nous donne que l'énergie magnétique est petite comparée avec le terme dominant dans (1.3.8). Théorème 1.3.4 (Estimation de l'énergie magnétique.). Avec les notations et l'hypothèse du Théorème 1.3.1, l'énergie magnétique du minimiseur satisfait Ce théorème a été prouvé par Helffer et Kachmar [27] quand H vérifie (1.3.9). Nous allons donner un développement asymptotique de l'énergie locale du minimiseur E(ψ, A; D) définie dans (1.2.8) où D ⊂ Ω est un ensemble ouvert. Théorème 1.3.5. Supposons que B 0 vérifie (1.1.3) et D un ouvert régulier tel que D ⊂ Ω, nous avons, 1. Si (ψ, A) est un minimiseur de (1.1.2) et H(κ) vérifie (1.3.1), alors,

	(κH) 2	Ω	| curl A -B 0 | 2 dx = o κH ln	κ H	,	lorsque κ -→ +∞ .	(1.3.11)

  Dans le théorème suivant, pour des champs appliqués qui vérifient (1.3.1) et pour des configurations (ψ, A) d'énergie minimale, on définit des vortex en s'inspirant des méthodes de Sandier et Serfaty

.13) où C 1 min est une constante positive, alors, E(ψ, A, D) ≤ κ 2 D f H κ |B 0 (x)| dx + o κH ln κ H , lorsque κ -→ +∞ . (1.3.14) Helffer et Kachmar ont étudié le même problème quand H vérifie (1.3.9). Pour être plus précis, le reste sous la condition (1.3.9) était κ 3 H o(1), il devient maintenant κH ln κ H o (1).

  3 , où dx est la mesure de Lebesgue de R 2 restreint à Ω .Lethéorème 1.3.6 nous donne que si (ψ, A) est un minimiseur et B 0 un champ magnétique variable vérifie (1.1.3), alors ψ a des vortex qui sont distribués partout dans Ω mais avec une densité non uniforme. Mentionnons ici que dans [46], Sandier et Serfaty ont obtenu (lorsque H vérifie (1.3.4) et B 0 est constant) que ψ a des vortex qui sont distribués uniformement dans Ω. 1.3.2 Méthodes de démonstration. première étape du travail est d'analyser l'état fondamental avec conditions aux limites de Neumann et avec conditions aux limites de Dirichlet. Nous démontrons l'existence d'une constante positive C, telle que

	Une

  We will see in Remark 3.3.4 that the second term in the right hand side of (3.1.10), which is actually more simply o(κH ln κ) when (3.1.7) is satisfied, is of lower order compared with the leading term. Actually (see in Theorem 3.2.1), the function f satisfies f

	3.1.1. Under Assumptions (3.1.2) and (3.1.7), the ground state energy in (3.1.6)
	satisfies, as κ -→ +∞
	E g (κ, H) = κ 2	Ω	f H κ	|B 0 (x)| dx + o κH ln	κ H	.	(3.1.10)

  The estimate in (3.2.11) and (3.2.12) are obtained by Fournais-Kachmar (see [18, Theorem 2.1 and Proposition 2.8]) and by Kachmar (see [31, Theorem 2.4]) respectively. The lower bound in (3.2.15) is a consequence of [31, Theorem 2.1 and (2.9)].The next proposition gives us a lower bound of the ground state energy e N (b, R) which is needed it in the proof of Proposition 3.5.1.

	Proposition 3.2.2. There exists a positive constant C, such that if
	R ≥ 1	and	0 < b < 1 ,	(3.2.16)

  1[ and R ≥ 1. This assumption is satisfied because R ≥ √ κHρ > 1 (see Remark 3.3.2). Therefore, we get from (3.3.7) and (3.3.8) the estimate

  The choice of the parameters δ in (3.3.12) and in(3.3.11) implies that all error terms are of lower order compared to κH ln κ H .Remark 3.3.4. The remainder term in(3.3.20) is small compared with the leading order term.

	where	r(κ, H, ) = O		κ + δ -1 4 (κH) 2 .	(3.3.20)
	Having in mind Property (3) of the function f established in Theorem 3.2.1, we recognize the
	lower Riemann sum and notice that ∪ γ∈J ,ρ Q γ, ⊂ Ω, then, we get by collecting (3.3.18)-(3.3.19)
	that	E κ,H (s, F, Ω) ≤ (1 + δ)κ 2	Ω	f H κ	|B 0 (x)| dx + r(κ, H, ) .	(3.3.21)
						.19)

  Energy and vorticity of the Ginzburg-Landau model with variable magnetic field Moreover, we can replace D by Ω and get E 0 (ψ, A; Ω) ≥ κ 2

	Ω	f H κ	|B 0 (x)| dx + o κH ln	κ H	.	(3.5.24)
	Proof of Corollary 3.1.4
	Having in mind (3.1.16), we write

5.

Notice that E 0 (ψ, A; D) ≥ E 0 (ψ, A; D ,ρ ). Using

(3.5.19

) and (3.5.21) with the same choices of δ, ρ and as in Remark 3.3.2, we obtain

E 0 (ψ, A; D) ≥ κ 2 D f H κ |B 0 (x)| dx + o κH ln κ H . (

3

.5.23)

  3 establishes an upper bound on the ground state energy. Section 4.4 contains useful estimates on minimizers. The estimates in Section 4.4 are used in Section 4.5 to establish a lower bound of the ground state energy and to finish the proof of Theorem 4.1.2, Corollary 4.1.3 and Theorem 4.1.4. In Section 4.6, we discuss the conclusion in Theorem 4.1.2 by providing various examples of pinning terms obeying Assumption 4.1.1. Section 7 is devoted to the proof of Theorem 4.1.5. Section 4.8 generalizes a theorem of Giorgi-Phillips concerning the breakdown of superconductivity under a large applied magnetic field. Sections 4.9 and 4.10 are devoted to the proof of Theorem 4.1.6. The proof of Theorem 4.1.7 is the purpose of Sections 4.11 and 4.12.

  Proposition 4.2.1. For all M > 0, there exist universal constants C M and R M such that

.

2.11)

The next proposition was proved in [6, Lemma 2.2, Proposition 2.4] in the case α = 1. It's present form can be deduced immediately from (4.2.8).

  .3.5) Using the bound | w ,x 0 , x 0 | ≤ 1, which is immediately deduced from the bound of | u R |, we get from (4.3.5), |w ,x 0 , x 0 | 2 ≤ a + ( x 0 , κ) .

	(4.3.6)
	Proposition 4.3.2. Under Assumptions (4.1.4)-(4.1.7), there exist positive constants C and κ 0

  4 4 a 2 , (4.3.17) which finishes the proof of Proposition 4.3.2.

	= κ -7 12 ,	ρ = κ -17 24 ,	L(κ) ≤ C κ	1 2 .	(4.3.18)
	and
		δ = κ -1 12			(4.3.19)

Application 4.3.3. We select , ρ, δ and the constraint on L(κ) as follows : Under Assumption (4.1.15), this choice permits to verify the assumptions in Proposition 4.3.2 and to obtain error terms of order o(κ 2 ). We have indeed as κ -→ ∞ κ = κ 19 12

  .4.13) 104 Pinning with a variable magnetic field of the two dimensional Ginzburg-Landau model Proof. Under Assumption (4.1.15), Theorem 4.3.4 yields curl

  .23) (ψ, A; a, Ω) is defined in(4.1.19). Using (4.1.17) and (4.5.22) (with D = Ω), then under Assumption (4.1.15) as κ -→ +∞

	4.6 Study of examples	109
	where E 0

  ) . ≤ C |x -x| , ∀t ∈ R 2 , ∀x, x ∈ D, s.t. |x -x| < 0 . (4.6.10)

	Lemma 4.6.4. Let D ⊂ R 2 be a bounded open set and φ : R 2 × D -→ R 2 be a continuous
	function satisfying :
	φ(t + T, x) = φ(t, x) ,	∀T ∈ T 1 Z × T 2 Z ,	(4.6.9)
	and uniformly Lipschitz, i.e. with the property that there exist constants C > 0 and 0 , such
	that,
	|φ(t, x) -φ(t, x)|

  .6.[START_REF] Fournais | On the third critical field in Ginzburg-Landau theory[END_REF] Hence we get that Ω ρ x → φ(t, x) is uniformly Lipschitz for t ∈ I -. Now, we apply Lemma 4.6.4 with D = Ω ρ and M = κ

	1 2 and we obtain,

  Proposition 4.6.7 (Verification of (A 4 )). Suppose that the function α defined in Proposition 4.6.5 satisfies |α| + |∇α| > 0 in R 2 .

		(4.6.19)
	Then Assumption (A 4 ) is satisfied.
	Proof. Using (4.6.19), a change of variable y = κ	1 2

  Remark 4.7.1. We can replace D by D c such that the estimate in (4.7.11) is still true. That is :

	4.8 Extension of the Giorgi-Phillips Theorem	121
	Notice that,
				a(x, κ) 2 dx =			a(x, κ) 2 dx +	a(x, κ) 2 dx .
				D		D∩{a(x,κ)≤0}	D∩{a(x,κ)>0}
	Therefore,
	-	κ 2 2 D	|ψ| 4 dx ≥ κ 2	D∩{a(x,κ)>0}	a(x, κ) 2 f H κ	|B 0 (x)| a(x, κ)	dx -	κ 2 2 D∩{a(x,κ)>0}	a(x, κ) 2 dx
										+ o κ 2 . (4.7.10)
	Dividing both sides by -κ 2 2 , we obtain, as κ -→ +∞ ,
			D	|ψ| 4 dx ≤ -	D∩{a(x,κ)>0}	a(x, κ) 2 2 f H κ	|B 0 (x)| a(x, κ)	-1 dx + o (1) .	(4.7.11)
										.7.8)
	Using (4.5.22) with h = χ and taking the choice of defined in (4.3.18), we get, as κ → +∞,
	κ 2 2 D	(a 2 -|ψ| 4 ) dx ≥ κ 2	D∩{a(x,κ)>0}	a(x, κ) 2 f H κ	|B 0 (x)| a(x, κ)	dx+	κ 2 2 D∩{a(x,κ)≤0}	a(x, κ) 2 dx
										+ o κ 2 . (4.7.9)

  .10.3) 132 Pinning with a variable magnetic field of the two dimensional Ginzburg-Landau model By choosing C such that C M > C 1 , we get, µ 1 (κ, H) < 0 .

	Case 2. Here, we suppose that
	sup x∈∂Ω	a(x) Θ 0 B 0 (x)	≥ sup x∈Ω	a(x) B 0 (x)	.

  Pinning with a variable magnetic field of the two dimensional Ginzburg-Landau model First, we let χ ∈ C ∞ (R) be a standard cut-off function such that

	134 χ = 1 in [1, ∞]	and	χ = 0 in ] -∞, 1/2] .	(4.10.12)
			3 2 .	(4.10.11)

.10.10)

Proof.

The techniques that will be used in this proof are similar with the ones in

[START_REF] Fournais | The ground state energy of the three dimensional Ginzburg-Landau functional Part I : Bulk regime[END_REF] Lemma 2.6

]. If H satisfies (4.10.6) for some C > 0, then, for any x ∈ Ω, we have. κ H B 0 (x) -κ 2 a(x) ≥ C κ

  iκHA)χ κ ψ| 2 -|∇χ κ | 2 |ψ| 2 dx = κ 2 Ω |χ κ | 2 (a(x) -|ψ| 2 )|ψ| 2 dx . (4.10.14) We estimate Ω |(∇ -iκHA)χ κ ψ| 2 dx from below. As in [27, Proposition 6.2], we can prove that,

	.10.13)
	Referring to (4.7.6), we have
	Ω

Ω |(∇ -

  By Proposition 4.10.1 and Theorem 4.10.4, we get the existence of positive constants C and κ 0 , 138 Pinning with a variable magnetic field of the two dimensional Ginzburg-Landau model such that for κ ≥ κ 0 ,

	κ max sup
	x∈Ω
			.10.25)
	We note that H loc C 3 ≤ H	loc C 3 and H cp C 3 ≤ H	cp C 3 . Therefore, all the critical fields are contained in the
	interval [H loc C 3 , H

cp C 3 ].

  4.11 Asymptotics of µ 1 (κ, H) : the case with vanishing magnetic fieldIn this section we give an estimate for the lowest eigenvalue µ 1 (κ, H) of the operator P Ω

						κHF,-κ 2 a
	B = κH	and	σ =	H κ 2 .	(4.11.1)
	We observe that,
	P Ω κHF,-κ 2 a = P Ω BF,-( B σ )	3 a 2	.
	We will give an estimate for the lowest eigenvalue µ B, σ of P Ω BF,-( B σ )	2 3 a

(see

(4.1.26)

) in the case when Γ = ∅ with a κ-independent pinning, i.e. a(κ, x) = a(x). The results in this section are valid under the assumption Γ = ∅, where the set Γ is introduced in (4.1.3). Let

  |(∇ -iBF)χ j h 2 ψ| 2 dx ≥ Θ 0 B |B 0 (ω j )| 2Since this is true for all x 0 ∈ Γ, we deduce that,Λ 1 (B 0 , a, σ) + CB -1/18 ,where Λ 1 (B 0 , a, σ) is introduced in (4.11.2).Theorem 4.11.4. Let λ max > 0. Suppose that Γ = ∅ and a ∈ C 1 (Ω). There exist two constants C 1 > 0 and κ 0 > 0 such that, if, κ ≥ κ 0 , and κ 0 κ -1 < H < λ max κ 2 (4.11.17)then µ 1 (κ, H) ≤ κ 2 Λ 1 B 0 , a, H κ 2 + C 1 κ

		µ B, σ ≤	σ B	2 3
		Proposition 4.11.3 permits to obtain :
						11 6 ,	as κ → +∞ .
						|χ j h 2 ψ| 2 dx -C B 2 4 1	|χ j h 2 ψ| 2 dx
	j	Ω			int Ω		int Ω
		≥	M Θ 0 2	B -CB 2 4 1	j	Ω	|χ j h 2 ψ| 2 dx
		=	M Θ 0 2	B -CB 2 4 1	Ω	|h 2 ψ| 2 dx .	(4.11.12)

  Proof. Let λ = κ -1 2 and Ω λ = {x ∈ Ω : dist(x, ∂Ω) > λ & dist(x, Γ) > λ}. We introduce a function h ∈ C ∞ c (Ω) satisfying 0 ≤ h ≤ 1 in Ω , h = 1 in Ω λ , supp h ⊂ Ω λ/2, Pinning with a variable magnetic field of the two dimensional Ginzburg-Landau model Now, the Cauchy-Schwarz inequality yields,c κ ψ L 2 (Ω) hψ 2 L 4 (Ω) ≤ c 2 ψ 2 L 2 (Ω) + κ 2 hψ 4 L 4 (Ω) , |B 0 (x)| -κ 2 a(x) |hψ| 2 dx ≤ Ω |(∇ -iκHA)hψ| 2 dx -κ 2 Ω a(x) |hψ| 2 dx + c 2 ψ 2 L 2 (Ω) + κ 2 hψ 4 L 4 (Ω) .We may use a localization formula as the one in (4.10.14) (but withχ κ = h) to write, |B 0 (x)| -κ 2 a(x) |hψ| 2 dx ≤ c 2 Ω |ψ| 2 dx + Ω |∇h| 2 |ψ| 2 dx + κ 2 Ω (h 4 -h 2 )|ψ| 4 dx

	which implies that
		≤ c 2	|ψ| 2 dx +
		Ω
					1 4 ψ 2 L 4 (Ω) .	(4.12.7)
	and	|∇h| ≤	C λ	in Ω ,

where C is a positive constant. Using (4.8.2), we can prove that (see the detailed proof in [27, Eq. (6.6)] when a is constant),

κ H Ω |B 0 (x)| |hψ| 2 dx -c κ ψ L 2 (Ω) hψ 2 L 4 (Ω) ≤ Ω |(∇ -iκHA)hψ| 2 dx . 148 Ω κ H Ω κ H Ω |∇h| 2 |ψ| 2 dx .

  -|u| 2 | 2 dx ≥ Applying Proposition A.2.6 with ω = Ω t , we have, H 1 (γ t ) ≥ 2 r(Ω t ) ≥ 2 r(γ t ) .Next, we write a lower bound of the integral of |u| 2 |∇ϕ -A| 2 . We will use the co-area formula two times two write, P4) : If (0 ≤ r i ≤ R i ) 1≤i≤k are 2k positive real numbers, (d i ) 1≤i≤k are integers, and

	∀i, j,	R i r i	=	R j r j	= α ,	α > 1 ,
	then
	k			k						k	k
	|d i |f (r i , R i ) ≥		|d i | f	r i ,	R i .	(A.3.17)
	i=1			i=1					i=1	i=1
						√	2		0	+∞	|1 -t 2 | ε	H 1 (γ t ) dt .
	It follows that,
	K	1 2ε 2 (1 -|u| 2 ) 2 dx ≥	√	2	0	+∞	|1 -t 2 | ε	r(γ t ) dt .	(A.3.7)
		t∈(0,∞)	dt	γt∩K	t 2 |∇ϕ -A| 2 | ∇|u| | 2 dl
	=	t∈(0,∞)	-t 2 d dt t			γs∩K	|∇ϕ -A| 2 | ∇|u| | 2 dl ds dt
	=	t∈(0,∞)	-t 2 d dt	K\Ωt	|∇ϕ -A| 2 dx dt
	= -	t 2 θ (t) dt ,			(A.3.8)
		t∈(0,∞)
	where θ(t) is introduced in (A.3.1).

K |∇|u|| 2 dx + K |u| 2 |∇ϕ -A| 2 dx = Knowing that (0, 1) ⊂ (0, ∞) and θ ≤ 0, we infer from (A.3.8), K |u| 2 |∇ϕ -A| 2 dx ≥ -1 0 t 2 θ (t) dt .

(

  A Theorem à la Sandier-Serfaty K \ Ω tε . Now, (A.5.10) and (A.5.11) yield,J K (u, A) ≥ t 2Since h is constant in every connected component of Ω tε , then ∇ h = 0 in Ω tε and consequently, (A.5.13) becomes,

		ε	1 2 K\Ωt ε	|∇ϕ -A| 2 dx +	1 2 K	| curl A -h ex | 2 dx
	≥ t 2 ε	1 2 K\Ωt ε	|∇ ĥ| 2 dx +	1 2 K	| ĥ -h ex | 2 dx .
	Using (A.1.5), we get,
	K\Ωt ε	|∇ ĥ| 2 dx +	K	| ĥ -h ex | 2 dx ≤	2 h ex t 2 ε	2	log	ε	1 √ h ex	.	(A.5.13)
						2 h ex t 2 ε	2	log	ε	1 √ h ex	.	(A.5.14)
					1 √ 2 hex .
	Then, for any r(ω) ≤ σ ≤ 1 2 √ hex , there exists a family (B i ) of disjoint disks of radii r i
	such that
	1.

i r i = σ 2. ω ⊂ ∪ i B i K |∇ ĥ| 2 dx + K | ĥ -h ex | 2 dx ≤

  Theorem à la Sandier-SerfatyTo see this, we write by the co-area formula,

	Now, consider the set
	T =	   β ∈ T :	∂K β	∂ ∂n ĥ	2	<	8 h ex α t ε 2 log 2	ε	1 √ h ex	  	.
	We claim that,				|T | ≥	α 4	.
				|∇ ĥ| 2 dx =			α		|∇ ĥ| 2 dβ .
		K\Kα						0		∂K β
	T \T	8 h ex α t ε 2 log 2	ε	1 √ h ex	dβ ≥ (|T | -|T |)	8 h ex α t ε 2 log 2	ε	1 √ h ex	.
	Now, using (A.5.14), we get,
	(|T | -|T |)	8 h ex α t ε 2 log 2	ε	1 √ h ex	≤	2 h ex t ε 2 log 2	ε	1 √ h ex	.
	Since |T | ≥ α 2 , we deduce that |T | ≥ α 4 .
					∂K β 0	∂ ∂n ĥ	2	<	8 h ex α t ε 2 log 2	ε	1 √ h ex	.	(A.5.21)

Since T ⊂ T ⊂ [0, α] and |∇ h| = |∂ h/∂n|, we get further, K\Kα |∇ ĥ| 2 dx ≥ Now, since |T | > 0, then T = ∅ and there exists β 0 ∈ [0, α] such that ∂K β 0 ∩ i B i = ∅ and

  5.22), use (A.5.14), (A.5.21) and expand ( -α) 2 to obtain, 2πd(t) ≥ h ex ( -α) 2 -

				C	3 2	h ex L(ε)	1 2
				t ε	α
	≥ 2 h ex 1 -	4α -	C t ε	L(ε) h ex α	.	(A.5.23)
	Now, we choose
							1
		α =	L(ε) h ex	2	3	.
	Under Assumption (A.1.4), we get, for ε small enough,
			α	4	,
	and,
	L(ε) √ h ex	,	L(ε) h ex 2	1 L(ε)	.
	Thus,
	α	L(ε) √ h ex 1 2 -1 3			√	1 h ex	.

Therefore this choice of α respects (A.5.20). Moreover, (A.5.23) becomes,

2π d(t) ≥ 2 h ex 1 -C t ε ∆ ,

1.1 Motivations et notations

µκ est faiblement convergente vers µ c'est-à-dire :µκ(f ) -→ µ(f ) , ∀f ∈ C0(Ω) .1.4 Énergie pour un modèle de Ginzburg-Landau avec un champ magnétique variable et un terme de "pinning"

1.4 Énergie pour un modèle de Ginzburg-Landau avec un champ magnétique variable et un terme de "pinning"

The ground state energy of the two dimensional Ginzburg-Landau functional with variable magnetic field

, (2.5.12)

.(2.5.24)

≤ H κ as κ -→ +∞ , (3.1.19)

µκ converge weakly to µ means that :µκ(f ) -→ µ(f ) , ∀f ∈ C0(Ω) .

Remerciements

Energy and vorticity of the Ginzburg-Landau model with variable magnetic field Using (3.7.49), we obtain

(3.7.51) Here, we have used the fact that B 0 ( γ) ≤ β 0 to estimate the errors terms, where β 0 is introduced in (3.7.6). Now it is time to determine γ∈J B 0 ( γ) 2 . We will do this in two steps : Upper bound : Notice that till now γ was an arbitrary point in Q (γ), but that our estimates are independent of this choice. We now select γ ∈ Q (γ) such that B 0 ( γ) = B γ, with B γ, satisfying (3.3.14), and get

We recognize in the right hand side above the lower Riemann sum of x -→ B 0 (x) and we use that S ⊂ S to obtain γ∈J B 0 ( γ) 2 ≤ S B 0 (x) dx .

(3.7.52)

Lower bound : We select γ ∈ Q (γ) such that B 0 ( γ) = B γ, with B γ, satisfies (3.5.17).

Similarly to what we did in the upper bound above, we get γ∈J B 0 ( γ) 2 ≥ S B 0 (x) dx .

Notice that using the regularity of ∂S and (3. In light of (3.7.56), we can easily show that µ κ converge weakly to µ = B 0 (x) dx, which means that :

Proof of (1) : We will prove that the sum of the radii of the disks (D( a k,γ , r k,γ )) k∈K ,γ γ∈J is less than (κH)

In fact, remembering the choice of δ(κ) in (3.7.3), (3.7.49) and that Q (γ) ⊂ Ω ∩ {B 0 > 0}, we have , we get γ∈J r k,γ ≤ (κH)

End of the proof of Theorem 3.1.6 : In {B 0 < 0} ∩ Ω, we apply Proposition 3.7.3 with

.7.58) So we get that, the convergence of mesure µ κ in (3.7.56) is still true when S ⊂ Ω ∩ {B 0 < 0}. Similarly, we can control the convergence of |µ κ |(S). Now we observe that the support of µ κ does not meet {B 0 = 0}. Hence µ κ (S) = µ κ (S ∩ {B 0 < 0}) + µ κ (S ∩ {B 0 > 0}) and we can apply the previous arguments to S -= S ∩ {B 0 < 0} and

In the next theorem, we give a simple proof of the result which says that (0, F) is the unique minimizer of the functional when H is sufficiently large and when the magnetic field B 0 is constant with pinning term. Then, there exist positive constants C and κ 0 , such that, if

then (0, F) is the unique solution to (4.1.12).

Proof. We assume that we have a non normal critical point (ψ, A) for E κ,H,a,B 0 . This means that (ψ,

Since ψ satisfies (4.8.15), this implies by assumption that the lowest Neumann eigenvalue

Thanks to Theorem 4.8.2, we get the existence of a constant C > 0, such that, if H ≥ C κ, then (0, F) is the unique solution to (4.1.12).

4.8.3

The case Γ = ∅.

We recall the definition of λ 0 in (4. 1.31), the definition of Γ in (4.1.3) and for any θ ∈ (0, π) we we have,

2 ) , as B → +∞ . (4.9.13)

Here, Θ 0 is introduced in (4.8.10).

Proof. We recall the definition of µ B,σ as follows :

The first term in the right hand side is studied by Helffer-Morame (see [START_REF] Helffer | Magnetic bottles in connection with superconductivity[END_REF]Theorem 9.1] with h = B -1 and µ B,σ = µ (1) (h) h 2 ) or Fournais-Helffer (see [14, Section 9.2.1]). They proved for any x 0 ∈ ∂Ω the existence of B 0 such that for B ≥ B 0 one can construct a trial function u such that,

2 ) , as B → +∞ .

The estimates of the second term in the right hand side are just as in (4.9.10) and this achieves the proof of the proposition.

Remark 4.9.6. ∂Ω being compact, we can choose x 0 ∈ ∂Ω, such that,

and we apply Proposition 4.9.5 with

which implies under Assumption 4.9.11 that, Suppose that (4.9.11) hold and a ∈ C 1 (Ω), we have

Here, Λ 1 is introduced in (4.9.1).

Notice that the conclusion in Theorem 4.9.7 is valid under the assumption κH ≥ B 0 with B 0 > 0 sufficiently large. Lemma 4.9.8 below takes care of the regime where κH = O(1). Lemma 4.9.8. Let C max > 0. Suppose that {a > 0} = ∅. There exists a constant κ 0 > 0 such that, if κ ≥ κ 0 and 0 ≤ H ≤ C max κ -1 , Putting (4.11.12) into (4.11.11), we obtain

(4.11.13)

We choose 1 = B -ρ and 9 22 < ρ < 11 29 . We observe that,

In this way, we infer from (4.11.13), that there exists a constant c > 0 such that, for B sufficiently large, 6 ) , as κ → +∞ .

Here, Λ 1 is introduced in (4.11.2).

Proof. We apply Proposition 4.11.1 with

Let us verify that the conditions of the proposition are satisfied for this choice. Thanks to (4.11.15), σ ∈ I. Now, as κ → +∞, we have,

This implies that, as κ → +∞, 6 ) . This finish the proof of the theorem.

Upper bound

The next theorem is a generalization of the results in [START_REF] Pan | Schrödinger operators with non-degenerately vanishing magnetic fields in bounded domains[END_REF] and [START_REF] Miqueu | Équation de Schrödinger avec champ magnétique qui s'annule[END_REF] valid when the pinning term a(κ, x) = a(x) is independent of κ and non-constant.

We denote by µ B, σ the lowest eigenvalue of the operator

.

Proposition 4.11.3. Suppose that Γ = ∅ and λ max > 0. There exist positive constants C and B 0 such that, for σ ∈ (0, λ max ], a ∈ C 1 (Ω) and B ≥ B 0 , we have,

Proof. Let x 0 ∈ Γ. In [START_REF] Pan | Schrödinger operators with non-degenerately vanishing magnetic fields in bounded domains[END_REF][START_REF] Miqueu | Équation de Schrödinger avec champ magnétique qui s'annule[END_REF], a quasi-mode u(B,

) and,

where B 0 and C are constants independent of the point x 0 and the parameter B, and

Using the smoothness of the function a(•), we get in the support of u(B,

Thus, we deduce that,

Thanks to the min-max principle, we deduce that,

Annexe A

A Theorem à la Sandier-Serfaty

In this chapter, we present a detailed proof of Theorem A.1.1 that will imply the conclusion in Proposition 3.7.3. Theorem A.1.1 and its proof are given by Sandier-Serfaty in [START_REF] Sandier | On the energy of type-II superconductors in the mixed phase[END_REF].

A.1 Statement of the Theorem

Let us consider two positive parameters h ex and ε.

), then we define the energy J K (u, A) as follows,

In this chapter, we will use the notation that if a(ε) and b(ε) are two non-negative functions of ε, then a b means that a(ε) = δ(ε)b(ε) and δ(ε) → 0 as ε → 0 + .

The aim of this chapter is to present a detailed proof of

There exists two positive constants ε 0 and C > 0, such that if

, as ε → 0 , (A.1.5)

then, there exist disjoint disks B 1 , ..., B k with the following properties :

• The sum of the radii of the disks B 1 , ..., B k is less than h

.

Notice that, in Theorem A.1.1, the configuration (u, A) is not a critical point of the functional J K . Furthermore, under the assumptions on ε and h ex , it is clear that,

and lim

The proof of Theorem A.1.1 will occupy the rest of this chapter.

A.2 Preliminaries

In this section, we collect some notions and theorems from [START_REF] Sandier | Vortices in the magnetic Ginzburg-Landau model[END_REF]Chapter 4].

Notation.

We will use the following notation :

• The letter C denotes a positive constant that is independent of the parameter ε, and whose value may change from a formula to another.

• If B is a ball, r(B) denotes its radius and if B is a collection of balls, then r(B) is the sum of the radii of the balls in the collection.

• For λ ≥ 0 the ball λB is the ball with same center as B and radius multiplied by λ. If B is a collection of balls, then λB = {λB : B ∈ B}

• Given > 0 and

denotes the square of side length centered at x and we write K = Q (0).

• For t ∈ R + and u ∈ C 1 (K, C), we denote by

Definition A.2.1.

A.2 Preliminaries

153

• [Radius of a compact set] : Let ω ⊂ R 2 be a compact set. We define the radius of ω,

where the infimum is taken over all finite coverings of ω by closed balls B 1 , ..., B k .

• Degree of S 1 valued functions :

Here, τ denote the unit tangent vector to ∂Ω compatible with this orientation.

Remark A.2.2.

2. The infimum which defines the radius is not necessarily achieved.

3. The notion of the radius is a subadditive set-function, i.e. r(ω 1 ∪ ω 2 ) ≤ r(ω 1 ) + r(ω 2 ).

4. As we shall see in Proposition A.2.6 below, there is a relation ship between r(ω) and the perimeter of ω. To do this, we merge every two intersecting balls in the collection (B k ) into a single ball as in Lemma A.2.3. This process is repeated a finite number of times until one arrives to the collection (B i ).

A Theorem à la Sandier-Serfaty Theorem A.2.5 (Ball growth). Let B 0 be a finite collection of disjoint closed balls. There exists a family {B(t)} t∈R + of collections of disjoint closed balls such that B(0) = B 0 and 1. For every s ≥ t ≥ 0,

2. There exists a finite set T ⊂ R + such that if [t 0 , t 1 ] ⊂ R + \ T , then B(t 1 ) = e t 1 -t 0 B(t 0 ).

r(B(t))

= e t r(B 0 ) for every t ∈ R + .

Next, we recall a relationship between the notion of the radius and the perimeter :

Here, the notion d B is introduced in Definition A.2.1.

A.3 Useful inequalities via the co-area formula

In the sequel, K is a square of side-length ∈ (0, 1), u : Ω → C, A : Ω → R 2 are C 1 , and for all t > 0,

Notice that for all t > 0, |u| > 0 in Ω t . We write u = |u|e iϕ and define for any t > 0,

Clearly θ is a decreasing function, hence almost everywhere differentiable and

where

Now, an integrating by parts yields,

Putting (A.3.7) and (A.3.9) into (A.3.6), we obtain,

(A.3.10) This finishes the proof of Lemma A.3.1.

In the next lemma, B R denotes the ball of center 0 and radius R.

(A.3.11) Here, h = curl A and d is the winding number of v restricted to the circle ∂B R , i.e.

Remark A.3.3. For all r ≤ t ≤ R, let d Bt be the winding number of v restricted to the circle ∂B t , i.e.

The obvious inequality |∇ϕ -A| ≥ | ∂ϕ ∂τ -A • τ | and the co-area formula yield, for all r ≤ t ≤ R,

A.3 Useful inequalities via the co-area formula 157

Since, for all r ≤ t ≤ R, B t ⊆ B R , then we find,

Now, by definition of d Bt , we get,

Thanks to the Cauchy-Schwarz inequality, we write,

Now, returning back to (A.3.12), we can bound e(t) from below as follows,

We will minimize the right-hand side of (A. 3.14) with respect to λ t . Define

where a, b, c and d are real numbers.

Clearly,

and it is easy to see that

is the minimizer of g(x), i.e., g(x) ≥ g(x min ) .

A Theorem à la Sandier-Serfaty We compute g(x min ) and obtain

This implies that

By this choose, we get from (A.3.15),

Simplifying and using that, for all 0 < t < 1, (1

We insert this into (A.3.16) then we integrating between r and R. In view of (A.3.13) and (A.3.16), we obtain,

which finishes the proof of Lemma A.3.2.

Define the function

Lemma A.3.4. The following properties are true :

(P1) : f (r, .) is increasing on 0, min

(P2) : f (r, r) = 0, and, for any r ≤ R ≤ min 1,

A Theorem à la Sandier-Serfaty 3. Letting h = curl A and v = e iϕ , then,

where C is a positive constant and d B i is the winding number of v restricted to ∂B i if B i V, and zero otherwise.

Proof. By definition of r(ω) and Remark A.2.4, there exists a finite collection of disjoint closed disks

Thanks to Theorem A.2.5, for all t > 0, there exists a family of disks

Furthermore, there exists a finite set

Let us define t * > 0 by the relation,

We will treat the harder case where this set is non-empty and 0 < t 1 < t n = t * . We let t 0 = 0.

We have

.

Now, we have the inclusion,

A.4 Construction of vortex disks for S 1 valued functions 161

In light of the obvious inclusion,

we get,

By Theorem A.2.5,

We can apply Lemma A.3.2 to write a lower bound for every

then we sum over i and use Lemma A.3.4 to write,

where f is the function introduced in (A.3.17), and for all t > 0,

Thanks to Lemma A.2.7, we have,

We insert this into (A.4.2) then sum over k ∈ {1, • • • , n}. In that way, we get, Now, since the term on the left hand side of (A.4.3) is independent of ε > 0, we take the limit as ε → 0 + and obtain that, Note that, by our choice of r(B(t * ) = σ and r(B(0) ∈ [r(ω), 2r(ω)], and the definition of the f , we infer now from (A.4.4)

where C is a constant. If the term log σ r(ω) -C ≤ 0, then we have the obvious estimate,

In that way, we finish the proof of the proposition.

A.5 Proof of Theorem A.1.1

This section contains the lengthy proof of Theorem A.1.1. We will work under the assumptions in Theorem A.1.1.

We will split the proof into seven steps.

Step 1 : Locating the set of vortices.

Recall that (u, A) is a configuration satisfying (A.1.5). We will prove that there exists

where γ tε = ∂Ω tε and Ω tε = {x ∈ K : |u(x)| > t ε }.

A.5 Proof of Theorem A.1.1

163

Having in mind the definition of a and b in (A.3.4) and (A.3.5), thanks to (A.3.3), we find that,

Now, the assumption in (A.1.5) ensures that, for ε ∈ (0, ε 0 ) and ε 0 sufficiently small,

Using the mean value theorem, there exists Here, we have used the fact that x log 1

x 4 -→ 0 as x -→ 0 + , and ε √ h ex -→ 0 as ε -→ 0 + . Therefore, for ε small enough, we obtain,

Step 2 : Construction of vortex disks.

We will apply Proposition A.4.1 with V = K, v = u |u| and ω = Ω tε .

Thanks to (A.5.1), for ε small enough, we observe that,

Thus, we can take σ = 1 2 √ hex in Proposition A.4.1. We find a finite collection of closed disks (B i ) i∈I , such that, every disk B i has radius r i ,

and,

(A.5.4)

Step 3 : Upper bound on the total degree.

We insert this into (A.5.4) and use the upper bound in (A.1.5) to obtain, Thanks to the condition in (A.1.4) and the estimate in (A.5.2), we have, log r(γ tε ) ε ≤ C log L(ε) .

We insert this into (A.5.5)

After simplification, we obtain,

Step 4 : Refinement of the family of disks.

Recall that u = |u|e iϕ and the definition of the term b(t) in (A. Notice that, proving (A.5.8) finishes the proof of Theorem A.1.1. We will prove (A.5.8) in four steps.

Step 4.1 : An auxiliary field. Let A be a minimizer of the following problem :

(A.5.9)

Starting from a minimizing sequence, it is standard to prove the existence of a minimizer A. Let ĥ = curl A. We will prove that ĥ satisfies, -∇ ⊥ ĥ = ∇ϕ -A in K \ Ω tε (A.5.10) ĥ = cst in each connected component of Ω tε (A.5.11) ĥ = h ex on ∂K \ Ω tε . (A.5.12)

Let B ∈ H 1 (K, R 2 ) and divB = 0. Therefore, for all s, A+sB ∈ H 1 (K, R 2 ) and div( A + sB) = 0. Let i(s) = I( A + sB) .

Since A is a minimizer of I, then i has a minimum at s = 0 and i (0) = 0. By straightforward computations, we find, Integrating by parts the second integral in the l.h.s. yields that h = curl A is a solution to (A.5.10)-(A.5.12).

Step 4.2 : Estimating the auxiliary field.

Here, we are going to prove that the auxiliary field ĥ is close to h. Recall that |(∇-iA)u| 2 = |∇|u|| 2 + |u| 2 |∇ϕ -A| 2 , A is a minimizer of the problem (A.5.9) and |u| > t ε = 1 -L(ε) -1 in