Keywords: periodic homogenization, nonlinear homogenization, porous medium, elliptic PDEs

PRÉAMBULE

La théorie de l'homogénéisation traite de l'étude des matériaux multi-échelles, c'est-à-dire de matériaux dont certaines propriétés physiques dépendent d'une micro-échelle, supposée très petite devant l'échelle d'observation. L'objectif principal est alors de remplacer ce matériau par un autre, plus simple, dont les propriétés physiques sont moyennées, ou homogénéisées. Bien que le cadre général de l'homogénéisation demande assez peu de propriétés sur la microstructure, une difficulté majeure est d'obtenir des quantités homogénéisées explicites et facilement calculables en pratique. Un cadre particulièrement agréable est le cadre périodique, pour lequel la description fine du matériau peut se limiter à son étude sur une cellule de périodicité. Une autre approche, qui est en fait apparue assez tôt, est l'approche stochastique. On ne connaît le matériau qu'en moyenne statistique et l'on impose des conditions sur la loi qu'il suit. Signalons qu'il existe d'autres cadres permettant d'obtenir des résultats précis d'homogénéisation, par exemple le cadre quasi-périodique. Une question intéressante, en ayant notamment en vue une modélisation plus proche de la réalité, est le traitement des défauts. Dans cette thèse, nous nous intéressons au cas de défauts locaux dans un fond périodique, que nous illustrons sur différents problèmes d'équations aux dérivées partielles elliptiques. Dans une première partie, on traite le cas de l'homogénéisation en milieu perforé. Puis, dans une seconde partie, on s'intéresse à l'homogénéisation d'équations non-linéaires dans lesquelles le coefficient oscillant est périodique et localement perturbé. Nous décrivons brièvement le contenu de chaque chapitre.

Dans le chapitre 1, nous présentons le contexte scientifique général et nous introduisons les résultats principaux exposés dans les chapitres suivants. Nous finissons par une brève section de perspectives. Ces perspectives s'appuient sur les résultats obtenus dans le corps de la thèse.

Dans le chapitre 2, nous étudions l'homogénéisation de l'équation de Poisson en milieu perforé. Les trous sont espacés d'une distance ε ≪ 1 et sont de taille ε. Nous rappelons d'abord les résultats bien connus du cas périodique puis nous introduisons la classe de domaines non périodiques étudiée dans cette thèse. Cette géométrie s'appuie fortement sur le cadre périodique : lorsque la microéchelle converge vers zéro, le domaine non-périodique coïncide avec le domaine périodique loin de l'origine. Nous construisons ensuite les outils classiques de l'homogénéisation : développement à deux échelles et correcteurs. Nous finissons enfin par des taux de convergence assez précis. Ces résultats redonnent, pour le cas sans défaut, ceux, bien connus, du cas périodique.

Dans le chapitre 3, nous travaillons sur le système de Stokes dans le même cadre que celui du chapitre précédent. Le comportement limite quand ε → 0 est donné par la loi de Darcy. Comme pour l'équation de Poisson, nous construisons d'abord les correcteurs d'ordre 1 puis nous obtenons des taux de convergence dans un cadre cette fois-ci plus restreint.

 dans le cas linéaire. Nous traitons d'abord le cas d'une équation semi-linéaire. Nous présentons dans un premier temps des résultats d'homogénéisation dans le cas iii périodique pour cette équation. Nous constatons ensuite que les outils principaux nécessaires à l'homogénéisation, tels que les correcteurs, sont couverts par le problème linéaire associé, déjà étudié dans des travaux précédents.

Dans le chapitre 5, nous nous concentrons sur une équation quasi linéaire. Contrairement au cas semi linéaire, l'équation du correcteur est non linéaire. Nous commençons par des rappels sur le cas périodique puis nous exhibons un cadre variationnel (qui correspond à des défauts suffisamment localisés, par exemple à support compact) permettant de construire les correcteurs. Nous étendons ensuite les résultats périodiques au cadre non périodique. Une hypothèse importante dans ce chapitre sera la non dégénérescence des correcteurs périodiques. Nous illustrons ces résultats par une description complète de la situation dans le cas unidimensionnel.

Articles écrits durant cette thèse

 for the linear case. We first present homogenization results for periodic coefficients. We then notice that the main tools allowing for homogenization, such as the correctors, are obtained by studying the associated linear problem. Such problems have already been studied in previous works. v In Chapter 5, we focus on a quasilinear equation. Contrary to the semilinear case, the corrector equation is nonlinear. We start by recalling the main results of the periodic case. We then exhibit a variational framework (that corresponds to sufficiently fast decreasing defects, for example compactly supported) and we build the corresponding correctors. This allows to extend the convergence results from the periodic setting to the nonperiodic one. An important assumption in this chapter will be the non degeneracy of the periodic correctors. We also give a precise description of the situation in the one dimensional setting.

FOREWORD

The homogenization theory deals with the study of multiscale materials. In these materials, some physical properties depend on a microscale that is assumed to be very small compared to the observation scale. The main purpose is to replace these media by simpler ones for which the physical properties are averaged (or homogenized). Though the general theory of homogenization requires few conditions on the microstructure, one major challenge is to obtain explicit homogenized quantities that are easily computable. The periodic case is such a fundamental example: the fine description of the material can be achieved simply by considering a single periodic cell. Another approach, that appeared early in the mathematical development of the homogenization theory, is to consider a stochastic setting. The material is only statistically known and we impose conditions on the law it follows. Let us mention that there exist other settings allowing precise homogenization results, for example the quasiperiodic setting. A relevant question, having particularly in mind a more realistic modeling, is the treatment of defects. In this thesis, we are interested in local defects that may appear in a periodic background. We examine different problems. In the first part, we study the case of perforated domains. In the second part, we are interested in nonlinear homogenization with oscillating coefficients that are local perturbations of a periodic background. We now briefly describe the content of each chapter.

In Chapter 1, we present the general scientific context and we introduce the main contributions of the thesis, that are detailed in subsequent chapters. We end by a brief perspective section which is based on the results obtained in this thesis.

In Chapter 2, we study the homogenization of Poisson equation in porous medium. The holes are separated by a distance ε ≪ 1 and are of size ε. We first recall the well known results of the periodic case. We then introduce the class of nonperiodic domains that are considered in this part. The geometry strongly relies on the periodic framework: when the microscale tends to zero, the nonperiodic porous medium coincides with the periodic one far from the origin. We build the classical objects of the homogenization theory: two scale expansion and correctors. We end by proving precise convergence rates. We recover the results of the periodic case in this new setting.

In Chapter 3, we work on the Stokes system in the same framework as in the previous chapter. The behaviour of the system converges when ε → 0 to Darcy's law. As for the Poisson equation, we build the first order correctors and we obtain convergence rates when imposing some conditions on the source term.

TABLE DES MATIÈRES

Contexte scientifique

Cette thèse traite de la théorie de l'homogénéisation. L'objectif principal de cette théorie est d'étudier des équations très oscillantes à une échelle microscopique ε ≪ 1. Il peut s'agir d'une équation aux dérivées partielles (EDP) posée dans un matériau très hétérogène comme présenté sur la figure 1.1. En ingénierie, la résolution numérique d'une telle EDP (par exemple l'équation de la chaleur stationnaire) nécessite la discrétisation de l'espace en une maille de taille H ≪ ε. En particulier, ε ≪ 1 devient une contrainte très forte sur H et donc sur le temps de calcul. Pour se libérer de cette contrainte, l'idée est d'approcher théoriquement le milieu hétérogène par un milieu homogène. L'équation décrivant ce milieu homogène peut ensuite se discrétiser facilement. On adopte formellement la stratégie suivant T ε ≈

(1)

T * ≈ (2) (T * ) H où
• T ε est la température du milieu hétérogène et T * la température dans le milieu homogénéisé ;

• (T * ) H est une discrétisation de T * .

La seconde approximation (2) est numérique (et fournie par un théorème de convergence d'un schéma numérique quand H → 0) tandis que la première (1) est théorique (et fournie par un théorème de convergence de la solution oscillante vers la solution homogénéisée quand ε → 0). L'homogénéisation permet précisément d'effectuer l'approximation (1). D'autres motivations à l'homogénéisation sont ε → 0 Celle-ci est fortement hétérogène et l'on note ε sa taille caractéristique d'oscillation. On aimerait remplacer le matériau de gauche par le matériau de droite, homogénéisé. issues de la physique et en particulier du passage d'une loi microscopique à une loi macroscopique. On peut par exemple citer la loi de Darcy, que nous étudierons dans le chapitre 3. Nous rappelons brièvement le contexte historique. Nous renvoyons à [START_REF] Schweizer | Darcy's law and groundwater flow modelling[END_REF] pour une description plus détaillée et à [START_REF] Sanchez-Palencia | Fluid flow in porous media[END_REF][START_REF] Allaire | One-phase newtonian flow[END_REF] pour une approche mathématique. En 1856, l'ingénieur Henry Darcy, en charge des conduites d'eau à Dijon, postule que la vitesse d'un fluide à travers un milieu poreux est linéaire en la différence de pression : -→ v = -κ∇p où κ est une constante appelée perméabilité du milieu. Cette loi phénoménologique a été observée en faisant couler de l'eau dans un tube de sable vertical, ceci en vue de la filtrer [START_REF] Darcy | Les fontaines publiques de la ville de Dijon: exposition et application[END_REF]Note D]. Plus d'un siècle plus tard, une explication théorique, issue de l'homogénéisation, a été donnée à la loi de Darcy [START_REF] Tartar | Incompressible fluid flow in a porous medium. Convergence of the homogenization process[END_REF][START_REF] Sanchez-Palencia | Fluid flow in porous media[END_REF]. En régime permanent, l'équation vérifiée par la vitesse du fluide est donnée par un système de Stokes. Le problème peut alors se reformuler en un problème d'homogénéisation : en notant ε la taille caractéristique d'un grain de sable, le système de Stokes vérifié par le fluide est posé dans un domaine Ω ε qui comporte des "trous" (les grains de sable) de taille ε. On souhaite alors passer à la limite ε → 0 dans l'EDP. Nous renvoyons à la sous-section 1.2.1.3 pour des résultats mathématiques précis.

Du point de vue mathématique, la théorie de l'homogénéisation déterministe s'est développée dans les années 1970 avec les travaux de De Giorgi-Spagnolo [START_REF] De | Sulla convergenza degli integrali dellŠenergia per operatori ellittici del secondo ordine[END_REF], Babuska [START_REF] Babuška | Homogenization approach in engineering[END_REF], Sanchez-Palencia [START_REF] Sanchez-Palencia | Fluid flow in porous media[END_REF], Bensoussan-Lions-Papanicolaou [START_REF] Bensoussan | Asymptotic Analysis for Periodic Structures[END_REF], Murat-Tartar [START_REF] Murat | Compacité par compensation[END_REF][START_REF] Tartar | Compensated compactness and applications to partial differential equations[END_REF][START_REF] Tartar | The general theory of homogenization: a personalized introduction[END_REF][START_REF] Tartar | H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations[END_REF] et, plus tard, N'Guetseng [START_REF] Nguetseng | A general convergence result for a functional related to the theory of homogenization[END_REF][START_REF] Nguetseng | Homogenization structures and applications i[END_REF], Allaire [START_REF] Allaire | Homogenization and two-scale convergence[END_REF] sur la convergence à deux échelles, Braides-Defranceshi pour la minimisation de fonctionnelles [START_REF] Braides | Homogenization of multiple integrals[END_REF], Cioranescu-Damlamian-Griso [START_REF] Cioranescu | The periodic unfolding method in homogenization[END_REF] pour la méthode d'éclatement périodique. Dans [START_REF] Bensoussan | Asymptotic Analysis for Periodic Structures[END_REF], une hypothèse importante est la périodicité de la micro-structure. Cela permet dans de nombreux cas d'obtenir des grandeurs homogénéisées explicites. Pour la simple équation de diffusion thermique stationnaire div a

• ε ∇T ε = f dans Ω, T ε = 0 sur ∂Ω (1.1.1)
où a est un coefficient périodique, on a les résultats bien connus1

T ε -→ ε→0 T * , -div a * ∇T * = f dans Ω, T * = 0 sur ∂Ω (1.1.2)
où a * se calcule très simplement en résolvant d problèmes de cellule, appelés problèmes des correcteurs (ici, d est la dimension de l'espace ambiant). Ces résultats se généralisent au cas d'une suite de coefficients (a ε ) ε abstraite, uniformément bornée et coercive par rapport à ε (voir par exemple [START_REF] Tartar | The general theory of homogenization: a personalized introduction[END_REF]Chapitre 6] ou encore [START_REF] Nguetseng | Homogenization structures and applications i[END_REF] pour un autre cadre général abstrait dans un espace de fonctions admettant une moyenne). On a alors l'existence d'une sous-suite (ε k ) k∈N , ε k -→ k→+∞ 0, et d'un coefficient homogénéisé a * (indépendant du second membre f mais dépendant a priori de la sous-suite (ε k ) k∈N ) tel que T ε k -→ k→+∞ T * , -div a * ∇T * = f dans Ω, T * = 0 sur ∂Ω.

(1. 1.3) La question du calcul pratique de a * se pose. Comme nous l'avons dit, ce calcul est remarquablement aisé dans le cas périodique. Pour autant, dans le cas général, il s'avère très compliqué à mener, notamment car a * est défini par la limite (faible) d'une suite de fonctions, cf [101, Chapitre 13].

Un enjeu important de l'homogénéisation déterministe au delà du cas périodique est de trouver des classes de coefficients dont les coefficients homogénéisés se calculent facilement en pratique. Une telle classe (assez proche du cas périodique) est par exemple le cadre quasi-périodique [START_REF] Blanc | Local profiles for elliptic problems at different scales: defects in, and interfaces between periodic structures[END_REF] qui correspond, dans (1.1.1), à des coefficients a de la forme a(y) = b(y 1 , y 2 ) où b est périodique en chacune de ses variables (avec des périodes éventuellement non commensurables).

Parallèlement au développement de l'homogénéisation déterministe est apparue l'homogénéisation stochastique avec les travaux de Kozlov, Papanicolaou, Varadhan [START_REF] Jikov | Homogenization of differential operators and integral functionals[END_REF][START_REF] George | Boundary value problems with rapidly oscillating random coefficients[END_REF]. Dans ce cas, le coefficient a ε = a(./ε, ω) dépend d'un paramètre ω ∈ Ω où (Ω, F, P) est un espace de probabilité. On peut en fait voir Ω comme un ensemble de coefficients. Les hypothèses imposées sur P sont la stationnarité (c'est-à-dire l'invariance de la loi par l'opération de translation d'un vecteur x ∈ R d ) et l'ergodicité (qui peut être vue comme une décorrélation aux larges échelles). Dans ce cadre, on peut montrer, à

′ ε ) ′ = f , T ε (-1 2 ) = T ε ( 1
2 ) = 0 où f (x) = 1, a(y) = 2 + cos(2πy), ε = 0.1 et ε = 0.05. La fonction u ε est à gauche, sa dérivée est à droite. nouveau dans le cas de la diffusion linéaire, que la suite (T ε (•, ω)) ε>0 s'homogénéise avec un coefficient limite déterministe pour lequel il existe une formule explicite. Nous renvoyons à [START_REF] Clozeau | Quantitative estimates in stochastic homogenization of elliptic equations and systems[END_REF]Introduction] pour une synthèse à ce sujet.

Une fois la limite de la suite oscillante (T ε ) ε>0 trouvée se pose la question du mode de convergence vers la solution homogénéisée T * . Pour avoir une intuition des phénomènes en jeu, nous représentons sur la figure 1.2 un exemple en dimension 1. Nous constatons sur la figure de gauche que la solution exacte T ε est proche de sa limite quand ε est petit. En revanche, ce n'est pas le cas de sa dérivée T ′ ε , qui oscille autour de (T * ) ′ . Revenant au problème de référence (1.1.1), cela suggère que la convergence dans (1.1.2) peut être forte, mais que ce n'est pas le cas de la convergence de ∇T ε , qui n'est en fait que faible. Pour obtenir la convergence forte des gradients, il est nécessaire d'introduire une famille de fonctions qui corrige les oscillations de T ε à l'échelle ε. Ces fonctions sont appelées correcteurs. Dans (1.1.2)-(1.1.3), une approximation forte de ∇T ε est typiquement de la forme (à, éventuellement, extraction d'une sous-suite près)

∇T ε = ∇T * + d i=1 ∇w ε i ∂ i T * + termes d'ordre plus élevé, (1.1.4)
où w ε i est une fonction appelée correcteur dans la direction2 e i à l'échelle ε. Sans rentrer dans les détails, signalons que les correcteurs peuvent être définis à l'aide d'une EDP ayant pour coefficient a ε . Le calcul pratique des fonctions w ε i est un enjeu majeur de l'homogénéisation. Il est d'ailleurs instructif de commenter ce calcul dans les différents cadres introduits ci-dessus. Dans le cas périodique, il est remarquablement aisé. En effet, deux simplifications de taille sont possibles :

(i) le coefficient oscillant a ε étant de la forme a(./ε), la fonction w ε i peut être écrite sous la forme w i (./ε) où, bien sûr, la fonction w i est indépendante de ε ;

(ii) le coefficient a étant périodique, la fonction w i peut être cherchée périodique. L'EDP définissant w i est alors posée sur le cube unité, voir (1.2.33).

Si l'on résume la situation, nous sommes passés, dans le cas périodique, d'un nombre infini de problèmes (w ε i ) ε∈(0,1) 1≤i≤d à un nombre fini de problèmes (w i ) 1≤i≤d posés sur le cube unité : le gain pratique est considérable. Dans le cadre quasi-périodique, on peut à nouveau chercher w ε i sous la forme w i (./ε).

La situation est alors similaire au cas périodique, à ceci près que l'équation donnant w i est posée sur tout l'espace R d , ce qui rend son étude plus difficile. Un théorème d'existence est montré dans [20, theorème 5.8]. Dans le cadre déterministe général, la situation est plus délicate, voir [101, Theorème 13.1]. La faible structure imposée sur la suite de coefficients oscillants (a ε ) ε impose de résoudre, pour chaque indice k de la sous-suite (ε k ) k∈N , ε k -→ k→+∞ 0, une équation dépendant, entre autre, de a ε k . Cela est donc, d'un point de vue pratique, inutilisable. Finissons ce paragraphe par le cas stochastique (stationnaire et ergodique). La forme de l'approximation de la solution oscillante est analogue à (1.1.4) avec w ε i = w i (./ε, ω), ω ∈ Ω. L'équation du correcteur w i (./ε, ω) est alors posée sur tout l'espace. Il est montré dans [89, theorème 2], en utilisant la structure imposée sur l'ensemble de coefficients (voir aussi [65, lemme 1]), que cette équation est, pour presque tout coefficient a(•, ω), bien posée.

Dans l'idée de mieux décrire le comportement de la solution oscillante, il est naturel d'essayer de démontrer des taux de convergence dans l'approximation (1.1.4). Cette question est parfois appelée homogénéisation quantitative, par opposition à l'homogénéisation qualitative, décrite ci-dessus, qui consiste à trouver la limite de la suite de fonctions oscillantes (T ε ) ε>0 quand ε → 0. Cette question est un champ de recherche actif dans le cas stochastique. Elle ne sera pas abordée ici. Nous mentionnons cependant les travaux fondamentaux de A. Gloria, F. Otto, S. Neukamm d'une part et de S. Amstrong, T. Kuusi, J.-C. Mourrat d'autre part et nous donnons [START_REF] Armstrong | Quantitative stochastic homogenization and large-scale regularity[END_REF][START_REF] Gloria | A regularity theory for random elliptic operators[END_REF] pour références. En ce qui concerne l'homogénéisation déterministe, les premiers résultats remontent aux années 1990 avec les contributions de M. Avellaneda et F. Lin [START_REF] Avellaneda | Lp bounds on singular integrals in homogenization[END_REF][START_REF] Avellaneda | Compactness methods in the theory of homogenization[END_REF] (notons que l'on peut légèrement relaxer l'hypothèse de périodicité dans ce travail, voir la discussion 5.1 p. 846 dans [START_REF] Avellaneda | Compactness methods in the theory of homogenization[END_REF]). On fait également référence à [START_REF] Shen | Periodic homogenization of elliptic systems[END_REF] pour un panorama sur l'homogénéisation quantitative périodique. En quelques mots, ces travaux montrent que, dans le cas périodique, et à nouveau pour l'équation (1.1.1), une approximation très simple de T ε est disponible :

T ε = T * + ε d i=1 w i • ε ∂ i T * + reste, (1.1.5)
où, grossièrement, le reste et son gradient se contrôlent par une puissance strictement positive de ε dans les normes de Lebesgue L q , q ∈ [1, +∞]. Pour autant, le cadre périodique, bien que très satisfaisant d'un point de vue des résultats mathématiques, semble bien restrictif du point de vue de la modélisation. À l'inverse, le cadre stochastique permet de traiter une classe de coefficients plus large mais semble plus coûteux à mettre en place d'un point de vue pratique. Dans cette thèse, nous souhaitons étudier un ensemble de problèmes non périodiques, tout en gardant un procédé d'homogénéisation simple à mettre en place et donc proche du cadre périodique. Nous aimerions, par exemple, prendre en compte la modélisation de défauts qui pourraient apparaître dans une microstructure supposée périodique. Notre réflexion s'appuie sur les travaux antérieurs de X. Blanc, C. Le Bris, P.-L. Lions et, plus tard, M. Josien, dans le cas de l'équation modèle (1.1.1) (et, par la suite, pour des équations qui ne sont pas sous forme divergence [START_REF] Blanc | On correctors for linear elliptic homogenization in the presence of local defects: the case of advection-diffusion[END_REF]). Les résultats obtenus dans ces articles seront détaillés dans la section 1.2.2.1 de cette introduction. Ils pointent dans deux directions (complémentaires) :

(A) construire une classe de coefficients non périodiques pour laquelle l'équation du correcteur admet une solution [START_REF] Blanc | A possible homogenization approach for the numerical simulation of periodic microstructures with defects[END_REF][START_REF] Blanc | Local profiles for elliptic problems at different scales: defects in, and interfaces between periodic structures[END_REF][START_REF] Blanc | On correctors for linear elliptic homogenization in the presence of local defects[END_REF] ;

(B) construire un cadre général (au-delà du cas périodique) déterministe permettant d'étendre les résultats quantitatifs de [START_REF] Avellaneda | Compactness methods in the theory of homogenization[END_REF][START_REF] Avellaneda | Lp bounds on singular integrals in homogenization[END_REF], voir [START_REF] Blanc | Precised approximations in elliptic homogenization beyond the periodic setting[END_REF].

Du premier point a émergé l'idée d'une classe de coefficients périodiques avec défauts. S'appuyant sur le cas d'un coefficient périodique, pour lequel on dispose de nombreux résultats mathématiques, les auteurs de [START_REF] Blanc | A possible homogenization approach for the numerical simulation of periodic microstructures with defects[END_REF][START_REF] Blanc | Local profiles for elliptic problems at different scales: defects in, and interfaces between periodic structures[END_REF][START_REF] Blanc | On correctors for linear elliptic homogenization in the presence of local defects[END_REF] traitent le cas d'une perturbation de ce coefficient. A l'échelle macroscopique, le défaut se concentre autour d'un point quand ε → 0. Dans ce cadre, ils parviennent à résoudre l'équation du correcteur. En ce qui concerne le second point, des taux de convergence sont obtenus dans [START_REF] Blanc | Precised approximations in elliptic homogenization beyond the periodic setting[END_REF] sous certaines hypothèses dont, surtout, l'existence des correcteurs et une quantification de leur sous-linéarité à l'infini. Ce cadre contient bien sûr le cadre périodique avec défauts introduit dans le premier point. Cette thèse vise à aborder d'autres problèmes d'homogénéisation au-delà du cas périodique, tout en gardant la même précision dans l'approximation de la solution oscillante. En s'appuyant sur les travaux de X. Blanc, C. Le Bris et P.-L. Lions, nous proposons des perturbations du cas périodique pour différents problèmes d'EDP elliptiques. Nous discuterons systématiquement, pour chacun des problèmes abordés dans ce manuscrit, les deux points (A) et (B) ci-dessus. Les problèmes que nous considérons ici sont de deux types :

• Équations oscillantes : il s'agit d'équations de la forme

-div a • ε , u ε , ∇u ε = f dans Ω (1.1.6)
où Ω est un domaine borné et f est le terme source. Nous imposerons des données au bord de type Dirichlet homogène. Ce sont des équations de conservation où le flux a peut potentiellement être non linéaire. Le cas a(y, u, ξ) = a(y)ξ correspond bien sûr à l'équation (1.1.1). On regardera des cas où la conductivité dépend de ξ, par exemple le cas du p-Laplacien a(y, ξ) = a(y)|ξ| p-2 ξ, 1 < p < +∞. Dans le cas scalaire, cela modélise, quand |∇u ε | est petit, un milieu très diffusif dans le cas 1 < p < 2 et peu diffusif quand p > 2. Dans le cas vectoriel, de telles lois de puissance sont utilisées pour modéliser l'écoulement de fluides non-Newtoniens. Le cas p > 2 correspond à des fluides qui se rigidifient quand ils sont accélérés et le cas p < 2 à des fluides qui se liquéfient.

• Domaines perforés : il s'agit d'équations posées dans un domaine Ω ε qui dépend de la microstructure ε. Nous considérerons des domaines Ω ε perforés par des trous de taille ε. Cela peut par exemple modéliser l'écoulement d'un fluide à travers un milieu poreux. Ce problème est alors relié à la loi de Darcy comme expliqué au début de l'introduction.

Beaucoup de références sont disponibles pour chacune des deux familles de problèmes ci-dessus. Nous en donnerons dans la section 1.2 ainsi qu'au cours des chapitres. Nous décrivons maintenant les résultats obtenus dans cette thèse.

Présentation de la thèse

Cette thèse se compose de deux parties distinctes. La première partie (chapitres 2 et 3) correspond aux articles [START_REF] Blanc | Homogenization of the Poisson equation in a non periodically perforated domain[END_REF][START_REF] Wolf | Homogenization of the Stokes system in a non periodically perforated domain[END_REF] (ainsi qu'au rapport [START_REF] Wolf | Homogenization of the poisson equation and the stokes system in some non periodically perforated domains[END_REF]) et traite d'homogénéisation dans les milieux perforés pour un cadre périodique avec défauts. Nous y étudions l'équation de Poisson et le système de Stokes. La seconde partie concerne l'homogénéisation d'équations non linéaires à coefficients oscillants périodiques et localement perturbés. Nous étudions une équation semi linéaire dans le chapitre 4 ainsi qu'une classe d'équations quasi linéaires dans le chapitre 5. Le contenu de ce dernier chapitre fournira [START_REF] Wolf | Homogenization of p-laplace type equations in a periodic setting with defects[END_REF]. Nous détaillons dans la suite de cette section les résultats obtenus dans ces deux parties.

Homogénéisation en milieu perforé avec défaut

Dans les chapitres 2 et 3 (voir aussi [START_REF] Blanc | Homogenization of the Poisson equation in a non periodically perforated domain[END_REF][START_REF] Wolf | Homogenization of the Stokes system in a non periodically perforated domain[END_REF]), nous introduisons une classe de domaines perforés qui permettent l'homogénéisation de l'équation de Poisson et du système de Stokes en s'appuyant sur les résultats connus du cas périodique [START_REF] Jikov | Homogenization of differential operators and integral functionals[END_REF][START_REF] Lions | Asymptotic expansions in perforated media with a periodic structure[END_REF][START_REF] Cioranescu | Homogenization in open sets with holes[END_REF][START_REF] Sanchez-Palencia | Fluid flow in porous media[END_REF][START_REF] Tartar | Incompressible fluid flow in a porous medium. Convergence of the homogenization process[END_REF]2]. Nous commençons par introduire cette classe de domaines.

Perturbation locale d'un domaine périodiquement perforé

Soit d ≥ 1, on note Q := -1 2 , 1 2 d le cube unité de R d . Soit k ∈ Z d , on note

Q k := Q + k = d i=1 k i - 1 2 , k i + 1 2 d .
Soit Notons également que, dans (1.2.1), on a supprimé les perforations qui intersectent le bord macroscopique, ceci afin de garantir la connexité. Cette hypothèse sera systématique dans l'étude du système de Stokes (chapitre 3) mais n'est pas supposée dans l'étude de l'équation de Poisson (chapitre 2). On définit maintenant les perturbations de Ω per Comme pour le cas périodique, on définit le domaine perforé Ω ε par

Ω ε := Ω \ ε k∈Yε O k , Y ε := {k ∈ Z d , εQ k ⊂ Ω} (1.2.3)
où, comme pour le cas périodique, on peut, dans certaines situations, inclure les perforations intersectant ∂Ω, ou à l'inverse, imposer une certaine couche limite sans perforation autour du bord macroscopique. Dans la suite, nous aurons régulièrement besoin (pour des questions de régularité elliptique) de supposer que les perforations non périodiques sont régulières uniformément en k ∈ Z d . Une formalisation de cette hypothèse est donnée dans le chapitre 3 pour les régularités Lipschitz et C 2 (mais, bien sûr, on peut formuler cela pour toute régularité). Nous présentons maintenant les résultats obtenus dans les chapitres 2 et 3.

Étude de l'équation de Poisson

Dans cette section, nous présentons les résultats de [START_REF] Blanc | Homogenization of the Poisson equation in a non periodically perforated domain[END_REF]. Nous y étudions l'équation suivante :

-∆u ε = f dans Ω ε u ε = 0 sur ∂Ω ε .
(1.2.4)

où Ω ε est défini par Ω ε := Ω \ εO et O est un ensemble de perforations périodiques et localement perturbées, tel qu'introduit dans la sous-section 1.2.1.1. Dans (1.2.4), le terme source vérifie pour le moment f ∈ L 2 (Ω). Le cas Ω ε = Ω per ε a été étudié dans [START_REF] Lions | Asymptotic expansions in perforated media with a periodic structure[END_REF]. Nous souhaitons généraliser ces résultats au cas d'un domaine périodique avec défauts. Nous commençons par étudier le comportement macroscopique de la suite (u ε ) ε>0 .

Une estimation a priori donne que, à extraction près, u ε /ε 2 -⇀ ε→0 u * dans L 2 (Ω). Notons que ceci utilise la version suivante de l'inégalité de Poincaré-Friedrichs à l'échelle ε : ∃C > 0, ∀ε > 0, ∀u ∈ H 1 0 (Ω ε ), ∥u∥ L 2 (Ωε) ≤ Cε∥∇u∥ L 2 (Ωε) .

(1.2.5)

Cette inégalité est prouvée dans [START_REF] Tartar | Incompressible fluid flow in a porous medium. Convergence of the homogenization process[END_REF] pour le cas périodique. Nous la généralisons au cas périodique avec défaut dans [START_REF] Blanc | Homogenization of the Poisson equation in a non periodically perforated domain[END_REF][START_REF] Wolf | Homogenization of the Stokes system in a non periodically perforated domain[END_REF] 3 . L'étape suivante est d'effectuer un développement à deux échelles sous la forme u ε (x) = ε 2 u 2 (x, y) + ε 3 u 3 (x, y) + ε 4 u 4 (x, y)

+ • • • , y = x ε , ( 1.2.6)
où les fonctions u i sont supposées régulières. Notons que, dans (1.2.6), les fonctions u i (x, •) sont des fonctions variant à l'échelle microscopique et traduisent ainsi les propriétés microscopiques du matériau. Par conséquent, dans le cas périodique, elles seront supposées périodiques. Dans le cas périodique avec défaut, elles seront supposées périodiques et localement perturbées dans un sens précisé ci-dessous. Insérant l'expression (1.2.6) dans (1.2.4), on trouve, formellement, où j ∈ 1, d . Nous nous concentrons dans la suite sur le premier correcteur w (0) , que l'on note w, et dont on admet momentanément l'existence dans l'espace

∀x ∈ Ω, ∀y ∈ R d , u i (x, y) = |j|=i-2 w (j) (y)D j f (x), (1.2.7) où D j = ∂ j1 • • • ∂ j d , j 1 + • • • + j d =
H 1 unif (R d ) := u ∈ H 1 loc (R d ), sup k∈Z d ∥u∥ H 1 (Q+k) < +∞ .
Dans [START_REF] Cioranescu | Homogenization in open sets with holes[END_REF], ce correcteur est utilisé dans le cas périodique pour trouver u * par la méthode des fonctions test oscillantes dont on rappelle brièvement le principe.

On commence par prolonger u ε par zéro dans les trous, cela fournit une fonction u ε ∈ H 1 0 (Ω). Soit ϕ ∈ D(Ω), on choisit ε assez petit pour que supp(ϕ) ⊂ k∈Yε εQ k (voir (1.2.3) pour la notation).

On introduit v ε := w(./ε)ϕ ∈ H 1 0 (Ω ε ) et on teste v ε contre (1.2.4). On obtient

ε -1 ¢ Ω ∇u ε • ∇w • ε ϕ = ¢ Ω f v ε - ¢ Ω w • ε ∇u ε • ∇ϕ.
(1.2.9)

De même, on teste l'équation du correcteur contre u ε ϕ :

ε -1 ¢ Ω ∇w • ε • ∇u ε ϕ = -ε -1 ¢ Ω u ε ∇w • ε • ∇ϕ + ε -2 ¢ Ω u ε ϕ. (1.2.10)
En faisant la différence entre (1.2.9) et (1.2.10), on trouve

¢ Ω f v ε -ε -2 ¢ Ω u ε ϕ = ¢ Ω w • ε ∇u ε • ∇ϕ -ε -1 ¢ Ω u ε ∇w • ε • ∇ϕ. (1.2.11)
On utilise alors l'estimation a priori

∥u ε ∥ L 2 (Ω) + ε∥∇u ε ∥ L 2 (Ω) ≤ Cε 2 ,
où C est une constante indépendante de ε, ainsi que la sous-linéarité du correcteur pour obtenir que (1.2.13)

En particulier, w (prolongée par zéro dans les perforations), vérifie w ∈ H 1 unif (R d ). Avant d'exposer la suite des résultats obtenus dans le Chapitre 2, nous faisons deux commentaires sur le théorème 1. D'abord, on peut établir et prouver ce théorème pour un second membre plus général de la forme g per + g avec g per ∈ L ∞,per (Q). Cela permet de construire les correcteurs d'ordre plus élevé. Ensuite, nous signalons que la preuve du théorème 1 repose sur des techniques variationnelles. Nous établissons d'abord une inégalité de Poincaré-Friedrichs puis nous résolvons (1.2.12) par un problème de minimisation.

Revenons maintenant aux résultats du chapitre 2. Après avoir montré l'existence de correcteurs, nous obtenons des taux de convergence. Les théorèmes suivants sont prouvés dans [START_REF] Blanc | Homogenization of the Poisson equation in a non periodically perforated domain[END_REF]. Ils sont énoncés avec la convention Ω ε = Ω \ εO. Nous commençons par un théorème de convergence

H 1 . Pour u ∈ W 1,q (Ω ε ), on définit |u| W 1,q (Ωε) := ∥∇u∥ L q (Ωε) . Théorème 2. Soit (O k ) k∈Z d une configuration non-périodique de trous et Ω ε = Ω \ εO le domaine perforé associé. Soit f ∈ D(Ω). Il existe une constante C > 0 indépendante de ε telle que ε -1 u ε -ε 2 w(./ε)f L 2 (Ωε) + u ε -ε 2 w(./ε)f H 1 (Ωε) ≤ Cε 2 .
Notons que les hypothèses du théorème 2 peuvent être affaiblies en f ∈ H 2 (Ω) et f |∂Ω = 0 si l'on suppose que les perforations sont uniformément de classe C 1,γ . Si f ne s'annule pas sur ∂Ω alors une couche limite apparait et dégrade le taux de convergence, phénomène usuel en homogénéisation. Dans le théorème ci-dessous, nous obtenons un taux de convergence L ∞ pour des seconds membres f très réguliers. Théorème 3. Soit (O k ) k∈Z d une configuration non-périodique de trous. On suppose que les perforations sont uniformément de classe C 1,γ et on note Ω ε = Ω \ εO le domaine perforé associé. Soit f ∈ D(Ω). Il existe une constante C > 0 indépendante de ε telle que

u ε -ε 2 w(./ε)f L ∞ (Ωε) ≤ Cε 3 .
Dans l'annexe 2.5.2 du chapitre 2, nous améliorons ces résultats en atteignant toutes les normes de Sobolev W 1,q (Ω ε ), q ∈ (1, +∞). Nous nous appuyons pour cela sur [START_REF] Masmoudi | Some uniform elliptic estimates in a porous medium[END_REF][START_REF] Jikov | Homogenization of differential operators and integral functionals[END_REF]. Pour des raisons de régularité, nous modifions légèrement la définition de Ω ε . On définit

Ω ε := Ω \ ε k∈Yε O k , Y ε := {k ∈ Z d , εQ k ⊂ Ω}. (1.2.14)
Nous prouvons alors le résultat suivant, qui donne des taux de convergence optimaux.

O k , k ∈ Z d sont uniformément de classe C 2 . Soit u ε la solution de (1.2.4). 1. Si f ∈ W 2,q ∩ L ∞ (Ω), alors il existe une constante C > 0 indépendante de ε telle que ε -1 ∥u ε -ε 2 w(•/ε)f L q (Ωε) + u ε -ε 2 w(•/ε)f W 1,q (Ωε) ≤ Cε 1+1/q . (1.2.15) 2. Si q ≥ 2, f ∈ W 2,q (Ω), Ω ′ ⊂⊂ Ω et Ω ′ ε := Ω ε ∩ Ω ′ , il existe une constante C > 0 indépendante de ε telle que ε -1 ∥u ε -ε 2 w(•/ε)f L q (Ω ′ ε ) + u ε -ε 2 w(•/ε)f W 1,q (Ω ′ ε ) ≤ Cε 2 .
(1.2.16)

L'ensemble des résultats obtenus dans le chapitre 2 montre que le cadre introduit dans la section 1.2.1.1 est traitable par les outils issus de l'homogénéisation périodique. Par ailleurs, en utilisant les correcteurs ad hoc, nous pouvons obtenir une qualité d'approximation aussi fine que dans le cas périodique. Il est à ce stade important de souligner que le correcteur non périodique est plus compliqué à calculer, puisqu'il résout une EDP sur un domaine non borné. Il est alors naturel de se demander si, en pratique, il est réellement nécessaire de calculer la fonction w pour chaque situation. Nous avons vu que cela n'était pas utile si l'on souhaite seulement calculer u * , qui ne dépend que de w per . Cela n'est également pas avantageux si l'on souhaite une approximation assez grossière de u ε , par exemple en norme H 1 (Ω). Cette affirmation provient du théorème 2.5.6 (iii) (et du théorème 4, (2)), voir aussi la remarque 2.5.11. Nous nous contentons ici de résumer sur la figure 1.5 les cas où la construction pratique du correcteur non périodique se justifie.

q +∞ 1 d
w n'améliore pas la convergence w améliore la convergence Figure 1.5 : Utilité du correcteur sur l'échelle des exposants de Lebesgue. En norme L q (ou en norme W 1,q ), le taux de convergence de u ε -ε 2 w(./ε)f vers zéro est meilleur que celui de u ε -ε 2 w per (./ε)f vers zéro quand q < d.

Étude du système de Stokes

Dans cette section, nous introduisons les résultats du chapitre 3 en les replaçant dans leur contexte. Pour f ∈ L 2 (Ω) d , nous étudions le système de Stokes ci-dessous :

     -∆u ε + ∇p ε = f dans Ω ε div u ε = 0 u ε = 0 sur ∂Ω ε , (1.2.17)
où Ω ε est un domaine périodiquement ou non perforé et f ∈ L 2 (Ω) d . Dans la suite, nous fixons un représentant quelconque de p ε dans L 2 (Ω ε )/R. L'étude de (1.2.17) dans le cas Ω ε = Ω per ε a été effectuée dans [START_REF] Tartar | Incompressible fluid flow in a porous medium. Convergence of the homogenization process[END_REF][START_REF] Sanchez-Palencia | Fluid flow in porous media[END_REF] puis dans [2] pour le cas d'une partie solide connexe (voir aussi [START_REF] Lipton | Darcy's law for slow viscous flow past a stationary array of bubbles[END_REF]). Le premier résultat donne le comportement qualitatif de la suite (u ε , p ε ).

Théorème 5 ([91, 99]

). Soit f ∈ L 2 (Ω) d et Ω per ε défini par (1.2.1). Il existe un prolongement de u ε et p ε à Ω tel que l'on ait les convergences    u ε ε 2 -⇀ ε→0 u * dans L 2 (Ω) p ε -→ ε→0 p 0 dans L 2 (Ω)/R, (1.2.18) où le couple (u * , p 0 ) est défini par u * = A(f -∇p 0 ) et ∇p 0 est donné par l'EDP -div A(f -∇p 0 ) = 0 dans Ω A(f -∇p 0 ) • n = 0 sur ∂Ω. (1.2.

19)

La matrice A se calcule à l'aide des problèmes de correcteurs périodiques . On a alors

     -∆w per
∀(i, j) ∈ 1, d 2 , A i,j = ¢ Q w per j • e i = ¢ Q w per i • e j .
Le théorème 5 montre que (u * , p 0 ) suit la Loi de Darcy. On rappelle brièvement la stratégie utilisée pour prouver le Théorème 5, en remarquant qu'elle s'adapte au cas périodique avec défaut. En utilisant l'inégalité de Poincaré [99, Lemma 1] (et [START_REF] Wolf | Homogenization of the Stokes system in a non periodically perforated domain[END_REF]Lemma 3.1] pour le cas non périodique), on a l'estimation a priori

∥u ε ∥ L 2 (Ω) ≤ Cε 2
(avec extension de u ε par zéro dans les perforations). Ainsi,

u ε /ε 2 -⇀ ε→0 u * dans L 2 (Ω)
H := {u ∈ L 2 (Ω) d , div u = 0, u • n |∂Ω = 0}
est fermé pour la norme L 2 (Ω) d . Cela donne en particulier u * • n |∂Ω = 0. La difficulté de la preuve est le prolongement de la pression p ε dans les trous. En effet, contrairement à u ε , il n'y a pas d'extension naturelle dans les perforations. Dans [START_REF] Tartar | Incompressible fluid flow in a porous medium. Convergence of the homogenization process[END_REF], une extension de p ε à Ω est construire à l'aide d'un opérateur de restriction.

Lemme 6 (Lemma 4 dans [START_REF] Tartar | Incompressible fluid flow in a porous medium. Convergence of the homogenization process[END_REF]). Il existe un opérateur R ε :

H 1 0 (Ω) d -→ H 1 0 (Ω ε ) d satisfaisant les propriétés suivantes : pour tout u ∈ H 1 0 (Ω) d , si u = 0 dans Ω \ Ω ε alors R ε u = u, si div u = 0 alors div R ε u = 0, (1.2.21)
∥∇R ε u∥ L 2 (Ωε) + ε -1 ∥R ε u∥ L 2 (Ωε) ≤ C ∥∇u∥ L 2 (Ω) + ε -1 ∥u∥ L 2 (Ω) , (1.2
F ε ∈ H -1 (Ω) d par ⟨F ε , u⟩ := ⟨∇p ε , R ε u⟩ pour u ∈ H 1 0 (Ω) d .
∀ϕ ∈ H 1 0 (Ω), |⟨∇P ε , ϕ⟩| ≤ C ∥ϕ∥ L 2 (Ω) + ε∥∇ϕ∥ L 2 (Ω) . En particulier, ∥P ε ∥ L 2 (Ω)/R ≤ C
   u ε /ε 2 -⇀ ε→0 u * dans L 2 (Ω) d P ε -→ ε→0 p 0 dans L 2 (Ω)/R. (1.2.23)
L'enjeu est maintenant d'identifier le couple (u * , p 0 ). Pour cela, comme pour l'équation de Poisson, nous introduisons les correcteurs associés au problème. Pour identifier ces correcteurs, on peut effectuer un développement à deux échelles :

u ε (x) = ε 2 u 2 (x, y) + ε 3 u 3 (x, y) + ε 4 u 4 (x, y) + • • • , y = x/ε p ε (x) = p 0 (x) + εp 1 (x, y) + ε 2 p 2 (x, y) + • • • , y = x/ε, (1.2.24)
u ε (x) ≈ ε 2 d j=1 w j (y)(f j -∂ j p 0 )(x) et p ε ≈ p 0 (x) + ε d j=1 p j (y)(f j -∂ j p 0 )(x), y = x/ε, (1.2.25) où f = (f 1 , • • • , f d ) et (w j , p j ) sont les correcteurs, solution de      -∆w j + ∇p j = e j dans R d \ O div w j = 0 w j = 0 sur ∂O, ( 1
w j = w per j + w j , p j = p per j + p j où ( w j , p j ) ∈ H 1 × L 2 loc (R 3 \ O) et (w per j , p per j ) est le correcteur périodique i.e. solution de (1.2.20). Par ailleurs, il existe une constante C > 0 indépendance de ε telle que ∥ p j -⟨ p j ⟩∥ L 2 ( 1 ε Ωε) ≤ Cε -1 , où p j est prolongée par sa valeur moyenne sur 1 ε Ω \ O dans 1 ε Ω ε et ⟨ p j ⟩ := 1 | 1 ε Ω ε | ¢ 1 ε Ωε p j .
On peut ensuite passer à la limite faible ε → 0 dans (1.2.25). On trouve que, toujours formellement,

u * = A(f -∇p 0 ) avec A = (w 1 , • • • , w d ) 1≤i≤d ∈ M d (R) et w j := L 2 -lim faible ε→0
w j (./ε), si cette limite existe. Dans les cas périodique et non périodique, on a, en utilisant le théorème 7 et un lemme classique sur les fonctions périodiques, que w j = ¢ Q w per j (y)dy. Cela montre formellement le théorème 5 dans le cadre (non) périodique. On peut alors suivre [99, Preuve du théorème 1], qui ne repose pas sur la périodicité, (et qui est essentiellement la même preuve que pour l'équation de Poisson) pour rendre cela rigoureux. Cela donne la convergence de (u ε , p ε ) vers la solution à la loi de Darcy dont la matrice de perméabilité A se calcule avec les correcteurs périodiques. Comme pour l'équation de Poisson, la limite macroscopique est inchangée par rapport au cas périodique.

Dans la suite, nous aimerions préciser la convergence de (u ε , p ε ). La première amélioration est fournie par [3] dans le cas périodique : [START_REF] Jankowiak | Non-conforming multiscale finite element method for Stokes flows in heterogeneous media[END_REF] en O(ε 1/2 ) fournissant ainsi un taux optimal. Récemment, ce taux optimal a été montré pour toute dimension dans [START_REF] Shen | Sharp convergence rates for Darcy's law[END_REF]. Dans tous ces travaux, un élément central est la construction de fonctions permettant de corriger les données aux bords (boundary correctors). Une façon de s'affranchir des difficultés liées au bord macroscopique est, dans le cas périodique, de choisir ε = 1/m, m ∈ N * et d'imposer des données au bord périodique sur un cube macroscopique, voir [START_REF] Marušić-Paloka | Asymptotic expansion for a flow in a periodic porous medium[END_REF]. Signalons, à ce sujet, que des taux de convergences à tout ordre, et donc la construction de correcteurs d'ordre élevé, ont été proposés dans ce cadre [START_REF] Feppon | High order homogenization of the stokes system in a periodic porous medium[END_REF]. Dans [START_REF] Wolf | Homogenization of the Stokes system in a non periodically perforated domain[END_REF], nous nous plaçons dans un cadre permettant de s'affranchir du bord ∂Ω et nous supposons d = 3. On rappelle que, formellement,

u ε ε 2 - d j=1 w j • ε (f j -∂ j p 0 ) -→ ε→0 0 dans L 2 (Ω) d , ( 1
u ε = ε 2 d j=1 w j • ε (f j -∂ j p 0 ) + reste.
La difficulté majeure provient de ce que (f j -∂ j p 0 ) |∂Ω ̸ = 0. On impose alors les conditions suivantes :

div Af = 0 f ∈ W 3,∞ c (Ω), ( 1
′′ ⊂⊂ Ω. Il existe une constante C > 0 indépendante de ε tel que pour tout ε < ε 0 (f, Ω ′′ , Ω), D 2 u ε -ε 2 w j • ε f j [L 2 (Ω ′′ ∩Ωε)] +ε -1 ∇ u ε -ε 2 w j • ε f j [L 2 (Ωε)] + ε -2 u ε -ε 2 w j • ε f j [L 2 (Ωε)] ≤ Cε (1.2.29) et ∇ p ε -ε p j • ε -λ j ε f j L 2 (Ω ′′ ∩Ωε) + p ε -ε p j • ε -λ j ε f j L 2 (Ωε)/R ≤ Cε, (1.2.30) où λ j ε = 1 |Ω ε | ¢ Ωε p j • ε .
Dans l'énoncé ci-dessus, nous avons utilisé la convention de sommation sur les indices répétés. Remarquons que la conclusion (1.2.30) peut se réduire à

∇ p ε -ε p j • ε -λ j ε f j L 2 (Ω ′′ ∩Ωε) + ∥p ε ∥ L 2 (Ωε)/R ≤ Cε
sans perte d'information. Les correcteurs p j permettent de corriger les oscillations de p ε à l'échelle microscopique : ils améliorent donc la convergence du gradient. Le théorème 8, limité pour l'instant à des estimations Hilbertiennes, s'appuie sur [START_REF] Masmoudi | Some uniform elliptic estimates in a porous medium[END_REF]Theorem 4.1]. La majeure partie de la preuve du Théorème 8 consiste en la construction de fonctions auxiliaires permettant de corriger la divergence

du reste R ε := u ε -ε 2 3 j=1 w j • ε f j .
Nous notons que, dans le théorème 8, l'hypothèse d'uniforme régularité de classe C 2 permet d'obtenir des constantes de régularité elliptique uniformes en k. Nous vérifions cela dans l'annexe 3.4.2. La technique de preuve pourrait s'adapter à l'équation de Poisson, ou à d'autres types de résultats de régularité. Nous ne détaillons pas ces passages, les preuves étant assez fastidieuses.

PRÉSENTATION DE LA THÈSE CHAPITRE 1. INTRODUCTION

Nous finissons cette section en discutant l'utilité pratique des correcteurs non-périodiques. Remarquons d'abord l'estimation

ε -2 ε 2 w j • ε f j L 2 (Ωε) ≤ Cε 3 2 = O(ε).
Cela signifie que, utiliser w per j à la place de w j dans (1.2.29), n'améliore par le taux de convergence en norme L 2 (et aussi en norme H 1 ). Pour exhiber des situations où le correcteur non périodique apporte, en pratique, une amélioration sur le taux de convergence, il faudrait obtenir des estimations L q , q > d. Ce point sera discuté dans les perspectives, section 1.3 et pourrait donner lieu à un travail futur.

Homogénéisation d'équations non-linéaires à coefficients oscillants

Dans cette section, nous présentons les résultats obtenus dans les chapitres 4 et 5. L'objectif est de généraliser le cadre introduit dans [START_REF] Blanc | On correctors for linear elliptic homogenization in the presence of local defects[END_REF][START_REF] Blanc | A possible homogenization approach for the numerical simulation of periodic microstructures with defects[END_REF][START_REF] Blanc | Local profiles for elliptic problems at different scales: defects in, and interfaces between periodic structures[END_REF] à des équations non-linéaires. Nous commençons par présenter ce cadre dans la section 1.2.2.1, que nous illustrons sur l'équation de diffusion linéaire.

Coefficients périodiques avec défauts

Nous présentons le cadre périodique avec défauts qui sera le coeur des chapitres 4 et 5. On fixe une application A per : R d -→ M d (R) où M d (R) désigne l'ensemble des matrices réelles de taille d × d. On suppose que A per est périodique et que

∃λ > 0, ∀y ∈ Q, ∀ξ ∈ R d , λ -1 |ξ| 2 ≤ A per (y)ξ • ξ ≤ λ|ξ| 2 .
(1.2.31)

Pour simplifier, on suppose que A per est symétrique i.e. A per (y) = A per (y) T pour tout y ∈ Q. On considère alors, pour ε > 0, le problème

-div A per (./ε)∇u ε = f dans Ω u ε ∈ H 1 0 (Ω), (1.2.32)
où le second membre f ∈ L 2 (Ω). L'homogénéisation périodique [START_REF] Bensoussan | Asymptotic Analysis for Periodic Structures[END_REF] montre que u ε -⇀

ε→0 u * dans H 1 (Ω)
où u * se calcule à l'aide de la matrice homogénéisée A * . Cette matrice est définie par le problème de cellule

-div A per (y)(p + ∇w per p (y)) = 0 dans Q w p ∈ H 1,per (Q), (1.2.33) où p ∈ R d . On a alors A * j,i := ¢ Q A per (y) (e i + ∇w ei (y)) dy • e j (1.2.34) et u * résout l'EDP -div A * ∇u * = f dans Ω u * ∈ H 1 0 (Ω). (1.2.35)
Si l'on suppose la régularité de A per (par exemple Hölderienne), les résultats fondamentaux [START_REF] Avellaneda | Compactness methods in the theory of homogenization[END_REF] permettent d'obtenir des taux de convergence du reste [START_REF] Blanc | A possible homogenization approach for the numerical simulation of periodic microstructures with defects[END_REF][START_REF] Blanc | Local profiles for elliptic problems at different scales: defects in, and interfaces between periodic structures[END_REF][START_REF] Blanc | On correctors for linear elliptic homogenization in the presence of local defects[END_REF]. Pour cela, on prend appui sur le problème périodique et l'on cherche w p sous la forme w p = w per p + w p où, formellement, w p tend vers zéro à l'infini. L'équation vérifiée par w p s'écrit sous la forme

R ε := u ε -u * -ε d j=1 w ei (./ε)∂ i u * vers zéro dans différentes normes (H 1 , W 1,q , C 0,α
< r < +∞ et A ∈ L r ∩ C 0,α (R d ) d×d . On pose A := A per + A et l'on considère le problème d'homogénéisation -div A(./ε)∇u ε = f dans Ω u ε ∈ H 1 0 (Ω).
-div A(y)(p + ∇w p (y)) = 0 sur R d , ( 1
-div A∇ w p = div g sur R d , g := A(p + ∇w per p ) ∈ L r (R d ) d , (1.2.38)
en utilisant le problème périodique. La résolution de (1.2.37) se fait en deux étapes :

• résoudre (1.2.38) pour des second membres agréables. Cela est fait dans [START_REF] Blanc | A possible homogenization approach for the numerical simulation of periodic microstructures with defects[END_REF]

quand g ∈ L 2 (R d ) d (ce qui correspond à un défaut A ∈ L 2 (R d
)) par des techniques variationnelles ;

• établir un résultat de continuité L r -L r pour l'application g -→ ∇v, où v résout l'équation div A∇v = g, voir [START_REF] Blanc | On correctors for linear elliptic homogenization in the presence of local defects[END_REF].

Cette analyse fournit un correcteur non périodique

w p ∈ W 1,∞ + • W 1,r (R d ), où • W 1,r
est l'espace de Sobolev homogène standard. On obtient alors (en dimension d ≥ 3) la quantification suivante de la sous-linéarité à l'infini de w p (r ̸ = d) :

∀ρ > 1, sup |x-y|≤ρ |w p (x) -w p (y)| ≤ Cρ 1-νr , ν r = min(1, d/r).
Cela permet d'obtenir des taux de convergence : dans [START_REF] Blanc | Precised approximations in elliptic homogenization beyond the periodic setting[END_REF], les taux de convergence connus en périodique sont généralisés au cadre présenté ici (et, à vrai dire, à un cadre plus général pourvu que l'on sache construire des correcteurs ayant la bonne sous-linéarité à l'infini). Nous finissons cette section par deux remarques. D'abord, la perturbation est, comme pour le cas des domaines perforés, locale. En particulier, cela n'affecte pas le comportement macroscopique de la solution, ni même la convergence forte du reste. Ensuite, la question de l'utilité pratique du correcteur se pose. À nouveau comme pour les domaines perforés, la construction de w p , p ∈ R d se justifie si l'on souhaite une approximation fine de u ε (ou plutôt de son gradient).

Étude d'une équation semi-linéaire

Dans un premier temps, nous étudions une équation semi-linéaire dans le cadre périodique avec défauts présenté ci-dessus :

-div A(./ε)∇u ε + u ε |u ε | p-2 = f dans Ω u ε = 0 sur ∂Ω, (1.2.39) où f ∈ H 1 0 ∩ L p (Ω) ′ , 1 < p < +∞ et A est de la forme A = A per + A,
∃λ > 0, ∀ξ ∈ R d , ∀y ∈ R d , λ -1 |ξ| 2 ≤ A(y)ξ • ξ ≤ λ|ξ| 2 et λ -1 |ξ| 2 ≤ A per (y)ξ • ξ ≤ λ|ξ| 2 A per , A ∈ C 0,α (R d ) d×d et A ∈ L r (R d ) d×d , 1 < r < +∞. ( 1
≈ u * + ε d j=1 w ei (./ε)∂ i u * , où w ei , 1 ≤ i ≤ d sont les correcteurs linéaires donnés par (1.2.33) et u * est solution de -div A * ∇u * + u * |u * | p-2 = f dans Ω u * = 0 sur ∂Ω, ( 1
u * dans H 1 0 ∩ L p (Ω) et la convergence forte u ε -→ ε→0 u * dans L r (Ω), 1 ≤ r < 2d d-2 . Si l'on suppose de plus que ∇ 2 u * ∈ L 2 (Ω), on a la convergence ∇R ε -→ ε→0 0 dans L 2 (Ω) où R ε := u ε -u * -ε d i=1 w ei (./ε)∂ i u * .

Remarquons que, sous les hypothèses

f ∈ L 2 (Ω) et p < 1 + d d-2 pour d ≥ 3 et sans restriction sur p pour d < 3, on a ∇ 2 u * ∈ L 2 (Ω)
. La difficulté du théorème 9 est le passage à la limite dans le terme non-linéaire u ε |u ε | p-2 puisque la suite (u ε ) ε>0 ne converge que faiblement dans L p (Ω) quand p > 2d/(d -2). Le théorème 9 montre que l'homogénéisation du problème (1.2.39) est essentiellement couverte par l'homogénéisation du problème (1.2.32). Le cas d'un coefficient périodique avec défauts se traite alors avec les mêmes outils que le cas linéaire [START_REF] Blanc | A possible homogenization approach for the numerical simulation of periodic microstructures with defects[END_REF][START_REF] Blanc | Local profiles for elliptic problems at different scales: defects in, and interfaces between periodic structures[END_REF][START_REF] Blanc | On correctors for linear elliptic homogenization in the presence of local defects[END_REF], tout du moins pour ce qui concerne la construction des correcteurs. Dans la suite, nous souhaitons nous concentrer sur des problèmes non linéaires dont l'équation du correcteur diffère du cas linéaire.

Étude d'une équation quasi-linéaire

Le Chapitre 5 propose l'homogénéisation de l'équation de p-Laplace dans le cadre périodique avec défauts introduit dans la section 1.2.2.1. Nous y construisons les correcteurs non périodiques et nous obtenons un théorème de convergence forte de la solution oscillante et de son gradient. Pour fixer les idées, on considère, pour p ≥ 2, l'équation

   -div a • ε ∇u ε |∇u ε | p-2 = f dans Ω u ε = 0 sur ∂Ω, (1.2.42) où f ∈ W -
tel que λ -1 < a per < λ où λ > 0. Soit f ∈ W -1,p ′ (Ω) et u ε satisfaisant (1.2.42). On a la convergence u ε -⇀ ε→0 u * dans W 1,p (Ω) où u * est l'unique solution de -div a * (∇u * ) = f dans Ω u * ∈ W 1,p 0 (Ω).
(1.2.43)

L'application a * : R d -→ R d est définie par :

∀ξ ∈ R d , a * (ξ) = ¢ Q a per (y)(ξ + ∇w per ξ (y))|ξ + ∇w per ξ (y)| p-2 dy, (1.2.44) où les fonctions w per ξ , ξ ∈ R d , sont les correcteurs périodiques et vérifient l'EDP    -div a per (y) ξ + ∇w per ξ ξ + ∇w per ξ p-2 = 0 dans Q w per ξ ∈ W 1,p per (Q). (1.2.45)
Par ailleurs, on a la convergence forte

∇u ε -∇u * -∇w per Mε∇u * • ε -→ ε→0 0 dans L p (Ω), (1.2.46)
où l'on a noté M ε : L p (Ω) -→ L p (Ω) l'opérateur de discrétisation défini par

M ε ϕ := k, ε(Q+k)⊂Ω 2 ε(Q+k) ϕ 1 ε(Q+k) . (1.2.47)
Une manière de deviner la forme de l'équation vérifiée par les correcteurs (1.2.45) est à nouveau d'effectuer un développement à deux échelles. Nous nous plaçons dans un cadre périodique, possiblement avec défaut. Pour alléger les notations, on pose

a per (y, ξ) := a per (y)ξ|ξ| p-2 , a(y, ξ) := a(y)ξ|ξ| p-2 et a(y, ξ) := a(y)ξ|ξ| p-2 pour y ∈ R d , ξ ∈ R d . On suppose que u ε se développe sous la forme u ε = u 0 (x, y) + εu 1 (x, y) + ε 2 u 2 (x, y) + • • • , y = x/ε, (1.2.48)
où les fonctions u i sont supposées régulières et y → u i (x, y) est périodique quand le coefficient a est périodique i.e. a = a per . On note alors u i sous la forme u per i . Dans le cas périodique avec défaut, on suppose que u i (x, •) est une perturbation à l'échelle microscopique de u per i

: pour tout x ∈ Ω et y ∈ R d , u i (x, •) := u per i (x, •) + u i (x, •) où ∇ y u i (x, •) ∈ L p ∩ C 0,α (R d ). On a ∇u ε = ε -1 ∇ y u 0 + ∇ x u 0 + ∇ y u 1 + ε (∇ x u 1 + ∇ y u 2 ) + • • • , y = x/ε, x ∈ Ω. ( 1
) = ε 1-p a(y, ∇ y u 0 ) + ε 2-p ∂ ξ a(y, ∇ y u 0 ) • ∇ x u 0 + ∇ y u 1 + ε (∇ x u 1 + ∇ y u 2 ) + • • • . (1.2.50) 1.2. PRÉSENTATION DE LA THÈSE CHAPITRE 1. INTRODUCTION Cela fournit div a(./ε, ∇u ε ) = ε -p div y a(y, ∇ y u 0 ) + ε 1-p div x a(y, ∇ y u 0 ) + ε 1-p div y ∂ ξ a(y, ∇ y u 0 ) • ∇ x u 0 + ∇ y u 1 + ε (∇ x u 1 + ∇ y u 2 ) + • • • + O(ε 2-p ).
En insérant ce développement dans (1.2.42) et en identifiant les puissances les plus élevées de 1/ε, on trouve quediv y a(y, ∇ y u 0 ) = 0 à x fixé. Dans le cas périodique, une estimation d'énergie et la monotonie de l'opérateur a per (y, •) fournissent ∇ y u per 0 = 0. Dans le cas non périodique, il faut montrer de plus que ∇ y u 0 = 0. On écrit que, formellement, -div y a(y, ∇ y u 0 ) = 0.

(1.2.51)

En utilisant4 la monotonie de l'opérateur a, on obtient que

∇ y u 0 = 0 et donc ∇ y u 0 = 0. D'où u per 0 (x, y) = u per 0 (x) et u 0 (x, y) = u 0 (x). Retournant à (1.2.49), on trouve que div a(./ε, ∇u ε ) = ε -1 div y a(y, ∇ x u 0 + ∇ y u 1 ) + div x a(y, ∇ x u 0 + ∇ y u 1 ) + div y [∂ ξ a(y, ∇ x u 0 + ∇ y u 1 ) • (∇ x u 1 + ∇ y u 2 + • • • )] + O(ε). (1.2.52)
A nouveau en insérant dans (1.2.42) et en identifiant les puissances les plus élevées de 1/ε, on obtient que div y a(y, ∇ x u 0 + ∇ y u 1 ) = 0. On en déduit que u 1 (x, y) = w ∇u0(x) (y) où, pour ξ ∈ R d , w ξ vérifie l'équation du correcteur

-div a(y) (ξ + ∇w ξ ) |ξ + ∇w ξ | p-2 = 0 dans R d , ( 1
, avec ∇ w ξ ∈ L p ∩ L ∞ (R d )). De même, u per 1 (x, y) = w per ∇u per 0 (x) (y). En exploitant à nouveau (1.2.52), on trouve que, nécessairement, -div x a(y, ∇ x u 0 + ∇ y u 1 ) -div y [∂ ξ a(y, ∇ x u 0 + ∇ y u 1 ) • (∇ x u 1 + ∇ y u 2 )] = f.
On peut réécrire cette égalité sous la forme

-div x a(y, ∇u 0 (x) + ∇w ∇u0(x) (y)) -div y [∂ ξ a(y, ∇ x u 0 + ∇ y u 1 ) • (∇ x u 1 + ∇ y u 2 )] = f, où ∇w ∇u0(x) (y) = ∇w per ∇u0(x) (y) + ∇ w ∇u0(x) (y) et ∇ w ∇u0 (x) ∈ L p ∩ L ∞ (R d
), toujours à x fixé. En moyennant sur la variable rapide y (ce qui revient à moyenner sur un grand cube Q R puis à faire R → +∞), on déduit5 que u 0 est solution de l'équation macroscopique (1.2.43) où a * est donné par (1.2.44). En particulier, ∇u 0 = ∇u per 0 et donc u 0 = u per 0 . Les calculs ci-dessus sont formels et ne constituent en rien une preuve. Notre objectif est maintenant de rendre ces observations rigoureuses.

La première étape est de résoudre (1.2.53). Pour cela, on s'appuie sur le cas linéaire et l'on exploite le problème périodique. On cherche w ξ sous la forme w ξ = w per ξ + w ξ et l'on écrit l'équation que doit vérifier w ξ :

-div a ξ + ∇w per ξ + ∇ w ξ ξ + ∇w per ξ + ∇ w ξ p-2 -ξ + ∇w per ξ ξ + ∇w per ξ p-2 = div h
+ ∇ w ξ ξ + ∇w per ξ + ∇ w ξ p-2 -ξ + ∇w per ξ ξ + ∇w per ξ p-2 •∇ w ξ = - ¢ h•∇ w ξ .
(1.2.56) En utilisant des estimations sur le p-Laplacien (voir [START_REF] Iwaniec | Projections onto gradient fields and L p -estimates for degenerated elliptic operators[END_REF]), on déduit l'estimation

∥∇ w ξ ∥ p-1 L p (R d ) ≤ C ∥h∥ L p ′ (R d ) .
Cela suggère de travailler dans l'espace

V := {w ∈ W 1,1 loc (R d ), ∇w ∈ L p (R d )}.
Cependant, un calcul rapide (par exemple quand p = 4) montre que le membre de gauche de (1.2.56) n'est pas nécessairement défini sur V . Il faut donc restreindre cet espace. Pour cela, on fait appel à l'inégalité

(ξ|ξ| p-2 -ζ|ζ| p-2 ) • (ξ -ζ) ≥ c |ξ| p-2 + |ζ| p-2 |ξ -ζ| 2 , et l'on constate que, nécessairement, ∇ w ξ ∈ L 2 ξ + ∇w per ξ p-2
dλ où dλ désigne la mesure de Lebesgue. On introduit alors l'espace A ce stade, deux remarques s'imposent. D'abord, les résultats sont présentés pour l'équation de p-Laplace mais ils peuvent s'étendre à des opérateurs a(y, ξ) plus généraux, voir la section 5.8.3. Ensuite, la méthode de preuve mise en place dans le chapitre 5 permet en fait de traiter le cas d'un coefficient a ∈ L q (R d ), q ∈ [p ′ , 2]. On obtient alors l'existence d'un correcteur dont le gradient est dans l'espace à poids L 2 ξ + ∇w per ξ p-2 dλ , mais on ne sait pas montrer que ∇ w ξ ∈ L 2 (R d ). En revanche, le cas a ∈ L 2+δ (R d ) est ouvert, voir la section 1.3 ci-dessous.

W ξ+∇w per ξ = w ∈ W 1,1 loc (R d ), ∇w ∈ L p (R d ) ∩ L 2 ξ + ∇w per ξ p-2 dλ /R
Après avoir construit les correcteurs, nous aimerions obtenir l'équivalent du théorème 10 dans le cadre périodique avec défauts. Dans le cadre périodique, un outil important pour obtenir la convergence forte des gradients (1.2.46) est la continuité de l'application

R d -→ L p (Q) ξ -→ ∇w per ξ .
(

(H) Pour ξ ∈ R d , il existe une constante C > 0 et un rayon r min > 0 tels que pour tout R > r min et tout w ∈ H 1 (Q \ Q 1/2
), on ait l'inégalité de Poincaré-Wirtinger à poids suivante :

ξ + ∇w per ξ (R•) p-2 2 w - 2 Q\Q 1/2 w L 2 (Q\Q 1/2 ) ≤ C ξ + ∇w per ξ (R•) p-2 2 ∇w L 2 (Q\Q 1/2 )
.

(

a • ε ∇u ε |∇u ε | p-2 -⇀ ε→0 a * (∇u * ) dans L p ′ (Ω).
(ii) On a la convergence forte

∇u ε -∇u * -∇w Mε∇u * • ε -→ ε→0 0 dans L p (Ω), (1.2.61)
où M ε est défini dans (1.2.47).

(iii) On a la convergence forte

∇u ε -∇u * -∇w per Mε∇u * • ε -→ ε→0 0 dans L p (Ω), (1.2
, η ∈ R d , ∥∇ w ξ -∇ w η ∥ L p (R d ) ≤ C |ξ| 1-β + |η| 1-β |ξ -η| β , β := 1 p -1 min(1, p -2).
La preuve du théorème 13 repose sur une linéarisation de l'équation du correcteur non périodique autour de ξ + ∇w per ξ . On fait ensuite appel à [START_REF] Blanc | Local profiles for elliptic problems at different scales: defects in, and interfaces between periodic structures[END_REF]. L'hypothèse de non dégénérescence est très forte : elle est satisfaite en dimensions d = 1, 2 [START_REF] Kirill | On full two-scale expansion of the solutions of nonlinear periodic rapidly oscillating problems and higher-order homogenised variational problems[END_REF] et pour les "faux" cas multidimensionnels tels que les matériaux lamellés, voir Section 5.2.4. La difficulté dans l'homogénéisation du p-Laplacien provient essentiellement de ce que le coefficient linéarisé ∂ ξ a(•, ξ + ∇w ξ ), qui se comporte comme

ξ + ∇w per ξ p-2
, peut être dégénéré. Des résultats récents étudient le cas non dégénéré dans le cadre de l'homogénéisation stochastique quantitative [START_REF] Clozeau | Quantitative nonlinear homogenization: control of oscillations[END_REF][START_REF] Fischer | Optimal homogenization rates in stochastic homogenization of nonlinear uniformly elliptic equations and systems[END_REF]. Il peut s'agir d'opérateurs a(y, ξ) à croissance quadratique et lipschitziens en ξ [START_REF] Fischer | Optimal homogenization rates in stochastic homogenization of nonlinear uniformly elliptic equations and systems[END_REF] ou d'opérateurs de la forme a(y, ξ) = a(y)ξ(1 + |ξ| p-2 ). Dans ces travaux, un outil important est l'équation du correcteur linéarisé [START_REF] Fischer | Optimal homogenization rates in stochastic homogenization of nonlinear uniformly elliptic equations and systems[END_REF][START_REF] Clozeau | Quantitative nonlinear homogenization: control of oscillations[END_REF], dans le cadre périodique avec défauts étudié dans cette thèse.

-div ∂ ξ a(•, ξ + ∇w ξ ) • (h + ∇w ξ,h ) = 0, h ∈ R d (
Nous terminons cette section par un résumé du chapitre 5. Nous y présentons l'homogénéisation de l'équation de p-Laplace dans un cadre périodique avec défauts. Nous prouvons que l'équation du correcteur admet, dans un cadre variationnel (qui correspond en fait à des défauts tendant suffisamment rapidement vers zéro à l'infini), une solution dans un espace de Sobolev à poids. Sous une hypothèse de non dégénérescence de ce poids, nous montrons, en utilisant des techniques issues du cas linéaire, la continuité de l'application ξ -→ ∇w ξ . Cela permet de reconstruire les oscillations de ∇u ε à l'aide de la solution homogénéisée et des correcteurs périodiques et non périodiques. Remarquons que, comme dans le cas linéaire, le comportement macroscopique de u ε n'est pas affecté par l'introduction du défaut local a. De même, l'utilisation du correcteur périodique dans la convergence forte du gradient n'affecte pas le mode de convergence, pourvu que l'on ne requiert pas une approximation de u ε dans des normes très fines. Dans la section 5.3, nous illustrons cela dans le cas de la dimension 1. En effet, des calculs explicites permettent de montrer que

u ′ ε -(u * ) ′ -w ′ (u * ) ′ • ε -→ ε→0 0 et u ′ ε -(u * ) ′ -w per (u * ) ′ ′ • ε -→ ε→0 0 dans L p . En revanche, u ′ ε -(u * ) ′ -w ′ (u * ) ′ • ε -→ ε→0 0 dans L ∞ , tandis que u ′ ε -(u * ) ′ -(w per (u * ) ′ ) ′ • ε ne tend pas nécessairement vers zéro en norme L ∞ .
Comme pour les cas linéaires, le correcteur non périodique améliore l'approximation fine de la solution oscillante. En dimension supérieure, ces problèmes sont hors de portée.

Perspectives

Dans cette section, nous présentons quelques questions ouvertes issues des travaux précédents.

Homogénéisation du système de Stokes. Une première piste possible concerne l'amélioration des taux de convergence obtenus dans le chapitre 3. Ils sont pour l'instant limités à l'exposant q = 2. Pourtant, les résultats obtenus pour l'équation de Poisson (mais aussi [START_REF] Masmoudi | Some uniform elliptic estimates in a porous medium[END_REF]) suggèrent que l'on pourrait, au moins sous l'hypothèse simplificatrice (1.2.28), généraliser ces taux au cas q > 2. Dans le cas q > d, on s'attend à ce que ces taux soient optimaux quand on utilise le correcteur non périodique. Au contraire, ils devraient se dégrader (quand q devient grand) lorsque l'on considère le reste "périodique"

u ε -ε 2 3 j=1
w per j (./ε)f j . Cela justifierait la construction pratique du correcteur non périodique lorsque l'on souhaite une approximation fine de la solution. L'obtention de taux de convergence demanderait deux étapes :

• évaluer le second membre de l'EDP satisfaite par le reste (R ε , π ε ). Cette étape est bien sûr liée à une propriété de sous-linéarité des correcteurs. Nous montrons un tel résultat dans l'annexe 3.4.3 pour les correcteurs d'ordre (w j , p j ) ;

• établir des estimées de régularité elliptique uniformes en ε pour un système de Stokes posé dans Ω ε . De tels résultats ont été énoncés dans [START_REF] Masmoudi | Some uniform elliptic estimates in a porous medium[END_REF] pour le cas périodique. Une preuve de ces résultats a été soumise récemment [START_REF] Shen | Compactness and large-scale regularity for Darcy's law[END_REF] dans le cas d'un domaine perforé dans l'espace complet. Il serait intéressant de voir si cette preuve s'étend au cas périodique avec défauts étudié ici.

Une autre possibilité serait d'essayer de comprendre le comportement (dans l'intérieur du domaine) de (u ε , p ε ) dans le cas où f ne satisfait plus (1.2.28). On pourrait pour cela commencer par des simulations numériques.

PERSPECTIVES CHAPITRE 1. INTRODUCTION

Homogénéisation d'équations quasi-linéaires. La première piste que nous pourrions explorer est la résolution de l'équation du correcteur (5.1.7) dans le cas d'un défaut qui décroit moins rapidement à l'infini vers zéro : cela se traduirait ici par la condition a ∈ L q (R d ), q > 2. Dans le cas linéaire, l'équation est résolue en étudiant la continuité L q -L q de l'opérateur de Riesz. L'obtention d'un tel résultat dans le cas non linéaire semble plus délicate. Cela constituerait une amélioration ambitieuse des résultats précédents.

Une autre possibilité est d'étudier le cas 1 < p < 2 dans le cadre du Chapitre 5. Dans le cas périodique, la restriction p ≥ 2 n'est pas nécessaire pour prouver les théorèmes de convergence de u ε et de son gradient. Cela devrait rester vrai pour le cadre non périodique. En revanche, la construction du correcteur non périodique, de même que les preuves des résultats de continuité sur l'application ξ -→ ∇w ξ , nécessitent, au moins sous leur forme actuelle, la condition p ≥ 2. Il serait pertinent de reprendre les méthodes de preuve sous l'hypothèse p < 2.

Première partie

Homogénéisation en milieu perforé avec défauts locaux

CHAPTER 2

HOMOGENIZATION OF THE POISSON EQUATION IN A NON-PERIODICALLY PERFORATED DOMAIN

This chapter is based on [START_REF] Blanc | Homogenization of the Poisson equation in a non periodically perforated domain[END_REF], written in collaboration with Xavier Blanc. We have added Subsection 2.5.2.

We study the Poisson equation in a perforated domain with homogeneous Dirichlet boundary conditions. The size of the perforations is denoted by ε > 0 and is proportional to the distance between neighbouring perforations. In the periodic case, the homogenized problem (obtained in the limit ε → 0) is well understood (see [START_REF] Lions | Asymptotic expansions in perforated media with a periodic structure[END_REF]). We extend these results to a non-periodic case which is defined as a localized deformation of the periodic setting. We propose geometric assumptions that make precise this setting and we prove results which extend those of the periodic case: existence of a corrector, convergence to the homogenized problem and two-scale expansion.

Introduction

In this chapter, we study the following problem:

-∆u ε = f in Ω ε u ε = 0 on ∂Ω ε , (2.1.1)
where f is a given smooth, compactly-supported function (this assumption may be relaxed, as we will see below in Remarks 2.1.4 and 2.2.4), and Ω ε is a perforated domain that we make precise in the following. Our aim is to study the asymptotic behaviour of u ε as ε → 0, deriving a two-scale expansion and proving convergence estimates. In [START_REF] Lions | Asymptotic expansions in perforated media with a periodic structure[END_REF], these results were obtained in the periodic case (that is, if the perforations are a periodic array of period ε). Here, we adapt this work to a non-periodic setting. Using Assumptions (A1) and (A2) below, which are inspired from the setting developed in [START_REF] Blanc | A possible homogenization approach for the numerical simulation of periodic microstructures with defects[END_REF][START_REF] Blanc | Local profiles for elliptic problems at different scales: defects in, and interfaces between periodic structures[END_REF][START_REF] Blanc | On correctors for linear elliptic homogenization in the presence of local defects[END_REF], we first prove the existence of a corrector (Theorem 2.2.1 below). While this result is trivial in the periodic case, it is not in the present setting. Then, we prove the convergence result stated in Theorem 2.2.2, which is a generalization of [START_REF] Lions | Asymptotic expansions in perforated media with a periodic structure[END_REF]Theorem 3.1] to the present setting.

We also prove such a convergence in L ∞ norm (Theorem 2.2.3 below), a result which was not proved in [START_REF] Lions | Asymptotic expansions in perforated media with a periodic structure[END_REF]. As it is noticed in [START_REF] Tartar | Incompressible fluid flow in a porous medium. Convergence of the homogenization process[END_REF][START_REF] Donato | Convergence of Dirichlet problems for monotone operators in a class of porous media[END_REF] for the periodic case, the crucial point in order to prove such results is a Poincaré inequality with an explicit scaling in ε, for functions vanishing in the perforations (see Lemma 2.1.1 below in the periodic case and Lemma 2.3.2 in the non-periodic case).

To our knowledge, the first contributions on the homogenization of elliptic problems in perforated domains are [START_REF] Cioranescu | Homogenization in open sets with holes[END_REF][START_REF] Tartar | Cours Peccot au collège de france[END_REF] in which qualitative results are obtained. The setting is periodic, the holes are "large" (i.e. they have the same size as the period), the equation is elliptic in divergence form and

INTRODUCTION CHAPTER 2. POISSON EQUATION

various boundary conditions are considered, such as constant Dirichlet boundary conditions on the holes and Neumann conditions. This implies that the limit is not necessarily trivial, in contrast to [START_REF] Lions | Asymptotic expansions in perforated media with a periodic structure[END_REF], where, as we will see below, u ε (x) ≈ ε 2 f (x)w(x/ε), for some periodic function w. The case of Robin boundary conditions along the holes is addressed in [START_REF] Cioranescu | Homogénéisation du problème de Neumann non homogène dans des ouverts perforés[END_REF]. In this work, an important tool to study the problem is the so-called extension operator, which is introduced in [START_REF] Cioranescu | Homogenization in open sets with holes[END_REF]. In [START_REF] Cioranescu | Periodic unfolding and Robin problems in perforated domains[END_REF][START_REF] Cioranescu | The periodic unfolding method in perforated domains[END_REF], the case of operators in divergence form with periodic coeffciients and with Robin boundary conditions is addressed, with the help of the periodic unfolding method (see [START_REF] Cioranescu | Periodic unfolding and homogenization[END_REF][START_REF] Cioranescu | The periodic unfolding method in domains with holes[END_REF]). The case of eigenvalue problems is considered in [START_REF] Vanninathan | Homogenization of eigenvalue problems in perforated domains[END_REF]. In [START_REF] Cioranescu | Un terme étrange venu d'ailleurs[END_REF][START_REF] Cioranescu | Un terme étrange venu d'ailleurs. II. In Nonlinear partial differential equations and their applications[END_REF], the geometry is periodic, the boundary conditions are of Dirichlet type but the holes are assumed to be asymptotically small compared to the period. The case of small holes with non-homogeneous Neumann boundary conditions is also developed in [START_REF] Conca | Non-homogeneous Neumann problems in domains with small holes[END_REF]. In [START_REF] Donato | Homogenization of the Poisson equation in a porous medium with double periodicity[END_REF], a perforated domain mixing large and small holes with a double periodicity structure is studied for the Poisson Equation with homogeneous Dirichlet boundary conditions. In [START_REF] Damlamian | Which sequences of holes are admissible for periodic homogenization with Neumann boundary condition?[END_REF], sufficient conditions on periodic holes are given which allow for homogenization. Sufficient conditions on the perforated domain are also provided in [START_REF] Donato | Convergence of Dirichlet problems for monotone operators in a class of porous media[END_REF] for the homogenization of non-linear operators under divergence form with homogeneous Dirichlet boundary conditions, see also [START_REF] Donato | On the homogenization of some nonlinear problems in perforated domains[END_REF] for nonlinear problems.

In [START_REF] Briane | H-convergence for perforated domains[END_REF], a formalization in link with the H-convergence was proposed under general assumptions on the perforations. However, the computations are less explicit than in our setting. A general (non-periodic) perforated domain was also considered in [START_REF] Nguetseng | Homogenization in perforated domains beyond the periodic setting[END_REF]: this setting requires that, among other assumptions, the same perforation is reproduced in some cells of a periodic grid (but not necessarily all of them).

In the following subsection, we recall the results proved in the periodic setting in [START_REF] Lions | Asymptotic expansions in perforated media with a periodic structure[END_REF]. Then, in Subsection 2.1.2, we study the case of a locally perturbed periodic geometry. We give conditions on the perforations (inspired from [START_REF] Blanc | A possible homogenization approach for the numerical simulation of periodic microstructures with defects[END_REF][START_REF] Blanc | Local profiles for elliptic problems at different scales: defects in, and interfaces between periodic structures[END_REF][START_REF] Blanc | On correctors for linear elliptic homogenization in the presence of local defects[END_REF]), which imply that, away from the defect, the perforations become periodic and which allow to prove convergence results similar to those of the periodic case. In Section 2.2, we give the main results of the article, together with some remarks and comments. Section 2.3 is devoted to a Poincaré-type inequality which is crucial in our proof. Finally, Section 2.4 is devoted to the proofs of the results stated in Section 2.2.

The periodic case

We start with some notations. We consider the d-dimensional unit cube Q = -

If k ∈ Z d , we have O per k ⊂⊂ Q k where Q k := Q + k. Given ε > 0, denote by O per ε the set of ε-perforations: O per ε := k∈Z d εO per k = k∈Z d ε(O per 0 + k) = εO per . (2.1.3)
We now define some useful functional spaces: (Q \ O per 0 ) will naturally be extended to R d \ O per by periodicity. All along the paper, we will denote the H 1 -semi-norm on a set V by

H 1,per (Q \ O per 0 ) := u ∈ H 1 loc (R d \ O per ) s.t
| • | H 1 (V ) : |u| H 1 (V ) := ¢ V |∇u| 2 1/2 .
Let Ω be a bounded, open and connected domain of R d that is locally Lipschitz regular. For ε > 0, denote by Ω ε := Ω \ O per ε . Note that Ω ε is open and bounded but may not be connected. One has

Ω ε = Ω ∩   k∈Z d ε(Q k \ O per k )   . (2.1.6)
Figure 2.1 shows the set Ω ε for two values ε 0 and ε 1 satisfying ε 0 > ε 1 . The set Ω ε is colored in light grey. We are interested in the Poisson problem (2.1.1). As we already mentioned, the source term f is supposed, as in [START_REF] Lions | Asymptotic expansions in perforated media with a periodic structure[END_REF], to be smooth and compactly supported in Ω. In fact (see Remark 3.3 of [START_REF] Lions | Asymptotic expansions in perforated media with a periodic structure[END_REF]), it is sufficient to assume that f ∈ C m-2 (Ω) and that D p f |∂Ω = 0 for all |p| ≤ m -2, where m is the order of the two scale expansion of u ε (that is, the order in ε that we seek in the approximation on u ε ). As pointed out in [START_REF] Le Bris | An MsFEM type approach for perforated domains[END_REF], the assumptions on f can be weakened further (see Remark 2.1.4 below).

By a simple application of the Lax-Milgram Lemma, we have the existence and the uniqueness of a solution u ε to (2.1.1). In order to study the dependence of u ε on ε, we will need the following Lemma which is a Poincaré-type inequality in H 1 0 (Ω ε ). It is proved in [99, Lemma 1] (see also [START_REF] Bourgeat | Homogenization in a perforated domain including a thin full interlayer[END_REF]Proposition 3.1]). A crucial point in the non-periodic case will be to have a similar result, with the same scaling in ε. This is done in Lemma 2.3.2 below. Note that a stronger result has in fact been independently proven in [START_REF] Donato | Convergence of Dirichlet problems for monotone operators in a class of porous media[END_REF]. Lemma 2.1.1 (Lemma 1 of [START_REF] Tartar | Incompressible fluid flow in a porous medium. Convergence of the homogenization process[END_REF]). There exists a constant C 0 > 0 independent of ε such that

∀u ∈ H 1 0 (Ω ε ), ∥u∥ L 2 (Ωε) ≤ C 0 ε∥∇u∥ L 2 (Ωε)
. This allows to prove Lemma 2 of [99]: Lemma 2.1.2 (Lemma 2 of [START_REF] Tartar | Incompressible fluid flow in a porous medium. Convergence of the homogenization process[END_REF]). The solution u ε of Problem (2.1.1) satisfies the estimates

∥u ε ∥ L 2 (Ωε) ≤ Cε 2 and ∥u ε ∥ H 1 0 (Ωε) ≤ Cε, (2.1.7)
where C is a constant independent of ε.

INTRODUCTION CHAPTER 2. POISSON EQUATION

Using a two-scale expansion of the form (see [START_REF] Lions | Asymptotic expansions in perforated media with a periodic structure[END_REF]Section 2])

u ε (x) = u per 0 x, x ε + εu per 1 x, x ε + ε 2 u per 2 x, x ε + ε 3 u per 3 x, x ε + • • • , (2.1.8)
where the functions u per i are defined on Ω × (Q \ O per 0 ), smooth and Q-periodic in the second variable, one proves that, at least formally,

u ε (x) = ε 2 w per x ε f (x) + • • • , (2.1.9)
where w per is the periodic solution of -∆w per = 1,

w per ∈ H 1,per 0 (Q \ O per 0 ). ( 2
(Q \ O per 0 ), ¢ Q\O per 0 ∇w per • ∇v = ¢ Q\O per 0 v.
The following convergence result is proved in [77, Theorem 3.1] (take m = 2 there). Then there exists a constant C > 0 independent of ε such that

ε -1 u ε -ε 2 w per (•/ε) f L 2 (Ωε) + u ε -ε 2 w per (•/ε) f H 1 (Ωε) ≤ Cε 2 , (2.1.11)
where w per is the unique solution to (2.1.10).

Remark 2.1.4. Since O per 0 is of class C 1,γ for some 0 < γ < 1, then Theorem 2.1.3 still holds true under the weaker hypotheses f ∈ H 2 ∩ L ∞ (Ω) and f |∂Ω = 0 (in the trace sense), see [START_REF] Le Bris | An MsFEM type approach for perforated domains[END_REF]Appendix A.2]. If we do not assume that f vanishes on ∂Ω, u ε -ε 2 w(•/ε)f does not vanish on ∂Ω either and we have the weaker estimate

∥u ε -ε 2 w(•/ε)f ∥ H 1 (Ωε) ≤ Cε 3/2 N (f ), where N (f ) = ∥f ∥ L ∞ + ∥∇f ∥ L 2 + ∥∆f ∥ L 2 .

The non-periodic case

We aim at extending the previous results to non-periodically perforated medium, in the special case of local perturbations of the periodic structure. More precisely, we define a reference periodic configuration by (2.1.2)-(2.1.3)-(2.1.6) and, for each k ∈ Z d , we denote by O k the (non-periodic) perforation in the cell Q k . We recall that O per 0 is of class C 1,γ for some 0 < γ < 1. Our first assumption reads as follows:

(A1) For all k ∈ Z d such that O k ̸ = ∅, O k is a locally Lipschitz open set such that O k ⊂⊂ Q k and Q k \ O k is connected.
In [START_REF] Blanc | Homogenization of the Poisson equation in a non periodically perforated domain[END_REF], it is assumed for simplicity that each periodic cell contains a non-periodic perforation. In this chapter, we slightly adapt some proofs to cover the case O k = ∅ for a finite number of k ∈ Z d . We next introduce geometric tools. For α > 0, define the Minkowski content of ∂O per 0 (i.e a widened boundary of O per 0 ) by the set Similarly, if k ∈ Z d and α > 0, denote the set

U per 0 (α) := {x ∈ R d s.t. dist(x, ∂O per 0 ) < α}. O per,- k α α O per k O per,+ k U per k (α) α k α k O k
U per k (α) := {x ∈ R d s.t. dist(x, ∂O per k ) < α} = U per 0 (α) + k. Now (see Figure 2
k (α) := O per k \ U per k (α) and O per,+ k (α) := O per k ∪ U per k (α).
The following properties hold true:

O per,- k (α) ⊂ O per,+ k (α) and U per k (α) = O per,+ k (α) \ O per,- k (α).
Finally, we clearly have

O per,+ k (α) = {x ∈ R d s.t. dist(x, O per k ) < α}, ( 2
α k > 0, (α k ) k∈Z d ∈ ℓ 1 (Z d ) and ∀k ∈ Z d , O per,- k (α k ) ⊂ O k ⊂ O per,+ k (α k ) (2.1.14)
i.e. O k is between the enlargement and the reduction of Note that these domains are not necessarily connected. We split the boundary of the domain O per \O into two parts (the one surrounding O per and the one surrounding O). For k ∈ Z d , we define

O per k . Note that if α k is sufficiently large, O per,- k (α k ) = ∅ and Q k ⊂ O per,+ k (α k ).
Γ k 1 = ∂O per k \ O k and Γ k 2 = ∂O k ∩ O per k s.t. ∂(O per k \ O k ) = Γ k 1 ∪ Γ k 2 .
(2.1.16)

O per k \ O k O k Γ k 1 Γ k 3 Γ k 2 O per k \ O k O k Γ k 1 Γ k 3
O k ∩ O per k ̸ = ∅. Right: O k ∩ O per k = ∅ We denote by Γ 1 (resp. Γ 2 ) the union of the Γ k 1 (resp. Γ k 2 ), k ∈ Z d : Γ 1 = k∈Z d Γ k 1 and Γ 2 = k∈Z d Γ k 2 .
(2.1.17)

We also split the boundary of 2.3 explains the above definitions. From Assumption (A2), we deduce Lemmas 2.5.1, 2.5.2 and 2.5.3, which are stated and proved in Appendix 2.5.1.

R d \ O ∪ O per into two parts. Write ∂(R d \ O ∪ O per ) = ∂(O ∪ O per ), and define for k ∈ Z d Γ k 3 = ∂O k \ O per k s.t. ∂(O ∪ O per ) = Γ k 1 ∪ Γ k 3 . (2.1.18) Note that ∂O k = Γ k 2 ∪ Γ k 3 . Γ 3 denotes the union of the Γ k 3 over k ∈ Z d . Figure
Remark 2.1.5. Assumption (A2) is a way to impose that the defect is localized. In [START_REF] Blanc | A possible homogenization approach for the numerical simulation of periodic microstructures with defects[END_REF][START_REF] Blanc | Local profiles for elliptic problems at different scales: defects in, and interfaces between periodic structures[END_REF][START_REF] Blanc | On correctors for linear elliptic homogenization in the presence of local defects[END_REF], such an assumption is written as a = a per + a, with a ∈ L q (R d ), and a per is periodic, where a is the coefficient of the considered elliptic equation. Here, writing a similar condition, we impose that the characteristic function of the perforations is a perturbation (i.e, a function in L q (R d )) of the periodic case. For a characteristic function, being in L q (R d ) is equivalent to being in L 1 (R d ), hence the condition (A2).

We will sometimes require the following uniform regularity assumption on the non-periodic perforations (O k ) k∈Z d : (A3) There exists γ ∈ (0, 1), r > 0 and M > 0 such that for all k ∈ Z d and x ∈ ∂O k , there exists an open set U x ⊂ R d-1 , 0 ∈ U x and a C 1,γ application Ψ x : U x -→ R such that, after eventually translating and/or rotating the coordinate system, we have that

Q k \ O k ∩ B(x, r) = {x ∈ B(x, r), x d > Ψ x (x 1 , ..., x d-1 ), (x 1 , ..., x d-1 ) ∈ U x }, ∂O k ∩ B(x, r) = {x ∈ B(x, r), x d = Ψ x (x 1 , ..., x d-1 ), (x 1 , ..., x d-1 ) ∈ U x }, and ∥Ψ x ∥ C 1,γ (Ux) ≤ M .
In the above Assumption (A3), we will, if necessary, replace C 1,γ by any C ℓ , C ℓ,γ , ℓ ∈ N, γ ∈ (0, 1). In this case, we say that the perforations O k are of class C ℓ , C ℓ,γ uniformly in k.

Results

In order to state our main result, we first need to prove that a corrector exists:

Theorem 2.2.1 (Existence and uniqueness of the corrector). Let (O k ) k∈Z d be a sequence of open sets satisfying Assumptions (A1)-(A2).

Let O be defined by (2.1.15), and

g = 1 R d \O per + g, with g ∈ L 2 (R d ).
There exists a unique w ∈ H 1

(R d \ O) such that w := w per + w ∈ H 1,per (Q) + H 1 (R d \ O) is solution in the sense of distributions to -∆w = g in R d \ O w |∂O = 0, (2.2.1)
where w per ∈ H 1,per 0 (Q\O per 0 ) is the unique solution of the periodic corrector problem (2.1.10) extended by zero to R d . Using Theorem 2.2.1 and a two-scale expansion, as it is done in the periodic case, we have the following result, which is the generalization of Theorem 2.1.3 to the present setting:

Theorem 2.2.2 (Convergence theorem in H 1 -norm). Let (O k ) k∈Z d be

a sequence of open sets satisfying Assumptions (A1)-(A2)

and assume that O is defined by (2.1.15). Let Ω ⊂ R d be a bounded domain and define for ε > 0 the perforated set Ω ε := Ω \ εO. Let f ∈ D(Ω) and u ε be the solution of Problem (2.1.1). Then there exists a constant C > 0 independent of ε such that

ε -1 u ε -ε 2 w (•/ε) f L 2 (Ωε) + u ε -ε 2 w (•/ε) f H 1 (Ωε) ≤ Cε 2 , (2.2.2)
where w = w per + w ∈ H 1,per (Q) + H 1 (R d \ O) is the unique solution of the corrector Problem (2.2.1) with g = 1.
We note that the constant C appearing in Theorem 2.2.2 is independent of ε but depends on f , on the non-periodic corrector constructed in Theorem 2.2.1 and on the Poincaré-

Friedrichs constant of Ω ε (denoted C in Lemma 2.3.2 below). Theorem 2.2.2 provides an error estimate of u ε -ε 2 w(•/ε)f in H 1 (Ω ε )-norm.
However, for this choice of norm, the use of a non-periodic corrector appears to be irrelevant, which means that we could also have used the periodic corrector w per in (2.2.2) without changing the rate of convergence. Indeed, we have

ε 2 w • ε f H 1 (Ωε) = O ε 2 . (2.2.3)
In order to prove (2.2.3), we only deal with the leading order term of the above quantity, that is, the L 2 -norm of the gradient. One has

¢ Ωε ∇ ε 2 w • ε f (x) 2 dx ≤ 2ε 2 ¢ Ωε ∇ w x ε 2 |f (x)| 2 dx + 2ε 4 ¢ Ωε w x ε 2 |∇f (x)| 2 dx ≤ Cε 2 ¢ Ωε ∇ w x ε 2 dx + Cε 4 ¢ Ωε w x ε 2 dx.
Thus, after the change of variable y = x/ε,

¢ Ωε ∇ ε 2 w • ε f (x) 2 dx ≤ Cε d+2 ¢ R d \O |∇ w(y)| 2 dy + Cε d+4 ¢ R d \O | w(y)| 2 dy,
where C depends on the L ∞ -norms of f and ∇f . We thus have (2.2.3), which implies (since d ≥ 2)

u ε -ε 2 w per • ε f H 1 (Ωε) ≤ u ε -ε 2 w • ε f H 1 (Ωε) + ε 2 w • ε f H 1 (Ωε) = O ε 2 .
Thus, using w per instead of w in convergence Theorem 2.2.2 does not change the order O(ε 2 ) of the error.

The following Theorem states that the use of w instead of w per improves the rate of convergence in L ∞ -norm for a non-periodic domain.

Theorem 2.2.3 (Convergence Theorem in L ∞ -norm). Let (O k ) k∈Z d be a sequence of open sets satisfying Assumptions (A1)-(A2)

and assume that O is defined by (2.1.15). Assume that the perforations O k are uniformly in k of class C 1,γ in the sense of Assumption (A3). Let Ω ⊂ R d be a bounded domain and define for ε > 0 the perforated set Ω ε := Ω \ εO. Let f ∈ D(Ω) and u ε be the solution of (2.1.1). Then there exists a constant C > 0 independent of ε such that

u ε -ε 2 w (•/ε) f L ∞ (Ωε) ≤ Cε 3 , where w = w per + w ∈ H 1 per (Q) + H 1 (R d \ O) is the unique solution of (2.2.1) with g = 1 R d \O . Note that ε 2 w(•/ε)f L ∞ (Ωε)
is generally of order ε 2 exactly. We fix K ⊂ Ω. One has

ε 2 w • ε f (ε•) L ∞ (K) ∼ ε→0 ε 2 f (0)∥ w∥ L ∞ (K) .
Besides, Theorem 2.2.3 implies

u ε -ε 2 w • ε f (ε•) L ∞ (K) ≤ Cε 3 .
Thus,

u ε -ε 2 w per • ε f (ε•) L ∞ (K) ∼ ε→0 Cε 2 . (2.2.4)
We have the same estimate as (2.2.4) when the L ∞ (K)-norm is replaced by the L 2 (K)-norm. This proves that the convergence of

u ε /ε 2 -w(•/ε)f holds at the microscale in L 2 -
∥g ε ∥ L 2 (Ωε) ≤ 2∥∇w∥ L ∞ ∥∇f ∥ L 2 + ∥w∥ L ∞ ∥∆f ∥ L 2 ≤ C∥f ∥ H 2 (Ω)
for ε < 1. We deduce by integration by parts that ∥u ε -

ε 2 w(•/ε)f ∥ H 1 (Ωε) ≤ Cε 2 .
If f does not vanish on ∂Ω, we can prove that there exists a constant C independent of ε such that

∥u ε -ε 2 w(•/ε)f ∥ H 1 (Ωε) ≤ Cε 3/2 N (f ).

The proof is analogous to [75, Appendix A.2] provided we use Lemma 2.4.11 below. This requires

f ∈ H 2 ∩ L ∞ (Ω).

Poincaré-Friedrichs inequalities

The main ingredient of the proof of Theorem 2.2.1 is the following Poincaré-type inequality.

Theorem 2.3.1. Let Q be the unit cube of R d and let U be an open subset of Q containing a box R = d i=1 [a i , b i ]. Then ∀v ∈ H 1 (Q \ U ) s.t v |∂U = 0, ¢ Q\U |v| 2 ≤ d |R| ¢ Q\U |∇v| 2 . (2.3.1)
Similarly,

∀v ∈ H 1 (Q) s.t v |U = 0, ¢ Q |v| 2 ≤ d |R| ¢ Q |∇v| 2 .
An important point in (2.3.1) is that the constant is explicit and depends only on R. This crucial point will allow us, with the help of Assumption (A2), to prove Lemma 2.3.2 below, in which the fundamental point is that the constant does not depend on ε. We thus have an explicit scaling with respect to ε, similarly to the periodic case. This allows us to adapt the proofs of [START_REF] Lions | Asymptotic expansions in perforated media with a periodic structure[END_REF].

Proof. By density, it is enough to show the result for v ∈ C 1 (Q) satisfying v = 0 on U . Fix x ∈ Q and write v(x) -v(x) = ¢ 1 0 ∇v((1 -t)x + tx) • (x -x)dt, where x = ( ai+bi 2 + x i (b i -a i )) 1≤i≤d ∈ R. Note that v(x) = 0 and |x -x| 2 ≤ d. Thus by the Cauchy-Schwarz inequality |v(x)| 2 ≤ d ¢ 1 0 |∇v((1 -t)x + tx)| 2 dt. Integrating with respect to x ∈ Q
and exchanging the two integrals yields

¢ Q |v(x)| 2 dx ≤ d ¢ 1 0 ¢ Q |∇v((1 -t)x + tx)| 2 dx dt Fix t ∈ [0, 1] and define the diffeomorphism ϕ t : Q ∋ x → (1 -t)x + tx. Note that ϕ t (Q) ⊂ Q and that | det J(ϕ t )| = d i=1 [(1 -t)(b i -a i ) + t] ≥ d i=1 (b i -a i ).
Thus by a change of variables,

¢ Q |∇v((1 -t)x + tx)| 2 dx ≤ 1 d i=1 (b i -a i ) ¢ Q |∇v| 2 .
Integrating with respect to t concludes the proof.

Theorem 2.3.1 and Assumption (A2) allow to prove the following, which is a generalization to the present setting of Lemma 2.1.1. Note that the proof of the following Lemma slightly differs from the published version [START_REF] Blanc | Homogenization of the Poisson equation in a non periodically perforated domain[END_REF] since it is implicitely assumed there that each cell Q k contains a perforation.

Lemma 2.3.2 (Poincaré-type inequality in

H 1 0 (Ω ε )). Let (O k ) k∈Z d be a sequence of open sets such that O k ⊂⊂ Q k . Suppose that the sequence (O k ) k∈Z d satisfies Assumption (A2). Let Ω be an open subset of R d . Define for ε > 0, Ω ε = Ω \ εO = Ω ∩   k∈Z d ε(Q k \ O k )   .
There exists a constant C > 0 independent of ε such that

∀u ∈ H 1 0 (Ω ε ), ¢ Ωε u 2 ≤ Cε 2 ¢ Ωε |∇u| 2 .
Proof. We first recall (see Lemma 2.5.3 in the appendix) that the set

K := {k ∈ Z d , O k ∩ O per k = ∅} is finite. • Step 1 : we define A := max k∈K |k|. By definition of K, the set A := |k|≤A+1 Q k contains a
perforation that we denote by O k . Thus, by the standard Poincaré-Friederichs inequality, we get the existence of a constant

C 0 = C A, O k , d such that ∀w ∈ H 1 0 (R d \ O), ¢ A |w| 2 ≤ C 0 ¢ A |∇w| 2 . (2.3.2)
In particular,

∀k ∈ K, ∀w ∈ H 1 0 (R d \ O), ¢ Q k \O k |w| 2 ≤ C 0 ¢ A |∇w| 2 .
(2.3.3)

•

Step 2 : we show that there exists ρ > 0 such that for all k ∈ K c , there exists a box

R k ⊂ O k satisfying |R k | ≥ ρ. Fix k ∈ Z d . By Lemma 2.5.2, there exists a ball B k ⊂ O k such that |B k | ≥ ρ with ρ independent of k. Thus, there exists a box R k ⊂ O k such that |R k | ≥ C(d)ρ
where C(d) is a constant depending only on d. We define ρ := C(d)ρ > 0. We next use Theorem 2.3.1 to obtain that

∀k ∈ K c , ∀w ∈ H 1 0 R d \ O , ¢ Q k \O k w 2 ≤ d ρ ¢ Q k \O k |∇w| 2 . ( 2
∀w ∈ H 1 0 (R d \ O), ¢ R d \O w 2 ≤ C ¢ R d \O |∇w| 2 . (2.3.5)
Now, we fix u ∈ H 1 0 (Ω ε ). We extend u by zero to R d \ εO and we define v

:= u(ε•). It is clear that v ∈ H 1 0 (R d \ O) and that ∀y ∈ R d \ O, ∇v (y) = ε∇u(εy). (2.3.6) Applying (2.3.5) with w = v ∈ H 1 0 (R d \ O) and using (2.3.6) yields ¢ 1 ε Ωε u 2 (εy)dy = ¢ R d \O u 2 (εy) dy ≤ Cε 2 ¢ R d \O |∇u| 2 (εy) dy = Cε 2 ¢ 1 ε Ωε |∇u| 2 (εy)dy.
Making the change of variables x = εy in each integral finally concludes the proof.

Remark 2.3.3. We deduce from the previous method of proof the following result that will be useful in Appendix 2.5.2. Let Ω ′ and Ω ′′ be two domains such that Ω ′ ⊂⊂ Ω ′′ ⊂⊂ Ω. We assume that

Ω ε := Ω\ε k∈Yε O k , where Y ε := {k ∈ Z d , εQ k ⊂ Ω}.
Then there exists a constant C > 0 independent of ε such that for all ε < ε 0 (Ω ′ , Ω ′′ , O) small enough and u ∈ H 1 0 (Ω ε ),

u L 2 (Ωε∩Ω ′ ) ≤ Cε ∇u L 2 (Ωε∩Ω ′′ ) . (2.3.7)
We briefly prove (2.3.7). We choose ε < ε 0 (Ω ′ , Ω ′′ , A) so that

k∈Y ′ ε εQ k ⊂ Ω ′′ , Y ′ ε := {k ∈ Z d , εQ k ∩ Ω ′ ̸ = ∅} and ε |k|≤A+1 Q k (= εA) ⊂ Ω ′′ . (2.3.8) Let u ∈ H 1 0 (Ω ε ). We extend u by zero in R d \ Ω ε . By (2.3.
3) and a scaling argument, we have that

∀k ∈ Y ′ ε ∩ K, ¢ ε(Q k \O k ) u 2 ≤ C 0 ε 2 ¢ εA |∇u| 2 . (2.3.9)
Similarly, by (2.3.4) and a scaling argument, we have that

∀k ∈ Y ′ ε ∩ K c , ¢ ε(Q k \O k ) u 2 ≤ d ρ ε 2 ¢ εQ k |∇u| 2 . (2.3.10) We sum (2.3.9) over k ∈ Y ′ ε ∩ K and (2.3.10) over k ∈ Y ′ ε ∩ K c .
Adding the results and using (2.3.8) gives

¢ Ωε∩Ω ′ u 2 ≤ Cε 2 ¢ Ωε∩Ω ′′ |∇u| 2 . (2.3.11)
This proves (2.3.7).

Proofs

Two-scale expansion

The aim of this section is to find an asymptotic equivalent of u ε as ε goes to zero. We begin by the two scale expansion of u ε . Write

u ε (x) = u 0 x, x ε + εu 1 x, x ε + ε 2 u 2 x, x ε + ε 3 u 3 x, x ε + • • • ,
where the functions u i are now defined on Ω × (R d \ O) and are of the form u per i

+ u i . Suppose that u i (x, •) ∈ H 1 (R d \ O
) and use the u per i 's defined in Section 2.1.1 and extended by zero to R d . Because of the homogeneous Dirichlet boundary conditions on u ε , we impose that u i (x, y) = 0 for y ∈ ∂O and any x ∈ Ω. The calculations leading to (2.1.9) (see [START_REF] Lions | Asymptotic expansions in perforated media with a periodic structure[END_REF]Section 2]) are still valid, so we have:

               -∆ y u 0 = 0 -∆ y u 1 -2(∇ x • ∇ y )u 0 = 0 -∆ y u 2 -2(∇ x • ∇ y )u 1 -∆ x u 0 = f -∆ y u 3 -2(∇ x • ∇ y )u 2 -∆ x u 1 = 0 • • • , (2.4.1)
where all these equations are posed on Ω × (R d \ O). These equations imply that u 0 and u 1 are constantly equal to zero. Indeed, fix x ∈ Ω. Since u per 0 ≡ 0, we get that u 0 (x, •) satisfies the PDE

-∆ y u 0 = 0 in R d \ O, u 0|∂O = 0. Multiplying by u 0 (x, •) ∈ H 1 0 (R d \ O
) and integrating by parts2 yields u 0 (x, •) ≡ 0. Thus u 0 ≡ 0. Similarly, u 1 ≡ 0. We are now left with the following equation on u 2 :

-∆ y u 2 (x, y) = f (x) in R d \ O u 2 (x, y) = 0, x ∈ Ω, y ∈ ∂O. (2.4.2)
According to (2.4.2), u 2 (x, y) = f (x)w(y), where w is a solution to the corrector equation (2.2.1) with g ≡ 1. This is why we introduced the corrector equation.

Proof of the existence of a corrector

The aim of this section is to prove Theorem 2.2.1. The difficulty of this theorem is that equation (2.2.1) is posed on an unbounded domain. We search for w in the form w per + w, where we impose that

w ∈ H 1 (R d \ O).
We write the equation on w and prove by energy minimization that there is a solution. The equation we want to solve for w is

-∆ w = 1 R d \Oper + g + ∆w per , (2.4.3)
where g ∈ L 2 (R d ) and w per ∈ H 1,per (Q) is the solution to (2.1.10) defined in Section 2.1. We recall that w per is extended by zero in O per . We impose that w = -w per on ∂O. It is worth noticing that w per / ∈ H 2 (Q) and thus the right-hand side of (2.4.3) cannot be in L 2 (R d \ O). Thus the linear form of the weak formulation of (2.4.3) is not of the form v → ¡ f v. In fact, we will have to deal with boundary terms along ∂O per . These terms express the fact that ∆w per is a Dirac measure on ∂O per (or that w per has normal derivative jumps along ∂O per ). In this section, we use that, since O per 0 is of class C 1,γ , then [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]Theorem 8.34] gives that w per ∈ C 1,γ (R d \ O per ).

PROOFS CHAPTER 2. POISSON EQUATION

Notation. We denote by ∂u ∂n ext (resp. ∂u ∂n int ) the exterior normal derivative of u on the outside (resp. inside) of a piecewise smooth closed surface Γ (when it is defined i.e u is H 2 on each side of the boundary).

Definition 2.4.1. We say that w ∈ H 1 (R d \ O) is a weak solution of (2.4.3) if ∀v ∈ C 1 c (R d \ O), ¢ R d \O ∇ w • ∇v + ¢ Γ1 ∂w per ∂n ext v - ¢ R d \O gv = 0 (2.4.4)
and w |∂O = -w per in the trace sense.

Note that, in the above Definition 2.4.1, we can replace Γ 1 by ∂O per after extension of v by zero in O. We check easily that w is solution to (2.4.3) in the sense of Definition 2.4.1 if and only if w := w per + w is solution in the distribution sense to (2.2.1).

Remark 2.4.2. We could also have written equation (2.4.3) as a system of PDEs coupled by transmission conditions:

           -∆ w = g in R d \ (O ∪ O per ) -∆ w = g in O per \ O w = -w per on Γ 2 ∪ Γ 3 ∂ w ∂n ext + ∂ w ∂n int = ∂w per ∂n ext on Γ 1 .
(2.4.5)

The three first equations are obviously necessary. The last equation is necessary to guarantee that

w = w per + w ∈ H 2 loc (R d \ O).
Using standard tools of the calculus of variations, one easily proves the following:

Lemma 2.4.3. Assume that w ∈ H 1 R d \ O . It is a weak solution of (2.4.3) in the sense of Definition 2.4.

1, if and only if it is a solution to the following minimization problem:

inf w∈V 1 2 ¢ R d \O |∇ w| 2 + ¢ Γ1 ∂w per ∂n ext w - ¢ R d \O g w , (2.4.6)
where the minimization space V is defined by 2.4 shows a function w ∈ V (extended to O by -w per ). In order to study the minimization problem (2.4.6), we will need the following Poincaré type inequality in V .

V := w ∈ H 1 (R d \ O) s.t w |∂O = -w per . (2.4.7) Definition 2.4.4. Let w ∈ V . We denote by W its extension to R d defined by W = -w per in O. The extension W of w satisfies W ∈ H 1 (R d ) under Assumptions (A1)-(A2) on the sequence (O k ) k∈Z d . Figure

Lemma 2.4.5 (Poincaré-type inequality in V ). Let (O k ) k∈Z d be a sequence of sets satisfying Assumptions (A1)-(A2). Define

O = k∈Z d O k . Let w per be the periodic corrector solution to (2.

1.10).

There exist constants C 0 > 0 and

C 1 > 0 such that for any w ∈ V , ¢ R d \O w 2 ≤ C 0 ¢ R d \O |∇ w| 2 + C 1 .
(2.4.8)

Denoting by W the extension of w (see Definition 2.4.4), we also have

¢ R d W 2 ≤ C 0 ¢ R d |∇ W | 2 + 2C 1 .
(2.4.9)

O per k \ O k O k ∩ O per k O k Γ k 1 Γ k 3 Γ k 2 W = 0 W = 0 W = -w per W = 0 O per k O k W = -w per Γ k 1 Γ k 3 Figure 2.4:
The function w (and its extension W ) on a perforated cell with and without overlapping

Proof. The proof is very similar to the proof of Lemma 2.3.2. We fix w ∈ V and we extend w by -w per in O. This gives a function

W ∈ H 1 (R d ). Note that ∀k ∈ Z d , W = 0 in O k ∩ O per k .
(2.4.10)

• Step 1. We recall that A := max k∈K |k| and K := {k ∈ Z d , O k ∩ O per k = ∅}. We define A := |k|≤A+1 Q k . There exists k ∈ Z d such that |k| ≤ A + 1 and O k ∩ O per k ̸ = ∅. Thus, the classical Poincaré inequality gives the existence of C 2 = C(O, O per , A, d) such that ¢ A W + w per 2 ≤ C 2 ¢ A ∇ W + ∇w per 2 .
We recall that thanks to elliptic regularity theory, w per ∈ W 1,∞ (Q). This yields

¢ A W 2 ≤ 2C 2 ¢ A ∇ W 2 + 2C 2 ∥w per ∥ 2 H 1 (A) ≤ 2C 2 ¢ R d |∇ W | 2 + 2C 2 |A|∥w per ∥ 2 W 1,∞ (Q) . (2.4.11) Now, the fact that W = -w per on O implies ¢ A\O | w| 2 ≤ 2C 2 ¢ A\O |∇ w| 2 + 2C 2 ¢ A∩O |∇w per | 2 + 2C 2 |A|∥w per ∥ 2 W 1,∞ (Q) ≤ 2C 2 ¢ A\O |∇ w| 2 + 4C 2 |A|∥w per ∥ 2 W 1,∞ (Q) .
(2.4.12)

• Step 2. We assume that |k| ≥ A so that k / ∈ K. In particular, O k ∩ O per k ̸ = ∅. We use Lemma 2.5.2: there exists a ball B k ⊂ O k ∩ O per k such |B k | ≥ ρ and thus a box R k ⊂ O k ∩ O per k such that |R k | ≥ C(d)
¢ Q k W 2 ≤ C 3 ¢ Q k |∇ W | 2 .
(2.4.13)

Recall that

¢ O k |∇ W | 2 ≤ ∥∇w per ∥ 2 L ∞ (Q) |O k \ O per k | ≤ ∥w per ∥ 2 W 1,∞ (Q) |O k \ O per k |. We thus have ¢ Q k \O k w 2 ≤ C 3 ¢ Q k \O k |∇ w| 2 + C 3 ∥w per ∥ 2 W 1,∞ (Q) |O k \ O per k |. (2.4.14) 2.4. PROOFS CHAPTER 2. POISSON EQUATION Define C 0 := max (2C 2 , C 3 ) and C 1 := C 0 ∥w per ∥ 2 W 1,∞ (Q)   2|A| + k∈Z d |O k \ O per k |   < +∞.
Summing (2.4.14) over k such that |k| ≥ A and adding (2.4.12) to the result, we get

¢ R d \O w 2 ≤ C 0 ¢ R d \O |∇ w| 2 + C 1 ,
which is (2.4.8). The proof of the inequality (2.4.9) is straightforward using (2.4.8).

Using Lemma 2.4.5, we prove the following:

Lemma 2.4.6. Suppose that the sequence Definition 2.4.4). Then, one has the following estimates:

(O k ) k∈Z d satisfies Assumption (A2). Let w ∈ V and denote by W ∈ H 1 (R d ) its extension (see
¢ Γ1 ∂w per ∂n ext W ≤ C + 1 4 ∥∇ W ∥ 2 L 2 (R d ) , (2.4.15)
where C is a constant independent of w,

¢ R d \O g W ≤ ∥ g∥ L 2 (R d ) ∥ W ∥ L 2 (R d ) (2.4.16)
and

¢ O per \O W ≤ |O per \ O| 1 2 ∥ W ∥ L 2 (R d ) .
(2.4.17)

Proof. Fix w ∈ V . Let us first show that W ∈ H 1 (R d ). We clearly have that W ∈ H 1 loc (R d ). We then write ¢ O W 2 + ¢ O |∇ W | 2 = ¢ O\O per |w per | 2 + ¢ O\O per |∇w per | 2 ≤ ∥w per ∥ 2 W 1,∞ (Q) k∈Z d |O k \ O per k |. ( 2
¢ Γ1 ∂w per ∂n ext w ≤ k∈Z d ¢ Γ k 1 ∂w per ∂n ext w ≤ k∈Z d ¢ ∂O per k ∂w per ∂n ext W ≤ ∂w per ∂n ext L ∞ (∂O per 0 ) k∈Z d ¢ ∂O per k | W | ≤ C(w per , O per ) k∈Z d ¢ O per k | W | + ¢ O per k |∇ W | . (2.4.19)
Now, recall that W = 0 in O k ∩ O per k , so that using successively the Cauchy-Schwarz inequality and trace continuity (see [52, Theorem 1, p 272] with p = 2), we have

¢ Γ1 ∂w per ∂n ext W ≤ C k∈Z d |O per k \ O k | 1/2 ∥ W ∥ L 2 (O per k ) + ∥∇ W ∥ L 2 (O per k ) .
We use the inequality ab ≤ D a 2 2 + b 2 2D with D to be chosen later:

¢ Γ1 ∂w per ∂n ext W ≤ CD 2 k∈Z d |O per k \ O k | + C D k∈Z d ∥ W ∥ 2 L 2 (O per k ) + ∥∇ W ∥ 2 L 2 (O per k ) .
(2.4.20)

Thus,

¢ Γ1 ∂w per ∂n ext W ≤ CD 2 k∈Z d |O per k \ O k | + C D ∥ W ∥ 2 L 2 (R d ) + ∥∇ W ∥ 2 L 2 (R d ) .
Lemma 2.4.5 implies

C D ∥ W ∥ 2 L 2 (R d ) + ∥∇ W ∥ 2 L 2 (R d ) ≤ 2CC 0 D ∥∇ W ∥ 2 L 2 (R d ) + CC 1 D .
Choosing D = 8CC 0 yields finally

¢ ∂O per ∂w per ∂n ext W ≤ C k∈Z d |O per k \ O k | + C + 1 4 ∥∇ W ∥ 2 L 2 (R d ) , ( 2
¢ Γ1 ∂w per ∂n ext v ≤ C O per \ O + C∥v∥ 2 H 1 0 (R d \O) .
Thus, the linear form v →

¢ Γ1 ∂w per ∂n ext v is continuous on H 1 0 (R d \ O).
First, we prove below that the minimization space V is not empty: Proof. We want to build a function ϕ ∈ H 1 (R d \ O) satisfying the boundary conditions ϕ = -w per on ∂O. We will first build ϕ on each cell

Q k . Let k ∈ Z d . If O k = ∅, we set ϕ k (x) = 0 in Q k .
We next treat the other cases.

Recall that δ per 0 = dist(O per k , ∂Q k ) and that δ 0 is defined in Lemma 2.5.3 of the Appendix. Set

ε per k := min(2α k , δ per 0 /2) and ε k := min(α k , δ 0 /2)
and note that since

α k -→ |k|→+∞ 0, there exists k 0 such that ∀|k| ≥ k 0 , ε per k = 2α k and ε k = α k . Define U per k (ε per k ) (resp. U k (ε k )) to be the ε per k (resp. ε k ) Minkowski content of ∂O per k (resp. ∂O k ), that is, U per k (ε per k ) := x ∈ R d s.t dist(x, ∂O per k ) < ε per k ⊂ Q k and U k (ε k ) := x ∈ R d s.t dist(x, ∂O k ) < ε k ⊂ Q k . Denote O per,+ k (ε per k ) := O per k ∪ U per k (ε per k ) = x ∈ R d s.t dist(x, O per k ) < ε per k ⊂ Q k and O + k (ε k ) := O k ∪ U k (ε k ) = x ∈ R d s.t dist(x, O k ) < ε k ⊂ Q k . 2.4. PROOFS CHAPTER 2. POISSON EQUATION Now, let χ k ∈ C ∞ c (Q k ) be a cut-off function satisfying      0 ≤ χ k ≤ 1 and χ k ≡ 1 in O k supp(χ k ) ⊂ O + k , supp(∇χ k ) ⊂ U k (ε k ) |∇χ k | ≤ C/ε k . We define ϕ k := -χ k w per . It is clear that ϕ k ∈ H 1 (R d ) and that ϕ k = -w per on ∂O k . One defines ϕ(x) = k∈Z d ϕ k (x) = k∈Z d ϕ k (x)1 Q k (x).
Note that since supp(ϕ k ) ⊂ Q k , all terms but one (which depends on x) vanish in the above sum. Thus ϕ = -w per on ∂O. Our goal is to prove that

ϕ ∈ H 1 (R d \ O) to conclude the proof. By Lemma 2.4.5, it is sufficient to show that ∇ϕ ∈ L 2 (R d \ O). Showing this is equivalent to prove that k∈Z d ∥∇ϕ k ∥ 2 L 2 (U k (ε k )) < +∞.
We are thus left to estimate each term

∥∇ϕ k ∥ L 2 (U (ε k ))
where k ∈ Z d . We study these terms only when |k| ≥ k 0 and k / ∈ K where K is defined in Lemma 2.5.1 of the Appendix (there is only a finite number of terms

k such that k ∈ K and |k| < k 0 ). Let k ∈ Z d such that |k| ≥ k 0 and k / ∈ K that is O k ∩ O per k ̸ = ∅. One has, using Assumption (A2), the inclusions O k ⊂ O + k (α k ) ⊂ O per,+ k (2α k ) and U k (α k ) ⊂ U per k (2α k ). (2.4.22)
We write

¢ U k (α k ) |∇ (χ k w per )| 2 ≤ 2 ¢ U k (α k ) |∇w per | 2 |χ k | 2 + 2 ¢ U k (α k ) |w per | 2 |∇χ k | 2 ≤ 2∥∇w per ∥ 2 L ∞ (U k (α k )) |U k (α k )| + 2∥w per ∥ 2 L ∞ (U k (α k )) C 2 α 2 k |U k (α k )|. Using that ∇w per ∈ L ∞ (R d ), d (U k (α k ), O per k ) ≤ α k and w per = 0 in O per k , we infer ∥w per ∥ L ∞ (U k (α k )) ≤ 2α k ∥∇w per ∥ L ∞ (Q) .
We conclude that

¢ U k (α k ) |∇ϕ k | 2 ≤ C|U k (α k )| + C|U k (α k )|α 2 k /α 2 k ≤ C|U k (α k )|.
Using (2.4.22), this yields

¢ U k (α k ) |∇ϕ k | 2 ≤ C|U per k (2α k )|.
We deduce that for k large enough, Proof. Let ( w n ) n∈N ⊂ V be a minimizing sequence of Problem (2.4.6) which exists by Lemma 2.4.8, that is,

¡ U k (α k ) |∇ϕ k | 2 ≤ 2Cα k (see (2.5.3)). Since (α k ) k∈Z d ∈ ℓ 1 (Z d ), one concludes that ϕ ∈ H 1 (R d \ O).
1 2 ¢ R d \O |∇ w n | 2 + ¢ Γ1 ∂w per ∂n ext w n - ¢ R d \O g w n -→ n→+∞ inf u∈V J(u).
We extend each w n by -w per in the perforations and denote by W n the extension (see Definition 2.4.4).

The sequence 1 2

¢ R d |∇ W n | 2 + ¢ Γ1 ∂w per ∂n ext W n - ¢ R d \O g W n
admits an upper bound independent of n. We first prove that ∥∇ W n ∥ L 2 (R d ) is bounded independently of n. We use Lemma 2.4.5 and Lemma 2.4.6 to bound each term:

¢ Γ1 ∂w per ∂n ext W n ≤ C + 1 4 ∥∇ W n ∥ 2 L 2 (R d ) , ¢ R d \O g W n ≤ C∥ W n ∥ L 2 (R d ) ≤ Lemma 2.4.5 C + C∥∇ W n ∥ L 2 (R d ) ,
where C denotes various constants independent of n. Hence, one gets

C ≥ 1 2 ¢ R d |∇ W n | 2 + ¢ ∂O per ∂w per ∂n ext W n - ¢ R d \O g W n ≥ 1 4 ∥∇ W n ∥ 2 L 2 (R d ) -C∥∇ W n ∥ L 2 (R d ) -C,
and thus

∥∇ W n ∥ 2 L 2 (R d ) ≤ C∥∇ W n ∥ L 2 (R d ) + C. This proves that ∥∇ W n ∥ L 2 (R d ) is bounded independently of n. With Lemma 2.4.5, one deduces that ∥ W n ∥ H 1 (R d )
is also bounded independently of n. Thus, by weak compactness, there exists up to the extraction of a sunsequence a weak limit

W ∈ H 1 (R d ) such that W n -⇀ H 1 W and W n -→ L 2 loc W . Denote w = W |R d \O .
We first show that w ∈ V . The strong convergence of ( W n ) n∈N in L 2 loc and the equality W n = -w per in O imply that W = -w per in O. For the boundary ∂O k , recall that the trace operator T k (see [START_REF] Evans | Partial differential equations[END_REF]

, Theorem 1, p 272] ) is weakly continuous from H 1 (O k ) to L 2 (∂O k ). Thus w| ∂O k = T k W = -T k w per = -w per | ∂O k . Since this is true for all k ∈ Z d , we have proved that w |∂O = -w per . Moreover, w ∈ H 1 (R d \ O).
Thus w ∈ V . We can now pass to the limit n → +∞. Since

w ∈ H 1 (R d \ O) → ¢ |∇w| 2 is convex
and continuous (in the strong norm), it is weakly lower semi-continuous and thus

¢ R d \O |∇ w| 2 ≤ lim inf n→+∞ ¢ R d \O |∇ w n | 2 .
(2.4.23)

By weak

H 1 -convergence, since g ∈ L 2 (R d \ O), ¢ R d \O g w n -→ n→+∞ ¢ R d \O g w. ( 2
2 ¢ R d |∇ w| 2 + ¢ Γ1 ∂w per ∂n ext w - ¢ R d \O g w ≤ inf u∈V J(u).
This finishes the proof of the existence.

To conclude the proof, we prove uniqueness: let w 1 and w 2 be two functions of

H 1 (R d \ O) such that w i := w per + w i , i = 1, 2 solve in the distribution sense (2.2.1). In particular, w 1 -w 2 ∈ H 1 0 (R d \O) satisfies in the distribution sense -∆ ( w 1 -w 2 ) = 0. Thus, ∀v ∈ D(R d \ O), ¢ R d \O ∇( w 1 -w 2 ) • ∇v = 0
By density of the test functions, we get that

∀v ∈ H 1 0 (R d \ O), ¢ R d \O ∇( w 1 -w 2 ) • ∇v = 0
We may choose v = w 1 -w 2 in the previous expression. This gives w 1 -w 2 = Cst. a.e. and finally w 1 -w 2 = 0 by the boundary conditions.

Remark 2.4.10. We could also have applied Lax-Milgram lemma to show that Problem (2.4.3) admits a weak solution. The ingredients are basically the same. Coercivity of the bilinear form is a direct consequence of Lemma 2.3.1 (see (2.3.5)). Continuity is proved using the same method as in the proof of Proposition 2.4.9, when passing to the limit in the minimizing sequence. This approach will be employed to prove the existence of a corrector for Stokes system in Chapter 3.

Proof of the convergence results

H 1 convergence

Proof of Theorem 2.2.2. We first define the first order approximation of u ε . Let g = 1 R d \O . With this choice of g, one has

g = 1 O per \O -1 O\O per = k∈Z d 1 O per k \O k -1 O k \O per k .
Moreover, Lemma 2.5.1 implies that g ∈ L 2 (R d ). Thus we can apply Theorem 2.2.1 and get the existence of a unique function w ∈ H 1 (R d \ O) such that w := w per + w satisfies

-∆w = 1 in R d \ O w |∂O = 0. in the sense of distribution. Note that w ∈ H 1 loc (R d \ O). Now, set ϕ ε := u ε -ε 2 w(•/ε)f. Since f ∈ D(Ω), w = 0 on ∂O and w ∈ H 1 loc (R d \ O), one gets that ϕ ε ∈ H 1 0 (Ω ε ). We have, in the distribution sense, -∆ϕ ε = f + ∆w • ε f + 2ε∇w • ε • ∇f + ε 2 w • ε ∆f = f -f + εg ε = εg ε , (2.4.26)
where

g ε = 2∇w • ε • ∇f + εw • ε ∆f. Note that ∥g ε ∥ L 2 (
¢ Ωε |∇ϕ ε | 2 = ε ¢ Ωε g ε ϕ ε ≤ Cε ¢ Ωε ϕ 2 ε 1/2 .
Thanks to Lemma 2.3.2, one concludes that

¢ Ωε |∇ϕ ε | 2 1/2 ≤ Cε 2 and ¢ Ωε ϕ 2 ε 1/2 ≤ Cε 3 ,
which finishes the proof.

L ∞ convergence

We first prove the following Lemma:

∈ L ∞ R d \ O . Proof. Let us first prove that w ∈ L ∞ (R d \ O). Fix k ∈ Z d and recall that -∆w = 1 in Q k \ O k w |∂O k = 0. ( 2

.4.27)

There exists a constant C independent of k such that

∥w∥ L ∞ (∂Q k ) ≤ C. ( 2
∈ Z d , O k ∪ O per k ⊂ d i=1 [k i -1 2 + δ, k i + 1 2 -δ] d
. By translation invariance and since ∂Q is compact, there exists x 1 , x 2 , ..., x ℓ ∈ ∂Q such that

∀k ∈ Z d , ∂Q k ⊂ ℓ i=1 B(x i + k, δ/2).
(2.4.29)

On each ball B(x i + k, δ), w satisfies -∆ w = 0. De Giorgi-Nash-Moser Theory (see [START_REF] Zhong | De Giorgi-Nash-Moser theory[END_REF], Theorem 4.22, p. 155) implies that there exists a constant C = C(d, δ) independent of x i and k such that sup

B(xi+k,δ/2) | w| ≤ C(d, δ) ¢ B(xi+k,δ) | w(x)| 2 dx 1 2 ≤ C∥ w∥ L 2 (R d \O) . (2.4.30)
The inclusion (2.4.29) together with (2.4.30) proves (2.4.28). We now apply the Maximum principle on w for each domain

Q k \ O k . Let R be such that Q k ⊂ B(k, R). The functions w + (x) := w(x) + |x -k| 2 2d + ∥w∥ L ∞ (∂Q k ) and w -(x) = w(x) + |x -k| 2 -R 2 2d -∥w∥ L ∞ (∂Q k )
are respectively supersolution and subsolution of (2.4.27). Thus, thanks to (2.4.28), ∥w∥

L ∞ (Q k \O k ) is bounded independently of k. Hence w ∈ L ∞ R d \ O .
For ∇w, we use Hölder Regularity results for the first derivatives. First recall that Assumption

(A1) implies that R d \ O is connected. For all x ∈ R d \ O such that dist(x, ∂O) > δ/2,
there exists a ball B x centered at x such that dist(B x , ∂O) = δ/2. Interior estimates (see [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF], Theorem 8.32, p. 210) give the existence of a constant C = C(δ, d) independent of x such that

∥w∥ C 1,γ (Bx) ≤ C ∥w∥ L ∞ (R d \O) + 1 ≤ C.
We have proved that ∇w is bounded at a distance δ/2 of ∂O. For the proof up to the boundary ∂O, we use Corollary 8.36 p. 212 of [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF] with the sets

Ω k = {x s.t dist(x, ∂O k ) < δ} \ O k , Ω ′ k = {x s.t dist(x, ∂O k ) < δ/2} \ O k and T k = ∂O k . We have d ′ = δ/2 which is independent of k and thus ∥w∥ C 1,γ (Ω ′ k ) ≤ C(T k , δ, d) ∥w∥ L ∞ (R d \O) + 1
where the dependence on T k appears through the C 1,γ -norms of the charts that flatten T k (see [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF], p.210). By hypothesis, we get that C(T k ) ≤ C 0 . This concludes the proof.

Proof of Theorem 2.2.3. Fix ε > 0 and define

v ε = u ε (ε•). Then v ε ∈ H 1 0 ( 1 ε Ω ε ) and satisfies        -∆v ε = ε 2 f (ε•) in 1 ε Ω ε v ε = 0 on ∂ 1 ε Ω ε . (2.4.31) Define ψ ε := v ε -ε 2 wf (ε•) ∈ H 1 0 ( 1 ε Ω ε ) and note that -∆ψ ε = ε 3 [2∇w • ∇f (ε•) + εw∆f (ε•)] =: ε 3 h ε .
Lemma 2.4.11 and the fact that

f ∈ D(Ω) imply that ∥h ε ∥ L ∞ ( 1 ε Ωε) ≤ C for all 0 < ε < 1. Define ψ + ε = ψ ε + ε 3 ∥h ε ∥ L ∞ ( 1 ε Ωε) (w + ∥w∥ L ∞ ) . Then ψ + ε is a supersolution of
+ ε ≥ 0 on 1 ε Ω ε . Similarly, ψ - ε = ψ ε -ε 3 ∥h ε ∥ L ∞ ( 1 ε Ωε) (w + ∥w∥ L ∞ )
is a subsolution of (2.4.31) and thus ψ -

ε ≤ 0 on 1 ε Ω ε . Finally, -ε 3 ∥h ε ∥ L ∞ ( 1 ε Ωε) (w + ∥w∥ L ∞ ) ≤ ψ ε ≤ ε 3 ∥h ε ∥ L ∞ ( 1 ε Ωε) (w + ∥w∥ L ∞ ) . The bound ∥h ε ∥ L ∞ ( 1 ε Ωε) ≤ C and Lemma 2.4.11 imply ∥ψ ε ∥ L ∞ ( 1 ε Ωε) ≤ Cε 3 .
Rescaling back concludes the proof.

Appendices

Proof of technical lemmas

where A∆B = (A ∪ B) \ (A ∩ B) = (A \ B) ∪ (B \ A) stands for the symmetric subset difference. Moreover, if K := {k ∈ Z d s.t O k ∩ O per k = ∅}, then |K| < +∞.
Proof. First note that, using (2.1.14),

O k \ O per k ⊂ U per k (α k ) and O per k \ O k ⊂ U per k (α k ). ( 2
∈ Z d , B k := B + k satisfies B k ⊂ B k ⊂ O per k and δ = dist(B k , ∂O per k ). Since (α k ) k∈Z d ∈ ℓ 1 (Z d ), there exists k 0 such that for all |k| ≥ k 0 , α k ≤ δ/2. Equation (2.1.13) implies B k ⊂ O per,- k (α k ) for |k| ≥ k 0 . This proves that ∀|k| ≥ k 0 , B k ⊂ O per k ∩ O k . If |k| < k 0 and O per k ∩ O k ̸ = ∅, there exists a ball B k such that B k ⊂ O per k ∩ O k . Defining ρ = min min |k|<k0 |B k |, |B| > 0 concludes the proof. Lemma 2.5.3. Let (O k ) k∈Z d be

a sequence of open sets satisfying Assumptions (A1)-(A2).

There exists

δ 0 > 0 such that ∀k ∈ Z d , dist(O k , ∂Q k ) ≥ δ 0 . Proof. Recall that for all k ∈ Z d , O per k ⊂⊂ Q k . Thus, by translation invariance, there exists a constant δ per 0 > 0 independent of k such that ∀k ∈ Z d , dist(O per k , ∂Q k ) = δ per 0 .
One has, using Assumption (A2) and in particular the inclusion

O k ⊂ O per,+ k (α k ), δ k := dist(O k , ∂Q k ) ≥ δ per 0 -α k ≥ δ per 0 /2
for k large enough, say |k| ≥ k 0 . Since for all |k| < k 0 , Assumption (A1) gives

δ k = dist(O k , ∂Q k ) > 0,
the Lemma is proved by defining δ 0 := min

δ per 0 2 , min |k|<k0 δ k . Q ′ 1,k O per k O k Q ′ 2,k Q k ∼ δ 0 ∼ δ 0 (

Improved rates of convergence

This section does not appear in the published version [START_REF] Blanc | Homogenization of the Poisson equation in a non periodically perforated domain[END_REF] of this chapter.

The goal of this section is to generalize the convergence Theorems 2.2.2 and 2.2.3 to Sobolev norms W m,q (Ω ε ) for 1 < q < +∞ and m ≥ 0 in the non-periodic setting introduced in Section 2.1.2. This will be made precise in Theorem 2.5.6 below, see also Remark 2.5.11.

We fix two smooth domains

Q ′ 1 and Q ′ 2 such that - 1 2 + δ 0 2 , 1 2 - δ 0 2 d ⊂⊂ Q ′ 1 ⊂⊂ Q ′ 2 ⊂⊂ Q, (2.5.4)
where δ 0 is given by Lemma 2.5.3. In particular,

Q ′ i ⊂ Q, i = 1, 2, d(Q ′ 1 , ∂Q ′ 2 ) > 0 and d(Q ′ 2 , ∂Q) > 0.
We slightly change in this section the definition of the perforated domain. We set

Ω ε := Ω \ k∈Yε εO k , ( 2.5.5)
where

Y ε := {k ∈ Z d , εQ k ⊂ Ω} and (O k ) k∈Z d is a set of non-periodic perforations satisfying (A1)-(A2).
With this definition, the set Ω ε has the same regularity as Ω and the perforations

O k , k ∈ Z d . For k ∈ Z d , we define Q ′ i,k = Q ′ i +k, k ∈ Z d .
We first state and prove Theorem 2.5.4 which is a generalization of [START_REF] Masmoudi | Some uniform elliptic estimates in a porous medium[END_REF]Theorem 2.1] to the present non-periodic setting. Note that the proof of this Theorem is sketched in the periodic case in [START_REF] Masmoudi | Some uniform elliptic estimates in a porous medium[END_REF]. We detail it here, since the non-periodic case introduced in this chapter contains of course the periodic case. Theorem 2.5.4. We fix 1 < q < +∞. Let Ω be a bounded smooth domain and Ω ε be defined by (2.5.5). Let us assume in addition that the perforations O k , k ∈ Z d are uniformly in k of class C 2 in the sense of Assumption (A3). Let f ∈ L q (Ω ε ). There exists a unique function u ε ∈ W 2,q (Ω ε ) solution to (2.1.1). Moreover, there exists a constant C > 0 independent of ε and f such that the following estimate holds true:

D 2 u ε L q (Ωε) + ε -1 ∇u ε L q (Ωε) + ε -2 u ε L q (Ωε) ≤ C f L q (Ωε) .
(2.5.6)

If the perforations O k , k ∈ Z d are in addition uniformly in k of class C m+2 , m ≥ 0 and f ∈ W m,q (Ω ε ), then u ε ∈ W m+2,q (Ω ε )
and there exists a constant C > 0 independent of ε and f such that We introduce

D m+2 u ε L q (Ωε) ≤ C m i=0 ε i-m D i f L q (Ωε) . ( 2
v ε := ε -2 u ε (ε•) and f ε := f (ε•). We immediately check that    -∆v ε = f ε v ε ∈ W 1,q 0 1 ε Ω ε .
For k ∈ Z d , we note that, thanks to (2.5.4) and Lemma 2.5.3 (see also Figure 2.5 left),

Q ′ i,k ∩ ℓ∈Z d O ℓ = O k , i = 1, 2.
Since the perforations O k are uniformly in k regular, we have (see [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]Theorem 9.13] for a constant depending a priori on k and also [107, Appendix B] for a proof that this constant is in fact independent of k in the case of the Stokes system 3 ) that there exists a constant C > 0 independent of k such that for all k ∈ Y ε , the following estimate holds true:

v ε q W 2,q (Q ′ 2,k \O k ) ≤ C q f ε q L q (Q k \O k ) + v ε q L q (Q k \O k ) .
(2.5.8) Summing (2.5.8) over k, we get that

k∈Yε v ε q W 2,q (Q ′ 2,k \O k ) ≤ C q f ε q L q ( 1 ε Ωε) + ∥v ε q L q ( 1 ε Ωε) .
In particular, writing

Ω ′ ε := k∈Yε ε(Q ′ 2,k \ O k ) ⊂⊂ Ω ε , (2.5.9) this proves that v ε q W 2,q ( 1 ε Ω ′ ε ) ≤ C q f ε q L q ( 1 ε Ωε) + ∥v ε q L q ( 1 ε Ωε) .
(2.5.10)

Scaling back (2.5.10) and taking the 1/q-th power provides

D 2 u ε L q (Ω ′ ε ) + ε -1 ∇u ε L q (Ω ′ ε ) + ε -2 u ε L q (Ω ′ ε ) ≤ C f L q (Ωε) + ε -2 ∥u ε L q (Ωε) .
(2.5.11)

We now show that

u ε L q (Ωε) ≤ Cε 2 f L q (Ωε) , (2.5.12)
where C is independent of f and ε. We first assume that q ≥ 2. Multiplying (2.1.1) by

u ε |u ε | q-2 ∈ W 1,q ′ 0
(Ω ε ) and integrating by parts, we obtain that

¢ Ωε ∇u ε • (q -1)|u ε | q-2 ∇u ε = ¢ Ωε f • u ε |u ε | q-2 .
3 The adaptation to Poisson equation is similar.

Thus,

(q -1) 2 q 2 ¢ Ωε ∇ |u ε | q 2 2 = ¢ Ωε f • u ε |u ε | q-2 .
Applying Hölder inequality gives

¢ Ωε ∇ |u ε | q 2 2 ≤ C(q) f L q (Ωε) u ε q/q ′ L q (Ωε) .
We use the L 2 -Poincaré inequality (see Theorem4 2.3.1) applied to

|u ε | q 2 ∈ H 1 0 (Ω ε ) that ¢ Ωε |u ε | q ≤ Cε 2 f L q (Ωε) u ε q/q ′ L q (Ωε) .
Noticing that q -q/q ′ = 1, we conclude that (2.5.12) holds true for q ≥ 2. The case 1 < q < 2 is treated by duality: let f ∈ L q (Ω ε ) and u ε ∈ W 1,q 0 (Ω ε ) solution to (2.1.1). For g ∈ L q ′ (Ω ε ), we introduce z ε ∈ L q ′ (Ω ε ) the solution to (2.1.1) with source term g. We have, using the equations solved by u ε and z ε , and integration by parts, that

¢ Ωε gu ε = ¢ Ωε ∇z ε • ∇u ε = ¢ Ωε ∇u ε • ∇z ε = ¢ Ωε f z ε
By Hölder inequality together with (2.5.12) for q ′ ≥ 2, we get that

∀g ∈ L q ′ (Ω ε ), ¢ Ωε gu ε ≤ z ε L q ′ (Ωε) f L q (Ωε) ≤ Cε 2 g L q ′ (Ωε) ∥f L q (Ωε) .
Taking the supremum over g, we get that

u ε L q (Ω) = sup ¢ Ωε gu ε , g L q ′ (Ωε) = 1 ≤ Cε 2 f L q (Ωε) .
Thus (2.5.12) is satisfied for all 1 < q < +∞ and (2.5.11) gives

D 2 u ε L q (Ω ′ ε ) + ε -1 ∇u ε L q (Ω ′ ε ) + ε -2 u ε L q (Ω ′ ε ) ≤ C f L q (Ωε) .
(2.5.13)

It remains to show (2.5.13) in Ω ε \ Ω ′ ε . We introduce a cut-off function χ ∈ D(Q ′ 2 ) such that χ = 1 in Q ′ 1 . We define χ ε k := χ • -k ε . By Lemma 2.5.3, the support of χ ε k contains strictly εO k . Besides, there exists a constant C = C(d, Q ′ 1 , Q ′ 2 ) > 0 such that |∇χ ε k | ≤ Cε -1 and |D 2 χ ε k | ≤ Cε -2
and the support of the χ ε k , k ∈ Z d are disjoint. We define

χ := 1 - k∈Yε χ ε k and w ε := u ε χ.
We have that

5 χ = 1 in Ω \ k∈Yε εQ ′ 2,k and that χ = 0 around each perforation εO k , k ∈ Y ε . In particular, w ε ∈ W 2,q ∩ W 1,q 0 (Ω). We have that ∆w ε L q (Ω) = -f χ + 2∇u ε • ∇χ + u ε ∆χ L q (Ω) ≤ C ∥f ∥ L q (Ω) + ε -1 ∥∇u ε ∥ L q (Ωε) + ε -2 ∥u ε ∥ L q (Ωε) ,
Using elliptic regularity theory in Ω, see e.g. [60, Theorem 9.13], we get that

D 2 w ε L q (Ω) ≤ C(Ω) ∥f ∥ L q (Ω) + ε -1 ∥∇w ε ∥ L q (Ω) + ε -2 ∥w ε ∥ L q (Ωε) . (2.5.14)
Using the form of the function w ε , we obtain that

D 2 w ε L q (Ω) ≤ C(Ω) ∥f ∥ L q (Ω) + ε -1 ∥∇u ε ∥ L q (Ωε) + ε -2 ∥u ε ∥ L q (Ωε) .
(2.5.15)

Noticing that

w ε = u ε in Ω ε \ Ω ′ ε , we get D 2 u ε L q (Ωε\Ω ′ ε ) ≤ C(Ω) ∥f ∥ L q (Ω) + ε -1 ∥∇u ε ∥ L q (Ωε) + ε -2 ∥u ε ∥ L q (Ωε) .
(2.5.16)

We next apply the following Gagliardo-Nirenberg inequality6

∥∇u ε ∥ L q (Ωε) ≤ C ∥D 2 u ε ∥ 1/2 L q (Ωε) ∥u ε ∥ 1/2 L q (Ωε) + ε -1 ∥u ε ∥ L q (Ωε)
where the constant C is independent of ε. This yields

D 2 u ε L q (Ωε\Ω ′ ε ) ≤ C ∥f ∥ L q (Ω) + ε -1 ∥D 2 u ε ∥ 1/2 L q (Ωε) ∥u ε ∥ 1/2 L q (Ωε) + ε -2 ∥u ε ∥ L q (Ωε) .
(2.5.17) Summing (2.5.13) and (2.5.17) gives

D 2 u ε L q (Ωε) ≤ C ∥f ∥ L q (Ω) + ε -1 ∥D 2 u ε ∥ 1/2 L q (Ωε) ∥u ε ∥ 1/2 L q (Ωε) + ε -2 ∥u ε ∥ L q (Ωε) . (2.5.18) Thus D 2 u ε L q (Ωε) ≤ C ∥f ∥ L q (Ω) + ε -2 ∥u ε ∥ L q (Ωε) ≤ (2.5.12) C∥f ∥ L q (Ωε) ,
where the constant C is independent of ε. This proves (2.5.6). The second part of the Theorem is proved similarly (by induction) using the scaling relations:

ε d/q D m+2 v ε L q ( 1 ε Ωε) = ε m D m+2 u ε L q (Ωε) and ε d/q f ε W m,q ( 1 ε Ωε) ∼ m i=0 ε i D i f L q (Ωε)
and higher elliptic regularity estimates, see [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]Theorem 9.15].

Remark 2.5.5. In Theorem 2.5.4, the proof of the inequality u ε L q (Ωε) ≤ Cε 2 f L q (Ωε) relies on an integration by parts and Poincaré inequality and requires no more regularity assumptions on the perforations than Lipschitz continuity. We can also obtain this estimate when the perforated domain

Ω ε has the form Ω ε = Ω \ ε k∈Z d O k .
Theorem 2.5.6. We fix ε > 0, q ∈ (1, +∞) and we denote by w the solution given by Theorem 2.2.1. We assume that the perforations O k , k ∈ Z d are of class C 2 uniformly in k . Let Ω ε be defined by (2.5.5).

1. Let f ∈ W 2,q (Ω) be compactly supported in Ω and u ε be the solution to (2.1.1). Then there exists a constant C > 0 independent of ε such that for all ε < ε 0 (Ω, f ) small enough,

ε -1 ∥u ε -ε 2 w(•/ε)f L q (Ωε) + u ε -ε 2 w(•/ε)f W 1,q (Ωε) ≤ Cε 2 . (2.5.19) 2. Let f ∈ W 2,q ∩ L ∞ (Ω)
and u ε be the solution to (2.1.1). Then there exists a constant C > 0 independent of ε such that for all ε < ε(Ω, Ω ′ )

ε -1 ∥u ε -ε 2 w(•/ε)f L q (Ωε) + u ε -ε 2 w(•/ε)f W 1,q (Ωε) ≤ Cε 1+1/q .
(2.5.20)

3. Let q ≥ 2, f ∈ W 2,q (Ω), u ε be the solution to (2.1.1) and Ω ′ ⊂⊂ Ω. We write

Ω ′ ε := Ω ′ ∩ Ω ε . Then there exists a constant C > 0 independent of ε such that for all ε < ε 0 (Ω, Ω ′ , O, d), ε -1 ∥u ε -ε 2 w(•/ε)f L q (Ω ′ ε ) + u ε -ε 2 w(•/ε)f W 1,q (Ω ′ ε ) ≤ Cε 2 .
(2.5.21)

4. Let f ∈ W m+2,q (Ω), m ≥ 2, be compactly supported in Ω and u ε be the solution to (2.1.1).

We assume that the perforations O k , k ∈ Z d are uniformly of class C m+2 . Then there exists a constant C independent of ε such that for all ε < ε 0 (Ω, f ),

D m+2 u ε -ε 2 w(•/ε)f ≤ Cε 1-m . (2.5.22)
Proof. We prove the first point. We introduce, as in the proof of Theorem 2.2.2, the function

ϕ ε := u ε -ε 2 w(./ε)f ∈ W 1,q 0 (Ω ε ).
We choose ε > 0 small enough such that

supp(f ) ⊂ k∈Yε εQ k . Since Ω ε ∩ supp(f ) ⊂ k∈Yε ε(Q k \ O k ),
we need to compute ∆ϕ ε in the set

k∈Yε ε Q k \ O k .
We have, as in the proof of Theorem 2.2.2,

-∆ϕ ε = εg ε , where g ε := 2∇w • ε • ∇f + εw • ε ∆f. (2.5.23)
By Lemma 2.4.11, we have that w(./ε) ∈ C 1,γ (R d \ εO). Thus, since f ∈ W 2,q (Ω), we get that g ε ∈ L q (Ω) and

∥g ε ∥ L q (Ω) ≤ ∥w∥ W 1,∞ (R d ) ∥∇f ∥ W 1,q (Ω) ≤ C.
Theorem 2.5.4 allows to conclude the first point. Note that this argument also gives point 4. Indeed, since the perforations are uniformly of class C m+2 , we can prove as in Lemma 2.4.11 that w ∈ W m+1,∞ (R d \ O). Thus, by Leibniz rule, we have that g ε ∈ W m,q (Ω ε ) and

∥D i g ε ∥ L q (Ωε) ≤ Cε -i .
Theorem 2.5.4 and in particular (2.5.7) allow to deduce (2.5.22).

We now prove the second point. The proof is adapted from the case (m, q) = (1, 2) of [START_REF] Le Bris | An MsFEM type approach for perforated domains[END_REF]. We fix a cut-off function

χ ∈ D(Ω) such that the support of χ is included in {x ∈ Ω, d(x, ∂Ω) > √ dε}. Note that the support of χ is included in k∈Yε εQ k . We assume that χ = 1 in x ∈ Ω, d(x, ∂Ω) ≥ 2 √ dε .
We have the bounds

|supp(1 -χ)| ≤ Cε, |supp(∇χ)| ≤ Cε, |∇χ| ≤ Cε -1 and |D 2 χ| ≤ Cε -2 ,
where C is independent of ε. In particular, ∥1 -χ∥ L q (Ωε) ≤ Cε 1/q , ∥∇χ∥ L q (Ωε) ≤ Cε 1/q-1 and ∥D 2 χ∥ L q (Ωε) ≤ Cε 1/q-2 .

(2.5.24)

We have the inequality

u ε -ε 2 w(./ε)f W 1,q (Ωε) ≤ u ε -ε 2 w(./ε)f χ W 1,q (Ωε) + ε 2 w(./ε)(1 -χ)f W 1,q (Ωε) . (2.5.25)
The same inequality is obviously true with the W 1,q -norm replaced by the L q -norm. We note that, since

w ∈ W 1,∞ (R d ) and f ∈ W 1,q ∩ L ∞ (Ω), ε -1 ε 2 w(./ε)(1 -χ)f L q (Ωε) + ε 2 w(./ε)(1 -χ)f W 1,q (Ωε) ≤ (2.5.24)
Cε 1+1/q , (2.5.26)

In (2.5.25), it remains to estimate u ε -ε 2 w(./ε)f χ W 1,q (Ωε) . We adopt the same strategy as in the proof of the first point. We introduce

R ε := u ε -ε 2 w(./ε)f χ ∈ W 1,q 0 (Ω ε ).
The support of f χ is included in

k∈Yε εQ k . Since k∈Yε εQ k ∩ Ω ε ⊂ R d \ εO, we have that R ε ∈ W 2,q ∩ W 1,q 0 (Ω ε ) and -∆R ε L q (Ωε) = f (1 -χ) + 2ε∇w(./ε) • ∇(f χ) + ε 2 w(./ε)∆(f χ) L q (Ωε) ≤ f L ∞ (Ω) 1 -χ L q (Ω) + ε ∇w L ∞ (R d ) ∇f ∥ L q (Ω) + f L ∞ (R d ) ∇χ L q (Ω) + ε 2 w L ∞ (R d ) ∆f L q (Ω) + 2 ∇f L q (Ω) ∇χ L ∞ (Ω) + f L ∞ (R d ) D 2 χ L q (Ω) ≤ (2.5.24)
Cε 1/q .

(2.5.27) Applying Theorem 2.5.4 to R ε , gathering (2.5.26) and (2.5.25) allows to deduce (2.5.20).

We now prove the third point. We fix Ω ′′ and Ω ′′′ such that Ω ′ ⊂⊂ Ω ′′ ⊂⊂ Ω ′′′ ⊂⊂ Ω. Let χ be a cut-off function (independent of ε) such that χ = 1 in Ω ′′ and supp(χ) ⊂ Ω ′′′ . For ε > 0 small enough, we have that supp(χ) ⊂ k∈Yε εQ k . Besides, we have the bounds ∥D j χ∥ L ∞ (Ω) ≤ C, j ≥ 0 where C is independent of ε. We introduce u χ ε and u 1-χ ε , the solutions to

-∆u χ ε = f χ in Ω ε u χ ε = 0 on ∂Ω ε and -∆u 1-χ ε = (1 -χ)f in Ω ε u 1-χ ε = 0 on ∂Ω ε .
We obviously have

u ε = u χ ε + u 1-χ ε . By the first point, we have for ε < ε 0 (Ω, f, Ω ′ ), ε -1 ∥u χ ε -ε 2 w(•/ε)f χ L q (Ωε) + u ε -ε 2 w(•/ε)f χ W 1,q (Ωε) ≤ Cε 2 .
(2.5.28)

We now use Lemma 2.5.7 below to bound for ε > 0 small enough

ε -1 u 1-χ ε -ε 2 w(./ε)f (1 -χ) L q (Ωε∩Ω ′ ) + u 1-χ ε -ε 2 w(./ε)f (1 -χ) W 1,q (Ωε∩Ω ′ ) ≤ Cε 2 (2.5.29)
Gathering together (2.5.28) and (2.5.29) gives (2.5.21).

Lemma 2.5.7. Let q ≥ 2, f ∈ L q (Ω) such that f = 0 in Ω ′′ ⊂⊂ Ω. Let u ε be the solution to (2.1.1).

Then for all Ω ′ ⊂⊂ Ω ′′ , there exist constants c, C, δ > 0 independent of ε, f and u ε such that

ε -1 u ε L q (Ωε∩Ω ′ ) + u ε W 1,q (Ωε∩Ω ′ ) ≤ Cε exp -δε -1 .
(2.5.30)

Note that Lemma 2.5.7 is adapted from [START_REF] Vasilievitch | Homogenization of differential operators and integral functionals[END_REF]Lemma 3.25]. The proof is made for q = 2 there and we extend it to arbitrary q ∈ [2, +∞). The fact that the domain is not periodically perforated does not play a role in this proof.

Proof of Lemma 2.5.7. Let 0 ≤ τ ≤ 1 be a cut-off function such that τ = 1 in Ω ′ 0 with Ω ′ ⊂⊂ Ω ′ 0 ⊂⊂ Ω ′′ and supp(τ ) ⊂ Ω ′′ . We introduce ϕ := (e µτ -1)

u ε |u ε | q-2 ∈ W 1,q ′ 0
(Ω ε ) with µ > 0 to be determined later. Multiplying (2.1.1) by ϕ and integrating by parts yields (q -1)

¢ Ωε |∇u ε | 2 |u ε | q-2 (e µτ -1) + µ ¢ Ωε (∇u ε • ∇τ ) e µτ u ε |u ε | q-2 = ¢ Ωε f (e µτ -1)u ε |u ε | q-2 .
Using that f e µτ = f in Ω, we get that (q -1)

¢ Ωε |∇u ε | 2 |u ε | q-2 e µτ = -µ ¢ Ωε (∇u ε • ∇τ ) e µτ u ε |u ε | q-2 + (q -1) ¢ Ωε |∇u ε | 2 |u ε | q-2 . (2.5.31)
Applying the Cauchy-Schwarz inequality gives

¢ Ωε |∇u ε | 2 |u ε | q-2 e µτ ≤ cµ∥∇τ ∥ L ∞ ¢ Ωε |∇u ε | 2 |u ε | q-2 e µτ 1/2 ¢ Ωε |u ε | q e µτ 1/2 + ¢ Ωε |∇u ε | 2 |u ε | q-2 .
(2.5.32) Thanks to the Poincaré inequality, we have that

¢ Ωε |u ε | q e µτ ≤ Cε 2 ¢ Ωε ∇ |u ε | q 2 e µτ 2 2 ≤ C 0 ε 2 ¢ Ωε |∇u ε | 2 |u ε | q-2 e µτ + C 1 µ 2 ∥∇τ ∥ 2 L ∞ ε 2 ¢ Ωε |u ε | q e µτ , (2.5.33)
where we have used that

∇ |u ε | q 2 2 = q 2 2 ∇u ε |u ε | q-4 2 u ε 2 = q 2 2 |∇u ε | 2 |u ε | q-2 .
(2.5.34)

We now choose

µ := min 1 √ 2C 1 ∥∇τ ∥ L ∞ ε , 1 2c∥∇τ ∥ L ∞ √ C 0 ε . (2.5.35)
From (2.5.33), we get that

1 2 ¢ Ωε |u ε | q e µτ ≤ Cε 2 ¢ Ωε |∇u ε | 2 |u ε | q-2 e µτ . ( 2
¢ Ωε |∇u ε | 2 |u ε | q-2 e µτ ≤ ¢ Ωε |∇u ε | 2 |u ε | q-2 .
Thus,

¢ Ωε∩Ω ′ 0 |∇u ε | 2 |u ε | q-2 ≤ 2e -µ ¢ Ωε |∇u ε | 2 |u ε | q-2 .
Applying Theorem 2.5.4 and in particular (2.5.6), we obtain the bound

¢ Ωε∩Ω ′ 0 |∇u ε | 2 |u ε | q-2 ≤ Ce -µ ε 2q-2 = Cε 2q-2 e -δε -1 . ( 2
¢ Ωε∩Ω ′ 2 |u ε | q ≤ Cε 2 ¢ Ωε∩Ω ′ 0 |∇u ε | 2 |u ε | q-2 ≤ Cε 2 ε 2q-2 e -δε -1 = Cε 2q e -δε -1 ,
where

Ω ′ 2 is chosen such that Ω ′ ⊂⊂ Ω ′ 2 ⊂⊂ Ω ′ 0 .
The Lemma is proved using Remark 2.5.8 (take Ω ′ = Ω ′ 2 there).

Remark 2.5.8. The method of proof of Theorem 2.5.4 shows the following result: let f ∈ L q (Ω ε ) and u ε ∈ W 1,q 0 (Ω ε ) be solution to (2.1.1), 1 < q < +∞. Then for all Ω ′ ⊂⊂ Ω ′ 2 ⊂⊂ Ω, there exists a constant C > 0 independent of f , u ε and ε such that for ε > 0 small enough,

ε -1 ∇u ε L q (Ωε∩Ω ′ ) ≤ C f L q (Ωε∩Ω ′ 2 ) + ε -2 u ε L q (Ωε∩Ω ′ 2 ) .
(2.5.38)

Remark 2.5.9. In Theorem 2.5.4, the estimates (2.5.19), (2.5.20) and (2.5.21) can be proved for ε > ε 0 up to changing the constant C. In contrast, (2.5.22) requires ε to be small because of the perforations that are removed along the boundary.

Remark 2.5.10. Theorem 2.5.6 is obviously true for q = 1 by using Hölder inequality and the statement of Theorem 2.5.6 for q = 1 + δ. It nevertheless requires to assume that f ∈ W 2,1+δ (Ω).

Remark 2.5.11. This remark is to be compared to the discussion after Theorem 2.2.2. Theorem 2. 5.4 shows that the non-periodic corrector w improves the convergence rates as long as we measure the remainder

u ε -ε 2 w(./ε)f in W 1,q -norm, q > d and f ∈ W 2,q ∩ L ∞ (Ω). Indeed, suppose that |f | ≥ α > 0 in Ω ′ ⊂⊂ Ω. Then lim inf ε→0 ε -(2+d/q) ε 2 w(./ε)f L q (Ωε∩Ω ′ ) ≥ α lim inf ε→0 ε -d/q ∥ w(./ε) L q (Ωε∩Ω ′ ) = α w L q (R d ) =: β.
We deduce that

lim inf ε→0 ε -2-d/q u ε -ε 2 w per (./ε)f L q (Ωε∩Ω ′ ) ≥ lim inf ε→0 ε -2-d/q ε 2 w(./ε)f L q (Ωε∩Ω ′ ) -u ε -ε 2 w(./ε)f L q (Ωε∩Ω ′ ) ≥ (2.5.20) β > 0.
(2.5.39) This computation shows that u ε -ε 2 w per (./ε)f L q (Ωε∩Ω ′ ) is of order O ε 2+d/q . Consequently, using the non-periodic corrector w in place of w per allows to improve the convergence rates from O ε 2+d/q to O ε 3 when f ∈ W 2,q ∩ L ∞ (Ω), q > d. This analysis was restricted to f ∈ D(Ω) in Theorem 2.2.3.

CHAPTER 3 HOMOGENIZATION OF THE STOKES SYSTEM IN A NON-PERIODICALLY PERFORATED DOMAIN

This chapter is based on [START_REF] Wolf | Homogenization of the Stokes system in a non periodically perforated domain[END_REF]. We have added Subsection 3.4.3.

In the previous chapter, we have studied the homogenization of the Poisson equation in a class of non periodically perforated domains. In this chapter, we examine the case of the Stokes system. We consider a porous medium in which the characteristic distance between two holes, denoted by ε, is proportional to the characteristic size of the holes. It is well known (see [2], [START_REF] Sanchez-Palencia | Fluid flow in porous media[END_REF] and [START_REF] Tartar | Incompressible fluid flow in a porous medium. Convergence of the homogenization process[END_REF]) that, when the holes are periodically distributed in space, the velocity converges to a limit given by the Darcy's law when the size of the holes tends to zero. We generalize these results to the setting of [START_REF] Blanc | Homogenization of the Poisson equation in a non periodically perforated domain[END_REF]. The non-periodic domains are defined as a local perturbation of a periodic distribution of holes. We obtain classical results of the homogenization theory in perforated domains (existence of correctors and regularity estimates uniform in ε) and we prove H 2 -convergence estimates for particular force fields. This chapter is based on [START_REF] Wolf | Homogenization of the Stokes system in a non periodically perforated domain[END_REF].

Introduction

In this paper, we study the three dimensional Stokes system in a perforated domain for an incompressible fluid with homogeneous Dirichlet boundary conditions:

     -∆u ε + ∇p ε = f in Ω ε , div u ε = 0, u ε = 0 on ∂Ω ε . (3.1.1)
In Equation (3.1.1), Ω ε ⊂ R 3 denotes the perforated domain, the vector valued function f is the force field, the unknowns u ε and p ε refer respectively to the velocity and the pressure of the fluid. The distance between two neighbouring holes is denoted by ε. We assume that the characteristic size of the holes is ε. Our purpose is to understand the limit of (u ε , p ε ) when ε → 0. We construct classical objects of the homogenization theory such as correctors (Theorem 3.2.1) and we give new rates of convergence of u ε to its limit when f is smooth, compactly supported and div(Af ) = 0 where A is the so-called permeability tensor (see Theorem 3.2.3).

To our knowledge, the first paper on the homogenization of the Stokes system in perforated domains is [START_REF] Tartar | Incompressible fluid flow in a porous medium. Convergence of the homogenization process[END_REF]. In this work, Equation (3.1.1) is studied for a periodic distribution of perforations in the macroscopic domain Ω (that is, each cell of a periodic array of size ε contains a perforation). It is in particular proved that (u ε /ε 2 , p ε ) converges in some sense to a couple (u 0 , p 0 ) given by the Darcy's law. This result can be guessed by performing a standard two scale expansion of (u ε , p ε ), see [START_REF] Sanchez-Palencia | Fluid flow in porous media[END_REF].

INTRODUCTION CHAPTER 3. STOKES SYSTEM

Error estimates between u ε and its first order term in ε are proved in [START_REF] Marušić-Paloka | Asymptotic expansion for a flow in a periodic porous medium[END_REF][START_REF] Marusic | An error estimate for correctors in the homogenization of the Stokes and Navier-Stokes equations in a porous medium[END_REF] for particular situations namely the two-dimensional case in [START_REF] Marusic | An error estimate for correctors in the homogenization of the Stokes and Navier-Stokes equations in a porous medium[END_REF] and the case of a periodic macroscopic domain in [START_REF] Marušić-Paloka | Asymptotic expansion for a flow in a periodic porous medium[END_REF]. Sharp error estimates under general assumptions on f have been obtained in [START_REF] Shen | Sharp convergence rates for Darcy's law[END_REF]. The case of boundary layers in an infinite two-dimensional rectangle has been addressed in [START_REF] Jäger | On the flow conditions at the boundary between a porous medium and an impervious solid[END_REF]. The results of [START_REF] Tartar | Incompressible fluid flow in a porous medium. Convergence of the homogenization process[END_REF] have been extended in [2] to porous medium in which both solid and fluid parts are connected. The case of holes that scale differently as ε is examined in [START_REF] Allaire | Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes I. Abstract framework, a volume distribution of holes[END_REF], see also the contributions [START_REF] Jing | A unified homogenization approach for the Dirichlet problem in perforated domains[END_REF][START_REF] Lu | Homogenization of Stokes equations in perforated domains: a unified approach[END_REF]. High order models for the homogenization of the Stokes system have been proposed in [START_REF] Feppon | High order homogenization of the stokes system in a periodic porous medium[END_REF]. Homogenization in periodically perforated domains for non-linear problems has been developed in [START_REF] Ansini | Separation of scales and almost-periodic effects in the asymptotic behaviour of perforated periodic media[END_REF][START_REF] Ansini | Asymptotic analysis of periodically-perforated nonlinear media[END_REF] via variational techniques.

The homogenization in perforated domains beyond the periodic setting has been considered in a deterministic setting in [START_REF] Donato | Convergence of Dirichlet problems for monotone operators in a class of porous media[END_REF], see also [START_REF] Sergueıĺ | Asymptotics at infinity of solutions to the Neumann problem in a sieve-type layer[END_REF] for a problem of homogenization with defects in the scalar case. The case of non-zero boundary conditions for the Stokes system in a non-periodically perforated domain with small holes is presented in [START_REF] Hillairet | On the homogenization of the Stokes problem in a perforated domain[END_REF]. The homogenization in random porous medium has been considered in [START_REF] Giunti | Convergence rates for the homogenization of the Poisson problem in randomly perforated domains[END_REF] for the Poisson equation and in [START_REF] Beliaev | Darcy equation for random porous media[END_REF][START_REF] Giunti | Homogenisation for the Stokes equations in randomly perforated domains under almost minimal assumptions on the size of the holes[END_REF][START_REF] Giunti | Derivation of Darcy's law in randomly punctured domains[END_REF], and the references therein, for the Stokes system. In this paper, we adapt the results of [START_REF] Tartar | Incompressible fluid flow in a porous medium. Convergence of the homogenization process[END_REF] to the setting of [START_REF] Blanc | Homogenization of the Poisson equation in a non periodically perforated domain[END_REF], that is to perforated domains that are defined as a local perturbation of the periodically perforated domain considered in [START_REF] Tartar | Incompressible fluid flow in a porous medium. Convergence of the homogenization process[END_REF]. This framework is inspired by the papers [START_REF] Blanc | A possible homogenization approach for the numerical simulation of periodic microstructures with defects[END_REF][START_REF] Blanc | Local profiles for elliptic problems at different scales: defects in, and interfaces between periodic structures[END_REF][START_REF] Blanc | On correctors for linear elliptic homogenization in the presence of local defects[END_REF] (see [START_REF] Blanc | Homogenization of the Poisson equation in a non periodically perforated domain[END_REF]Remark 1.5]). The purpose of these works is to study the homogenization of elliptic PDEs with coefficients that are periodic and perturbed by a defect which belongs to L r , 1 < r < +∞.

The paper is organized as follows. We recall in subsection 3.1.2 the main results of the homogenization of the Stokes system in the periodic case. We introduce in subsection 3.1.3 the non-periodic setting. We state in Section 3.2 the main results of this paper and we make some remarks. These results are proved in Section 3.3. Some technical Lemmas are given in Appendix 3.4.1. In Appendix 3.4.2, we give more specific geometric assumptions on the non-periodic perforations that allow to obtain the results of Section 3.2.

General notations

The canonical basis of R 3 is denoted e 1 , e 2 , e 3 . We denote the euclidian scalar product between two vectors u and v by u • v. The euclidian distance to a subset A ⊂ R 3 will be written d(•, A). The diameter of A will be denoted by diam(A). If A is a Lipschitz domain, we denote by n the outward normal vector. | • | will be the Lebesgue measure on R 3 .

If A, B are two real matrices, we write A : B := 3 i,j=1 A i,j B i,j . If X is a vector or a matrix, its transpose will be denoted by X T . If A ⊂ R 3 , the complementary set of A will be written A c . We define

Q :=] -1 2 , 1 2 [ 3 and, for k ∈ Z 3 , Q k := 3 j=1 -1 2 + k j , 1 2 + k j 3 = Q + k. If x ∈ R 3 and r > 0,
we denote by B(x, r) the open ball centered in x of radius r.

The gradient operator of a real or vector valued function will be denoted ∇• and the second order derivative of a real or vector valued function will be written D 2 •. The divergence operator will be denoted div • and the scalar or vectorial Laplacian ∆•.

Functional spaces.

If ω is an open subset of R 3 , m ∈ N * and 1 ≤ q ≤ +∞, we denote by L q (ω) the standard Lebesgue spaces and H m (ω), W m,q (ω) the standard Sobolev spaces. We denote by [L q (ω)]

3 ,

[H m (ω)]
3 and [W m,q (ω)] 3 the spaces of vector valued functions whose components are respectively elements of L q (ω), H m (ω) and W m,q (ω). The space L q (ω)/R corresponds to the equivalence classes for the relation ∼ defined by: for all f, g ∈ L q (ω), f ∼ g if and only if f -g is a.e constant in ω. D(ω) will be the set of smooth and compactly supported functions in ω. We denote by C ∞ (ω) (resp. C ∞ (ω)) the set of smooth functions defined on ω (resp. ω).

Review of the periodic case

In this subsection, we recall the results of the homogenization of the Stokes system in periodically perforated domains with large holes. For more details, see [START_REF] Tartar | Incompressible fluid flow in a porous medium. Convergence of the homogenization process[END_REF][START_REF] Sanchez-Palencia | Fluid flow in porous media[END_REF]2]. k . We define some periodic functional spaces that will be used in the sequel. Using the notations of our problem, we set for 1 ≤ q ≤ +∞, L q,per Q\O per 0

:= u ∈ L q loc (R 3 \O per ) s.t. u is Q -periodic and H 1,per Q\O per 0 := u ∈ H 1 loc (R 3 \O per ) s.t. u is Q -periodic and ∂ i u are Q -periodic, i = 1, 2, 3 ,
where the subscript loc means locally in R 3 . The space of H 1 -periodic vector valued functions will be H 1,per Q\O per 0 3

. The space of H 1 -periodic functions that vanish on the perforations is

H 1,per 0 Q\O per 0 := u ∈ H 1,
For f ∈ L 2 (Ω) 3 , there exists a unique couple (u ε , p ε ) ∈ H 1 0 (Ω per ε ) 3 × L 2 (Ω per ε )/R solution of System (3.1.1
). The Poincaré inequality in perforated domains (see e.g. [99, Lemma 1]) and standard energy estimates yield the bound

∥u ε ∥ [L 2 (Ω per ε )] 3 ≤ Cε 2
where C is a constant independent of ε. Thus, after extraction of a subsequence, u ε /ε 2 converges L 2 -weakly to some limit velocity u * . Besides, it can be proved (see [START_REF] Tartar | Incompressible fluid flow in a porous medium. Convergence of the homogenization process[END_REF]Theorem 1]) that the pressure p ε converges L 2 (Ω)/R-strongly to the macroscopic pressure p 0 which is defined up to the addition of a constant. The couple (u * , p 0 ) is determined by the Darcy's law which we recall here We note that for fixed j ∈ {1, 2, 3}, Problem (3.1.4) is well-posed in the space H 1,per 0 [START_REF] Sanchez-Palencia | Fluid flow in porous media[END_REF]). A central point in the proof of the convergence of p ε to p 0 is the construction of an extension of the pressure p ε in the periodic holes. This extension is constructed in [START_REF] Tartar | Incompressible fluid flow in a porous medium. Convergence of the homogenization process[END_REF] by a duality argument.

     div(u * ) = 0 in Ω, u * = A(f -∇p 0 ), u * • n = 0 on ∂Ω.
(Q \ O per 0 ) 3 × L 2,per (Q \ O per 0 )/R (see
The corrector equations (3.1.4) can be guessed by a standard two-scale expansion of u ε and p ε of the form

u ε = u exp 0 x, x ε + εu exp 1 x, x ε + ε 2 u exp 2 x, x ε + ε 3 u exp 3 x, x ε + • • • , p ε = p exp 0 x, x ε + εp exp 1 x, x ε + ε 2 p exp 2 x, x ε + ε 3 p exp 3 x, x ε + • • •
where the functions u exp i (x, •) and p exp i (x, •) are Q-periodic for fixed x ∈ Ω (see [START_REF] Sanchez-Palencia | Fluid flow in porous media[END_REF]Section 7.2]). It can be proved that the function p exp 0 is independent of the microscopic variable and that p exp 0 (x, x ε ) = p 0 (x) for all x ∈ Ω (where p 0 is given by (3.1.2)). Besides, the functions u 0 and u 1 vanish and (we use, as indicated above, the summation convention over repeated indices)

u exp 2 x, x ε = w per j x ε (f j -∂ j p 0 )(x) and p exp 1 x, x ε = p per j x ε (f j -∂ j p 0 )(x).
We define the remainders

R ε := u ε -ε 2 w per j • ε (f j -∂ j p 0 ) and π ε := p ε -p 0 -εp per j • ε (f j -∂ j p 0 ). The strong convergence R ε /ε 2 → 0 in L 2 (Ω per ε )-norm is proved in [3, Theorem 1.3].
An H 1 -quantitative estimate of this convergence is given in [START_REF] Shen | Sharp convergence rates for Darcy's law[END_REF], provided that Ω is of class C 2,α . We will provide a new H 2 -convergence estimate when div(Af ) = 0 and f is compactly supported in Ω (see Theorem 3.2.3 and Remark 3.

below).

In what follows, we extend w per j by zero in the periodic perforations. The pressure p per j is extended by a constant λ j (for example zero) in the perforations.

The non-periodic setting

We fix a periodic set of perforations as described in the previous subsection. We describe the nonperiodic setting (see [START_REF] Blanc | Homogenization of the Poisson equation in a non periodically perforated domain[END_REF] for more details). For k ∈ Z 3 and α > 0, we define (see figure 3.3a)

O per,+ k (α) := {x ∈ Q k , d(x, O per k ) < α}, and O per,- k (α) := {x ∈ O per k , d(x, ∂O per k ) > α}. For all k ∈ Z 3 , we fix an open subset O k of Q k .
We suppose that the sequence (O k ) k∈Z 3 satisfies Assumptions (A1)-(A5) below. We define the non periodic set of perforations by

O := k∈Z 3 O k . 3.1. INTRODUCTION (A1) For all k ∈ Z 3 , we have O k ⊂⊂ Q k and Q k \ O k is connected. (A2) For all k ∈ Z 3 , the perforation O k is Lipschitz continuous.
(A3) There exists a sequence (α k ) k∈Z 3 ∈ ℓ 1 (Z 3 ) such that for all k ∈ Z 3 , α k > 0 and we have the following chain inclusion:

O per,- k (α k ) ⊂ O k ⊂ O per,+ k (α k ).
We refer to figure 3.3a for an illustration of (A3).

The assumptions (A1)-(A2) are analogous to the one made on O per and guarantee connectedness and some regularity on the perforated domain. Assumption (A3) is the geometric assumption that makes precise that (O k ) k∈Z 3 is a perturbation of (O per k ) k∈Z 3 . It is worth mentioning that, in the whole paper, we fix a periodic distribution of holes that is peturbed in the sense of Assumption (A3). This models some local defects that could appear in this periodic setting. In particular, the translation of (O per k ) k∈Z 3 by a fixed vector ℓ 0 ∈ R 3 \ {0} does not satisfy Assumption (A3) but corresponds to another periodic set of perforations. We also point out that Assumption (A3) is linked to the method of proof which strongly uses that the non-periodic perforations are close to a periodic set of holes. We recall (see [22, Lemma A.1 and Lemma A.3]) that Assumptions (A1)-(A3) imply the following facts:

• There exists δ > 0 such that for all k ∈ Z 3 , d(O k , ∂Q k ) ≥ δ. In other words, O k is strictly included in Q k , uniformly with respect to k.

• We have

k∈Z 3 |O k ∆O per k | < +∞ (3.1.5)
where ∆ stands for the sets symmetric difference operator.

Using the first point above, we can introduce two smooth open sets Q ′ and Q ′′ such that (see Figure 3.2)

Q ′ ⊂⊂ Q ⊂⊂ Q ′′ and for all k ∈ Z 3 , (Q ′ + k) ∩ O = (Q ′′ + k) ∩ O = O k . We define, for k ∈ Z 3 , Q ′ k := Q ′ + k and Q ′′ k := Q ′′ + k. (3.1.6)
The sets Q ′ k and Q ′′ k , k ∈ Z 3 will be used several times in the sequel.

Q k Q ′ k O per k O k Q ′′ k Figure 3.2: A cell Q k , k ∈ Z 3 (A4)
This assumption is divided into two sub-assumptions (A4) 0 and (A4) 1 .

(A4) 0 For all 1 < q < +∞, there exists a constant C 0 q > 0 such that for all k ∈ Z 3 , the problem

div v = f in Q k \ O k , v = 0 on ∂ Q k \ O k (3.1.7) 3.1. INTRODUCTION CHAPTER 3. STOKES SYSTEM with f ∈ L q (Q k \ O k ) completed with the compatibility condition ¢ Q k \O k f = 0 (3.1.8) admits a solution v such that v ∈ W 1,q (Q k \O k ) 3 and ∥v∥ [W 1,q (Q k \O k )] 3 ≤ C 0 q ∥f ∥ L q (Q k \O k ) .
(3.1.9) (A4) 1 For all 1 < q < +∞, there exists a constant

C 1 q > 0 such that for all k ∈ Z 3 , Problem (3.1.7) with f ∈ W 1,q 0 (Q k \ O k ) completed with the compatibility condition (3.1.8) admits a solution v such that v ∈ W 2,q 0 (Q k \ O k ) 3 and ∥v∥ [W 2,q (Q k \O k )] 3 ≤ C 1 q ∥f ∥ W 1,q (Q k \O k ) .
(3.1.10) (A5) For all 1 < q < +∞, there exists a constant C q > 0 such that for all Assumptions (A4)-(A5) are the weakest possible given our method of proof. However, they are associated to PDEs and we would like a somewhat more geometric interpretation of these assumptions, in the spirit of (A3). In fact, we may replace (A4)-(A5) by the likely stronger (but geometric) Assumptions (A4)'-(A5)' below.

k ∈ Z 3 , if (v, p) ∈ W 1,q (Q ′′ k \O k ) 3 × L q (Q ′′ k \O k ) is solution to the Stokes problem      -∆v + ∇p = f in Q ′′ k \ O k , div v = 0, v = 0 on ∂O k (3.1.11) with f ∈ L q (Q ′′ k \O k ), then (v, p) ∈ W 2,q (Q k \O k ) 3 × W 1,q (Q k \O k ) and ∥v∥ [W 2,q (Q k \O k )] 3 + ∥p∥ W 1,q (Q k \O k ) ≤ C q ∥f ∥ L q (Q ′′ k \O k ) 3 + ∥v∥ [W 1,q (Q ′′ k \O k )] 3 + ∥p∥ L q (Q ′′ k \O k ) . ( 3
We suppose that there exist r > 0 and M > 0 such that for all k ∈ Z 3 and for all x ∈ ∂O k , there exists ζ x : U x → R where U x ⊂ R 2 , 0 ∈ U x and r x > r such that, after eventually rotating and/or translating the local coordinate system, we have that ζ x (0) = 0 and

Q k \ O k ∩ B(x, r x ) = {(y 1 , y 2 , y 3 ) ∈ B(x, r x ), y 3 > ζ x (y 1 , y 2 ) and (y 1 , y 2 ) ∈ U x }. (3.1.13)
We assume the following uniform regularity properties: In Assumptions (A4)'-(A5)' above, we emphasize that M is independent of k and x.

We prove in Appendix 3.4.2 that Assumptions (A3) and (A4)' imply Assumption (A4) and that Assumptions (A3) and (A5)' imply Assumption (A5). We also note that (A5)' implies (A4)'. • We remove a finite number of perforations;

O per,- k (α) α α O per k O per,+ k (α)
• We make ℓ 1 -translations of the periodic perforations that is we choose a sequence

(δ k ) k∈Z 3 such that δ k ∈ R 3 , k∈Z 3 |δ k | < +∞ and for all k ∈ Z 3 , O k ⊂⊂ Q k and O k = O per k + δ k . Remark 3.1.3. The assumption O k ⊂⊂ Q k is

automatically implied by (A3) except for a finite number of cells. Dropping it would change some technical details but not the results of the paper.

The perforated domain. We assume that Ω is a smooth bounded domain of R 3 . We denote

Y ε := {k ∈ Z 3 , εQ k ⊂ Ω} (3.1.14)
and define (see Figure 3.3b)

Ω ε := Ω \ k∈Yε εO k . (3.1.15)
We notice that Ω ε is a bounded, locally Lipschitz and connected open subset of

R 3 . For f ∈ L 2 (Ω) 3 , there is a unique solution (u ε , p ε ) ∈ H 1 0 (Ω ε ) 3 ×L 2 (Ω ε )/R to the Stokes system      -∆u ε + ∇p ε = f in Ω ε , div u ε = 0, u ε = 0 on ∂Ω ε . (3.1.16)
In the sequel, we study the homogenization of (u ε , p ε ).

Results

The first result concerns the existence of the first order correctors. We can perform a two scale expansion of the form

u ε (x) = ε 2 u exp 2 x, x ε + εu exp 3 x, x ε + • • • , p ε (x) = p exp 0 x, x ε + εp exp 1 x, x ε + • • • (3.2.1)

RESULTS

CHAPTER 3. STOKES SYSTEM to (3.1.16) and find that

p exp 0 x, x ε = p 0 (x), u exp 2 (x, y) = 3 j=1 w j (y)(f j -∂ j p 0 )(x) and p exp 1 (x, y) = 3 j=1 p j (y)(f j -∂ j p 0 )(x), (3.2.2)
where f 1 , f 2 , f 3 denote the components of the vector field f , (w j , p j ) is solution to the following Stokes system for j = 1, 2, 3:

     -∆w j + ∇p j = e j in R 3 \ O, div w j = 0, w j = 0 on ∂O. ( 3
) ∈ H 1 (R 3 \ O) 3 × L 2 loc (R 3 \ O)
. Moreover, we have the following estimate

∥ p j -⟨ p j ⟩∥ L 2 ( 1 ε Ωε) ≤ Cε -1 ,
where C is a constant independent of ε and ⟨ p j ⟩ denotes the mean value of p j on 1 ε Ω ε . Note that the obvious solution w j = 0, p j = e j • x does not satisfy the statement of Theorem 3.2.1. If it were the case, then there would exist

w j ∈ H 1 (R 3 \ O)
3 such that w per j + w j = 0. Necessarily, this implies that w per j is identically zero. However, recalling (3.1.4), this would imply that ∇p per j = e j in Q \ O per 0 . Thus, by connectedness p per j = e j • x in Q \ O per (up to a constant) which is impossible since p per j is periodic. We emphasize that the main point of Theorem 3.2.1 is to provide a solution (w j , p j ) to (3.2.3) where w j can be decomposed as a sum of the periodic corrector w per j (defined in (3.1.4)) and a perturbation w j ∈ H 1 (R 3 \ O) that, formally, vanish at infinity. As pointed out in the introduction of the thesis (see Subsection 1.2.1.3), we can prove that there exists an extension of (u ε , p ε ) to Ω such that

   u ε /ε 2 -⇀ ε u * in L 2 (Ω) p ε -→ ε→0 p 0 in L 2 (Ω)/R,
where (u * , p 0 ) is given by the Darcy's law (3.1.2)-(3.1.3). In particular, the macroscopic behaviour of (u ε , p ε ) remains unchanged. We define

R ε := u ε -ε 2 3 j=1 w j • ε (f j -∂ j p 0 ) and π ε := p ε -p 0 -ε 3 j=1 p j • ε (f j -∂ j p 0 ).
Following the ideas of the proof of [3, Theorem 1.3], we can prove under the assumption

f ∈ W 3,∞ (Ω) 3 that R ε /ε 2 -→ ε→0
0 in the non-periodic setting for the L 2 (Ω) 3 -norm (where it is understood that u ε and w j , j = 1, 2, 3 are extended by zero in the perforations). This fact, though relevant because it makes (3.2.1) rigorous, is not strong enough to justify the construction of the non-periodic correctors (w j , p j ), j = 1, 2, 3. Indeed, if we set

R per ε := u ε -ε 2 3 j=1 w per j • ε (f j -∂ j p 0 ), we notice that R ε = R per ε -ε 2 3 j=1 w j • ε (f j -∂ j p 0 ). Since w j ∈ L 2 (R 3 \ O)
3 , one has for j = 1, 2, 3:

w j • ε (f j -∂ j p 0 ) [L 2 (Ωε)] 3 = ε 3 2 ∥ w j (f j -∂ j p 0 )(ε•)∥ [L 2 ( 1 ε Ωε)] 3 ≤ ε 3 2 ∥ w j ∥ [L 2 (R 3 )] 3 ∥f j -∂ j p 0 ∥ L ∞ (Ω) = Cε 3 2 . Thus R per ε /ε 2 = R ε /ε 2 + O(ε 3/2
). This proves that R per ε /ε 2 -→ ε→0 0 for the L 2 (Ω) 3 -norm. So, using w per j instead of w j does not change the convergence of u ε to its first order asymptotic expansion.

Yet, since w j and p j , j = 1, 2, 3 are the ad hoc correctors for the non-periodic setting, there must be situations highlighting that the approximation of u ε (resp. p ε ) by

ε 2 w j (•/ε) (f j -∂ j p 0 ) (resp. p 0 + εp j (•/ε) (f j -∂ j p 0 ))
. Let f ∈ L 2 (Ω ε ) 3 and (u, p) ∈ H 1 0 (Ω ε ) 3 × L 2 (Ω ε )/R be solution of      -∆u ε + ∇p ε = f in Ω ε , div(u ε ) = 0, u ε = 0 on ∂Ω ε .
(3.2.4)

Then (u ε , p ε ) ∈ H 2 (Ω ε ) 3 × H 1 (Ω ε )/R
and there exists a constant C > 0 such that for any domain

Ω ′′ ⊂⊂ Ω and all ε < ε 0 (Ω ′′ ), ∥D 2 u ε ∥ [L 2 (Ω ′′ ∩Ωε)] 3 + ε -1 ∥∇u ε ∥ [L 2 (Ωε)] 3 + ε -2 ∥u ε ∥ [L 2 (Ωε)] 3 + ∥∇p ε ∥ L 2 (Ω ′′ ∩Ωε) 3 + ∥p ε ∥ L 2 (Ωε)/R ≤ C∥f ∥ [L 2 (Ωε)] 3 .
Furthermore, the couple

(u ε , p ε ) is unique in H 1 (Ω ε ) 3 × L 2 (Ω ε )/R.
Theorem 3.2.3 (Convergence Theorem). Suppose that assumptions (A1)-(A5) are satisfied. Let f ∈ W 3,∞ (Ω) 3 be such that div(Af ) = 0 and f is compactly supported in Ω. There exists a constant

C > 0 such that if Ω ′′ ⊂⊂ Ω and ε < ε 0 (Ω ′′ , Ω, f ), D 2 u ε -ε 2 w j • ε f j [L 2 (Ω ′′ ∩Ωε)] 3 +ε -1 ∇ u ε -ε 2 w j • ε f j [L 2 (Ωε)] 3 + ε -2 u ε -ε 2 w j • ε f j [L 2 (Ωε)] 3 ≤ Cε (3.2.5)
and

∇ p ε -ε p j • ε -λ j ε f j L 2 (Ω ′′ ∩Ωε) + p ε -ε p j • ε -λ j ε f j L 2 (Ωε)/R ≤ Cε, (3.2.6)
where

λ j ε = 1 |Ω ε | ¢ Ωε p j • ε .
Remark 3.2.4. We note that Theorem 3.2.2 and Theorem 3.2.3 are valid in the periodic case (that is in the framework of subsection 3.1.2). This provides a new situation in which quantitative error estimates can be obtained, besides the ones of [START_REF] Marušić-Paloka | Asymptotic expansion for a flow in a periodic porous medium[END_REF][START_REF] Shen | Sharp convergence rates for Darcy's law[END_REF].

Remark 3.2.5. The assumptions div(Af ) = 0 and f compactly supported in Ω make boundary effects disappear. Indeed, it is straightforward to see that in this case ∇p 0 = 0 in Ω (see (3.1.2)). Since f is compactly supported, we have ε 2 w j (•/ε)f j = 0 on ∂Ω, so u ε and its first order expansion coincide on ∂Ω. This explains why the O(ε

• ε / ∈ H 2 (Ω ε ) (unless of course Ω ε = Ω per ε ) for j = 1, 2, 3
. This is due to the normal derivative jumps of w per j (•/ε) along the parts of ε∂O per that are included in Ω ε . This shows that, in the non-periodic case, using the periodic corrector in (3.2.2) does not give the expected convergence rate, contrary to the non-periodic corrector. Remark 3.2.7. Theorem 3.2.2 and Theorem 3.2.3 can be proved up to the boundary of Ω with the same convergence rates when Ω is of class C 2 . The proof is rather technical and will be omitted here. Remark 3.2.8. Theorem 3.2.2 can be proved for the H m -norm, m > 0 in the periodic domain Ω per ε (see [START_REF] Masmoudi | Some uniform elliptic estimates in a porous medium[END_REF]Theorem 4.2]) and in the non-periodic domain Ω ε , provided that we require higher regularity of O k in (A5)' (typically that O k is uniformly with respect to k of class C m+2 , see [START_REF] Galdi | An introduction to the mathematical theory of the Navier-Stokes equations: Steady-state problems[END_REF]Theorem IV.5.1]

): if f ∈ [H m (Ω ε )] 3 , then (u ε , p ε ) ∈ H m+2 (Ω ε ) 3 × H m+1 (Ω ε )/R
and there exists a constant

C independent of ε such that ∥D m+2 u ε ∥ [L 2 (Ω ′′ ∩Ωε)] 3 + ∥D m+1 p ε ∥ L 2 (Ω ′′ ∩Ωε) ≤ C m i=0 1 ε i ∥D m-i f ∥ [L 2 (Ωε)] 3 .
Remark 3.2.9. This paper presents only the three dimensional case. All that follows is true in dimension greater than 3. As for the two dimensional case, Theorem 3.2.1 and Theorem 3.2.2 are valid.

The rest of the paper is devoted to proofs. In Section 3.3.1, we give the proof of Theorem 3.2.2 in both periodic and non periodic perforated domains. We next prove in Section 3.3.2 the existence of the non-periodic correctors. Finally, Section 3.3.3 is devoted to the proof of the convergence Theorem 3.2.3. Some technical Lemmas, especially concerning divergence problems, are postponed to Appendix 3.4.1.

Proofs

Proof of Theorem 3.2.2

We first state the following Poincaré-Friedrichs inequality: Lemma 3.3.1. Suppose that Assumptions (A1) and (A3) are satisfied. There exists a constant C > 0 independent of ε such that for all u ∈ H 1 0 (Ω ε ) 3 , one has

¢ Ωε |u| 2 ≤ Cε 2 ¢ Ωε |∇u| 2 .
Proof. For simplicity, we assume that each cube Q k contains a non-periodic perforation. We recall that Y ε is defined by (3.1.14) and we define Z ε := {k ∈ Z d , εQ k ∩∂Ω ̸ = ∅}. We have the decomposition

Ω ε = k∈Yε ε(Q k \ O k ) ∪ k∈Zε (εQ k ) ∩ Ω . (3.3.1)
Thanks to Assumption (A3) and the proof of [22, Lemma 3.2], we know that there exists a constant C > 0 independent of k and ε such that for all k ∈ Y ε , we have the inequality

¢ ε(Q k \O k ) u 2 ≤ Cε 2 ¢ ε(Q k \O k ) |∇u| 2 . (3.3.2)
We now fix k ∈ Z ε . Thanks to the proof of [99, Lemma 1], there exists a constant C > 0 which is independent of k and ε such that Let (u ε , p ε ) be the solution of (3.1.16). We have by classical energy estimates the following inequalities:

¢ (εQ k )∩Ω u 2 ≤ Cε 2 ¢ (εQ k )∩Ω |∇u| 2 . ( 3
¢ Ωε |∇u ε | 2 1 2 ≤ Cε∥f ∥ [L 2 (Ωε)] 3 and ¢ Ωε |u ε | 2 1 2 ≤ Cε 2 ∥f ∥ [L 2 (Ωε)] 3 (3.3.4)
which will be useful in the proof of Theorem 3.2.2.

Proof of Theorem 3.2.2. In this proof, C will denote various constants independent of ε that can change from one line to another. We fix Ω ′′ ⊂⊂ Ω. We first show the following estimate:

∥D 2 u ε ∥ [L 2 (Ω ′′ ∩Ωε) 3 ] 3×3 +∥∇p ε ∥ [L 2 (Ω ′′ ∩Ωε)] 3 ≤ C ε -1 ∥∇u ε ∥ [L 2 (Ωε)] 3×3 + ε -2 ∥u ε ∥ [L 2 (Ωε)] 3 + ∥f ∥ [L 2 (Ωε)] 3 . (3.3.5) Proof of (3.3.5): we study Problem (3.2.4) on each periodic cell Q k \ O k . Let k ∈ Y ε such that εQ ′′ k ⊂ Ω. We recall that Q ′′ k is introduced in (3.1.6) and we define in Q ′′ k \ O k the functions      U k ε := ε -2 u ε (ε•), P k ε := ε -1 p ε (ε•) -λ k , F k ε := f (ε•), where λ k ∈ R is chosen such that ¢ Q ′′ k \O k P k ε = 0. Then (U k ε , P k ε ) ∈ H 1 Q ′′ k \ O k 3 × L 2 Q ′′ k \ O k and (U k ε , P k ε ) is solution to the following Stokes system      -∆U k ε + ∇P k ε = F k ε in Q ′′ k \ O k , div(U k ε ) = 0, U k ε = 0 on ∂O k . (3.3.6)
By applying Assumption (A5) to System (3.3.6), we get the estimate

U k ε [H 2 (Q k \O k )] 3 + ∥P k ε ∥ H 1 (Q k \O k ) ≤ C U k ε [H 1 (Q ′′ k \O k )] 3 + P k ε L 2 (Q ′′ k \O k ) + F k ε [L 2 (Q ′′ k \O k )] 3 . (3.3.7) Assumption (A4) 0 and [58, Lemma III.3.2] applied with Ω 1 := Q k \ O k and Ω 2 := Q ′′ k \ Q ′ k give a function v ∈ H 1 0 (Q ′′ k \ O k ) 3 such that div(v) = P k ε and ∥v∥ [H 1 (Q ′′ k \O k )] 3 ≤ C∥P k ε ∥ L 2 (Q ′′ k \O k ) , (3.3.8)
where C is independent of k. Thus,

∥P k ε ∥ 2 L 2 (Q ′′ k \O k ) = ⟨∇P k ε , v⟩ H -1 ×H 1 0 (Q ′′ k \O k ) ≤ ∥∇P k ε ∥ [H -1 (Q ′′ k \O k )] 3 ∥v∥ [H 1 (Q ′′ k \O k )]
3 .

(3.3.9)

Gathering together (3.3.8) and (3.3.9) yields

∥P k ε ∥ L 2 (Q ′′ k \O k ) ≤ C∥∇P k ε ∥ [H -1 (Q ′′ k \O k )] 3 . (3.3.10)
The triangle inequality applied to the first equation of (3.3.6) then provides the inequality

∥∇P k ε ∥ [H -1 (Q ′′ k \O k )] 3 ≤ ∥∆U k ε ∥ [H -1 (Q ′′ k \O k )] 3 + ∥F k ε ∥ [H -1 (Q ′′ k \O k )] 3 ≤ ∥∇U k ε ∥ [L 2 (Q ′′ k \O k )] 3×3 + ∥F k ε ∥ [L 2 (Q ′′ k \O k )]
U k ε [H 2 (Q k \O k )] 3 + ∥P k ε ∥ H 1 (Q k \O k ) ≤ C U k ε [H 1 (Q ′′ k \O k )] 3 + F k ε [L 2 (Q ′′ k \O k )] 3 .
In particular, we deduce

D 2 U k ε [L 2 (Q k \O k ) 3 ] 3×3 + ∥∇P k ε ∥ [L 2 (Q k \O k )] 3 ≤ C U k ε [H 1 (Q ′′ k \O k )] 3 + F k ε [L 2 (Q ′′ k \O k )]
3 . (3.3.12)

Scaling back (3.3.12) gives

∥D 2 u ε ∥ [L 2 (εQ k \O k ) 3 ] 3×3 + ∥∇p ε ∥ [L 2 (εQ k \O k )] 3 ≤ C ε -1 ∥∇u ε ∥ [L 2 (εQ ′′ k \O k )] 3×3 + ε -2 ∥u ε ∥ [L 2 (εQ ′′ k \O k )] 3 + ∥f ∥ [L 2 (εQ ′′ k \O k )]
3 .

(3.3.13) Thus,

∥D 2 u ε ∥ 2 [L 2 (εQ k \O k ) 3 ] 3×3 + ∥∇p ε ∥ 2 [L 2 (εQ k \O k )] 3 ≤ C ε -2 ∥∇u ε ∥ 2 [L 2 (εQ ′′ k \O k )] 3×3 + ε -4 ∥u ε ∥ 2 [L 2 (εQ ′′ k \O k )] 3 + ∥f ∥ 2 [L 2 (εQ ′′ k \O k )]
3 .

(

We next sum (3.3.14) over k ∈ Y ε , where

Y ε := {k ∈ Z 3 , εQ ′′ k ⊂ Ω}.
We note that for ε < ε 0 (Ω ′′ ), we have the chain of inclusion

Ω ′′ ∩ Ω ε ⊂ k∈ Yε ε Q k \ O k ⊂ k∈ Yε ε Q ′′ k \ O k ⊂ Ω ε .
We get

∥D 2 u ε ∥ 2 [L 2 (Ω ′′ ∩Ωε) 3 ] 3×3 + ∥∇p ε ∥ 2 [L 2 (Ω ′′ ∩Ωε)] 3 ≤ C ε -2 ∥∇u ε ∥ 2 [L 2 (Ωε)] 3×3 + ε -4 ∥u ε ∥ 2 [L 2 (Ωε)] 3 + ∥f ∥ 2 L 2 (Ωε) .
(3.3.15) Estimate (3.3.5) is proved. We now conclude the proof of Theorem 3.2.2. We have, inserting (3.3.4) in the right hand side of (3.3.5),

∥D 2 u ε ∥ [L 2 (Ω ′′ ∩Ωε) 3 ] 3×3 + ε -1 ∥∇u ε ∥ [L 2 (Ωε)] 3×3 + ε -2 ∥u ε ∥ [L 2 (Ωε)] 3 + ∥∇p ε ∥ [L 2 (Ω ′′ ∩Ωε)] 3 ≤ C∥f ∥ [L 2 (Ωε)] 3 .
It remains to show that

∥p ε ∥ L 2 (Ωε)/R ≤ C∥f ∥ [L 2 (Ωε)] 3 . (3.3.16)
By Lemma 3.4.3 stated in the appendix and the first line of (3.3.6), we get

∥p ε ∥ L 2 (Ωε)/R ≤ Cε -1 ∥∇u ε ∥ [L 2 (Ωε)] 3×3 + C∥f ∥ [H -1 (Ωε)] 3 . We now show that ∥f ∥ [H -1 (Ωε)] 3 ≤ Cε∥f ∥ [L 2 (Ωε)] 3 . (3.3.17)
Indeed, for any ϕ ∈ H 1 0 (Ω ε ) 3 , we write that, using successively Cauchy-Schwarz inequality and

Poincaré inequality (see Lemma 3.3.1),

⟨f, ϕ⟩ = ¢ Ωε f • ϕ ≤ ∥f ∥ [L 2 (Ωε)] 3 ∥ϕ∥ [L 2 (Ωε)] 3 ≤ Cε∥f ∥ [L 2 (Ωε)] 3 ∥∇ϕ∥ [L 2 (Ωε)] 3×3 ≤ Cε∥f ∥ [L 2 (Ωε)] 3 ∥ϕ∥ [H 1 0 (Ωε)] 3
Thus (3.3.17). Finally, we conclude with the use of (3.3.4) that

∥p ε ∥ L 2 (Ωε)/R ≤ Cε -1 ∥∇u ε ∥ [L 2 (Ωε)] 3×3 + C∥f ∥ [L 2 (Ωε)] 3 ≤ C∥f ∥ [L 2 (Ωε)] 3 .
This proves (3.3.16) and concludes the proof of Theorem 3.2.2.

Proof of Theorem 3.2.1

We use the periodic correctors (w per j , p per j ) defined in (3.1.4) and we search w j and p j in the form w j = w per j + w j and p j = p per j + p j . We recall (see the last paragraph of Subsection 3.1.3) that w per j is extended by zero in O per and that p per j is extended by a constant λ j . The Stokes system defining (

w j , p j ) is      -∆ w j + ∇ p j = e j + ∆w per j -∇p per j in R 3 \ O, div w j = 0, w j = -w per j on ∂O. (3.3.18)
The proof consists in applying Lax-Milgram's Lemma to (3.3.18). We first need to prove some preparatory Lemmas. In the sequel, we will use the notation T j := e j + ∆w per j -∇p per j for j ∈ {1, 2, 3}.

Lemma 3.3.2. Suppose that Assumption (A3) is satisfied. For all 1 < q < +∞, we have that 3 , where q ′ = q/(q -1).

T j ∈ W -1,q ′ (R 3 \ O)
Proof. Let ϕ ∈ D(R 3 \ O) 3 . We extend ϕ by 0 in the perforations. We estimate ⟨T j , ϕ⟩ by an integration by parts:

⟨T j , ϕ⟩ = ⟨e j + ∆w per j -∇p per j , ϕ⟩ for i, j ∈ {1, 2, 3}. Thus,

= ¢ R 3 \O e j • ϕ - ¢ R 3 \O ∇w per j : ∇ϕ + ¢ R 3 \O (p per j -λ j )div(ϕ) = ¢ R 3 e j • ϕ - ¢ R 3 ∇w per j : ∇ϕ + ¢ R 3 p per j -λ j div(ϕ) = ¢ R 3 e j • ϕ - ¢ R 3 \O per ∇w per j : ∇ϕ + ¢ R 3 \O per (p per j -λ j )div(ϕ). Since w per j (resp. p per j -λ j ) is of class C 2,α (resp. of class C 1,α ) in R 3 \ O per (
: ∇ϕ = ¢ ∂O per ∂w per j ∂n • ϕ - ¢ R 3 \O per ∆w per j • ϕ, ¢ R 3 \O per (p per j -λ j )div(ϕ) = ¢ ∂O per (p per j -λ j )ϕ • n - ¢ R 3 \O per
⟨T j , ϕ⟩ = ¢ R 3 e j • ϕ + ¢ R 3 \O per ∆w per j -∇p per j • ϕ + ¢ ∂O per (p per j -λ j )ϕ • n - ¢ ∂O per ∂w per j ∂n • ϕ = ¢ R 3 e j • ϕ - ¢ R 3 \O per e j • ϕ + ¢ ∂O per (p per j -λ j )ϕ • n - ¢ ∂O per ∂w per j ∂n • ϕ = ¢ O per \O e j • ϕ + ¢ ∂O per (p per j -λ j )ϕ • n - ¢ ∂O per ∂w per j ∂n • ϕ. = (A) + (B) + (C)
We treat each term separetely.

Term (A)

. By Hölder inequality and Assumption (A3) (more precisely (3.1.5)), we obtain that

¢ O per \O e j • ϕ ≤ O per \ O 1 q ′ ∥ϕ∥ [L q (R 3 \O)] 3 ≤ C ∥ϕ∥ [W 1,q (R 3 \O)] 3 .

Term (B).

We have by standard regularity results (see [58, Theorem IV.7.1]) that p per j ∈ L ∞ (∂O per 0 ). We apply a Trace Theorem W 1,1 (O per 0 ) → L 1 (∂O per 0 ) (see e.g. [52, Theorem 1, p. 258]) that yields a constant C, which is by translation invariance independent of k, such that for all

k ∈ Z 3 , ∥ϕ∥ [L 1 (∂O per k )] 3 ≤ C∥ϕ∥ [W 1,1 (O per k )] 3 . (3.3.19)
By applying (3.3.19) in the second inequality, we get

¢ ∂O per (p per j -λ j )ϕ • n ≤ p per j -λ j L ∞ (∂O per ) ¢ ∂O per |ϕ| = C k∈Z 3 ¢ ∂O per k |ϕ| ≤ C k∈Z 3 ¢ O per k |ϕ| + |∇ϕ| = C ¢ O per \O |ϕ| + |∇ϕ|,
where we used in the last equality that ϕ = 0 in O. Using (3.1.5), we conclude thanks to Hölder inequality that

¢ ∂O per (p per j -λ j )ϕ • n ≤ C O per \ O 1 q ′ ∥ϕ∥ [L q (O per \O)] 3 + ∥∇ϕ∥ [L q (O per \O)] 3×3 ≤ C∥ϕ∥ [W 1,q (R 3 \O)] 3 .

Term (C).

The argument is similar to Term (B). This gives the existence of a constant C > 0 such that:

¢ ∂O per ∂w per j ∂n • ϕ ≤ C∥ϕ∥ [W 1,q (R 3 \O)] 3 ,
where C is independent of ϕ. We conclude that there exists a constant C > 0 such that

∀ϕ ∈ D R 3 \ O 3 , |⟨T j , ϕ⟩| ≤ C∥ϕ∥ [W 1,q (R 3 \O)] 3 .
This proves the Lemma.

Lemma 3.3.3. Suppose that Assumptions (A1) and (A3) are satisfied. For all 1 < q < +∞, there exists a function ϕ j ∈ W 1,q (R 3 ) 3 such that ϕ j = w per j on ∂O. Proof. By Assumption (A3), there exists a sequence (α k )

O per k Q k O k 2α k (a) First case O per k Q k O k 2α k (b) Second case
k∈Z 3 ∈ ℓ 1 (Z 3 ) such that for all k ∈ Z 3 , α k > 0 and {x ∈ O per k , d(x, ∂O per k ) > α k } ⊂ O k ⊂ {x ∈ Q k , d(x, O per k ) < α k }. Let k ∈ Z 3 . If O k ⊂ O per k , then we define the function χ k by χ k (x) = 0 for all x ∈ Q k . If O k ̸ ⊂ O per
k , there are two cases (see Figure 3.4). First case. We have

{x ∈ R 3 , d(x, O per k ) < 2α k } ⊂ Q k .
We consider a function χ k which is smooth and compactly supported such that

χ k = 1 in {x ∈ Q k , d(x, O per k ) < α k }, χ k = 0 in {x ∈ Q k , d(x, O per k ) < 2α k } c .
We can choose χ k such that the following estimates are satisfied:

|χ k | ≤ 1 ; |∇χ k | ≤ C α k and supp(χ k ) ∩ (Q k \O per k ) ≤ Cα k , (3.3.20)
where the constants C are independent of k.

Second case. We have

{x ∈ R 3 , d(x, O per k ) < 2α k } ̸ ⊂ Q k .
We consider a smooth and compactly supported function χ k such that

χ k = 1 in O k , χ k = 0 outside of Q k .
Because α k -→ |k|→+∞ 0 and because there exists δ > 0 such that

∀k ∈ Z 3 , d(O k , ∂Q k ) ≥ δ,
there are only a finite number of such configurations. After possible changes of the constant C, we can suppose that (3.3.20) is valid for all k ∈ Z 3 .

Conclusion. We define

ϕ j := k∈Z 3 χ k w per j ∈ W 1,q loc (R 3 ) 3 .
We study the W 1,q -local norm of ϕ j . We fix k ∈ Z 3 ; one has in Q k :

|∇ϕ j | = ∇ χ k w per j ≤ |∇χ k | w per j + ∇w per j |χ k | .
We now use that ∇w per j is bounded and the inequalities (3.3.20):

|∇ϕ j | ≤ Cα -1 k |w per j | + C.
To obtain that |∇ϕ j | is bounded on its support, it suffices to show a bound of the type

|w per j | ≤ Cα k in {x ∈ Q k , d(x, O per k ) < 2α k }.
Since w per j = 0 on O per k and ∇w per j ∈ L ∞ (Q), this estimate follows from a classical Taylor inequality. We conclude that

∃C > 0, ∀k ∈ Z 3 , ∀x ∈ Q k , |∇ϕ j (x)| ≤ C. Because ∀k ∈ Z 3 , |supp(ϕ j ) ∩ Q k | = supp(χ k ) ∩ Q k \ O per k = O(α k ),
and because of Assumption (A3), we conclude that |supp(ϕ j )| < +∞ and so ∇ϕ j ∈ L q (R 3 ) 3×3 .

Similarly, ϕ j ∈ L q (R 3 ) 3 . This concludes the Lemma.

We define, when R > 0,

Ω R := RΩ \ k s.t. Q k ⊂RΩ O k . If R = 1/ε, one has Ω R = 1 ε Ω ε . Lemma 3.3.4. Let T ∈ H -1 (R 3 \ O) 3 . The Stokes problem      -∆w + ∇p = T in R 3 \ O, div(w) = 0, w = 0 on ∂O (3.3.21) admits a solution (w, p) such that (w, p) ∈ H 1 0 (R 3 \ O) 3 × L 2 loc (R 3 \ O) and ∇p ∈ H -1 (R 3 \O) 3 .
Moreover, for all R > 0, we have the estimate

p -λ R L 2 (Ω R ) ≤ CR ∥∇w∥ [L 2 (R 3 \O)] 3×3 + ∥T ∥ [H -1 (R 3 \O)] 3 , λ R = 1 |Ω R | ¢ Ω R p, (3.3.22)
where C is a constant independent of T and R and p is extended by

1 |RΩ \ O| ¢ RΩ\O p in the perfo- rations.
Proof. We consider the space Using [7, Theorem 2.1], this implies that there exists a distribution p ∈ D ′ (R 3 \O) such that ∆w+T = ∇p. In particular, ∇p ∈ H -1 (R 3 \ O) 3 .

H := {v ∈ H 1 0 (R 3 \ O) 3 , div(v) = 0}.
Let 1 n ∈ N such that Ω ⊂⊂ (2n + 1)Q. We fix R > 0 and we define

Q R := 2 R 2 + 1 (2n + 1) Q \ O = [(2n R + 1) Q] \ O, n R := 2 R 2 n + R 2 + n,
where ⌈•⌉ denotes the ceiling function. We have that RΩ \ O ⊂ Q R and

Q R := |k|≤n R Q k \ O k . Since ∇p ∈ H -1 (R 3 \ O) 3 , we have ∇p ∈ H -1 (Q R ) 3 and ∥∇p∥ [H -1 (Q R )] 3 ≤ ∥∇p∥ [H -1 (R 3 \O)] 3 ≤ ∥∇w∥ [L 2 (R 3 \O)] 3×3 + ∥T ∥ [H -1 (R 3 \O)] 3
owing to the triangle inequality. Since n R = O(R), we have, arguing as in the proofs of Lemma 4.1.2 and Lemma 3.4.2, the existence of C > 0 independent of R such that p -

2 Q R p L 2 (Q R ) ≤ C(2n R + 1) ∥∇p∥ H -1 (Q R ) ≤ CR ∥∇p∥ H -1 (Q R ) .
This gives

p - 2 Q R p L 2 (Q R ) ≤ CR ∥∇w∥ [L 2 (R 3 \O)] 3×3 + ∥T ∥ [H -1 (R 3 \O)] 3 , ( 3.3.24)
where the constant C is independent of R. We now note that Proof of Theorem 3.2.1. We fix j ∈ {1, 2, 3}. Lemma 3.3.3 gives a function ϕ j ∈ H 1 (R 3 ) 3 such that

λ R = 1 |RΩ \ O| ¢ RΩ\O p and that p -λ R L 2 (Ω R ) = p - 2 RΩ\O p L 2 (RΩ\O) ≤ 2 p - 2 Q R p L 2 (RΩ\O) . ( 3
ϕ j = w per j on ∂O. The problem div( v j ) = div(ϕ j ) in R 3 \ O, v j = 0 on ∂O admits a solution v j ∈ H 1 (R 3 \ O)
3 thanks to Lemma 3.4.4. Indeed, we just have to check that

∀k ∈ Z 3 , ¢ ∂O k ϕ j • n = ¢ ∂O k w per j • n = ¢ O k div(w per j ) = 0.
Defining v j := v j -ϕ j yields a solution to the problem

div(v j ) = 0 in R 3 \ O, v j = -
∈ H -1 (R d \ O) 3 , there exists a pair ( v j , p j ) ∈ H 1 0 (R 3 \ O) 3 ×L 2 loc (R 3 \ O) solution of the Problem      -∆ v j + ∇ p j = T j + ∆v j in R 3 \ O, div( v j ) = 0, v j = 0 on ∂O. (3.3.27)
We set w j := v j + v j and p j = p j and we finish the proof of Theorem 3.2.1.

Remark 3.3.5. As for the homogenization of Poisson equation, see [START_REF] Blanc | Homogenization of the Poisson equation in a non periodically perforated domain[END_REF], we can prove the existence of w j using a minimization problem. This reads as finding a solution to

min v∈V 1 2 ¢ R 3 \O ∇v : ∇v -⟨T j , v⟩ ,
where the minimizing space is

V := v ∈ H 1 (R 3 \ O) 3 , div(v) = 0, v |∂O = -w per j .

Proof of Theorem 3.2.3

Strategy of the proof

We introduce

R ε := u ε -ε 2 3 j=1 w j • ε f j and P ε := p ε -ε 3 j=1 p j • ε - 1 | 1 ε Ω ε | ¢ 1 ε Ωε p j f j .
The strategy of the proof is to find a Stokes system satisfied by (R ε , π ε ) and then to apply Theorem 3.2.2. We need to compute the quantities

-∆R ε + ∇P ε and div(R ε ). (3.3.28)
The construction of auxiliary functions is necessary to correct the divergence equation satisfied by R ε , which doesn't have a suitable order in ε. This is done in subsection 3.3.3.2 below (Lemma 3.3.6). The proof of Theorem 3.2.3 is completed in subsection 3.3.3.3, in particular the computations (3.3.28).

Some auxiliary functions

We recall that the correctors w j , j ∈ {1, 2, 3} constructed in Theorem 3.2.1 are extended by zero in the non-periodic perforations. If i ∈ {1, 2, 3}, we denote w i j := w j • e i the i th -component of w j . Similarly, w i,per j (resp. w i j ) will be the i th -component of w per j (resp. w i j ). We recall that the definition of the matrix A is given in Equation (3.1.3).

Lemma 3.3.6. Suppose that Assumption (A4)

1 is satisfied. Let i, j ∈ {1, 2, 3} and χ be a function of class C ∞ with support in Q \ Q ′ such that ¡ Q χ = 1 where Q ′ is defined in (3.1.6
) (see also Figure 3.2). We extend χ by periodicity to R 3 \ O. The problem

-divz i j = w i j -χA i j in R 3 \ O, z i j = 0 on ∂O (3.3.29)
admits a solution z i j ∈ H 2 0,loc (R 3 \ O) 3 . Besides, we have the estimate

∥z i j ∥ H 2 k∈Yε (Qk\Ok) 3 ≤ Cε -3 2 ∥w i,per j ∥ [H 1 (Q)] 3 + Cε -1 ∥ w i j ∥ [H 1 (R 3 )] 3 (3.3.30)
for all ε > 0 where C is a constant independent of ε.

PROOFS

Proof. We fix i, j ∈ {1, 2, 3}. We search z i j under the form z i j = ∇Ψ i j + g i j .

Step 1. We build a function Ψ i j such that ∇Ψ i j ∈ H 2,per (Q) 3 + H 2 loc (R 3 ) 3 and

-∆Ψ i j = w i j -χA i j in R 3 .
The periodic part of Ψ i j is defined by solving the problem

-∆Ψ i,per j = w i,per j -χA i j in Q, Ψ i,per j ∈ H 1,per (Q). (3.3.31) Since ¡ Q w i,per j -χA i j = 0, Problem (3.3.31) is well posed in H 1,per (Q)/R. We choose Ψ i,per j such that ¡ Q Ψ i,per j = 0. Because w i,per j -χA i j ∈ H 1,per (
Q), standard elliptic regularity results state that ∇Ψ i,per j ∈ H 2,per (Q) 3 . Besides, there exists a constant C such that

∥∇Ψ i,per j ∥ [H 2 (Q)] 3 ≤ C∥w i,per j -χA i j ∥ [H 1 (Q)] 3 ≤ C∥w i,per j ∥ [H 1 (Q)] 3 . (3.3.32)
We now build the non-periodic part of Ψ i j . We extend w i j by -w i,per j in O. We note that, with this extension, w i j ∈ H 1 (R 3 ) 3 . We consider the problem

-∆ Ψ i j = w i j in R 3 ,
and its solution given formally by the Green function:

Ψ i j = C 3 1 | • | * R 3 w i j .
Thanks to the remarks after the proof of [61, Theorem 9.9] (see [61, p.235]), we have that

D 2 Ψ i j ∈ H 1 (R 3 )
3×3 and the Calderon-Zygmund estimates:

∥D 2 Ψ i j ∥ [L 2 (R 3 )] 3×3 = ∥ w i j ∥ L 2 (R 3 ) and ∥D 3 Ψ i j ∥ [L 2 (R 3 ) 3 ] 3×3 = ∥∇ w i j ∥ [L 2 (R 3 )] 3 . (3.3.33)
Using the Sobolev injection

• H 1 (R 3 ) → L 6 (R 3 ) for ∇ Ψ i j , we deduce that ∇ Ψ i j ∈ L 6 (R 3 )
3 and, using (3.3.33), that the estimate

∥∇ Ψ i j ∥ [L 6 (R 3 )] 3 ≤ C∥ w i j ∥ L 2 (R 3 )
holds true. In particular, ∇ Ψ i j ∈ L 2 loc (R 3 ) 3 and, thanks to Hölder inequality, we have

∥∇ Ψ i j ∥ [L 2 ( 1 ε Ωε)] 3 ≤ C ε ∥∇ Ψ i j ∥ [L 6 (R 3 )] 3 . We deduce that ∥∇ Ψ i j ∥ [L 2 ( 1 ε Ωε)] 3 ≤ C ε ∥∇ Ψ i j ∥ [L 6 (R 3 )] 3 ≤ C ε ∥ w i j ∥ [L 2 (R 3 )] 3 . ( 3
∥∇ Ψ i j ∥ [H 2 ( 1 ε Ωε)] 3 ≤ C ε ∥ w i j ∥ [H 1 (R 3 )] 3 . (3.3.35)
We define Ψ i j := Ψ i,per j + Ψ i j and verify that

-∆Ψ i j = w i,per j -χA i j + w i j = w i j -χA i j on R 3 .

PROOFS CHAPTER 3. STOKES SYSTEM

We use the periodicity of ∇Ψ i,per j and write that

∥∇Ψ i j ∥ [H 2 ( 1 ε Ωε)] 3 ≤ ∥∇Ψ i,per j ∥ [H 2 ( 1 ε Ωε)] 3 + ∥∇ Ψ i j ∥ [H 2 ( 1 ε Ωε)] 3 ≤ Cε -3 2 ∥∇Ψ i,per j ∥ [H 2 (Q)] 3 + ∥∇ Ψ i j ∥ [H 2 ( 1 ε Ωε)] 3 , (3.3.36)
where the constant C is independent of ε. We make use of (3.3.35) and (3.3.32) and deduce that

∥∇Ψ i j ∥ [H 2 ( 1 ε Ωε)] 3 ≤ Cε -3 2 ∥w i,per j ∥ [H 1 (Q)] 3 + Cε -1 ∥ w i j ∥ [H 1 (R 3 )] 3 . (3.3.37)
Step 2. We introduce a cut-off function χ 1 such that χ 1 = 1 in Q ′ and χ 1 = 0 out of Q (see Figure 3.2). We fix k ∈ Z 3 and define χ k

1 := χ 1 (• -k).
The goal of this step is to solve the following problem:

       div(g i,k j ) = 0 in Q k \ O k , g i,k j = -∇Ψ i j on ∂O k , g i,k j = 0 on ∂Q k . (3.3.38) We first solve div(h i,k j ) = div(χ k 1 ∇Ψ i j ) on Q k \ O k , h i,k j ∈ H 2 0 (Q k \O k ) 3 . (3.3.39)
The compatibility condition (3.1.8) is satisfied:

¢ Q k \O k div(χ k 1 ∇Ψ i j ) = ¢ ∂O k χ k 1 ∇Ψ i j • n + ¢ ∂Q k χ k 1 ∇Ψ i j • n = - ¢ O k ∆Ψ i j = 0. Since div(χ k 1 ∇Ψ i j ) ∈ H 1 0 (Q k \ O k ), we obtain by Assumption (A4) 1 a solution h i,k j ∈ H 2 0 (Q k \ O k ) 3 to (3.3.39) which satisfies the estimate ∥h i,k j ∥ [H 2 (Q k \O k )] 3 ≤ C∥div(χ k 1 ∇Ψ i j )∥ H 1 (Q k \O k ) ≤ C∥∇Ψ i j ∥ [H 2 (Q k \O k )] 3 .
We extend h i,k j by 0 to R 3 \ O. We then define g i,k j := h i,k j -χ k 1 ∇Ψ i j . We note that g i,k j = 0 out of Q k and that g i,k j ∈ H 2 (R 3 \ O) 3 . Besides, g i,k j solves Problem (3.3.38) and satisfies the estimate

∥g i,k j ∥ [H 2 (Q k \O k )] 3 ≤ C∥∇Ψ i j ∥ [H 2 (Q k )] 3 . (3.3.40)
Step 3. We set

g i j (x) := g i,k j (x) if x ∈ Q k . Then we have div(g i j ) = 0 in R 3 \ O, g i j = -∇Ψ i j on ∂O. Besides, g i j ∈ H 2 loc (R 3 \ O) 3 and summing (3.3.40) over k ∈ Y ε yields the estimate ∥g i j ∥ H 2 ( k∈Yε Q k \O k ) 3 ≤ C∥∇Ψ i j ∥ [H 2 ( 1 ε Ωε)] 3 . (3.3.41)
We define z i j := ∇Ψ i j + g i j . We have z i j ∈ H 2 loc (R 3 \O) 3 . Besides, z i j is a solution of (3.3.29) and, collecting (3.3.37) and (3.3.41), we prove the estimate (3.3.30):

∥z i j ∥ H 2 ( k∈Yε Q k \O k ) 3 ≤ C∥∇Ψ i j ∥ [H 2 ( 1 ε Ωε)] 3 ≤ Cε -3 2 ∥w i,per j ∥ [H 1 (Q)] 3 + Cε -1 ∥ w i j ∥ [H 1 (R 3 )] 3 . (3.3.42)
It remains to prove that z i j ∈ H 2 0,loc (R 3 \O) 3 . For that, we fix k ∈ Z 3 and we notice that in a neighbourhood of the perforation ∂O k , the equality

z j i = h i,k j + (1 -χ k 1 )∇Ψ i j = h i,k j is satsified. Since h i,k j ∈ H 2 0 (Q k \O k ) 3 , it proves that z i j ∈ H 2 0,loc (R 3 \O) 3 .
This ends the proof. Proof. We choose ε > 0 small enough such that supp(f ) ⊂ k∈Yε εQ k .

Proof of convergence

We define (see Figure 3.5) Ω ′ := {x ∈ Ω s.t. f (x) ̸ = 0}. We now set

u 1 ε := ε 2 w j • ε f j + ε 3 z i j • ε ∂ i f j and p 1 ε := ε p j • ε -λ j ε f j , λ j ε := 1 | 1 ε Ω ε | ¢ 1 ε Ωε p j .
We have u 1 ε ∈ H 1 0 (Ω ε ) 3 and p 1 ε ∈ L 2 (Ω ε ) and thus

-∆u 1 ε + ∇p 1 ε ∈ H -1 (Ω ε ) 3 .
Since (see Figure 3.5) f = 0 in Ω \ Ω ′ , we have that u 1 ε and p 1 ε are compactly supported in Ω. It is thus sufficient to compute -∆u 1 ε + ∇p 1 ε in Ω ε ∩ Ω ′ . We notice that, for ε > 0 small enough, Ω ε ∩ Ω ′ = Ω ′ \ εO. Besides, thanks to Lemma 3.3.6, we have

z i j (•/ε) ∈ H 2 (Ω ′ \ εO) 3 . We compute in Ω ′ \ εO : ∆u 1 ε = ∆w j • ε f j + 2ε∂ k w j • ε ∂ k f j +ε 2 w j • ε ∆f j + ε∆z i j • ε ∂ i f j + 2ε 2 ∂ k z i j • ε ∂ k ∂ i f j + ε 3 z i j • ε ∆∂ i f j . and ∇p 1 ε = ∇p j • ε f j + ε p j • ε -λ j ε ∇f j .
Thus,

∆u 1 ε -∇p 1 ε = ∆w j • ε f j + 2ε∂ k w j • ε ∂ k f j + ε 2 w j • ε ∆f j + ε∆z i j • ε ∂ i f j + 2ε 2 ∂ k z i j • ε ∂ k ∂ i f j + ε 3 z i j • ε ∆∂ i f j -∇p j • ε f j -ε p j • ε -λ j ε ∇f j = -f j e j + εf ε = -f + εf ε , (3.3.43)

PROOFS CHAPTER 3. STOKES SYSTEM

where

f ε := 2∂ k w j • ε ∂ k f j + εw j • ε ∆f j + ∆z i j • ε ∂ i f j + 2ε∂ k z i j • ε ∂ k ∂ i f j + ε 2 z i j • ε ∆∂ i f j -p j • ε -λ j ε ∇f j .
Equation (3.3.43) is still valid in Ω ε \ Ω ′ (the LHS and RHS vanish). We define

R ε := u ε -u 1 ε and π ε := p ε -p 1 ε . Thus (R ε , π ε ) ∈ H 1 0 (Ω ε ) 3 × L 2 (Ω ε ) and -∆R ε + ∇π ε = εf ε in Ω ε , f ε ∈ L 2 (Ω ε ) 3 .
Using that f ∈ W 3,∞ (Ω) 3 , we infer

∥f ε ∥ [L 2 (Ωε)] 3 ≤ ∇w j • ε [L 2 (Ω ′ \εO)] 3×3 ∥∇f j ∥ [L ∞ (Ω)] 3 + ε w j • ε [L 2 (Ω ′ \εO)] 3 ∥∆f j ∥ L ∞ (Ω) + ∆z i j • ε [L 2 (Ω ′ \εO)] 3 ∥∂ i f j ∥ L ∞ (Ω) + ε ∇z i j • ε [L 2 (Ω ′ \εO)] 3×3 ∥∇∂ i f j ∥ [L ∞ (Ω)] 3 + ε 2 z i j • ε [L 2 (Ω ′ \εO)] 3 ∥∆∂ i f j ∥ L ∞ (Ω) + p j • ε -λ j ε L 2 (Ω ′ \εO) ∥∇f j ∥ [L ∞ (Ω)] 3 . ≤ Cε 3 2 ∥w j ∥ [H 1 ( 1 ε Ω ′ \εO)] 3 + ∥z i j ∥ [H 2 ( 1 ε Ω ′ \εO)] 3 + ∥p j -λ j ε ∥ L 2 ( 1 ε Ω ′ \εO) = Cε 3 2 [(A) + (B) + (C)] .
(3.3.44) We treat each term separetely. For (A), we have

∥w j ∥ [H 1 ( 1 ε Ω ′ \εO)] 3 ≤ ∥w per j ∥ [H 1 ( 1 ε Ω ′ \εO)] 3 + ∥ w j ∥ [H 1 ( 1 ε Ω ′ \εO)] 3 ≤ Cε -3 2 ∥∇w per j ∥ H 1 (Q) + ∥ w j ∥ [H 1 (R 3 \O)] 3 . (3.3.45)
For (B), we apply Lemma 3.3.6 (and especially (3.3.30)):

∥z i j ∥ [H 2 ( 1 ε Ω ′ \εO)] 3 ≤ Cε -3 2 ∥w i,per j ∥ [H 1 (Q)] 3 + Cε -1 ∥ w i j ∥ [H 1 (R 3 )] 3 (3.3.46)
For (C), Theorem 3.2.1 gives

∥p j -λ j ε ∥ L 2 ( 1 ε Ω ′ \εO) ≤ ∥p per j -λ j,per ε ∥ L 2 ( 1 ε Ω ′ \εO) + ∥ p j -λ j ε ∥ L 2 ( 1 ε Ω ′ \εO) ≤ Cε -3 2 ∥p per j ∥ L 2 (Q) + Cε -1 . ( 3
∥f ε ∥ [L 2 (Ωε)] 3 ≤ C.
We now study div(R ε ). Using Lemma 3.3.6, we have in Ω ε :

div(R ε ) = -ε 2 χ • ε A i j ∂ i f j -ε 3 z i j • ε • ∇∂ i f j .
We recall that div(Af ) = A i j ∂ i f j = 0. Thus,

-div(R ε ) = ε 3 z i j • ε • ∇∂ i f j .
We have that ε

3 z i j • ε • ∇∂ i f j ∈ H 1 0 (Ω ε ) 3 and ¡ Ωε ε 3 z i j • ε • ∇∂ i f j = 0. By Lemma 3.4.5 stated in the appendix, there exists S ε ∈ H 2 0 (Ω ε ) 3 such that div(S ε ) = ε 3 z i j • ε • ∇∂ i f j and ∥S ε ∥ H 2 (Ωε) ≤ Cε 2 z i j • ε • ∇∂ i f j H 1 0 (Ωε)
.

Using that f ∈ W 2,∞ (Ω) 3 and Lemma 3.3.6, we get

z i j • ε • ∇∂ i f j [H 1 (Ωε)] 3 ≤ C ε . Thus ∥S ε ∥ H 2 (Ωε) ≤ Cε. (3.3.48) We now define R ε := R ε + S ε . The pair ( R ε , π ε ) ∈ H 1 0 (Ω ε ) 3 × L 2 (Ω ε ) is solution to the following Stokes sytem:        -∆ R ε + ∇π ε = εf ε -∆S ε , div( R ε ) = 0, R ε|∂Ω ε = 0. (3.3.49)
We notice that εf ε -∆S ε ∈ L 2 (Ω ε ) 3 thus we may apply Theorem 3.2.2: for all Ω ′′ ⊂ Ω, we have for ε < ε 0 (Ω ′′ ),

∥D 2 R ε ∥ L 2 (Ω∩Ω ′′ ε ) ≤ C∥εf ε -∆S ε ∥ L 2 (Ωε) ≤ Cε∥f ε ∥ L 2 (Ωε) + ∥S ε ∥ H 2 (Ωε) ≤ Cε, and ∥∇π ε ∥ L 2 (Ω ′′ ∩Ωε) ≤ Cε.
By the triangle inequality and (3.3.48), we conclude that

D 2 u ε -ε 2 w j • ε f j L 2 (Ω ′′ ∩Ωε) ≤ Cε and ∇ p ε -ε p j • ε -λ j ε f j L 2 (Ω ′′ ∩Ωε) ≤ Cε.
Proof. We first extend f by 0 in the perforations. We then solve the problem

   -div(v 1 ) = f in RΩ, v 1 ∈ W 1,q 0 (RΩ) 3 . (3.4.4)
By Lemma [58, Theorem III.3.1] and a simple scaling argument, Problem (3.4.4) admits a solution v 1 such that

∥v 1 ∥ W 1,q (RΩ) ≤ CR∥f ∥ L q (RΩ)
with the constant C being independent of R. For k ∈ Z 3 such that Q k ⊂ RΩ, we consider the problem

     div(v k 2 ) = 0 in Q k \ O k , v k 2 = 0 on ∂Q k , v k 2 = -v 1 on ∂O k . (3.4.5)
The compatibility condition for (3.4.5) is satisfied:

-

¢ ∂O k v 1 • n = ¢ O k div(v 1 ) = - ¢ O k f = 0.
Arguing as for Problem (3.3.38), we show that Problem (3.4.5) admits a solution

v k 2 ∈ W 1,q (Q k \ O k ) 3
such that (the constant C is independent of k thanks to Assumption (A4) 0 ):

∥v k 2 ∥ [W 1,q (Q k \O k )] 3 ≤ C∥v 1 ∥ [W 1,q (Q k \O k )] 3 . (3.4.6)
We extend v k 2 by zero to R 3 \ O. We define the function

v 2 := k,Q k ⊂RΩ v k 2 1 Q k \O k , Summing (3.4.6) over k such that Q k ⊂ RΩ yields ∥v 2 ∥ W 1,q (Ω R ) ≤ C∥v 1 ∥ W 1,q (RΩ) ≤ CR∥f ∥ L q (Q R ) .
We set v = v 1 + v 2 and notice that v satisfies the conclusion of Lemma 4.1.2.

Lemma 3.4.2. Suppose that Assumption (A4)

0 is satisfied. Let 1 < q < +∞ and R > 0. Let f ∈ D ′ (Ω R ) be such that ∇f ∈ W -1,q (Ω R ) 3 . Then f ∈ L q (Ω R )/R and ∥f ∥ L q (Ω R )/R ≤ CR∥∇f ∥ [W -1,q (Ω R )] 3 (3.4.7)
where C is a constant independent of f and R.

Proof. The fact that f ∈ L q (Ω R )/R follows from [7, Lemma 2.7]. We now show the estimate (3.4.7).

For u ∈ L 1 (Ω R ), we denote

λ u := 1 |Ω R | ¡ Ω R u. We prove that there exists a constant C independent of R such that ∥f -λ f ∥ L q (Ω R ) ≤ CR∥∇f ∥ [W -1,q (Ω R )] 3 . (3.4.8)
We argue by duality. We set q ′ = q/(q -1). We fix a function g ∈ L q ′ (Ω R ) and we define g := g -λ g .

We apply Lemma 4.1.2 to g: there exists a function

v g ∈ W 1,q ′ 0 (Ω R ) 3 such that -div(v g ) = g, ∥v g ∥ [W 1,q ′ (Ω R )] 3 ≤ CR∥g∥ L q ′ (Ω R ) . Since ∥g∥ L q ′ (Ω R ) ≤ 2∥g∥ L q ′ (Ω R ) , we have ∥v g ∥ [W 1,q ′ (Ω R )] 3 ≤ CR∥g∥ L q ′ (Ω R )
. We now write :

⟨∇f, v g ⟩ W -1,q ×W 1,q ′ 0 (Ω R ) 3 = - ¢ Ω R (f -λ f )div(v g ) = - ¢ Ω R (f -λ f )(g -λ g ) = - ¢ Ω R (f -λ f )g. Thus ¢ Ω R (f -λ f )g ≤ ∥∇f ∥ [W -1,q (Ω R )] 3 ∥v g ∥ W 1,q ′ 0 (Ω R ) 3 ≤ CR∥∇f ∥ [W -1,q (Ω R )] 3 ∥g∥ L q ′ (Ω R ) .
Taking the supremum over g, we conclude the proof of the Lemma.

Lemma 3.4.3 (Scaling). Suppose that Assumption (A4) 0 is satisfied. Let 1 < q < +∞. Let ε > 0 and Ω ε be defined by (3.1.15). There exists a constant C > 0 independent of ε such that for all

f ∈ D ′ (Ω ε ) such that ∇f ∈ W -1,q (Ω ε ), we have f ∈ L q (Ω ε )/R and the estimate ∥f ∥ L q (Ωε)/R ≤ Cε -1 ∥∇f ∥ [W -1,q (Ωε)] 3 .
Proof. We apply Lemma 3.4.2 with R = 1/ε and use a scaling argument.

Lemma 3.4.4. Suppose that Assumption (A4) 0 is satisfied. Let 1 < q < +∞ and F ∈ W 1,q (R 3 ) 3 .

Suppose that for all k ∈ Z 3 ,

¢ ∂O k F • n = 0. (3.4.9)
The problem

-div(v) = div(F ) in R 3 \ O, v = 0 on ∂O (3.4.10) admits a solution v ∈ W 1,q (R 3 \ O) 3 such that ∥v∥ [W 1,q (R 3 \O)] 3 ≤ C∥F ∥ [W 1,q (R 3 )] 3
where C is a constant independent of F .

Proof. As in the proof of Lemma 3.3.6, we search the function v under the form v = ∇Ψ + v 1 where

-∆Ψ = div(F ) in R 3 , that is, formally, Ψ(x) = C ¢ R 3 F (y) • (x -y) |x -y| 3 dy and div(v 1 ) = 0 in R 3 \ O, v 1 = -∇Ψ on ∂O.
Since F ∈ L q (R 3 ) 3 , we know that ∇Ψ ∈ L q (R 3 ) 3 and that there exists a constant

C > 0 such that ∥∇Ψ∥ [L q (R 3 )] 3 ≤ C∥F ∥ [L q (R 3 )] 3
(see e.g. [58, Exercice II.11.9]). Besides, since div(F ) ∈ L q (R 3 ), the estimate ∥D 2 Ψ∥ [L q (R 3 )] 3×3 ≤ C∥div(F )∥ L q (R 3 ) holds true (see e.g. [61, Theorem 9.9 & p. 235]). Thus, ∇Ψ ∈ W 1,q (R 3 )

3 and ∥∇Ψ∥ [W 1,q (R 3 )] 3 ≤ C∥F ∥ [W 1,q (R 3 )] 3 .
We define the function v 1 on each cell 3 . This proves the Lemma.

Q k \ O k as a solution to      -div(v k 1 ) = 0 in Q k \ O k , v k 1 = 0 on ∂Q k , v k 1 = -∇Ψ on ∂O k . ( 3
∥ [W 1,q (Q k \O k )] 3 ≤ C∥∇Ψ∥ [W 1,q (Q k )]
Lemma 3.4.5. Suppose that Assumption (A4) 1 is satisfied. Let g ∈ H 1 0 (Ω ε ) be such that

¢ Ωε g = 0.
The problem

-div(u) = g in Ω ε , u = 0 on ∂Ω ε (3.4.12) admits a solution u ∈ H 2 0 (Ω ε ) 3 such that ∥u∥ H 2 (Ωε) ≤ C ε ∥g∥ H 1 (Ωε) , (3.4.13)
where the constant C is independent of ε.

Proof. The proof is very similar to the proof of Lemma 3.3.6. We explain here only the main lines and refer to Subsection 3.3.3.2 for details. We first extend g by 0 in the perforations. We notice that g ∈ H 1 0 (Ω) and

¢ Ω g = 0.
We consider the problem We fix a cell Q k such that εQ k ⊂ Ω. We build a function

-div(v) = g in Ω, v = 0 on ∂Ω. ( 3
v k 1 ∈ H 2 (ε Q k \ O k ) 3 such that      div(v k 1 ) = 0 in ε Q k \ O k , v k 1 = -v on ε∂O k , ∇v k 1 = -∇v on ε∂O k . (3.4.16)
For that, we use a cut-off function

χ k ε := χ 1 • ε -k as in Step 2 of the proof of Lemma 3.3.6. We solve div(w k ) = div(χ k ε v) in ε Q k \ O k , w k = 0 on ε∂ Q k \ O k (3.4.17)
and then set

v k 1 := w k -χ k ε v. [
ε 2 ∥D 2 v k 1 ∥ L 2 (εQ k \O k ) + ε∥∇v k 1 ∥ L 2 (εQ k \O k ) + ∥v k 1 ∥ L 2 (εQ k \O k ) ≤ C(ε 2 ∥D 2 v∥ L 2 (εQ k ) + ε∥∇v∥ L 2 (εQ k ) + ∥v∥ L 2 (εQ k ) ), (3.4.18)
where the constant C is independent of k and ε. We extend v k 1 by zero to Ω ε . We define

v 1 := k∈Yε v k 1 .
Then, after summation of (3.4.18) over k, the estimate

ε 2 ∥D 2 v 1 ∥ [L 2 (Ωε) 3 ] 3×3 +ε∥∇v 1 ∥ [L 2 (Ωε)] 3×3 + ∥v 1 ∥ [L 2 (Ωε)] 3 ≤ C ε 2 ∥D 2 v∥ [L 2 (Ω) 3 ] 3×3 + ε∥∇v∥ [L 2 (Ω)] 3×3 + ∥v∥ [L 2 (Ω)] 3 (3.4.19)
holds true. We note that the function u := v+v 1 satisfies the conclusion of Lemma 3.4.5. Furthermore, using (3.4.15) and (3.4.19), we get

ε 2 ∥D 2 u∥ [L 2 (Ωε) 3 ] 3×3 + ε∥∇u∥ [L 2 (Ωε)] 3×3 + ∥u∥ [L 2 (Ωε)] 3 ≤ C ε 2 ∥D 2 v∥ [L 2 (Ω) 3 ] 3×3 + ε∥∇v∥ [L 2 (Ω)] 3×3 + ∥v∥ [L 2 (Ω)] 3 ≤ C ε 2 ∥g∥ H 1 (Ω) + ∥v∥ [H 1 (Ω)] 3 ≤ C ε 2 ∥g∥ H 1 (Ω) + ∥g∥ L 2 (Ω) ≤ C ε 2 ∥g∥ H 1 (Ω) + ε∥g∥ H 1 (Ω) ≤ Cε∥g∥ H 1 (Ω) . (3.4.20)
where we used Lemma 3.3.1 on g in the last inequality. Thus (3.4.13) is proved.

Geometric assumptions

We prove in this section that Assumptions (A3) and (A4)' imply Assumption (A4) and that Assumptions (A3) and (A5)' imply Assumption (A5). Appendix 3.4.2 follows the proofs of [58, Theorem III.3.1] and [58, Theorem IV.5.1] and makes precise the dependance of the constants appearing in these arguments. We begin by a covering Lemma.

Lemma 3.4.6. Suppose that Assumption (A3) is satisfied. Let 0 < ρ < d(∂O per 0 , ∂Q). There exists N ∈ N * such that for all k ∈ Z 3 , there exist 2N balls B k i , i = 1, ..., 2N such that (i) for all i = 1, ..., N , we have that

B k i = B(ξ k i , ρ), ξ k i ∈ ∂O k and {x ∈ Q k \ O k , d(x, ∂O k ) < 3ρ/16} ⊂ N i=1 B k i ;
(ii) for all i = N + 1, ..., 2N , we have that

B k i = B(ξ k i , ρ/32), ξ k i ∈ {x ∈ Q k \ O k , d(x, ∂O k ) > ρ/16} and {x ∈ Q k \ O k , d(x, ∂O k ) ≥ 3ρ/16} ⊂ 2N i=N +1 B k i .
Moreover, there exist 2N balls B 0,per i , i = 1, ..., 2N and η = η(ρ) > 0 such that

(iii) for all i = 1, ..., 2N , B 0,per i ⊂ Q ′′ and x s.t. d x, Q \ O per 0 < η ⊂ 2N i=1 B 0,per i .
(iv) there exists a bijection σ : {1, ..., 2N } → {1, ..., 2N } such that for all i ∈ {1, ..., 2N -1}, we have that We note that there exists ρ > 0 such that for all i ∈ {1, ..., N 0 }, there exist two points y i ∈ B 0,per i ∩O per 0 and z i ∈ B 0,per i \ O per 0 satisfying d(y i , ∂O per 0 ) > ρ and d(z i , ∂O per 0 ) > ρ. We define for each k ∈ Z 3 ,

Ω 0,per σ(i) ∩ 2N s=i+1 Ω 0,per σ(s) ̸ = ∅ and Ω 0,per j := B 0,per j ∩ Q \ O per 0 . (v) for all but a finite number of k ∈ Z 3 , we have that B k,per i ⊂ B k i for all i = 1, ..., 2N and x s.t. d x, Q k \ O k < η/2 ⊂ 2N i=1 B k,per i , where B k,per i := B 0,per i + k.
x k i := x i + k, y k i := y i + k, z k i := z i + k and B k,per i := B 0,per i + k = B(x k i , ρ/2)
. By translation invariance, we obviously have (3.4.21) with 0 replaced by any k ∈ Z 3 .

We consider k ∈ Z 3 such that α k < min(ρ/16, ρ) (where we recall that α k is introduced in (A3)). Then, by Assumption (A3) and (3.4.21), we have that

{x ∈ Q k , d(x, ∂O k ) < 3ρ/16} ⊂ {x ∈ Q k , d(x, ∂O per k ) < ρ/4} ⊂ N0 i=1 B k,per i . ( 3
y k i ∈ B k,per i ∩ O k . Similarly, we have that z k i ∈ B k,per i \ O k . Thus, there exists ξ k i ∈ [y k i , z k i ] ∩ ∂O k , proving that ∂O k ∩ B k,per i ̸ = ∅. We fix an arbitrary point ξ k i ∈ ∂O k ∩ B k,per i and we notice that B k,per i ⊂ B(ξ k i , ρ). By (3.4.22), we conclude that {x ∈ Q k , d(x, ∂O k ) < 3ρ/16} ⊂ N0 i=1 B k i . (3.4.23) It remains to cover {x ∈ Q k \ O k , d(x, ∂O k ) ≥ 3ρ/16}. By (A3), we have that {x ∈ Q k \ O k , d(x, ∂O k ) ≥ 3ρ/16} ⊂ {x ∈ Q k \ O per k , d(x, ∂O per k ) ≥ ρ/8}. (3.4.24)
By compactness and translation invariance, we can cover the right hand side of (3.4.24) by

N 1 balls B k,per i = B(x k i , ρ/32), i = N 0 + 1, ..., N 0 + N 1 where x k i is of the form x k i = x i + k and x i ∈ {x ∈ Q \ O per 0 , d(x, ∂O per 0 ) ≥ ρ/8}. We set B k i := B k,per i and ξ k i := x k i . By (A3), we get that ξ k i ∈ {Q k \ O k , d(x, ∂O k ) > ρ/16}
. With N to be fixed later, we have proved (i)-(ii) for k ∈ Z 3 such that α k < min(ρ/16, ρ).

We fix k ∈ Z 3 such that α k ≥ min(ρ/16, ρ). We take any covering of

{x ∈ Q k , d(x, ∂O k ) < 3ρ/16} with balls B k i = B(ξ k i , ρ), ξ k i ∈ ∂O k and i ∈ {1, ..., N k 0 }. We then take any covering of {x ∈ Q k \ O k , d(x, ∂O k ) ≥ 3ρ/16} with balls B k i = B(ξ k i , ρ/32), ξ k i ∈ {x ∈ Q k \ O k , d(x, ∂O k ) ≥ 3ρ/16} and i ∈ {N k 0 + 1, ..., N k 0 + N k 1 }. We set N := max k∈Z 3 {N k 0 , N k 1 } where N k 0 = N 0 and N k 1 = N 1 if α k < min(ρ/16, ρ).
Note that because of (A3), we have that N < +∞. If N k 0 < N or N k 1 < N , we duplicate one of the balls in order to define 2N balls B k i , i = 1, ..., 2N . We proceed similarly for B 0,per i , i = 1, ..., N 0 + N 1 . Assertions (i), (ii), (iii) and (v) are proved. We prove easily (iv) by connectedness of Q \ O per 0 .

Assumptions (A3) and (A4)' imply (A4)

Proof that (A4) 0 is satisfied

Let k ∈ Z 3 .
We formulate [58, Theorem III.3.1] in our particular setting: suppose that there exists

Ω k i , i = 1, ..., N k such that Q k \ O k = N k i=1 Ω k i , (3.4.25)
where Ω k i is star-shaped with respect to a ball B k i of radius ρ k i such that B k i ⊂⊂ Ω k i . We define for i = 1, ..., N k -1

F k i := Ω k i ∩ N k s=i+1
Ω k s and we assume that

F k i ̸ = ∅ for all i ∈ {1, ..., N k -1}. Then Problem (3.1.7) with f ∈ L q (Q k \ O k ) admits a solution v satisfying (3.1.9) with C 0 q (k) ≤ C(q) diam(Q k \ O k ) min N k i=1 ρ k i 3 1 + diam(Q k \ O k ) min N k i=1 ρ k i 1 + |Q k \ O k | 1-1/q min N k -1 i=1 |F i | 1-1/q N k . (3.4.26)
To bound C 0 q (k) uniformly in k, it is sufficient to show that Q k \ O k has a decomposition of the form (3.4.25) where N k is independent of k, ρ k i and |F k i | are uniformly bounded from below in k and i. We first explain how to find such a decomposition with N k and ρ k i independent of k and i. By making precise the dependance on the geometry of ∂O k at each step of the proof of [START_REF] Galdi | An introduction to the mathematical theory of the Navier-Stokes equations: Steady-state problems[END_REF]Lemma II.1.3], we can show that (A4)' implies that there exists ρ > 0 such that for all k ∈ Z 3 and

ξ k ∈ ∂O k , there exists an open set G ξ k such that Ω ξ k := G ξ k ∩ Q k \ O k is star-shaped with respect to a ball of radius ρ strictly included in Ω ξ k and B(ξ k , ρ) ⊂ G ξ k .
We next apply Lemma 3.4.6 with ρ given before and we denote by B k i , i = 1, ..., 2N the family of balls that we obtain. For i = 1, ..., N , we define

G k i := G ξ k i and Ω k i := Ω ξ k i . For i = N + 1, ..., 2N , we define Ω k i := B k i ∩ Q k \ O k . Since B k i ⊂ G k i for i = 1, ..., N and because B k i , i = 1, ..., 2N covers Q k \ O k , we have that (3.4.25) is satisfied with ρ k i ≥ ρ/32 and N k = 2N .
It remains to check that there exists a relabeling of the Ω k i 's such that we have that min

2N -1 i=1 |F k i | ≥ C
where C > 0 is independent of k. We use Lemma 3.4.6.(iii)-(v). According to Remark 3.4.7, we relabel the Ω k,per i (note that this also implies a relabeling of the

Ω k i 's) such that ∀i ∈ {1, ..., 2N -1}, F k,per i := Ω k,per i ∩ Ω k,per i+1 ∪ • • • ∪ Ω k,per 2N ̸ = ∅.
We then fix ρ ′ > 0 such that for all i ∈ {1, ..., 2N -1}, we have that

F k,per i contains a ball B k,per i ′ of radius ρ ′ such that B k,per i ′ ⊂⊂ Q k \ O per k .
We fix k ∈ Z 3 such that Lemma 3.4.6.(v) is satisfied and such that

α k < 2N -1 min i=1 d B k,per i ′ , ∂O per k .
(3.4.27)

Then, for all i ∈ {1, ..., 2N -1}, we have that

B k,per i ′ ⊂⊂ Q k \ O k . (3.4.28)
We then recall that

F k i = Ω k i ∩ 2N s=i+1 Ω k i = G k i ∩ 2N s=i+1 G k s ∩ Q k \ O k .
By Lemma 3.4.6.(v), we have that B k,per j ⊂ G k j for all j ∈ {1, ..., 2N }. Together with (3.4.28), this yields that

B k,per i ′ ⊂ F k i for all i ∈ {1, ..., 2N -1}. Thus, min 2N -1 i=1 |F k i | ≥ 4 3 πρ ′3 .
Since by (A3) there are only a finite number of indices k such that (3.4.27) is not satisfied, we conclude that, after eventually relabeling these F k i 's, we have that min

2N -1 i=1 |F k i | ≥ C > 0.

Proof that (A4) 1 is satisfied

We briefly sketch the proof of (A4) 1 and we refer to the proof of (A4) 0 for some ingredients. Let

k ∈ Z 3 and f ∈ W 1,q 0 (Q k \ O k ).
To solve Problem (3.1.7), we use a decomposition of the form (3.4.25) with N k uniform in k (= N ) and Ω k i that is star-shaped with respect to a ball of radius ρ uniformly bounded from below in k and i, as constructed in the proof of (A4) 0 . We then write

f = f 1 +• • •+f N where f i ∈ W 1,q 0 (Ω i ), ¡ Ωi f i = 0, ∥f i ∥ W 1,q (Ω k i ) ≤ C k i ∥f ∥ W 1,q (Q k \O k )
and we solve the Problem:

   div v i = f i in Ω k i v i ∈ W 2,q 0 (Ω k i ) 3 .
Thanks to the estimate (III.3.23) of [58, p. 168], we have that

∥v i ∥ [W 2,q (Ωi)] 3 ≤ C(q, ρ)∥f i ∥ W 1,q (Ω k i ) ≤ C(q, ρ)C k i ∥f ∥ W 1,q (Q k \O k ) .
Extending v i by zero to

Q k \ O k and setting v := v 1 + • • • + v N , we have that v solves Problem (3.1.7) with the estimate ∥v∥ [W 2,q (Q k \O k )] 3 ≤ C(q, ρ)N C k i ∥f ∥ W 1,q (Q k \O k ) .
We can conclude that (A4) 1 is satisfied if C k i is uniformly bounded in i and k. To prove that, we make precise the dependance of the constant controlling ∥f i ∥ W 1,q (Ω k i ) in the proof of [START_REF] Galdi | An introduction to the mathematical theory of the Navier-Stokes equations: Steady-state problems[END_REF]]. This constant depends on N and on the maximum of the W 1,∞ -norms of the functions Ψ k i , i = 1, ..., 2N and

χ k i , i = 1, ..., 2N -1 where {Ψ k 1 , ..., Ψ k 2N } is a partition of unity associated to {G k 1 , ..., G k 2N } and χ k i ∈ D(F k i ) satisfies ¡ F k i χ k i = 1.
Because of Lemma 3.4.6.(v), the family {Ψ k 1 , ..., Ψ k 2N } may be chosen independently of k (by using the periodic balls), except for a finite number of indices k. Besides, still after the exclusion of a finite number of indices k, we have shown in the proof of (A4) 0 that F k i contains a ball of radius ρ ′ which is uniformly bounded in i and k. Thus, χ k i may be chosen as the translation of a reference function χ satisfying χ ∈ D(B(0, ρ ′ )) and

¡ B(0,ρ ′ ) χ = 1. This proves that max 2N i=1 C k i ≤ C for all but a finite number of k ∈ Z 3 . Applying [58, Lemma III.3.4] for the remaining indices k, we conclude that max 2N i=1 C k i ≤ C for all k ∈ Z 3
. This concludes the proof of (A4) 1 .

Assumptions (A3) and (A5)' imply (A5)

We fix f ∈ L q (Q ′′ k \ O k ) 3 and we consider the pair (v, p) solution to (3.1.11). We want to prove the regularity estimate (3.1.12). The interior regularity property is given by the following result (see [58, Theorem IV.4.1]):

∥D 2 v∥ L q (Ω k ) 3×3×3 + ∥∇p∥ L q (Ω k ) 3 ≤ C ∥v∥ W 1,q (Ω ′ k ) 3 + ∥p∥ L q (Ω ′ k ) + ∥f ∥ L q (Ω ′ k ) 3 , (3.4.29)
where

Ω k ⊂⊂ Ω ′ k ⊂⊂ Q ′′ k \
O k and C depends only on q and on the distance between Ω k and (Ω ′ k ) c . The regularity up to the boundary follows from the discussion [58, pp.271-274]. By tracing the dependance of the constants in these arguments, we can show that, under Assumption (A5)', there exist a radius ρ > 0, a constant d > 1 and a constant C > 0 such that dρ < d(Q, ∂Q ′′ ) and for all k ∈ Z 3 and x ∈ ∂O k , we have that

∥D 2 v∥ L q ((Qk\Ok)∩B(x,ρ)) 3×3×3 + ∥∇p∥ L q ((Qk\Ok)∩B(x,ρ)) 3 ≤ C ∥v∥ W 1,q ((Qk\Ok)∩B(x,dρ)) 3 + ∥p∥ L q ((Qk\Ok)∩B(x,dρ)) + ∥f ∥ L q ((Qk\Ok)∩B(x,dρ)) 3 .
(3.4.30)

We combine estimates (3.4.29) and (3.4.30). We fix k ∈ Z 3 . Let B k i i=1,...,2N be the family of balls given by Lemma 3.4.6 (applied with ρ defined by (3.4.30)). Thanks to (3.4.30) and the inequality

∀ a 1 , ..., a p > 0, a q 1 + • • • + a q p ≤ (a 1 + • • • + a p ) q ≤ C p,q (a q 1 + • • • + a q p ), (3.4.31)
we have for all i ∈ {1, ..., N },

∥D 2 v∥ q L q ((Qk\Ok)∩B(ξ k i ,ρ)) 3×3×3 + ∥∇p∥ q L q ((Qk\Ok)∩B(ξ k i ,ρ)) 3 ≤ C ∥v∥ q W 1,q ((Qk\Ok)∩B(ξ k i ,dρ)) 3 + ∥p∥ q L q ((Qk\Ok)∩B(ξ k i ,dρ)) + ∥f ∥ q L q ((Qk\Ok)∩B(ξ k i ,dρ))
3 .

(3.4.32) Summing (3.4.32) over i ∈ {1, ..., N } and using that

U k := {x ∈ Q k \ O k , d(x, ∂O k ) < 3ρ/16} ⊂ N i=1 B(ξ k i , ρ) and Q k \ O k ∩ B(ξ k i , dρ) ⊂ Q ′′ k \ O k yield ∥D 2 v∥ q L q (U k ) 3×3×3 + ∥∇p∥ q L q (U k ) 3 ≤ CN ∥v∥ q W 1,q (Q ′′ k \O k ) 3 + ∥p∥ q L q (Q ′′ k \O k ) + ∥f ∥ q L q (Q ′′ k \O k ) 3 .
(3.4.33) We now apply (3.4.29) to

Ω k = {x ∈ Q k \ O k , d(x, ∂O k ) > ρ/8} and Ω ′ k = Q ′′ k \ O k . We have that d Ω k , (Ω ′ k ) c = min d(Q, ∂Q ′′ )
, ρ/8 is independent of k. Thus, using (3.4.29) and (3.4.31) yield

∥D 2 v∥ q L q (Ω k ) 3×3×3 + ∥∇p∥ q L q (Ω k ) 3 ≤ C ∥v∥ q W 1,q (Q ′′ k \O k ) 3 + ∥p∥ q L q (Q ′′ k \O k ) + ∥f ∥ q L q (Q ′′ k \O k ) 3 , (3.4.34)
where C is independent of k. Summing (3.4.32) and (3.4.34) and using that (3.4.31) proves that (A5) is satisfied.

U k ∪ Ω k = Q k \ O k together with

An additional result on the correctors

This section does not appear in the published version [START_REF] Wolf | Homogenization of the Stokes system in a non periodically perforated domain[END_REF].

In this section, we prove Theorem 3.4.8 below. This is the first step in obtaining improved convegence rates i.e. in W 1,q -norm for q > 2 for the homogenization of the Stokes system. Theorem 3.4.8 (L ∞ -bounds on the correctors). Assume that the perforations

(O k ) k∈Z 3 are of class C 2,α uniformly in k. Then (w j , p j ) ∈ W 1,∞ × L ∞ (R 3 \ O) for j = 1, 2, 3.
Note that the above Theorem gives the sublinearity of the correctors. In particular, we have that ∥εp j (./ε)∥ L q (Ωε)/R ∼ ε for any q ∈ (1, +∞). This was restricted to q = 2 in Theorem 3.2.1.

We give the proof of Theorem 3.4.8 in the case of compactly supported perturbations. We indicate in the following where we use this assumption and why the argument is still valid in the case of general perturbations.

Proof of Theorem 3.4.8. We fix j ∈ {1, 2, 3}. We extend w j by -w per j in O. We have in particular that w j ∈ H 1 (R 3 ). We show that there exists a constant C > 0 independent of k ∈ Z 3 such that for all k ∈ Z 3 , we have

∥w j ∥ W 1,∞ (Q k ) ≤ C and ∥p j ∥ L ∞ (Q k ) ≤ C.
We first prove that the velocity is bounded. We recall that [START_REF] Galdi | An introduction to the mathematical theory of the Navier-Stokes equations: Steady-state problems[END_REF]Theorem IV.4.1 and Remark IV.4.1], we have that for all m ≥ 0,

-∆ w j + ∇ p j = 0 div w j = 0 in Q ′′ k \ Q ′ k for some domains Q ′ k = Q ′ + k and Q ′′ k = Q ′′ + k such that Q ′ ⊂⊂ Q ⊂⊂ Q ′′ . We fix in what follows a smooth domain Q such that Q ⊂⊂ Q ⊂⊂ Q ′′ . We also define U := Q \ Q, where Q ′ ⊂⊂ Q ⊂⊂ Q. For k ∈ Z 3 , we define Q k := Q+k and U k := U +k and we note that U k ⊂⊂ Q ′′ k \Q ′ k . By
∥ w j ∥ H m+2 (U k ) ≤ C∥ w j ∥ H 1 (Q ′′ k \Q ′ k )
where the constant C is independent of k by translation invariance. Choosing m > 3/2 and applying the injection

H m+2 (U k ) → C 2,α (U k ) for some α > 0 gives that ∥ w j ∥ C 2,α (U k ) ≤ C∥ w j ∥ H 1 (Q ′′ k \Q ′ k )
, where the constant C is, here again, independent of k by translation invariance. In particular, since

∂ Q k ⊂ U k , we have that ∥ w j ∥ C 2,α (∂ Q k ) ≤ C∥ w j ∥ H 1 (Q ′′ k \Q ′ k ) ≤ C∥ w j ∥ H 1 (R 3 \O) ≤ C. This proves that ∥w j ∥ C 2,α (∂ Q k ) ≤ ∥w per j ∥ C 2,α (∂ Q k ) + ∥ w j ∥ C 2,α (∂ Q k ) ≤ C. (3.4.35)
Now, we recall that

-∆w j + ∇p j = e j in Q k \ O k div w j = 0.
Schauder estimates (see [START_REF] Galdi | An introduction to the mathematical theory of the Navier-Stokes equations: Steady-state problems[END_REF]Theorem IV.7.1]) provide the existence of a constant λ k j such that

∥w j ∥ C 2,α ( Q k \O k ) + ∥p j -λ k j ∥ C 1,α ( Q k \O k ) ≤ C k ∥e j ∥ C 0,α ( Q k \O k ) + ∥w j ∥ C 2,α ∂ Q k \O k ≤ C k ∥e j ∥ C 0,α ( Q k \O k ) + ∥w j ∥ C 2,α ∂ Q k .
The constant C k depends on the C 2,α -norms of the charts flattening ∂O k which are assumed to be bounded independently of k. Thus, C k is independent of k and finally, using also (3.4.35), we derive that .4.36) This proves that the velocity is bounded. The rest of the proof is devoted to the pressure. Note that, thanks to (3.4.36), we have

∥w j ∥ C 2,α (Q k \O k ) + ∥p j -λ k j ∥ C 1,α (Q k \O k ) ≤ C. ( 3
p j - 2 Q k \O k p j L ∞ (Q k \O k ) ≤ p j -λ k j L ∞ (Q k \O k ) + λ k j - 2 Q k \O k p j L ∞ (Q k \O k ) ≤ C + 2 Q k \O k λ k j -p j ≤ C + p j -λ k j L ∞ (Q k \O k ) ≤ C.
This means that we may choose

λ k j = 2 Q k \O k p j
in (3.4.36). We show in the following lines that the sequence (λ k j ) k∈Z 3 is bounded. This will prove that

∥p j ∥ L ∞ (Q k \O k ) ≤ ∥p j -λ k j ∥ L ∞ (Q k \O k ) + |λ k j | ≤ (3.4.36) C + max k∈Z 3 |λ k j | ≤ C(j),
concluding our proof. To show that (λ k j ) k∈Z 3 is bounded, we are going to quantify its growth on neighbouring cells. Since λ k j is a mean-value on Q k \ O k , it would be convenient that for all but a finite number of k ∈ Z 3 , the volume |Q k \ O k | is independent of k. Thus, we assume that there exists

K > 0 such that for all |k| ≥ K, |Q k \ O k | = |Q k \ O per k |. Since λ k j = 2 Q k \O k p per j + 2 Q k \O k p j , it is sufficient to show that λ k j := 2 Q k \O k p j
is bounded independently of k. We fix i ∈ {1, 2, 3}, we are going to show that

λ k j -λ k+ei j ≤ C ∥∇ w j ∥ L 2 (Q k ∪Q k+e i ) + √ α k + √ α k+ei , (3.4.37)
where C is independent of k and |k| ≥ K + 1 and the sequence (α k ) k∈Z 3 appears in Assumption (A3). We fix |k| ≥ K. We write the PDEs satisfied by ( w j , p j ) on the (connected domains)

Q k \ O k and Ω ei k := Int (Q k \ O k ) ∪ (Q k+ei \ O k+ei ) (Int(A) denotes the interior of A):      -∆ w j + ∇ p j = T j in Q k \ O k div w j = 0 w j = -w per j on ∂O k (3.4.38) and      -∆ w j + ∇ p j = T j in Ω ei k div w j = 0 w j = -w per j on ∂O k ∪ ∂O k+ei , (3.4.39)
where

∥T j ∥ H -1 (Qk\Ok) ≤ C √ α k and ∥T j ∥ H -1 (Ω e i k ) ≤ C √ α k + √ α k+ei .
By Assumption (A4) 0 , there exits

v k 1 ∈ H 1 0 (Q k \ O k ) 3 such that div(v k 1 ) = p j - 2 Q k \O k p j and v k 1 H 1 0 (Q k \O k ) ≤ C p j - 2 Q k \O k p j L 2 (Q k \O k )
, where the constant C is independent of k. Similarly, by Assumption (A4) 0 and [START_REF] Galdi | An introduction to the mathematical theory of the Navier-Stokes equations: Steady-state problems[END_REF] (for a bound on the constant appearing in (3.4.40)), there exists 2

v k 2 ∈ H 1 0 (Ω ei k ) 3 such that div(v k 2 ) = p j - 2 Ω e i k p j and v k 2 H 1 0 (Ω e i k ) ≤ C p j - 2 Ω e i k p j L 2 (Ω e i k ) . ( 3
Ω e i k p j = 1 2 λ k j + λ k+ei j yields p j -λ k j L 2 (Q k \O k ) ≤ C ∥∇ w j ∥ L 2 (Q k ) + √ α k (3.4.41)
and

p j - 1 2 λ k j + λ k+ei j L 2 (Ω e i k ) ≤ C ∥∇ w j ∥ L 2 (Q k ∪Q k+e i ) + √ α k + √ α k+ei .
In particular,

p j - 1 2 λ k j + λ k+ei j L 2 (Q k \O k ) ≤ C ∥∇ w j ∥ L 2 (Q k ∪Q k+e i ) + √ α k + √ α k+ei (3.
|Q k \ O k | 1 2 2 λ k j -λ k+ei j = 1 2 λ k j -λ k+ei j L 2 (Q k \O k ) ≤ p j - 1 2 λ k j + λ k+ei j L 2 (Q k \O k ) + p j -λ k j L 2 (Q k \O k ) ≤ C ∇ w j ∥ L 2 (Q k ∪Q k+e i ) + √ α k + √ α k+ei .
(3.4.43)

Hence we have proved (3.4.37). Defining

β k j := ∥∇ w j ∥ L 2 |k-ℓ|≤1 Q ℓ + |k-ℓ|≤1 √ α ℓ
gives the existence of a constant C independent of k and i such that for all |k| ≥ K and i = 1, 2, 3,

λ k j -λ k±ei j ≤ Cβ k j . (3.4.44)
If |k| ≤ K + 1, we define

β k j = max i=1,2,3 λ k j -λ k±ei j , so that (3.4.44) is valid for all k ∈ Z 3 . Since ∇ w j ∈ L 2 (R 3 ) 3×3 and (α k ) k∈Z 3 ∈ ℓ 1 (Z 3 ), we have that (β k j ) k∈Z 3 ∈ ℓ 2 (Z 3 ). We set for all k ∈ Z 3 , γ k j = max |k-ℓ|≤1 β ℓ j , then (γ k j ) k∈Z 3 ∈ ℓ 2 (Z 3 ) and if ℓ := (ℓ 1 , ℓ 2 , ℓ 3 ) satisfies |k -ℓ| ∞ ≤ 1, then | λ ℓ j -λ k j | ≤ λ ℓ j -λ (ℓ1,ℓ2,k3) j + λ (ℓ1,ℓ2,k3) j -λ (ℓ1,k2,k3) j + λ (ℓ1,k2,k3) j -λ k j ≤ 3γ k j .
Applying Lemma 3.4.9 below, we get that λ k j k∈Z 3 is bounded. This concludes the proof of the Theorem.

In the following Lemma 3.4.9 and its proof, we denote by

| • | the | • | ∞ norm on Z 3 .
Lemma 3.4.9. Let (v k ) k∈Z 3 be a real valued sequence such that there exists

(γ k ) k∈Z 3 ∈ ℓ 2 (Z 3 ) such that for all ℓ ∈ Z 3 such that |k -j| ≤ 1, |v j -v k | ≤ γ k . (3.4.45) Then (v k ) k∈Z 3 is bounded.
Proof of Lemma 3.4.9. Let ρ ∈ D B(0, 1 4 ) such that

¡ Q ρ = 1 and ϕ 0 := ρ * 1 Q . We have that supp(ϕ 0 ) ⊂ Q + B 0, 1 4 ⊂ - 3 4 , 3 4
3 ⊂ |ℓ|≤1 Q ℓ .
We define ϕ :=

j∈Z 3 v j ϕ 0 (• -j) = j∈Z 3 v j ρ * 1 Qj . We obviously have ϕ ∈ L 2 loc (R 3 ). Besides, ϕ = v k in 3 i=1 ]k i -1 4 , k i + 1 4 [. We have |∇ϕ| = k∈Z 3 |k-j|≤1 v j ∇ϕ 0 (• -j) 1 Q k .
On the other hand, for

x ∈ Q k , |k-j|≤1 v j ∇ϕ 0 (x -j) = v k |k-j|≤1 ∇ϕ 0 (x -j) + |k-j|≤1 (v j -v k )∇ϕ 0 (x -j).
We have

|k-j|≤1 ∇ϕ 0 (x -j) = |k-j|≤1 ∇ρ * 1 Qj (x) = ¢ B(0,1/4) ∇ρ = 0. Thus, for x ∈ Q k , |∇ϕ(x)| 2 ≤ 9 |k-j|≤1 |v j -v k | 2 |∇ϕ 0 (x -j)| 2 ≤ Cγ 2 k
This proves that ∇ϕ ∈ L 2 (R 3 ). Because of the inclusion

• H 1 (R 3 ) → L 6 (R 3 )/R, there exists M ∈ R such that ∥ϕ -M ∥ L 6 (R 3 ) ≤ C. Thus, ∥ϕ -M ∥ L 6 k∈Z 3 3 i=1 ]ki-1 4 ,ki+ 1 4 [ ≤ C
and finally

k∈Z 3 1 8 |v k -M | 6 ≤ C 6 .
This proves the Lemma.

Remark 3.4.10 (General configuration of holes). Theorem 3.4.8 is proved for compactly supported perturbations. For general configurations, the method of proof still applies. We can for example build a partition (R k ) k∈Z 3 of R 3 into rectangles such that for all

k ∈ Z 3 , Q ′ k ⊂⊂ R k ⊂⊂ Q ′′ k and for k large enough, R k \ O k = Q k \ O per k and R k+ei ∪ R k is connected for all i ∈ {1, 2,

3}. One way to build this partition is to translate the upper face of each cube column by column starting from each cube

Q (k1,k2,0) , (k 1 , k 2 ) ∈ Z.

INTRODUCTION CHAPTER 4. SEMILINEAR PDE

where the matrix-valued coefficient A ε is of the form A ε = A(•/ε) and the matrix-valued function

A : Q → M d (R) is Q-periodic, elliptic and bounded, that is, ∃α > 0, ∀x ∈ Q, ∀ξ ∈ R d , s.t. α -1 |ξ| 2 ≤ ξ T A(x)ξ ≤ α|ξ| 2 . (4.1.2)
We discuss in Section 4.3 the case of a periodic coefficient that is locally perturbed by a defect (in the sense presented in Subsection 1.2.2.1). We also assume for the sake of simplicity that A is symmetric (see Remark 4.3.1 for the non-symmetric case) and that A ∈ C 0,α (R d ), 0 < α ≤ 1. The weak form of (4.1.1) is

∀v ∈ V, ¢ Ω A ε ∇u ε • ∇v + ¢ Ω u ε |u ε | p-2 v = ⟨f, v⟩.
Problem (4.1.1) corresponds to the Euler-Lagrange equation of the minimization problem:

min v∈V 1 2 ¢ Ω ∇v • A ε ∇v + 1 p ¢ Ω |v| p -⟨f, v⟩ . (4.1.3)
We show that Problem (4.1.3), and thus Problem (4.1.1), admits a solution when ε is fixed. We define the functional

J ε (v) := 1 2 ¢ Ω ∇v • A ε ∇v + 1 p ¢ Ω |v| p -⟨f, v⟩, ∀v ∈ V.
Using (4.2.7) and the Young inequality, we have that

J ε (v) ≥ α∥v∥ 2 H 1 0 (Ω) + 1 p ∥v∥ p L p (Ω) -∥f ∥ V ′ ∥v∥ H 1 0 (Ω) + ∥v∥ L p (Ω) ≥ α 2 ∥v∥ 2 H 1 0 (Ω) + 1 2p ∥v∥ p L p (Ω) - 1 2α ∥f ∥ 2 V ′ + 2 1 p-1 p ′ ∥f ∥ p ′ V ′ .
Thus, J ε is coercive on V . Besides, J ε is clearly strictly convex and continuous on V (which is a reflexive Banach space). This proves the existence and uniqueness of a solution u ε ∈ V to (4.1.3). Note that the uniqueness can also be proved by a monotonicity argument.

We immediately notice that, by integration by parts and thanks to (4.1.2), the following a priori estimate is satisfied

∥u ε ∥ V ≤ C, (4.1.4)
where the constant C is independent of ε. This implies that there exists u * ∈ V such that, after extraction of a subsequence:

i ∇u ε -⇀ ε→0 ∇u * in L 2 -weakly; ii u ε -⇀ ε→0 u * in L p -weakly; iii u ε -→ ε→0 u * in L r -strongly for all r ∈ [1, 2 * ) where 2 * = 2d d-2 if d > 2 2 * = +∞ if d = 2.
This point is a consequence of Rellich's Theorem.

The rest of this work is devoted to the identification of u * (see Theorem 4.1.1).

Formal arguments and result

The formal asymptotic expansion of u ε under the form

u ε (x) = u 0 x, x ε + εu 1 x, x ε + ε 2 x, x ε + • • • , x ∈ Ω (4.1.5)
where the functions y -→ u i (x, y) are periodic is done in [START_REF] Bensoussan | Asymptotic Analysis for Periodic Structures[END_REF]Section 16.1, in a more general setting. The cascade of equations implies in particular that u 0 (x, y) = u 0 (x) and that

-div(A * ∇u 0 ) + u 0 |u 0 | p-2 = f u 0 ∈ H 1 0 ∩ L p (Ω), (4.1.6)
where A * = (A * 1 , ..., A * n ) is defined, as in the linear case, by

A * i = ¢ Q A(y)(e i + ∇w ei )dy,
and w q , q ∈ R d are the standard linear correctors. We recall that they are defined as the solution to -div(A(y)(q + ∇w q )) = 0

w q ∈ H 1,per (Q), (4.1.7)
where

H 1,per (Q) := u ∈ H 1 loc (R d ) s.t. u is Q -periodic /R
. By standard elliptic regularity theory (see e.g. [59, Theorem 7.12]), we have that w q ∈ W 1,∞ (Q), since the coefficient A is Hölder continuous. The two scale expansion (4.1.5) is made rigorous for some particular cases in [START_REF] Bensoussan | Asymptotic Analysis for Periodic Structures[END_REF]Section 16.2 and 16.3]. Since these cases do not cover Equation ( 4 Note that a sufficient condition to obtain

D 2 u 0 ∈ L 2 (Ω) is that u 0 |u 0 | p-2 ∈ L 2 (Ω) by elliptic regularity theory. It is the case if u 0 ∈ L 2(p-1) (Ω). The latter is true if d = 2. If d ≥ 3, it is true if 2(p -1) ≤ 2d d-2 i.e. p ≤ 1 + d d-2 .
The proof of Theorem 4.1.1 relies on Murat and Tartar's compensated compactess method (the Div-Curl Lemma, see [START_REF] Murat | Compacité par compensation[END_REF]Theorem 2,p. 478]) and follows the strategy of the linear case. The main difficulty is to pass to the limit in the non-linear term u ε |u ε | p-2 . This is done in the following Lemma: Lemma 4.1.2. Let 2 ≤ p < +∞ and f ∈ V ′ . Then, after extraction of a subsequence, the sequence

u ε |u ε | p-2 converges L p ′ -strongly to u * |u * | p-2 and u ε converges L p -strongly to u * when ε → 0.
The proof of Lemma 4.1.2 is firstly achieved when f ∈ L ∞ (Ω) using maximum principle type arguments for semi-linear equations (see [START_REF] Brezis | Semilinear equations in ∖ N without condition at infinity[END_REF]Lemma 2,p. 3]). The general case f ∈ V ′ is then deduced by density. Remark 4.1.3. We note that the proof of Lemma 4.1.2 is easy when p < 2 * . Indeed, we recall the standard inequality

∀x, y ∈ R, x|x| p-2 -y|y| p-2 ≤ C |x| p-2 + |y| p-2 |x -y|, (4.1.9)
which is a consequence of the fact that x → x|x| p-2 ∈ W 1,∞ loc (R) and that

d dx x|x| p-2 = (p-1)|x| p-2 .
Thanks to (4.1.9), we have that

u ε |u ε | p-2 -u * |u * | p-2 p ′ ≤ C |u ε | p ′ (p-2) + |u * | p ′ (p-2) |u ε -u * | p ′ .
Using the Hölder inequality with the exponent p/p ′ and the identities p ′ (p-2) = p-p ′ and p/p ′ p/p ′ -1 = p p -p ′ , we deduce that

u ε |u ε | p-2 -u * |u * | p-2 p ′ L p ′ (Ω) ≤ C ∥u ε ∥ p-p ′ L p (Ω) + ∥u * ∥ p-p ′ L p (Ω) ∥u ε -u * ∥ p ′ L p (Ω) . ( 4
∥u ε ∥ L ∞ (Ω) ≤ C (4.2.1)
where C is a constant independent of ε. Indeed, we recall that the

L 2 -strong convergence u ε -→ ε→0 u *
holds true after the extraction of a subsequence. We have, after another extraction of a subsequence, convergence almost everywhere to u * . Thus,

u ε |u ε | p-2 -→ ε→0 u * |u * | p-2 a.e. in Ω.
By the dominated convergence Theorem, we deduce the statement of Lemma 1. It is thus sufficient to prove (4.2.1). The proof requires two steps.

Step 1. We prove a Kato-type inequality which states as follows (see also [START_REF] Brezis | Semilinear equations in ∖ N without condition at infinity[END_REF]Lemma A.1,p. 13]): let g ∈ L p ′ (Ω) and u ∈ H 1 0 (Ω) such that -div(A ε ∇u) ≤ g in the weak sense i.e. for all ϕ ∈ V such that ϕ ≥ 0, we have

¢ Ω A ε ∇u • ∇ϕ ≤ ¢ Ω gϕ. (4.2.2)
Then, for all ϕ ∈ D(Ω) such that ϕ ≥ 0, we have the inequality

¢ Ω A ε ∇u + • ∇ϕ ≤ ¢ Ω g sgn + (u) • ϕ, ( 4.2.3)
where sgn + (x) := max(0, x |x| ) for x ̸ = 0, sgn + (0) = 1 2 and x + := x+|x| 2 = max(x, 0) for x ∈ R.

We prove (4.2.3). Let Ψ ∈ C 2 (R, R) be a convex, nondecreasing and positive function such that Ψ ′ and Ψ ′′ are bounded. We first prove that ∀ϕ ∈ D(Ω), ϕ ≥ 0,

¢ Ω A ε ∇Ψ(u) • ∇ϕ ≤ ¢ Ω gΨ ′ (u)ϕ. ( 4.2.4)
We will next approximate the function u + := max(0, u) by a sequence of functions (Ψ n (u)) n∈N where Ψ n ∈ C 2 (R, R) is convex, nondecreasing and positive. For Ψ ∈ C 2 (R, R) as above, it is easily proved that Ψ(u) ∈ H 1 (Ω) and that ∇Ψ(u) = Ψ ′ (u)∇u.

Similarly, Ψ ′ (u) ∈ H 1 (Ω) and ∇Ψ ′ (u) = Ψ ′′ (u)∇u.

We fix ϕ ∈ D(Ω) such that ϕ ≥ 0. We have that

¢ Ω A ε ∇Ψ(u) • ∇ϕ = ¢ Ω A ε Ψ ′ (u)∇u • ∇ϕ = ¢ Ω A ε ∇u • Ψ ′ (u)∇ϕ.
On the other hand, Ψ ′ (u)ϕ ∈ H 1 0 (Ω) and ∇(Ψ ′ (u)ϕ) = ϕΨ ′′ (u)∇u + Ψ ′ (u)∇ϕ. Thus,

¢ Ω A ε ∇Ψ(u) • ∇ϕ = - ¢ Ω A ε ∇u • [ϕΨ ′′ (u)∇u] + ¢ Ω A ε ∇u • ∇(Ψ ′ (u)ϕ).
The first term may be written in the form

¢ Ω A ε ∇u • [ϕΨ ′′ (u)∇u] = ¢ Ω ϕΨ ′′ (u)A ε ∇u • ∇u ≥ α ¢ Ω ϕΨ ′′ (u)|∇u| 2 ≥ 0,
by the ellipticity condition (4.1.2) and since ϕ ≥ 0 and Ψ is convex. The second term satisfies

¢ Ω A ε ∇u • ∇(Ψ ′ (u)ϕ) ≤ ¢ Ω gΨ ′ (u)ϕ, since Ψ ′ (u)ϕ ∈ L ∞ ∩ H 1 0 (Ω)
and by (4.2.2). Thus, (

We define for n ∈ N * , Ψ n (x) :=

1 2 x + 1 n + x 2 .
Then it is easily checked that Ψ n is positive, nondecreasing, convex, that Ψ ′ n is bounded uniformly in n and that Ψ ′′ n is bounded (but not uniformly in n). Besides, Ψ n converges when n → +∞ uniformly to x → max(x, 0) and Ψ ′ n converges pointly to sgn + . Thus, Ψ n (u) converges when n → +∞ uniformly to u + and Ψ ′ n (u) converges to sgn + (u) in L r for all 1 ≤ r < +∞ by the dominated convergence Theorem. Since ∇u ∈ L 2 (Ω), we have that

∇Ψ n (u) = Ψ ′ n (u)∇u -→ n→+∞ sgn + (u)∇u = ∇u + in L 1 (Ω)
We fix ϕ ∈ D(Ω) such that ϕ ≥ 0. We have for all n ∈ N * , by (4.2.4)

¢ Ω A ε ∇Ψ n (u) • ∇ϕ ≤ ¢ Ω gΨ ′ n (u)ϕ.
The above convergences allow to pass to the limit n → +∞ and find that

¢ Ω A ε ∇u + • ∇ϕ ≤ ¢ Ω gΨ ′ (u)ϕ.
This proves the inequality (4.2.3).

Step 2. We set

U := ∥f ∥ 1/(p-1) ∞
and we prove that u ε ≤ U in Ω. We have that

-div(A ε ∇U ) + U |U | p-2 = ∥f ∥ ∞ ≥ f.
We consequently deduce that, in the weak sense,

-div A ε ∇(u ε -U ) ≤ U |U | p-2 -u ε |u ε | p-2 and U |U | p-2 -u ε |u ε | p-2 ∈ L p ′ (Ω)
We are in the context of Step 1. We may thus deduce that for all ϕ ∈ D(Ω) such that ϕ ≥ 0, the following inequality holds true:

¢ Ω A ε ∇(u ε -U ) + • ∇ϕ ≤ ¢ Ω U |U | p-2 -u ε |u ε | p-2 sgn + (u ε -U )ϕ.
Since the function x → x|x| p-2 is non-decreasing, we get that

¢ Ω U |U | p-2 -u ε |u ε | p-2 sgn + (u ε -U )ϕ ≤ 0.
Finally,

∀ϕ ∈ D(Ω), ϕ ≥ 0, ¢ Ω A ε ∇(u ε -U ) + • ∇ϕ ≤ 0. ( 4.2.5)
By density of D(Ω) in H 1 0 (Ω), (4.2.5) is also true for all ϕ ∈ H 1 0 (Ω) such that ϕ ≥ 0 a.e in Ω. Thus, noticing that (u ε -U ) + = 0 on ∂Ω and taking ϕ = (u ε -U ) + in (4.2.5) gives

¢ Ω A ε ∇(u ε -U ) + • ∇(u ε -U ) + ≤ 0.
Thus, by ellipticity and the boundary conditions, (u ε -U ) + = 0 and finally u ε ≤ U in Ω. We have proved that u ε is bounded uniformly in ε from above. Replacing f by -f (and thus u ε by -u ε ), we also show that u ε is uniformly bounded from below. This concludes the proof of Lemma 4.2.1.

Proof of Lemma 4.1.2. In view of Remark 4.1.3, it is sufficient to show that u ε converges L p -strongly to u * when ε -→ 0, after extraction of a subsequence. We use Lemma 4.2.1 and we argue by density. We first prove that the (non-linear) application Ψ :

V ′ -→ V f -→ u ε
is continuous (and more precisely Hölder continuous). Note that the parameter ε is here fixed.

Let g, h ∈ V ′ and v ε , w ε ∈ V be the solutions to (4.1.1) associated to the source terms g, h respectively. We have that

-div A ε ∇(v ε -w ε ) + v ε |v ε | p-2 -w ε |w ε | p-2 = g -h. ( 4.2.6)
We next multiply (4.2.6) by v ε -w ε and we integrate by parts. The ellipticity condition (4.1.2) together with the following inequality (see [68, p. 298])

(x|x| p-2 -y|y| p-2 )(x -y) ≥ β|x -y| p , β > 0 (4.2.7) yield α ¢ Ω |∇(v ε -w ε )| 2 + β ¢ Ω |v ε -w ε | p ≤ ⟨g -h, v ε -w ε ⟩. Thus, α∥∇(v ε -w ε )∥ 2 L 2 (Ω) + β∥v ε -w ε ∥ p L p (Ω) ≤ ∥g -h∥ V ′ ∥v ε -w ε ∥ V .
We now apply the Young inequality:

α∥∇(v ε -w ε )∥ 2 L 2 (Ω) +β∥v ε -w ε ∥ p L p (Ω) ≤ 1 2α ∥g -h∥ 2 V ′ + α 2 ∥∇(v ε -w ε )∥ 2 L 2 (Ω) + β p ∥v ε -w ε ∥ p L p (Ω) + 1 p ′ β 1 p-1 ∥g -h∥ p ′ V ′ .
Recalling that p ≥ 2, this implies that there exists a constant

C = C(α, β, p) > 0 independent of ε such that ∥w ε -v ε ∥ V ≤ C∥g -h∥ γ V ′ , γ := 1 p -1 , ( 4.2.8)
when ∥g -h∥ V ′ ≤ 1. This proves that Ψ is (locally) Hölder continuous.

We now use a density argument. We fix

k ∈ N * , f ∈ V ′ and g k ∈ L ∞ (Ω) such that ∥f -g k ∥ V ′ ≤ 1/k. We denote by (u ε ) ε>0 ⊂ V (resp. (v k ε ) ε>0 ⊂ V
) the sequence of solutions to (4.1.1) with source term f (resp. g k ). We know that, up to a subsequence, we have that u ε and v k ε converge L p -weakly to some limits u * and (v * ) k as ε → 0. By (4.2.8), we have that for all ε > 0,

∥u ε -v k ε ∥ L p (Ω) ≤ Ck -γ (4.2.9)
and, since u ε -v ε converges L p -weakly to u * -(v * ) k , by weak semi-continuity of the norm, that

∥u * -(v * ) k ∥ L p (Ω) ≤ Ck -γ . (4.2.10)
Next, Lemma 4.1.2 applied to g k ∈ L ∞ (Ω) and to the sequence (v k ε ) ε>0 gives, up to extracting a subsequence, that lim

ε→0 ∥v k ε -(v * ) k ∥ L p (Ω) = 0. (4.2.11)
Thus, by the triangle inequality and gathering together (4.2.9), (4.2.10) and (4.2.11), we get that, along a subsequence of (u ε ) ε>0 (depending on k),

lim sup ε→0 ∥u ε -u * ∥ L p (Ω) ≤ 2Ck -γ . ( 4.2.12)
By a diagonal extraction procedure, we show that there exists a subsequence of (u ε ) ε>0 such that for all k ∈ N * , (4.2.12) is satisfied. In particular, u ε converges along this subsequence to u * strongly in L p (Ω).

Proof of Theorem 4.1.1

It is sufficient to prove that for all converging subsequence (u ε k ) k∈N of (u ε ) ε>0 , we have that The whole sequence (u ε ) ε>0 will then converge weakly to u 0 in V by uniqueness of the solution to the homogenized Problem (4.1.6).

u ε k -⇀ k→+∞ u 0 in V,
Proof of (4.2.13). We fix an extraction of (u ε ) ε>0 (that we still denote u ε ) and we write u * its weak limit in V . We define, for q ∈ R d , w q,ε := w q (•/ε) and we consider the equality (on Ω),

A ε ∇u ε • q + ∇w q,ε = ∇u ε • A ε q + ∇w q,ε , (4.2.14)
which is valid since A is symmetric. We pass to the limit on each side of the above equality. For the LHS of (4.2.14), we recall that, since p ≥ 2, we have that

V ′ ⊂ W -1,p ′ (Ω). This proves that f ∈ W -1,p ′ (Ω) and, together with Lemma 4.1.2, that f -u ε |u ε | p-2 is W -1,p ′ (Ω)-strongly convergent after extraction of a subsequence. Thus, div [A ε ∇u ε ] is W -1,p ′ (Ω)-strongly convergent. Besides, since A ε is a.e. bounded, p ′ ≤ 2 and ∥∇u ε ∥ L 2 (Ω) ≤ C (see (4.1. 4
)), we have that

∥A ε ∇u ε ∥ L p ′ (Ω) ≤ C∥∇u ε ∥ L p ′ (Ω) ≤ C∥∇u ε ∥ L 2 (Ω) ≤ C.
Thus, there exists by compactness

r * ∈ L p ′ (Ω) d such that A ε ∇u ε -⇀ ε→0 r * in L p ′ (Ω) d after another
extraction of a subsequence. Besides, because w q is periodic, we have that

¡ Q ∇w q = 0 and thus, because ∇w q ∈ L p (Q), that q + ∇w q • ε -⇀ ε→0 q in L p (Ω). Since q + ∇w q • ε = ∇ q • x + εw q • ε
, we deduce that curl q + ∇w q

• ε = 0. The Div-Curl Lemma (see e.g. [90, Theorem 2.1] -this is just an integration by parts in the present case) implies that

A ε ∇u ε • q + ∇w q,ε -⇀ ε→0 r * • q (4.2.15)
in the distribution sense. For the RHS of (4.2.14), we once again apply the Div-Curl Lemma, in the

L 2 -L 2 version: ∇u ε • A ε [q + ∇w q,ε ] -⇀ ε→0 ∇u * • A * q = A * ∇u * • q (4.2.16)
in the distribution sense. Collecting (4.2.15) and (4.2.16) gives the identity A * ∇u * = r * . This shows that

A ε ∇u ε -⇀ ε→0 A * ∇u * in L p ′ (Ω),
after extraction of a subsequence. Since ∥A ε ∇u ε ∥ L 2 (Ω) ≤ C, the convergence also holds L 2 -weakly (after another extraction). Applying Lemma 4.1.2, we have proved that there exists a subsequence (u ε ℓ ) ℓ∈N of (u ε ) ε>0 such that:

∀v ∈ V, ¢ Ω A ε ℓ ∇u ε ℓ • ∇v + ¢ Ω u ε ℓ |u ε ℓ | p-2 v -→ ℓ→+∞ ¢ Ω A * ∇u * • ∇v + ¢ Ω u * |u * | p-2 v.
Thus,

∀v ∈ V, ¢ Ω A * ∇u * • ∇v + ¢ Ω u * |u * | p-2 v = ⟨f, v⟩,
which is the weak formulation of (4.1.6). This concludes the proof of (4.2.13) since, necessarily, u * = u 0 . It remains to prove (4.1.8). We have that

∇R ε = R ε -εw i (./ε)∇∂ i u * , R ε := ∇u ε - d i=1 (e i + ∇w i (./ε)) ∂ i u * .
Because D 2 u * ∈ L 2 (Ω) and the correctors w i are bounded, it is sufficient to show that R ε tends strongly to zero in L 2 (Ω). By coercivity, we have that

∥R ε ∥ 2 L 2 (Ω) ≤ α ¢ Ω A ε R ε • R ε . (4.2.17)
An immediate computation shows that

1 α R ε 2 L 2 (Ω) ≤ ¢ Ω A ε ∇u ε • ∇u ε - d i=1 ¢ Ω A ε ∇u ε • (e i + ∇w i (./ε))∂ i u * - d i=1 ¢ Ω A ε (e i + ∇w i (./ε)) • ∇u ε ∂ i u * + d i,j=1 ¢ Ω A ε (e i + ∇w i (./ε)) • (e j + ∇w j (./ε))∂ i u * ∂ j u * .
(4.2.18) We treat each term of the RHS of (4.2.18) separately.

First term. Using (4.1.1) and testing against u ε gives

¢ Ω A ε ∇u ε • ∇u ε = - ¢ Ω |u ε | p + ⟨f, u ε ⟩ -→ ε→0 - ¢ Ω |u * | p + ⟨f, u * ⟩ = ¢ Ω A * ∇u * • ∇u * (4.2.19)
where we used Lemma 4.1.2 and the convergence of (u ε ) ε>0 . Second term. We apply the Div-Curl Lemma. We first notice that div

A ε ∇u ε = u ε |u ε | p-2 -f con- verges W -1,p ′ (Ω) strongly (we recall that V ′ ⊂ W -1,p ′ (Ω)). Besides, A ε ∇u ε converges L p ′ -
weakly to A * ∇u * and e i + ∇w i (./ε) converges L p -weakly to e i . Thus

A ε ∇u ε (e i + ∇w i (./ε)) -⇀ ε→0 A * ∇u * • e i in D ′ (Ω). (4.2.20) Since ∇w i ∈ W 1,∞ (Q), we have the bound ∥A ε ∇u ε (e i + ∇w i (./ε))∥ L 2 (Ω) ≤ C.
Consequently, we can upgrade (4.2.21), up to extracting a subsequence, into

A ε ∇u ε (e i + ∇w i (./ε)) -⇀ ε→0 A * ∇u * • e i in L 2 (Ω). (4.2.21)
Since ∂ i u * ∈ L 2 (Ω), we conclude that the second term tends to

¢ Ω A * ∇u * • ∇u * .
Third term. As for the previous term, we prove that, up to extracting a subsequence, it converges to

¢ Ω A * e i • ∇u * ∂ i u * = ¢ Ω A * ∇u * • ∇u * .
Fourth term. As a periodic function, we have that

A ε (e i + ∇w i (./ε)) • (e j + ∇w j (./ε)) converges in the weak-⋆ in the L ∞ toplogy to A * e i • e j . Since ∂ i u * ∂ j u * ∈ L 1 (Ω), we conclude that d i,j=1 ¢ Ω A ε (e i + ∇w i (./ε)) • (e j + ∇w j (./ε))∂ i u * ∂ j u * -→ ε→0 d i,j=1 ¢ Ω A * e i • e j ∂ i u * ∂ j u * = ¢ Ω A * ∇u * • ∇u * .
Collecting the above convergences, we conclude that, up to extracting a subsequence,

lim ε→0 R ε 2 L 2 (Ω) = 0.
Since the above arguments may be repeated for any subsequence of (R ε ), the whole sequence converges. This concludes the proof of the Theorem.

Remarks and extension to a periodic setting with defects

In this section, we discuss the homogenization of (4.1.1) in the periodic setting with defect introduced in [START_REF] Blanc | Precised approximations in elliptic homogenization beyond the periodic setting[END_REF][START_REF] Blanc | On correctors for linear elliptic homogenization in the presence of local defects[END_REF][START_REF] Blanc | A possible homogenization approach for the numerical simulation of periodic microstructures with defects[END_REF][START_REF] Blanc | Local profiles for elliptic problems at different scales: defects in, and interfaces between periodic structures[END_REF]. We first recall this setting: let A : R d -→ M d (R) be of the form A = A per + A, where A per is a periodic matrix valued tensor and A ∈ L r (R d ) d×d , 1 < r < +∞. We assume that A per and A are elliptic and bounded in the sense (4.1.2) and that

A ∈ C 0,α (R d ), A per ∈ C 0,α (R d ), 0 < α ≤ 1.
We consider the following semi-linear PDE

-div A • ε ∇u ε + u ε |u ε | p-2 = f, f ∈ V ′ . ( 4.3.1)
In (4.3.1), the matrix A represents a local defect that is inserted, at the microscale, in a periodic material. The exponent r will be the integrability of the defect. Performing a two-scale expansion similar to the periodic case, we find that the corrector equation associated to (4.3.1) is

-divA(y)(q + ∇w q ) = 0 in R d , q ∈ R d . ( 4.3.2)
This PDE, similar to the corrector equation of the associated linear problem (4.0.2), has been studied in [START_REF] Blanc | Local profiles for elliptic problems at different scales: defects in, and interfaces between periodic structures[END_REF][START_REF] Blanc | On correctors for linear elliptic homogenization in the presence of local defects[END_REF]. It is proved that (4.3.2) admits a solution of the form w per q + w q where w per q is the periodic corrector (i.e. solution to (4.3.2) for A = A per ) and ∇ w q ∈ L r ∩ L ∞ (R d ) d where r is the integrability of the defect. In particular, we have the fundamental sublinearity property at infinity (see [START_REF] Blanc | Precised approximations in elliptic homogenization beyond the periodic setting[END_REF]Proposition 5.1]):

∀ρ > 1, sup |x-y|≤ρ |w q (x) -w q (y)| ≤ cρ 1-νr , ν r = min 1, d r ,
where c is a constant independent of ρ. Using the non-periodic correctors w q , q ∈ R d , the proof of Theorem 4.1.1 extends verbatim to the periodic setting with defect introduced above. Let us note that, because of the integrabilty of the defect A and of the non-periodic corrector ∇ w q , we get that

A • ε q + ∇w q • ε -⇀ ε→0 ¢ Q A per (y)(q + ∇w q )dy in L 2 (Ω) and q + ∇w q • ε -⇀ ε→0 q in L p (Ω).
This proves that u ε -⇀ ε→0 u * in V with u * being the solution to (4.1.6). In particular, the macroscopic behaviour of u ε is the same as for the periodic case. Although we did not pursue in this direction, we believe that it is possible to obtain convergence rates in the homogenization of (4.1.1), as it is done in [START_REF] Blanc | Precised approximations in elliptic homogenization beyond the periodic setting[END_REF] for the linear case. We close this section with two Remarks on the case of non-symmetric coefficients on the one hand, and the case 1 < p < 2 on the other hand, both for the periodic and periodic with defect case: Remark 4.3.1 (The non-symmetric case). If A is non-symmetric, we use as in the linear case the adjoint cell corrector problem (see e.g. [START_REF] Cioranescu | An introduction to homogenization[END_REF] or [START_REF] Bensoussan | Asymptotic Analysis for Periodic Structures[END_REF]). We briefly sketch the proof: we consider, instead of (4.1.7), the PDE

-div A T (y)(q + ∇w T q ) = 0 w T q ∈ H 1,per (Q), (4.3.3)
where A T denotes the matrix transpose of A. We then write (4.2.14) under the form

A ε ∇u ε • q + ∇w T q,ε = ∇u ε • A T ε q + ∇w T q,ε .
As above, the Div-Curl Lemma allow to prove that, up to extracting a subsequence,

A ε ∇u ε -⇀ ε→0 (A * ) T ∇u * in the L 2 (Ω) topology,
where

A * := ¢ Q A T (y) e 1 + ∇w T e1 (y) , • • • , ¢ Q A T (y) e d + ∇w T e d (y) 1≤i≤d . Since A * = (A * ) T , we conclude that A ε ∇u ε converges L 2 -weakly to A * ∇u * .
The rest of the proof of Theorem 4.1.1 is identical to the symmetric case. Remark 4.3.2 (Case 1 < p < 2). This chapter only treats the case p ≥ 2. The case 1 < p < 2 can be easily addressed. In this case, V = H 1 0 (Ω) and u ε converges strongly in L p (Ω) to u * as ε → 0 by Rellich Theorem. We can thus pass directly to the limit in the nonlinear term. CHAPTER

5

HOMOGENIZATION OF THE p-LAPLACE EQUATION IN A PERIODIC SETTING WITH A LOCAL DEFECT

This chapter is based on [START_REF] Wolf | Homogenization of p-laplace type equations in a periodic setting with defects[END_REF].

In this chapter, we consider the homogenization of the p-Laplace equation with a periodic coefficient that is perturbed by a local defect. This setting has been introduced in [START_REF] Blanc | A possible homogenization approach for the numerical simulation of periodic microstructures with defects[END_REF][START_REF] Blanc | Local profiles for elliptic problems at different scales: defects in, and interfaces between periodic structures[END_REF] in the linear setting p = 2. We construct the correctors and we derive the limit of the oscillating function in the case p > 2 under the assumption that the periodic correctors are non degenerate. This chapter is based on [START_REF] Wolf | Homogenization of p-laplace type equations in a periodic setting with defects[END_REF].

Introduction

This chapter is concerned with the homogenization of non-linear degenerate elliptic equations in a periodic setting with defects. More precisely, we are interested in p-Laplacian type equations that are defined, for some p ≥ 2, as

   -div a • ε ∇u ε |∇u ε | p-2 = f in Ω u ε ∈ W 1,p 0 (Ω) (5.1.1)
for a fixed bounded domain Ω ⊂ R d , d ≥ 1 and f ∈ L p ′ (Ω). For p = 2, we recover the standard linear conductivity equation. In (5.1.1), the scalar-valued coefficient a is assumed to be of the form

a = a per + a, ( 5.1.2)
where a per is a periodic coefficient with standard coercivity and boundedness condition and a is a perturbation of a per such that a ∈ L q (R d ) for some 1 ≤ q ≤ p p-1 . We assume that the coefficient a itself is coercive and bounded and we choose λ > 0 such that ∀y ∈ R d , λ -1 < a per (y) < λ and λ -1 < a(y) < λ.

(

For fixed ε > 0, Problem (5.1.1) is well-posed and corresponds to the Euler-Lagrange equation of the minimization Problem min

v∈W 1,p 0 (Ω) 1 p ¢ Ω a • ε |∇v| p - ¢ Ω f v . (5.1.4)
The behaviour of (5.1.1) when ε → 0 has been studied in the absence of perturbation, i.e. when a = a per . It corresponds to a particular case of the homogenization of the equation

-divA • ε , ∇u ε = f (5.1.5)
under general growth and continuity conditions for the operator A(y, ξ) (in our case, we have that A(y) = a per (y)ξ|ξ| p-2 ). The homogenized limit of (5.1.5) is derived in [START_REF] Fusco | Further results on the homogenization of quasilinear operators[END_REF][START_REF] Fusco | On the homogenization of quasilinear divergence structure operators[END_REF]. It is proved that u ε converges in the W 1,p -weak topology, when ε → 0, to u * which is defined by the homogenized equation

-divA * (∇u * ) = f u * ∈ W 1,p 0 (Ω), (5.1.6)
where, for ξ ∈ R d , the homogenized operator is

A * (ξ) := ¢ Q A y, ξ + ∇w ξ (y) dy,
and the function w ξ ∈ W 1,p per (Q) is the corrector in the direction ξ given as the periodic solution (up to an additive constant) to the equation

-divA(•, ξ + ∇w ξ ) = 0.
(5.1.7)

The strong convergence of the gradient

∇u ε -∇u * -∇w ∇u * (./ε) -→ ε→0 0 in L p (Ω) (5.1.8)
has been obtained in [START_REF] Dal | Correctors for the homogenization of monotone operators[END_REF] with ∇u * replaced by its discretization at small scale ε, for measurability reasons, see Section 5.2 below for the details. The periodic homogenization of the integral functionals corresponding to (5.1.1) is exposed in e.g. [START_REF] Braides | Gamma-convergence for Beginners[END_REF]. The stochastic case has been studied qualitatively in [START_REF] Dal | Nonlinear stochastic homogenization[END_REF]. Recently, quantitative results for non-linear stochastic problems have been obtained in [START_REF] Fischer | Optimal homogenization rates in stochastic homogenization of nonlinear uniformly elliptic equations and systems[END_REF] with optimal convergence rates for non-degenerate non-linear operators with quadratic growth, see also [START_REF] Wang | Quantitative estimates on periodic homogenization of nonlinear elliptic operators[END_REF] for the deterministic case. The case of stochastic non-degenerate operators with p-growth, p > 2, is addressed in [START_REF] Clozeau | Quantitative nonlinear homogenization: control of oscillations[END_REF].

In this chapter, we study the equation (5.1.7) when the perturbation a belongs to the space L q (R d ) for 1 ≤ q ≤ p p-1 and to some Hölder space (see Theorem 5.2.3 below). We then derive the homogenized limit of the sequence (u ε ) ε>0 and we study the convergence of the two-scale expansion (5.1.8) when we use, on the one hand, the periodic corrector and, on the other hand, the non-periodic corrector (corresponding respectively to the solutions of (5.1.7) when A(y, ξ) = a per (y)ξ|ξ| p-2 and A(y, ξ) = a(y)ξ|ξ| p-2 ). We also illustrate the quantitative convergence of the two-scale expansion (5.1.8) in the one-dimensional setting and prove that, in this case, using the non-periodic corrector instead of the periodic corrector in fact improves the quality of convergence of (5.1.8). The main difficulty of this work is that Equation (5.1.7) is posed on the whole space R d . One major tool to obtain the strong convergence (5.1.8) in the non-periodic case is the continuity of the application ξ -→ ∇w ξ (see Theorem 5.2.4 below). This will be proved under one of the two Assumptions (A4) or (A4)' below.

Before stating our main results, we would like to comment on the special case p = 2 for the homogenization of the Problem (5.1.1). This problem is very standard since the 70's for a periodic coefficient a, see e.g. [START_REF] Bensoussan | Asymptotic Analysis for Periodic Structures[END_REF] for qualitative results and [START_REF] Avellaneda | Compactness methods in the theory of homogenization[END_REF] for quantitative results. It is worth mentioning that, in this case, the homogenization objects such as correctors, homogenized limits are explicit and very easy to compute. The setting (5.1.1)-(5.1.2) has first been introduced in [START_REF] Blanc | A possible homogenization approach for the numerical simulation of periodic microstructures with defects[END_REF] for q = 2. It models local defects that could appear, at the microscale, in a periodic background. The results obtained have been generalized to the case 1 ≤ q < +∞ in [START_REF] Blanc | Local profiles for elliptic problems at different scales: defects in, and interfaces between periodic structures[END_REF][START_REF] Blanc | On correctors for linear elliptic homogenization in the presence of local defects[END_REF] and convergence rates have been proved in [START_REF] Blanc | Precised approximations in elliptic homogenization beyond the periodic setting[END_REF]. In [START_REF] Goudey | A periodic homogenization problem with defects rare at infinity[END_REF], a new non-periodic setting has been introduced to model defects that are not local but rare at infinity. We stress that, in [START_REF] Blanc | A possible homogenization approach for the numerical simulation of periodic microstructures with defects[END_REF][START_REF] Blanc | Local profiles for elliptic problems at different scales: defects in, and interfaces between periodic structures[END_REF][START_REF] Blanc | On correctors for linear elliptic homogenization in the presence of local defects[END_REF][START_REF] Blanc | Precised approximations in elliptic homogenization beyond the periodic setting[END_REF][START_REF] Goudey | A periodic homogenization problem with defects rare at infinity[END_REF], as in the present work, the macroscopic behaviour of the oscillating solution remains the same as in the case of a periodic coefficient. This will be expressed, for the non-linear case, in Theorem 5.2.7 below.

The chapter is organized as follows. The main results of the paper are presented in Section 5.2. We develop in Section 5.3 explicit calculations in the one dimensional setting and obtain convergence results. We then turn in Section 5.4 to the existence of the non-periodic correctors in any dimension. The properties of the non-periodic corrector are proved in Section 5.5. We then derive qualitiative homogenization results in Section 5.6. We finally prove in Section 5.7 a weaker continuity result for the mapping ξ -→ ∇w ξ that is enough to derive qualitative homogenization. We recall in Appendix 5.8.1 the proof of classical results in the periodic case. Technical inequalities are gathered in Appendix 5.8.2.

Main results

Notations

In the whole paper, d ≥ 1 will be the dimension of the ambient space. The standard unit cube - 1 2 , 1 2 d will be denoted Q. The euclidian norm will be written | • | as well as the Lebesgue measure of a measurable subset of R d . Let Ω be a bounded domain of R d . If 1 < q < +∞ is an exponent, we define its conjugate by q ′ := q/(q -1). The euclidian open ball of R d centered in x and of radius r > 0 will be written B(x, r). If x = 0, we write B r := B(0, r). We use similar notations for cubes, namely Q(x, r) and Q r . We define the mean-value operation for a measurable and integrable function u by

2 B(x,r) u := 1 |B(x, r)| ¢ B(x,r) u.
The indicator function of a measurable set A is denoted 1 A .

The standard Lebesgue and Sobolev spaces are denoted by L q (Ω) and W 1,q (Ω). The associated norms are

∥u∥ L q (Ω) := ¢ Ω |u| q 1/q and ∥u∥ W 1,q (Ω) := ¢ Ω |u| q 1/q + ¢ Ω |∇u| q 1/q .
The space L q per (resp. W 1,q per ) denotes the set of functions that are periodic and locally belong to L q (resp. W 1,q ). Theses two spaces are endowed with the norms ∥u∥ L q per :=

¢ Q |u| q 1/q
and ∥u∥ W 1,q per :=

¢ Q |u| q 1/q + ¢ Q |∇u| q 1/q .
The space of uniformly L q (resp. W 1,q ) functions is denoted by L q unif (resp. W 1,q unif ). These spaces are endowed with the norms

∥u∥ L q unif (R d ) := sup x∈R d ∥u∥ L q (x+Q) and ∥u∥ W 1,q unif (R d ) := sup x∈R d ∥u∥ W 1,q (x+Q) .
For 0 < α < 1, the space C 0,α refers to the standard Hölder space endowed with the norm

∥u∥ C 0,α := ∥u∥ L ∞ + sup x̸ =y |u(x) -u(y)| |x -y| α .
We define, for δ > 0, the discretization operator M δ : L q (Ω) -→ L q (Ω) introduced in [START_REF] Dal | Correctors for the homogenization of monotone operators[END_REF][START_REF] Fusco | On the homogenization of quasilinear divergence structure operators[END_REF]. If ϕ ∈ L q (Ω), we set

M δ ϕ := k∈Z d s.t. δ(Q+k)⊂Ω 2 δ(Q+k) ϕ 1 δ(k+Q) . (5.2.1)
It is clear that M δ is linear and bounded over L q (Ω) and that M δ ϕ -→ δ→0 ϕ in L q (Ω).

The periodic case

We assume in this paragraph that a = 0 in (5.1.2). In this case, the corrector equation is, according to (5.1.7):

-div a per (y)(ξ + ∇w per ξ )|ξ + ∇w per ξ | p-2 = 0.

(5.2.

2)

The equation (5.2.2) admits a unique solution w per ξ in the space W 1,p per (Q)/R. Indeed, the weak formulation of (5.2.2) is

∀ϕ ∈ W 1,p per (Q)/R, ¢ Q a per (y)(ξ + ∇w per ξ )|ξ + ∇w per ξ | p-2 • ∇ϕ = 0, (5.2.3)
which is exactly the Euler-Lagrange equation of the minimization Problem min

v∈W 1,p per (Q)/R 1 p ¢ Q a per (y) ξ + ∇v p dy . (5.2.4)
It is easy to see that the functional appearing in Problem (5.2.4) is strictly convex, coercive and continuous with respect to ∇v. Thus, (5.2.4) admits a minimizer w per ξ , the gradient of which is unique. We impose that 1 Q w per ξ = 0 so that w per ξ is itself unique. Besides, we have the following Proposition (see [START_REF] Fusco | On the homogenization of quasilinear divergence structure operators[END_REF][START_REF] Fusco | Further results on the homogenization of quasilinear operators[END_REF][START_REF] Dal | Correctors for the homogenization of monotone operators[END_REF] or Appendix 5.8.1 below for a proof) gathering the main properties of the application ξ -→ ∇w per ξ :

Proposition 5.2.1. Let a per : R d -→ R d be a periodic and Lipschitz continuous coefficient satisfying (

(i) The map ξ -→ ∇w per ξ is homogeneous in the sense that for all ξ ∈ R d and t ∈ R, ∇w per tξ = t∇w per ξ .

(5.

2.5)

(ii) There exists a constant C = C(d, p, a per ) > 0 and an exponent α = α(d, p, a per ) > 0 such that for all ξ ∈ R d , ∇w per ξ ∈ C 0,α (R d ) and, moreover, we have the estimates

∥∇w per ξ ∥ L p unif (R d ) ≤ C|ξ| and ∥∇w per ξ ∥ C 0,α (R d ) ≤ C|ξ|. (5.2.6) (iii) There exists a constant C = C(d, p, a per ) > 0 such that for all ξ, η ∈ R d , ∥∇w per ξ -∇w per η ∥ L p unif (R d ) ≤ C |ξ| 1-β + |η| 1-β |ξ -η| β , β := 1 p -1 . (5.2.7) (iv) There exists a constant C = C(d, p, a per ) > 0 such that for all ξ, η ∈ R d , ∇w per ξ -∇w per η L ∞ (R d ) ≤ C |ξ| 1-γ + |η| 1-γ |ξ -η| γ , γ := βp p + d/α , (5.2.8)
where β is defined in (5.2.7) and α is given by (ii).

It is proved in [START_REF] Fusco | On the homogenization of quasilinear divergence structure operators[END_REF] that u ε converges weakly in W 1,p (Ω) to u * which is defined by (5.1.6). Note that (5.1.6) is well posed due to the monoticity of A * (see [START_REF] Dal | Correctors for the homogenization of monotone operators[END_REF] and [START_REF] Le | Nonlinear Elliptic Partial Differential Equations[END_REF]Corollary 8.1]). Convergence in the L ∞ -norm may be obtained in the one-dimensional setting, see Section 5.3 below. (ii) We have the strong convergence

∇u ε -∇u * -∇w Mε∇u * • ε -→ ε→0 0 in L p (Ω), (5.2

.21)

where M ε is defined by (5.2.1).

(iii) We have the strong convergence

∇u ε -∇u * -∇w per Mε∇u * • ε -→ ε→0 0 in L p (Ω), (5.2

.22)

where M ε is defined by (5.2.1).

We stress that, instead of assuming (A4), Theorem 5.2.7 can be proved under the assumption that the mapping

Φ p : R d -→ L p unif (R d ) ξ -→ ∇w ξ (5.2.23)
is continuous. This continuity can be obtained under the following Assumption (A4)' which is clearly weaker than Assumption (A4):

(A4)' For ξ ∈ R d , there exists a constant C > 0 that may depend on ξ and ∇w per ξ such that the following weighted Poincaré-Wirtinger inequality holds true: there exists r min > 0 such that for all R > r min and w ∈ H

1 Q \ Q 1/2 , |ξ + ∇w per ξ (R•)| p-2 2 w - 2 Q\Q 1/2 w L 2 (Q\Q 1/2 ) ≤ C |ξ + ∇w per ξ (R•)| p-2 2 ∇w∥ L 2 (Q\Q 1/2 ) . (5.2.24)
We comment in Subsection 5.2.4 on Assumption (A4)' and we will provide a sufficient condition on ξ + ∇w per ξ so that (5.2.24) is satisfied. We are able to prove the following Theorem: We close this section by mentioning that the results of Theorem 5.2.7 can be improved in the one-dimensional setting. We devote Section 5.3 to convergence results in this particular case. Remark 5.2.9. To derive Theorem 5.2.8, it is sufficient, instead of Assumption (A4)', to assume that the set of smooth functions with compact support over R d , denoted by

C ∞ 0 (R d ), is dense in W ξ+∇w per ξ .
We show in Lemma 5.8.3 (see Appendix 5.8.2) that, as pointed out in [START_REF] Vasilševich | Weighted sobolev spaces[END_REF], the density result is implied by Assumption (A4)'. In particular, the practical criterion given by Lemma 5.2.12 implies the density of C 0 ∞ (R d ). Note that, under Assumption (A4)', we can easily prove (by density) that (5.2.12)-(5.2.13) admits a unique solution in the distribution sense in W ξ+∇w per ξ . Remark 5.2.10. The method of proof of this paper allows to build the non-periodic correctors for a defect a that belongs to the dual space of W ξ+∇w per ξ , see Lemma 5.4.1 (iii). This is in particular the case if a ∈ L 2 (R d ). We are however not able to show that the non-periodic corrector satisfies ∇ w ξ ∈ L 2 (R d ) but only that ∇ w ξ ∈ L 2 (|ξ + ∇w per ξ | p-2 dλ), see Remark 5.2.11 below. More generally, building the non-periodic correctors for a defect a ∈ L 2+δ ∩ C 0,α (R d ) is a challenging problem that we are unable to address for now. In the linear setting p = 2, this was achieved in [START_REF] Blanc | On correctors for linear elliptic homogenization in the presence of local defects[END_REF] by studying the continuity from L q (R d ) to L q (R d ) for q > 2 of the Riesz operator associated to the coefficient a. Assume that there exists x 0 ∈ Q such that ξ + ∇w per ξ (x 0 ) = 0. We can assume by invariance translation that x 0 = 0. Owing to Proposition 5.2.1 (ii), we have that |ξ + ∇w per ξ (x)| ≤ C|x| α in Q. Let ϕ ∈ D(Q) be such that ϕ = 1 on B(0, 1/4). We define

Ψ := k∈Z d \{0} 1 |k| δ+ν ϕ(|k| ν (• -k))
, where δ, ν > 0 will be chosen later. We have

∇Ψ p L p (R d ) = k∈Z d \{0} 1 |k| pδ ¢ |k| -ν Q ∇ϕ(|k| ν x) p dx = k∈Z d \{0} ∥∇ϕ∥ p L p (Q) |k| pδ+dν .
(5.2.25)

Besides, we have that

∇Ψ 2 L 2 (|ξ+∇w per ξ | p-2 dλ) = k∈Z d \{0} 1 |k| 2δ ¢ |k| -ν Q ∇ϕ(|k| ν x) 2 |ξ + ∇w per ξ (x)| p-2 dx ≤ k∈Z d \{0} C∥∇ϕ∥ 2 L 2 (Q)
|k| αν(p-2)+dν+2δ .

(5.2.26)

Finally, ∇Ψ 2 L 2 (R d ) = k∈Z d \{0} 1 |k| 2δ ¢ |k| -ν Q ∇ϕ(|k| ν x) 2 = k∈Z d \{0} ∥∇ϕ∥ 2 L 2 (Q) |k| dν+2δ .
(5.2.27)

We fix ν ∈ ( d d+2 , 1) and δ ∈ max{ d(1-ν) p , d(1-ν)-αν(p-2) 2 }, d(1-ν) 2 so that ∇Ψ ∈ L 2 (|ξ+∇w per ξ | p-2 dλ)∩ L p (R d ) and ∇Ψ / ∈ L 2 (R d ).
Note that ∇Ψ ∈ C 0,δ/ν (R d ) so that this counter-example is consistent with the result of Theorem 5.2.4 (ii) since δ < ν.

Comments on the Assumptions

On Assumption (A4). Assumption (A4) is quite restrictive but is known to be true in dimension 1. Besides, it is proved in [START_REF] Kirill | On full two-scale expansion of the solutions of nonlinear periodic rapidly oscillating problems and higher-order homogenised variational problems[END_REF]Lemma 2,p. 404] that it is also satisfied in dimension d = 2.

We show here that Assumption (A4) is satisfied for laminate materials (in any dimension). Suppose that a per (x) = a 0 (x 1 ) where a 0 : R -→ R is a periodic function. Let ξ ̸ = 0. In this case, the periodic corrector w per ξ is a function of the first variable i.e. w per ξ (x) = w 0 ξ (x 1 ) and (5.2.2) becomes

- d dx 1   a0(x1) ξ 1 + dw 0 ξ dx 1 ξ 1 + dw 0 ξ dx 1 2 + ξ 2 2 + • • • + ξ 2 d p-2 2    = 0. (5.2.28)
If there exists i ≥ 2 such that ξ i ̸ = 0, then |ξ + ∇w per ξ | ≥ |ξ i | > 0. In the other cases, ξ i = 0 for i ≥ 2, thus ξ 1 ̸ = 0 and (5.2.28) reduces to:

- d dx 1   a 0 (x 1 ) ξ 1 + dw 0 ξ dx 1 ξ 1 + dw 0 ξ dx 1 p-2   = 0. ( 5

.2.29)

There exists a constant C(ξ) such that ξ 1 + dw 0 ξ dx1 p-1 = C(ξ)/a 0 (x 1 ), where z p-1 := sgn(z)|z| p-1 . If C(ξ) = 0, then w 0 ξ (x 1 ) = -ξ 1 x 1 which contradicts the periodicity of w 0 ξ . Hence C(ξ) ̸ = 0. Thus

ξ 1 + dw 0 ξ dx 1 ≥ (|C(ξ)|λ) 1 p-1 > 0.
We have proved that |ξ + ∇w per ξ | ≥ c(ξ) > 0 on Q: this easily implies (A4). Indeed, if by contradiction (A4) were not satisfied, then there would exist by homogeneity (ξ n ) n∈N ⊂ R On Assumption (A4)'. The Assumption (A4)' is the weakest possible given our method of proof. It is satisfied in dimension d = 1, 2 because (A4) is satisfied. For higher dimensions, we provide here a sufficient condition implying (A4)': Lemma 5.2.12 (see [START_REF] Cardone | Estimates in homogenization of degenerate elliptic equations by spectral method[END_REF] and [START_REF] Vasil | Homogenization of degenerate elliptic equations[END_REF]). Assume that d ≥ 2 and that |ξ

+ ∇w per ξ | 2-p ∈ L d/2 (Q), then (A4)' is satisfied.
Proof. For the sake of completeness, we reproduce the proof here, see also [START_REF] Cardone | Estimates in homogenization of degenerate elliptic equations by spectral method[END_REF]Lemma 8]. We fix R > 1. Then by periodicity, we immediately check that |ξ

+ ∇w per ξ (R•)| 2-p ∈ L d/2 (Q \ Q 1/2 ) and that |ξ + ∇w per ξ (R•)| 2-p L d/2 (Q\Q 1/2 ) ≤ 4 |ξ + ∇w per ξ | 2-p L d/2 (Q) .
(5.2.30)

We then write, using Proposition 5.2.1 (ii), that

|ξ+∇w per ξ (R•)| p-2 2 (w - 2 Q\Q 1/2 w) L 2 (Q\Q 1/2 ) ≤ C|ξ| p-2 2 w - 2 Q\Q 1/2 w L 2 (Q\Q 1/2 ) ≤ C|ξ| p-2 2 ∇w L 2d d+2 (Q\Q 1/2 ) ≤ C|ξ| p-2 2 ¢ Q\Q 1/2 |ξ + ∇w per ξ (R•)| p-2 |∇w| 2 1 2 ¢ Q\Q 1/2 |ξ + ∇w per ξ (R•)| 2-p d/2 1 d , ( 5.2.31)
where we used Poincaré-Sobolev embedding [58, Exercise II.5.9]) in the second inequality, Hölder inequality with exponents r = (d + 2)/d, r ′ = (d + 2)/2 and the fact that dr ′ /(d+2) = d/2 in the third inequality. Using (5.2.30) allows to conclude the proof.

• W 1,2d/(d+2) (Q \ Q 1/2 ) → L 2 (Q \ Q 1/2 ) (see
If we assume that {ξ + ∇w per ξ = 0} is a finite number of points (in the case d > 2) and that all critical points have finite order, denoting by m the maximum order of the corresponding zero points, we have that |ξ + ∇w per ξ | 2-p ∈ L d/2 (Q) if and only if m d 2 (p -2) < d i.e. p < 2 + 2/m. Thus, in this case, Assumption (A4)' can be replaced by assuming that p < 2 + 2/m. Note also that if ξ + ∇w per ξ vanishes at order m along a line (or a curve) in dimension d,

then |ξ + ∇w per ξ | 2-p ∼ |x| m(2-p) which is L d/2 (Q) if and only if d 2 m(p -2) < d -1 i.e. p < 2 + 2(d-1)
dm . Remark 5.2.13. The Assumption (A4)' is used in the proof of Lemma 5.7.1 which allows to pass from solutions in the distribution sense to solutions in the sense of Definition 5.2.2 for PDEs of the form (5.2.12). We then take advantage of Lemma 5.7.1 in the proof of Theorem 5.2.8 by working locally in a concentration-compactness method.

Extension to other non-linear operators

We have limited the presentation of the results to the simplest operator (5.1.1) in order to avoid some technicalities and the use of abstract existence Theorems for non-linear PDEs. However, the result of this paper extends to more general operators. We explain below the type of problems that we can address with the technique developed in this work.

The first direct extension concerns the equivalent of (5.1.1) when a is a matrix-valued coefficient. This corresponds to the following non-linear operator:

a(y, ξ) := ⟨A(y)ξ, ξ⟩ p-2 2 A(y)ξ, y ∈ R d , ξ ∈ R d , (5.2.32)
where A is of the form A = A per + A. We assume that the matrix A per is periodic and that A and A per are symmetric and positive definite, that is,

∃λ > 0, ∀y ∈ R d , λ -1 |ξ| 2 ≤ ⟨A(y)ξ, ξ⟩ ≤ λ|ξ| 2 and λ -1 |ξ| 2 ≤ ⟨A per (y)ξ, ξ⟩ ≤ λ|ξ| 2 .
The perturbation A satisfies A ∈ L p ′ ∩ C 0,1 (R d ) d×d . The periodic correctors can be defined thanks to variational techniques by considering the minimization problem min

w per ξ ∈H 1,per (Q) 1 p ¢ Q A(y)(ξ + ∇w per ξ ), ξ + ∇w per ξ p/2
.

The non-periodic equation corresponding to (5.2.12) is A per (•)ξ. It is easily proved that f ∈ L p ′ (R d ) d and that the method of proof of Section 5.4 extends to this case by studying the functional

-div a(•, ξ + ∇w per ξ + ∇ w ξ ) -a(•, ξ + ∇w per ξ ) = div(h), ( 5
F ξ (v) := 1 p ¢ R d A(y)(ξ + ∇w per ξ + ∇v), ξ + ∇w per ξ + ∇v p/2 -A(y)(ξ + ∇w per ξ ), ξ + ∇w per ξ p/2 -p⟨A(y)(ξ + ∇w per ξ ), ξ + ∇w per ξ ⟩ p-2 2 A(y)(ξ + ∇w per ξ ) • ∇v dy + ¢ R d h • ∇v.
Note that the inequalities given in Appendix 5.8.2 are valid for the matrix model (5.2.32). Concerning the continuity results for the application ξ -→ ∇ w ξ , the results proved in sections 5.5, 5.6 and 5.7 still hold true.

The second less direct extension corresponds to non-variational operators, that is, PDEs that cannot be written as a minimization problem. We consider operators a(y, ξ) that satisfy the following properties:

(1) for all ξ ∈ R d , a(•, ξ) is a measurable function and ξ -→ a(y, •) for fixed y ∈ R d is of class C 1 (R d ) and of class C 2 (R d \ {0}).

(2) the application ξ -→ a(y, ξ) is homogeneous i.e. a(y, tξ) = t p-1 a(y, ξ) for t ∈ R and y, ξ ∈ R d . We also assume that a(•, ξ) is a uniformly in ξ Lipschitz continuous function: there exists λ > 0 such that ∀y,

y ′ ∈ R d , ∀ξ ∈ R d , |a(y, ξ) -a(y ′ , ξ)| ≤ λ|y -y ′ ||ξ| p-1 . ∀y, y ′ ∈ R d , ∀ξ ∈ R d , |∂ ξ a(y, ξ) -∂ ξ a(y ′ , ξ)| ≤ λ|y -y ′ ||ξ| p-2 .
(3) we have that a(y, ξ) = a per (y, ξ) + a(y, ξ) where a per (•, ξ) is a periodic function satisfying the same homogeneity and regularity properties as a. We assume that the perturbation a satisfies:

∃b ∈ L p ′ ∩L ∞ (R d ), ∀ξ ∈ R d , ∀y ∈ R d , a(y, ξ) ≤ b(y)|ξ| p-1 and ∂ ξ a(y, ξ) ≤ b(y)|ξ| p-2 .
(4) There exists λ > 0 such that

{a(y, ξ) -a(y, ξ ′ )} • {ξ -ξ ′ } ≥ λ -1 |ξ| p-2 + |ξ ′ | p-2 |ξ -ξ ′ | 2 , {a per (y, ξ) -a per (y, ξ ′ )} • {ξ -ξ ′ } ≥ λ -1 |ξ| p-2 + |ξ ′ | p-2 |ξ -ξ ′ | 2 , and |a(y, ξ) -a(y, ξ ′ )| ≤ λ |ξ| p-2 + |ξ ′ | p-2 |ξ -ξ ′ | |a per (y, ξ) -a per (y, ξ ′ )| ≤ λ |ξ| p-2 + |ξ ′ | p-2 |ξ -ξ ′ |.
We also assume that sup

y∈R d sup |ξ|=1 ∂ 2 ξ a(y, ξ) ≤ λ. (5.2.35)
We define the operator

A :            W ξ+∇w per ξ -→ W ξ+∇w per ξ ′ ∇v -→    W ξ+∇w per ξ -→ R ∇h -→ ¢ R d a(•, ξ + ∇w per ξ + ∇v) -a(•, ξ + ∇w per ξ ) • ∇h.
(5.2.36) We can show that A is hemicontinuous, bounded, coercive and strongly monotone. By [76, Corollary 8.1], the PDE A(∇v) = F, where F := div a(•, ξ + ∇w per ξ ), admits a unique solution in W ξ+∇w per ξ . The results of Section 5.7, which are sufficient to prove the qualititative homogenization of Section 5.6 (which is in fact the main result of this paper), only use the PDE and are thus directly generalized. The results of Section 5.5 can be proved using the PDE instead of the minimization problem (5.4.24). These extensions are detailed in Appendix 5.8.3 of this chapter. Remark 5.2.14. A simple example of a non-variational operator satisfting the above assumptions is a(y, ξ) = A(y)ξ |ξ| p-2 , where A is a positive definite and bounded symmetric matrix that can be written under the form A = A per + A where A ∈ L p ′ ∩C 0,1 (R d ) d×d . We check that a is not variational: assume by contradiction that there exists a function

F : R d × R d → R such that a(y, ξ) = ∂ ξ F (y, ξ).
In particular, thanks to Schwartz Theorem, we should have that for all i, j ∈ {1, ..., d},

∂ ξj [a(y, ξ) i ] = ∂ ξi [a(y, ξ) j ] .
Expanding each term gives, for ξ ̸ = 0,

A(i, j)|ξ| p-2 + (p -2) [A(y)ξ] i ξ j |ξ| p-4 = A(j, i)|ξ| p-2 + (p -2) [A(y)ξ] j ξ i |ξ| p-4
In particular, for all ξ ̸ = 0 and (i, j) ∈ {1, ..., d} 2 ,

[A(y)ξ] i ξ j = [A(y)ξ] j ξ i .
This shows that A is a scalar matrix i.e. proportional to the identity.

The one-dimensional setting

We consider the homogenization of (5.1.1) in the one-dimensional case. This equation reads as:

-a(./ε)u ′ ε |u ′ ε | p-2 ′ = f u ε (-1 2 ) = u ε ( 1 2 ) = 0, (5.3.1)
where a is of the form a = a per + a with a ∈ L q ∩C 0,α (R), 1 < q < +∞ and a satisfies Assumption (A1).

In this section, we assume that f ∈ L p ′ (-1 2 , 1 2 ). Direct computations show that

u ′ ε = -F + C ε a(./ε) 1/(p-1) , F (x) = ¢ x -1 2 f, (5.3.2)
where

x 1 p-1 := sgn(x)|x| 1 p-1 for x ∈ R. The constant C ε is such that ¢ 1 2 -1 2 -F + C ε a(./ε) 1/(p-1) = 0. (5.3.3)
We note that the function F is bounded and thus the sequence (C ε ) ε>0 is bounded. Passing to the limit ε -→ 0 in (5.3.2) and (5.3.3), we get that u ε -⇀

ε→0 u * in W 1,p (-1 2 , 1 2 ) and C ε -→ ε→0 C * , where (u * ) ′ = -F + C * a * 1/(p-1)
,

¢ 1 2 -1 2 (-F + C * ) 1/(p-1) = 0.
and the homogenized coefficient is defined by

a * := L p -weaklim ε→0 a • ε -1 p-1 -(p-1)
.

We easily show with the ingredients used in Remark 5.3.2 below that

a * =   ¢ 1 2 -1 2 1 a per 1 p-1   -(p-1)
.

The homogenized equation solved by u * is

-a * (u * ) ′ |(u * ) ′ | p-2 ′ = f u * -1 2 = u * 1 2 = 0.
The corrector equations (5.2.9) and (5.2.12)-(5.2.13) in the direction ξ ∈ R are easy to solve (see Remark 5.3.2 below):

ξ + w ′ ξ = ξ a * a 1 p-1 and ξ + (w per ξ ) ′ = ξ a * a per 1 p-1 .
(

Let R ε := u ε -(u * ) ′ -w (u * ) ′ (./ε). be the remainder between u ε and its two scale expansion. When u * is regular enough, we have that

R ′ ε = (u 1 ε ) ′ -(u * ) ′ (1 + w ′ (./ε)) -εw(./ε)(u * ) ′′ = (-F + C ε ) 1/(p-1) -(-F + C * ) 1/(p-1) a(./ε) 1/(p-1) =:(u 1 ε ) ′ -εw(./ε)(u * ) ′′ , ( 5.3.5)
where w := w 1 . We concentrate in the sequel on the first term of (5.3.5), the second one being related to the regularity of u * on the one hand (which is not related to homogenization) and to the sublinearity of w on the other hand. We prove briefly that w is sublinear: indeed, we can write w ′ = (w per ) ′ + w ′ where, due to Remark 5.3.1 below, w ′ ∈ L q (R d ). By Hölder (or Morrey) inequality, we get immediately that w is sublinear. Since w per is periodic and bounded, it is in particular also sublinear. This proves that w is sublinear. We use Lemma 5.8.2 stated in Appendix 5.8.2 to obtain the bound

|u ′ ε -(u * ) ′ (1 + w ′ (./ε))| ≤ λ|C ε -C * | 1/(p-1) -→ ε→0 0 uniformly. (5.3.6)
We have obtained the L ∞ -strong convergence of (u 1 ε ) ′ to zero when we use the non-periodic corrector. Let us now introduce the "periodic" remainder R per ε which is defined by R per

ε := u ε -u * -εw per (u * ) ′ (./ε). We have that (R per ε ) ′ = (-F + C ε ) 1/(p-1) a(./ε) 1/(p-1) - (-F + C * ) 1/(p-1) a per (./ε) 1/(p-1) + εw per 1 (./ε)(u * ) ′′ = (u 1 ε ) ′ + (-F + C * ) 1/(p-1)
1 a(./ε) 1/(p-1) -1 a per (./ε) 1/(p-1) + εw per 1 (./ε)(u * ) ′′ .

(5.3.7)

The first term tends uniformly to zero while the second one does not tend to zero in L ∞ unless a = 0 or C * = 0. Indeed, testing (5.3.7) at the microscale gives:

(-F + C * ) 1/(p-1) 1 a(./ε) 1/(p-1) - 1 a per (./ε) 1/(p-1) (εx) ≥ c(p, λ)| -F (εx) + C * | 1/(p-1) | a(x)| -→ ε→0 c(p,λ)|C * | 1/(p-1) | a(x)|̸ =0
. This shows that the convergence of the remainder deteriorates when using w per ξ instead of w ξ . We close this section by commenting on the integrability of the correctors in the particular 1D setting. We show in Remark 5.3.1 that, in this case, the exponent given by Theorem 5.2.5 is optimal for q = p ′ , see also Remark 5.2.6.

Remark 5.3.1. Suppose that a ∈ L q (R d ) ∩ C 0,α (R d ), 1 < q < +∞. An explicit calculation shows that

w ξ ′ = -ξ + (w per ξ ) ′ + ξ + (w per ξ ) ′ 1 - a a 1 p-1 , ( 5
w ξ ′ ∼ x→±∞ - 1 p -1 a(ξ + (w per ξ ) ′ ) a .
Thus w ξ ′ ∈ L q (R d ), that is w ξ ′ has the same integrability as a and this exponent is optimal.

Remark 5.3.2. We show below that there exists a unique solution w ξ to (5.2.9) that is sublinear at infinity. This justifies, in dimension one, to search w ξ under the form w per ξ + w ξ where w ξ ′ ∈ L p (R).

Assume that w ξ is a sublinear solution to (5.3.4). Then, there exists a constant C such that ξ + w ′ ξ = (C/a) 1/(p-1) . We have by sublinearity that

ξ = lim x→+∞ 2 x 0 ξ + w ′ ξ = lim x→+∞ 2 x 0 C a 1 p-1 = C 1 p-1 lim x→+∞ 2 x 0 1 a 1 p-1 .
However, by Lemma 5.8.2, we have that

2 x 0 1 a 1 p-1 - 2 x 0 1 a per 1 p-1 ≤ 2 x 0 a 1 p-1 -(a per ) 1 p-1 a 1 p-1 (a per ) 1 p-1 ≤ Cst. 2 x 0 | a| 1 p-1 ,
where Cst. denotes a constant depending only on p and λ. Since a ∈ L p ′ (R d ), we get by Hölder inequality that

2 x 0 | a| 1 p-1 -→ x→+∞ 0.
This shows that

lim x→+∞ 2 x 0 1 a 1 p-1 = lim x→+∞ 2 x 0 1 a per 1 p-1 = 1 a * 1 p-1
and gives that C = ξ|ξ| p-2 a * . This shows that w ξ is necessarily of the form (5.3.4).

Numerical experiments. We have implemented for p = 3 the solution to (5.1.1) in the 1D setting for f (x) = 2x and a(y) := a per (y) + a(y) = 2 + cos(2πy) + 10e -|y| on the domain Ω := (-1 2 , 1 2 ). The boundary conditions are homogeneous Dirichlet conditions i.e. u ε (- 1 2 ) = u ε ( 1 2 ) = 0. The coefficient a satisfies of course Assumptions (A1)-(A3). The results are plotted on Figure 5.1. We comment on these results. We have plotted for different values of ε the function u ′ ε (which is labeled as 'exact solution'), the periodic two scale approximation (u * ) ′ + (w per ) ′ (./ε)(u * ) ′ (which is labeled as 'periodic two-scale approx.') and the non-periodic two scale approximation (u * ) ′ + w ′ (./ε)(u * ) ′ (which is labeled as 'non-periodic two-scale approx.'). Tables 5.1 and 5.2 give numerical values for the periodic and non-periodic remainders in L 2 and L ∞ -norm for different values of ε. We see that on Figure 5.1, qualitatively, the non-periodic twoscale approximation fits efficiently the exact solution for each chosen value of ε. The periodic two-scale approximation corresponds to the exact solution far from the defect, which, as ε -→ 0, concentrates aroung the origin. We notice that the non-periodic corrector is useful to reconstruct the oscillations of the exact solution locally around the defect. Tables 5.1 and 5.2 express the same idea: the L ∞ -norms of the periodic remainders remain unchanged as ε decreases whereas those of the nonperiodic remainder decrase with ε. For the L 2 -norm, which is weaker than the L ∞ -norm, both norms decrease as ε gets closer to zero although the nonperiodic approximation is more accurate than the periodic approximation. This means that, depending on the precision we want (and also on the regularity on f and a), we may use the periodic corrector, which is much easier to compute, or the non-periodic corrector, if we seek for a fine approximation of the exact solution. This can also be seen theoretically since R per ε -R ε = ε w ′ (./ε)(u * ) ′ and, for all q ≤ p,

ε w ′ (./ε)(u * ) ′ L q (0,1) ≤ Cε d/p ∥(u * ) ′ ∥ L ∞ (0,1) ∥ w ′ ∥ L p (R) .
In any case, we get that R per ε -R ε -→ ε→0 0 in L q -norm, q ≤ p but not in L ∞ -norm. Another way to reformulate the preceding remark is the following: the non-periodic corrector provides a better approximation at the microscale.

ε ∥R per ε ∥ L ∞ ∥R ε ∥ L ∞ 0.

Existence of the non-periodic correctors: proof of Theorem 5.2.3

We start this section with some preliminary results: (ii) Its topological dual space is

-div(g), g = g 1 + g 2 |ξ + ∇w per ξ | p-2 , g 1 ∈ L p ′ (R d ), g 2 |ξ + ∇w per ξ | p-2 2 ∈ L 2 (R d ) .
(iii) Each bounded sequence in W ξ+∇w per ξ admits a weakly converging subsequence. . This proves (i).

We denote dλ the Lebesgue measure over

R d , U := L p (R d , dλ) d and V := L 2 (R d , |ξ+∇w per ξ | p-2 dλ) d . It is clear that each function -div(g) where g = g 1 + g 2 |ξ + ∇w per ξ | p-2 , (g 1 , g 2 ) ∈ U ′ × V ′ defines an element of W ξ+∇w per ξ ′
by the following duality bracket:

⟨-div(g), v⟩ = ¢ R d g 1 • ∇v + ¢ R d g 2 • ∇v|ξ + ∇w per ξ | p-2 , v ∈ W ξ+∇w per ξ . Conversely, let T ∈ W ξ+∇w per ξ ′
. We consider the space U ×V endowed with the norm ∥(u, v)∥ U ×V := max(∥u∥ U , ∥v∥ V ). We introduce V := {∇v, v ∈ W ξ+∇w per ξ } and the following linear form T ′ (∇v, ∇v) := T (v) defined over D := V × V ⊂ U × V . By the Hahn-Banach Theorem, there exists

T ′ ∈ (U × V ) ′ such that T ′ = T ′ on D.
We have the decomposition

∀(u, v) ∈ U × V, T ′ (u, v) = T ′ (u, 0) + T ′ (0, v) =: T U (u) + T V (v),
with T U ∈ U ′ and T V ∈ V ′ . By the characterization of U ′ and V ′ , there exist when n -→ +∞. But this is straightforward with the characterization (ii) and the weak convergences obtained above.

g 1 ∈ L p ′ (R d ) d and g 2 ∈ L 2 (|ξ + ∇w per ξ | p-2 dλ) d such that ∀(u, v) ∈ U × V, T ′ (u, v) = ¢ R d g 1 • u + ¢ R d g 2 • v|ξ + ∇w per ξ | p-2 dλ.
We now fix ξ ∈ R d , h ∈ L p ′ (R d ) d , a coefficient a satisfying Assumptions (A1)-(A2)-(A3). We introduce the functional F ξ defined by

F ξ (v) := 1 p ¢ R d ag ξ+∇w per ξ (∇v) + ¢ R d h • ∇v, (5.4.2)
where the function g ξ is defined ny (5.8.8):

g ξ (x) := |ξ + x| p -|ξ| p -pξ|ξ| p-2 • x.
Since g ξ (x) ≥ 0 over R d , we immediately have that F ξ is defined over

V := v ∈ W 1,1 loc (R d ), ∇v ∈ L p (R d ) /R (5.4.3)
and takes its values in R ∪ {+∞}. Note that since F ξ (v) only depends on ∇v, F ξ is well-defined on the space of equivalence classes V . For R > 0, we define the mapping , we have thanks to (5.8.9) together with Hölder inequality that

F R ξ :    V -→ R v -→ 1 p ¢ B R ag ξ+∇w per ξ (∇v) + ¢ R d h • ∇v. ( 5
-∥h∥ L p ′ (R d ) ∥∇v∥ L p (R d ) + cλ -1 ¢ R d |∇v| p + |ξ + ∇w per ξ | p-2 |∇v| 2 ≤ F ξ (v) ≤ ∥h∥ L p ′ (R d ) ∥∇v∥ L p (R d ) + Cλ ¢ R d |∇v| p + |ξ + ∇w per ξ | p-2 |∇v| 2 .
(5.4.6)

We recall the following Young inequality

∥h∥ L p ′ (R d ) ∥∇v∥ L p (R d ) ≤ 1 p ′ ∥h∥ p ′ L p ′ (R d ) + 1 p ∥∇v∥ p L p (R d ) (5
F ξ (v) ≤ 1 p ′ ∥h∥ p ′ L p ′ (R d ) + (2Cλ + 1) v p W ξ+∇w per ξ .
This proves the right-most inequality of (5.4.5) after changing the constant C. For the left-most inequality, we write that, again by Young inequality

- λ 2c ∥h∥ p ′ L p ′ (R d ) + cλ -1 2 ¢ R d |∇v| p + cλ -1 ¢ R d |ξ + ∇w per ξ | p-2 |∇v| 2 ≤ F ξ (v).
Besides, noticing that

¢ R d |∇v| p 2/p ≤ ¢ R d |∇v| p + 1,
we get the lower bound

1 2 ∥v∥ 2 W ξ+∇w per ξ -1 ≤ ¢ R d |∇v| p + ¢ R d |ξ + ∇w per ξ | p-2 |∇v| 2 .
Thus,

- λ 2c ∥h∥ p ′ L p ′ (R d ) + cλ -1 2 + cλ -1 4 ∥v∥ 2 W ξ+∇w per ξ ≤ F ξ (v).
After changing the constant c, we get (5.4.5). This proves (i).

We prove (ii). Since the function z -→ |z| p is stricly convex, it is clear that the application x -→ |ξ + x| p is strictly convex. Thus, g ξ+∇w per ξ (see (5.8.8) for the definition) is strictly convex. This implies that for all R > 0,

F R ξ ((1 -t)v 1 + tv 2 ) ≤ (1 -t)F R ξ (v 1 ) + tF R ξ (v 2 ) ≤ (1 -t)F ξ (v 1 ) + tF ξ (v 2 ).
(5.4.9)

Letting R -→ +∞ in (5.4.9) gives by the monotone convergence Theorem that

F ξ ((1 -t)v 1 + tv 2 ) ≤ (1 -t)F ξ (v 1 ) + tF ξ (v 2 ) (5.4.10)
hence F ξ is convex. As for the strict convexity, suppose that F ξ (v i ) < +∞, i = 1, 2 and that equality holds in (5.4.10) for some t ∈ (0, 1). Then as all functions are integrable,

¢ R d (1 -t)g ξ+∇w per ξ (∇v 1 ) + tg ξ+∇w per ξ (∇v 2 ) -g ξ+∇w per ξ ((1 -t)∇v 1 + tv 2 ) = 0.
By positivity, we get that (1 -t)g ξ+∇w per . We have that We note that, using the definition of g ξ (5.8.8),

F ξ (v + u) -F ξ (v) = 1 p ¢ R d a g ξ+∇w per ξ (∇v + ∇u) -g ξ+∇w per ξ (∇v) + ¢ R d h • ∇u. ( 5
A = g ξ+∇w per ξ +∇v (∇u) ≤ C ∇u p + ξ + ∇w per ξ + ∇v| p-2 ∇u 2 ,
where we have used the right-most part of inequality (5.8.9). Thus, applying the inequality (b

1 + b 2 ) p-2 ≤ C(p)(b p-2 1 + b p-2 2 ) for b 1 , b 2 ≥ 0, we get that |A| ≤ C |∇u| p + |ξ + ∇w per ξ | p-2 |∇u| 2 + |∇v| p-2 |∇u| 2 .
(5.4.16)

We now note that, due to Hölder inequality and the fact that where the constant C is independent of v and u. We can now conclude the proof of Lemma 5.4.3: using (5.4.12) and the notations (5.4.14) and (5.4.15), we have that

p 2 ′ p 2 -1 = p 2 =⇒ p 2 ′ p -2 = p, we obtain ¢ R d |∇v| p-2 |∇u| 2 ≤ ¢ R d |∇v| p 1-2/p ¢ R d |∇u| p 2/p . ( 5
F ξ (v + u) -F ξ (v) - 1 p ¢ R d B + ¢ R d h • ∇u = 1 p ¢ R d A. (5.4.21) Defining L v (u) := 1 p ¢ R d B + ¢ R d h • ∇u = ¢ R d a (ξ + ∇w per ξ + ∇v)|ξ + ∇w per ξ + ∇v| p-2 -(ξ + ∇w per ξ )|ξ + ∇w per ξ | p-2 • ∇u + ¢ R d h • ∇u
and noting that, thanks to (5.4.20), L v is a bounded linear form on W ξ+∇w per ξ , we have, gathering (5.4.21) and (5.4.18) together, , it is in particular weakly lower semi-continuous. Thus

F ξ (v + u) -F ξ (v) -L v (u) = O u→0 ∥u∥
F ξ (v) ≤ lim inf n→+∞ F ξ (v n ) = inf W ξ+∇w per ξ F ξ .
This concludes the existence of a solution to (5.4.24). The uniqueness is given by the strict convexity of F ξ , see Lemma 5.4.2 (ii). We finally note that the convexity of F ξ together with its differentiability ensure that being a solution to Problem (5. We can prove that F ξ is weakly lower semi-continuous on V . Indeed, let us consider a sequence (∇v n ) n∈N that weakly converges to some ∇v in L p (R d ). We aim at proving that

F ξ (v) ≤ lim inf n→+∞ F ξ (v n ).
(5.4.25)

We recall the notation (5.4.4). It is straightforward to check that F R is convex and strongly continuous. Thus, F R is weakly lower semi-continuous. However, for fixed n ∈ N, we have

F R ξ (v n ) ≤ F ξ (v n ). Thus, F R ξ (v) ≤ lim inf n→+∞ F R ξ (v n ) ≤ lim inf n→+∞ F ξ (v n ). ( 5
∈ R d , Φ ′′ γ (Z)(h, h) ≥ p|ξ + T + Z| p-2 |h| 2 + p|η + T -Z| p-2 |h| 2 -γ p p|Z| p-2 |h| 2 -γ p p(p -2)|Z| p-4 (Z • h) 2 ≥ p |ξ + T + Z| p-2 + |η + T -Z| p-2 -γ p (p -1)|Z| p-2 |h| 2 .
(5.5.5) We next note that

|Z| p-2 = 1 2 (Z + ξ + T ) + 1 2 (Z -η -T ) + 1 2 (η -ξ) p-2 ≤ C(p) |ξ + T + Z| p-2 + |η + T -Z| p-2 + |ξ -η| p-2 ,
(5.5.6)

where we have used the triangle inequality together with the fact that for all m ≥ 1 and p ≥ 2, there exists a constant C(p, m) such that

∀a 1 , ..., a m ≥ 0, (a 1 + • • • + a m ) p-2 ≤ C(p, m) a p-2 1 + • • • + a p-2 m .
Estimate (5.5.6) together with inequality (5.5.5) give that

∀h ∈ R d , Φ ′′ γp (Z)(h, h) ≥ -p|ξ -η| p-2 |h| 2 for γ p := 1 C(p)(p -1)
.

The function Φ γp + p 2 |ξ -η| p-2 | • | 2 is convex, hence ∀Z ∈ R d , Φ γp (Z) + p 2 |ξ -η| p-2 |Z| 2 ≥ Φ γp (0) + ∇Φ γp (0) • Z.
We have thus proved that

Φ γp (Z) ≥ |ξ + T | p + |η + T | p + p (ξ + T )|ξ + T | p-2 -(η + T )|η + T | p-2 • Z - p 2 |ξ -η| p-2 |Z| 2 .
This proves estimate (5.5.4) if T = 0. If T ̸ = 0, it remains to prove that

(ξ + T )|ξ + T | p-2 -(η + T )|η + T | p-2 -ξ|ξ| p-2 + η|η| p-2 ≤ c p δ p-3 |ξ -η||T |. (5.5.7)
We want to apply the mean-value inequality to the function Ψ T defined by

Ψ T (x) := |x + T | p-2 (x + T ) -x|x| p-2 , x ∈ [ξ, η] ⊂ R d \ B(0, δ/2), which is differentiable over R d . We have that Ψ ′ T (x) = |x + T | p-2 -|x| p-2 I + (x + T ) ⊗ (x + T )|x + T | p-4 -x ⊗ x|x| p-4 .
We now note that there exists a constant C p > 0 such that for all

x ∈ R d \ B(0, δ/2), |x + T | p-2 -|x| p-2 ≤ C p δ p-3 |T | + |x| p-3 |T | (5.5.8) and (x + T ) ⊗ (x + T )|x + T | p-4 -x ⊗ x|x| p-4 ≤ C p δ p-3 |T | + |x| p-3 |T | .
(5.5.9)

Noting that |x| p-3 ≤ ( 1 2 ) p-3 |δ| p-3 since p ≤ 3, we have proved (5.5.7). The proof of Lemma 5.5.1 is completed up to the justification of (5.5.8)-(5.5.9).

Proof of (5.5.8) and (5.5.9). We concentrate on the first inequality: assume first that

|T | ≥ 1 2 |x| ≥ 1 4 |δ|, then |x + T | p-2 -|x| p-2 ≤ C p |T | p-2 ≤ C p δ p-3 |T |.
(5.5.10)

thus by [START_REF] Avellaneda | Un théorème de liouville pour des équations elliptiques à coefficients périodiques[END_REF], we get that ∇ w ξ ∈ L max(p ′ ,p/4) (R d ) d and we can prove, similarly to (5.5.23) that .5.24) where the constant on the right-hand side of (5.5.24) is potentially greater than the one on the righthand side of (5.5.23) but the dependance on the data remains the same. If p ′ ≥ p/4, the Theorem is proved. Otherwise, we iterate similarly. The procedure ends at step k for which p/2 k ≤ p ′ : we thus obtain that ∇ w ξ ∈ L p ′ (R d ) d and that there exists a constant

∥∇ w ξ ∥ L max(p ′ ,p/4) (R d ) ≤ C ∥ a∥ L p ′ (R d ) , λ, d, p, α, c, C per ∞ , C ∞ , C p , ( 5
C f inal := C a, λ, d, p, α, c, C per ∞ , C ∞ , C p such that ∥∇ w ξ ∥ L p ′ (R d ) ≤ C f inal .
Theorem 5.2.5 is proved.

Proof of Theorem 5.2.4

Proof of (i). This is due to Proposition 5.

¢ R d |∇ w ξ | p ≤ ∥f ∥ L p ′ (R d ) ∥∇ w ξ ∥ L p (R d ) ≤ ∥ a∥ L p ′ (R d ) ∥ξ + ∇w per ξ ∥ p-1 L ∞ (Q) ∥∇ w ξ ∥ L p (R d ) .
(5.5.25) Thus, by Proposition 5.2.1 (ii) and (5.5.25), we obtain the first estimate of (5.2.16).

We show that there exists α > 0 independent of ξ such that ∇w ξ ∈ C 0,α (R d ). We introduce the function w ξ := ξ • x + w ξ , then ∇w ξ solves the standard homogeneous p-Laplace equation with varying coefficient a. Applying [74, Theorem 1], we get that ∇w ξ is continuous over R d . Besides, by [START_REF] Kuusi | A nonlinear stein theorem[END_REF]Theorem 4], there exists a constant c ≥ 1 and a radius r > 0 depending only on d, p, λ and the Lipschitz constant of a, denoted a Lip such that for all x ∈ R d ,

|∇w ξ (x)| ≤ c 2 B(x,r) |∇w ξ | p ′ 1/p ′ ≤ c 2 B(x,r) |∇w ξ | p 1/p .
(5.5.26)

Due to the form of ∇w ξ , see also (5.2.14) and the first estimate of (5.2.16), we have that To specify the dependence of C 0 in ξ, we first take |ξ| = 1 and we then apply the homogeneity, Theorem 5.2.4 (i). This gives that C 0 = C 0 (p, d, λ, a Lip )|ξ| and concludes the proof of (ii), gathering (5.5.27) and (5.5.28) and the fact that

|∇w ξ (x)| ≤ c|ξ| + cr -d/p ∥∇ w ξ ∥ L p (R d ) ≤ C(d, p, λ, a Lip )|ξ|. ( 5
∥∇ w ξ ∥ C 0,α (R d ) ≤ |ξ| + ∥∇w ξ ∥ C 0,α (R d ) .
Proof of (iii). We assume that 2 ≤ p < 3. Let us fix ξ ∈ R d such that |ξ| = 1. In the proof, c > 0 will denote a universal constant given by (A4). We consider η ∈ R d such that ξ ̸ = η. In the sequel, we fix δ 0 ∈ (0, 1) such that C(1 + 2 1-γ )δ γ 0 + δ 0 ≤ c/2, where C and γ are given by (5.2.8). Case 1. We assume that |ξ -η| ≥ δ 0 . Then, thanks to Theorem 5.2.4 (ii), we have that

∥∇ w ξ -∇ w η ∥ L p (R d ) ≤ C p + C p |η|.
(5.5.29)

We now note that for all 0 < β ≤ 1,

C p + C p |η| ≤        C p δ β 0 |ξ -η| β (1 + |η|) ≤ 2 β C p δ β 0 |ξ -η| β (1 + |η| 1-β ) if |η| ≤ 2. C p C( β) |η| -1 β (1 + |η| 1-β ) ≤ C p C( β)|ξ -η| β (1 + |η| 1-β ) if |η| > 2,
(5.5.30)

where we used that the function x .5.31) This gives (5.2.17).

→ 1+x |x-1| β (1+x 1-β ) is bounded on [2, +∞[. Thus ∥∇ w ξ -∇ w η ∥ L p (R d ) ≤ C(δ 0 , β, C p )|ξ -η| β (1 + |η| 1-β ). ( 5
Case 2. We assume that |ξ -η| < δ 0 . Then, by the choice of δ 0 and Proposition 5.2.1 (iv), we have that

ξ + ∇w per ξ -η + ∇w per η L ∞ (Q) ≤ c 2 and |ξ + ∇w per ξ | ≥ c. ( 5.5.32)
Recalling the notation (5.4.2), we have that

F ξ (∇ w ξ ) + F η (∇ w η ) < F ξ ∇ w ξ + ∇ w η 2 + F η ∇ w ξ + ∇ w η 2 < +∞, (5.5.33)
where we have used that ξ ̸ = η, F z admits a unique minimizer for z ∈ R d and ∇

w ξ ∈ L 2 (R d ), ∇ w η ∈ L 2 (R d ). We recall that F R z (∇v) := ¢ B R ag z+∇w per z (∇v) + ¢ R d f z • ∇v, z ∈ R d , ∇v ∈ L p (R d ) (5.5.34)
and that R -→ F R z (∇v) is a non-decreasing function. Thus, for R large enough, we have the inequality

F R ξ (∇ w ξ ) + F R η (∇ w η ) -F R ξ ∇ w ξ + ∇ w η 2 -F R η ∇ w ξ + ∇ w η 2 ≤ 0. ( 5
G ξ+∇w per ξ ,η+∇w per η (∇ w ξ ,∇ w η ) ≥ γ p |∇ w ξ -∇ w η | p -c p |ξ + ∇w per ξ -(η + ∇w per η )| p-2 ||∇ w ξ -∇ w η | + c p-3 |ξ + ∇w per ξ -(η + ∇w per η )||∇ w ξ + ∇ w η | |∇ w ξ -∇ w η |.
(5.5.36) For all R > 0, we can integrate (5.5.36) over the ball B R . Using the notation (5.5.34) and the form of the map G ξ,η (X, Y ), see (5.5.1), this yields

F R ξ (∇ w ξ ) + F R η (∇ w η ) -F R ξ ∇ w ξ + ∇ w η 2 -F R η ∇ w ξ + ∇ w η 2 - 1 2 ¢ R d (h ξ -h η ) • (∇ w ξ -∇ w η ) ≥ γ p ¢ B R a|∇ w ξ -∇ w η | p -c p ¢ B R a |ξ + ∇w per ξ -(η + ∇w per η )| p-2 ||∇ w ξ -∇ w η | + c p-3 |ξ + ∇w per ξ -(η + ∇w per η )||∇ w ξ + ∇ w η | |∇ w ξ -∇ w η |, (5.5.37)
where h z = a(z + ∇w per z )|z + ∇w per z | p-2 for z ∈ R d . For R large enough, we get because of (5.5.35) that

- 1 2 ¢ R d (h ξ -h η ) • (∇ w ξ -∇ w η ) ≥ γ p ¢ B R a|∇ w ξ -∇ w η | p -c p ¢ B R a |ξ + ∇w per ξ -(η + ∇w per η )| p-2 • |∇ w ξ -∇ w η | + c p-3 |ξ + ∇w per ξ -(η + ∇w per η )||∇ w ξ + ∇ w η | |∇ w ξ -∇ w η |, (5.5
.38) Letting R -→ +∞ in (5.5.38) and using Theorem 5.2.5, we get by the monotone convergence Theorem that

- 1 2 ¢ R d (f ξ -f η )•(∇ w ξ -∇ w η ) ≥ γ p ¢ R d |∇ w ξ -∇ w η | p -c p ¢ R d |ξ + ∇w per ξ -(η + ∇w per η )| p-2 • |∇ w ξ -∇ w η | + c p-3 |ξ + ∇w per ξ -(η + ∇w per η )||∇ w ξ + ∇ w η | |∇ w ξ -∇ w η |.
Thus, applying the Hölder inequality, Proposition 5.2.1 (iv) and Theorem 5.2.5 under the form 2) . This gives (5.2.17) when |ξ| = 1. The case |ξ| ̸ = 1 is treated by homogeneity.

∥∇ w z ∥ L p ′ (R d ) ≤ C|z|, z ∈ R d , we get ¢ R d |∇ w ξ -∇ w η | p ≤ C ∥ a∥ L p ′ |ξ -η| γ ∥∇ w ξ -∇ w η ∥ L p (R d ) + c p C|ξ -η| γ(p-2) + c p-3 C|ξ -η| γ ∥∇ w ξ -∇ w η ∥ L p . Thus ∥∇ w ξ -∇ w η ∥ p-1 L p (R d ) ≤ C|ξ -η| γ(p-
Gathering Case 1 and Case 2, we have proved Theorem 5.2.4 (iii) for p ∈ [2, 3). The proof of the case p ≥ 3 is performed using the same method and (5.5.3).

Remark 5.5.3. As suggested by (5.5.3), the assumptions of Theorem 5.2.4(iii) may be weakened when p ≥ 3. In this case, it is sufficient to assume, instead of (A4), that ∇ w ξ ∈ L p ′ (R d ) , that ∇w η ∈ L p ′ (R d ) and that we have an estimate of the form (5.2.19).

Proof of (iv).

It is analogous to the proof of Proposition 5.2.1 (iv).

Qualitative Homogenization: proof of Theorem 5.2.7

The proof of Theorem 5.2.7 is an adaptation of [START_REF] Fusco | On the homogenization of quasilinear divergence structure operators[END_REF] and [START_REF] Dal | Correctors for the homogenization of monotone operators[END_REF]Theorem 2.1] to the present setting. We start with the following central Lemma: Lemma 5.6.1. For ξ ∈ R d , let us write ∇ w ξ the solution to (5.2.12)-(5.2.13) given by Theorem 5.2.3. Assume that the application

R d -→ L p unif (R d ) ξ -→ ∇ w ξ (5.6.1) is continuous. Then for all Ψ ∈ L p (Ω) d , lim sup ε→0 ¢ Ω ∇ w MεΨ • ε p = 0. (5.6.2)
Proof of Lemma 5.6.1. We first show the following assertion: We now turn to the proof of (5.6.2). By an immediate application of the Jensen inequality, we have that

∀δ > 0, ∃A > 0, ∀|x| ≥ A, ∀ξ ∈ R d , ∥∇ w ξ ∥ L p (x+Q) ≤ δ|ξ|. ( 5
∀B ∈ N ∪ {+∞}, |k|<B, ε(Q+k)⊂Ω ε d Ψ k ε p ≤ ¢ Ω∩B∞(0,εB) |Ψ| p , Ψ k ε := 2 ε(Q+k) Ψ, (5.6.4)
where B ∞ (x, r) denotes the ball centered in x and of radius r > 0 for the | • | ∞ -norm on R d . Let δ > 0 and A be given by (5.6.3). We have that (5.6.5)

¢ Ω ∇ w MεΨ • ε p = k∈Z d , ε(Q+k)⊂Ω ε d ¢ Q+k ∇ w Ψ k ε p ≤ |k|<A, ε(Q+k)⊂Ω ε d ¢ Q+k ∇ w Ψ k ε p + |k|≥A, ε(Q+k)⊂Ω ε d ¢ Q+k ∇ w Ψ k ε p ≤ (5.6.3),(5.2.16) C |k|<A, ε(Q+k)⊂Ω ε d |Ψ k ε | p + δ p |k|≥A, ε(Q+k)⊂Ω ε d |Ψ k ε | p ≤ ( 5
By the dominated convergence Theorem, we have that

lim sup ε→0 ¢ Ω ∇ w MεΨ • ε p ≤ δ p ¢ Ω |Ψ| p .
(5.6.6) Since (5.6.6) is true for all δ > 0, we haved proved (5.6.2).

We now state the analogous of [START_REF] Dal | Correctors for the homogenization of monotone operators[END_REF]Lemma 3.5] to the present non-periodic setting. Before that, we introduce for ξ, y ∈ R d the notations p per (y, ξ) := ξ + ∇w per ξ (y) and p(y, ξ) := ξ + ∇w ξ (y) = p per (y, ξ) + ∇ w ξ (y).

(5.6.7) Lemma 5.6.2. Assume that the Assumptions of Lemma 5.6.1 are satisfied. Let Ψ ∈ L p (Ω) and

Φ ∈ L p (Ω) such that Φ = m j=1 η j 1 Ωj where m j=1 Ω j ⊂⊂ Ω, Ω k ∩ Ω ℓ = ∅ for k ̸ = ℓ and |∂Ω j | = 0 for j ∈ {1, m}. Then there exists a constant C > 0 independent of ε, Ψ and Φ such that lim sup ε→0 p(•/ε, M ε Ψ) -p(./ε, Φ) L p (Ω) ≤ C ∥Ψ∥ 1-β L p (Ω) + ∥Φ∥ 1-β L p (Ω) ∥Ψ -Φ∥ β L p (Ω) , (5.6.8)
where β is given by Proposition 5.2.1 (iii).

Proof of Lemma 5.6.2. We first notice that

¢ Ω ∇ w Φ • ε p = m j=1 ¢ Ωj ∇ w ηj • ε p ≤ ε d m j=1 ¢ R d ∇ w ηj p -→ ε→0 0. ( 5
p(•/ε, M ε Ψ) -p(./ε, Φ) L p (Ω) ≤ lim sup ε→0 p per (•/ε, M ε Ψ) -p per (./ε, Φ) L p (Ω)
≤RHS of (5.6.8)

+ lim sup ε→0 ∇ w MεΨ (./ε) L p (Ω) + ∇ w Φ (./ε) L p (Ω)
=0 by Lemma 5.6.1 and (5.6.9)

.

With these tools, we can prove Theorem 5.2.7. The first point (i) is detailed in Appendix 5.8.4 since it is mainly a rewriting of [START_REF] Fusco | On the homogenization of quasilinear divergence structure operators[END_REF]. Note that for this point, the continuity of ξ -→ ∇ w ξ is not neeeded. The only result on the non-periodic correctors ∇w ξ , ξ ∈ R d that is used is Theorem 5.2.4 (ii). The proof of Theorem 5.2.7 (ii) follows the proof of [44, Theorem 2.1]. In the following, we sketch the proof of Theorem 5.2.7 (ii) by insisting on the points that differ from [START_REF] Dal | Correctors for the homogenization of monotone operators[END_REF]. The proof of Theorem 5.2.7 (iii) follows from Theorem 5.2.7 (ii) together with Lemma 5.6.1.

Sketch of proof of Theorem 5.2.7 (ii).

Since M ε ∇u * converges to ∇u * when ε → 0 in L p (Ω), it is sufficient to show, using the notation (5.6.7) that

R ε := ∇u ε -p(./ε, M ε ∇u * ) -→ ε→0 0 in L p (Ω).
(5.6.10)

During the proof, we introduce a step function Φ as in Lemma 5.6.2 satisfying ∥∇u * -Φ∥ L p (Ω) ≤ δ. By monotonicity of the p-Laplace operator, see (5.8.5), and Assumption (A1), we have that

λ -1 c∥R ε ∥ p L p (Ω) ≤ ¢ Ω a(./ε)|∇u ε | p-2 ∇u ε -a(./ε)|p(./ε, M ε ∇u * )| p-2 p(./ε, M ε ∇u * ), ∇u ε -p(./ε, M ε ∇u * ) = A ε -B ε -C ε + D ε , ( 5
.6.11) where

A ε := ¢ Ω a(./ε)|∇u ε | p , B ε := ¢ Ω a(./ε)|∇u ε | p-2 ∇u ε • p(./ε, M ε ∇u * ) C ε := ¢ Ω a(./ε)|p(./ε, M ε ∇u * )| p-2 p(./ε, M ε ∇u * ) • ∇u ε and D ε := ¢ Ω a(./ε)|p(./ε, M ε ∇u * )| p . The term A ε is obviously treated by the L p -weak convergence u ε -⇀ ε→0 u * : A ε = ¢ Ω f u ε -→ ε→0 ¢ Ω f u * = ¢ Ω a * (∇u * ) • ∇u * .
(5.6.12)

We study the term B ε when M ε ∇u * is replaced by Φ. This gives:

¢ Ω a(./ε)|∇u ε | p-2 ∇u ε • p(./ε, Φ) = m j=1 ¢ Ωj a(./ε)|∇u ε | p-2 ∇u ε • p(./ε, η j ).
We then apply the standard div-curl Lemma, keeping in mind that a(./ε)|∇u ε | p-2 ∇u ε converges L p ′ (Ω)-weakly to a * (∇u * ) when ε → 0, that p(./ε, η j ) converges L p -weakly to η j and that, thanks to Theorem 5.2.4 (ii), a(.

/ε)|∇u ε | p-2 ∇u ε • p(./ε, η j ) is bounded in L p ′ (Ω), uniformly with respect to ε. Thus, m j=1 ¢ Ωj a(./ε)|∇u ε | p-2 ∇u ε • p(./ε, η j ) -→ ε→0 m j=1 ¢ Ωj a * (∇u * ) • η j = ¢ Ω a * (∇u * ) • Φ.
In view of Lemma 5.6.

D ε -D per ε ≤ ¢ Ω a • ε p • ε , M ε ∇u * p + C ¢ Ω a per • ε p p • ε , M ε ∇u * p-1 + p per • ε , M ε ∇u * p-1 ∇ w Mε∇u * • ε .
(5.6.16) We show that each term of the RHS of (5.6.16) vanishes as ε -→ 0. We use Theorem 5.2.4 (ii) and (5.6.4) with B = +∞, which imply that there exists a constant C > 0 independent of ε such that

¢ Ω p • ε , M ε ∇u * p + p per • ε , M ε ∇u * p ≤ C∥∇u * ∥ p L p (Ω) .
(5.6.17)

With the Hölder inequality and Lemma 5.6.1, we prove that the second term of the RHS of (5.6.16) tends to zero as ε -→ 0. As for the first term, we write that

¢ Ω a • ε p • ε , M ε ∇u * p ≤ C ¢ Ω a • ε p • ε , Φ p + C ¢ Ω a • ε p • ε , M ε ∇u * -p • ε , Φ p ≤
(5.2.16),(5.6.17)

C a • ε ∥ L 1 (Ω) =O(ε d/p ′ ) ∥Φ∥ p L ∞ (R d ) + C∥ a∥ L ∞ (R d ) p • ε , M ε ∇u * -p • ε , Φ p L p (R d ) ,
(5.6.18) where we used that a ∈ L ∞ (R d ) and the bound |p(y, ξ)| ≤ C|ξ| where C > 0 is independent of y and ξ. Collecting (5.6.15), (5.6.16), (5.6.18) and Lemma 5.6.2, we have proved that lim sup

ε→0 D ε - ¢ Ω a * (∇u * ) • ∇u * = O(δ pβ ).
(5.6.19)

Finally, collecting (5.6.12), (5.6.13), (5.6.14), (5.6.19) and (5.6.11), we obtain that lim sup

ε→0 ∥R ε ∥ p L p (Ω) ≤ ¢ Ω |a * (∇u * ) • (∇u * -Φ)| + ¢ Ω |(a * (∇u * ) -a * (Φ)) • ∇u * | + O(δ β ). (5.6.20)
Using the following property of a * , see [START_REF] Dal | Correctors for the homogenization of monotone operators[END_REF]Remark 1.3],

a * (∇u * ) -a * (Φ) L p ′ (Ω) ≤ C ∥∇u * ∥ p-2 L p (Ω) + ∥Φ∥ p-2 L p (Ω) ∥∇u * -Φ∥ L p (Ω) ,
we conclude that lim sup ε→0 ∥R ε ∥ p L p (Ω) = O(δ β ) where the O is independent of δ. Since this is true for all δ > 0, we conclude that ∥R ε ∥ L p (Ω) -→ ε→0 0.

Remark 5.6.3. Using the same strategy as above, it is straightforward to show that Theorem 5.2.7 holds with the operator M ε replaced by M ε ν , 0 < ν < 1. In this case, the continuity of the application ξ → ∇ w ξ is not needed and we only use that ∇ w ξ ∈ L p (R d ).

5.7 Continuity of ξ → ∇w ξ : proof of Theorem 5.2.8

Preliminary Lemmas

We begin this section with the following lemma that defines weak solution of PDEs of the form (5. Proof of Lemma 5.7.1. We define u := ξ + ∇w per ξ in the proof. We fix w ∈ W u . In the following, χ will denote a smooth and compactly supported function with support in Q(0, 1) such that χ = 1 in Q(0, 1 2 ). We fix R > 0 and we introduce the function

Φ R := w - 2 Q R \Q R/2 w χ • R .
By the Poincaré-Wirtinger inequality, we have that Φ R ∈ W 1,p 0 (Q R ). By (5.8.7), we have the bound

a (u + ∇v)|u + ∇v| p-2 -u|u| p-2 ≤ λC |u + ∇v| p-2 + |u| p-2 |∇v| ≤ λC(p) |u| p-2 |∇v| + |∇v| p-1 , (5.7.3)
where we have used the inequality (b

1 + b 2 ) p-2 ≤ C(p)(b p-2 1 + b p-2 2 ) for b 1 , b 2 ≥ 0. Thus, since ∇v ∈ L p (Q R ) and p = p ′ (p -1), div a (u + ∇v)|u + ∇v| p-2 -u|u| p-2 ∈ W -1,p ′ (Q R ) and div(h) ∈ W -1,p ′ (Q R ).
Consequently, we can test (5.7.1) against Φ R and obtain, after expanding ∇Φ R ,

¢ Q R a (u + ∇v)|u + ∇v| p-2 -u|u| p-2 • χ • R ∇w + 1 R ¢ Q R a (u + ∇v)|u + ∇v| p-2 -u|u| p-2 • w - 2 Q R \Q R/2 w ∇χ • R = - ¢ Q R h • χ • R ∇w - 1 R ¢ Q R h • w - 2 Q R \Q R/2 w ∇χ • R .
(5.7.4)

We now recall the following Poincaré-Wirtinger inequality:

w - 2 Q R \Q R/2 w L p (Q R \Q R/2 ) ≤ CR∥∇w∥ L p (Q R \Q R/2 ) (5.7.5)
which is simply a rescaled version of the L p inequality on Q \ Q 1/2 . Besides, thanks to Assumption (A4)' (and its rescaled version), we have that

|u| p-2 2 w - 2 Q R \Q R/2 w L 2 (Q R \Q R/2 ) ≤ CR∥|u| p-2 2 ∇w∥ L 2 (Q R \Q R/2 ) .
This yields, together with (5.7.3), Hölder inequality and the inclusion supp(∇χ

) ⊂ Q R \ Q R/2 , that ¢ Q R \Q R/2 a (u + ∇v)|u + ∇v| p-2 -u|u| p-2 • w - 2 Q R \Q R/2 w ∇χ • R ≤ λCR∥∇χ∥ L ∞ ∇v p-1 L p (Q R \Q R/2 ) ∇w L p (Q R \Q R/2 ) + |u| p-2 2 ∇v L 2 (Q R \Q R/2 ) |u| p-2 2 ∇w L 2 (Q R \Q R/2 )
(5.7.6) and

¢ Q R h • w - 2 Q R \Q R/2 w ∇χ • R ≤ CR∥∇χ∥ L ∞ ∥h∥ L p ′ (Q R \Q R/2 ) ∥∇w∥ L p (Q R \Q R/2 ) . (5.7.7)
Collecting (5.7.4), (5.7.6) and (5.7.7) and recalling that v, w ∈ W u , we have that

¢ R d χ • R a (u + ∇v)|u + ∇v| p-2 -u|u| p-2 + h • ∇w -→ R→+∞ 0.
(5.7.8)

On the other hand, by the dominated convergence Theorem, again since v, w ∈ W u , we have that

¢ R d χ • R a (u + ∇v)|u + ∇v| p-2 -u|u| p-2 + h • ∇w -→ R→+∞ ¢ R d a (u + ∇v)|u + ∇v| p-2 -u|u| p-2 + h • ∇w.
(5.7.9) Thus (5.7.2) is satisfied.

The next lemma allows to pass to the limit in PDEs of the form (5.7.1).

Lemma 5.7.2. Let (∇ϕ

n ) n∈N ⊂ L ∞ (R d ) d , (a n ) n∈N ⊂ L ∞ (R d ), (h n ) n∈N ⊂ L p ′ (R d ) d and (v n ) n∈N ⊂ V (see (5.4. 3
)), such that v n ∈ W ∇ϕn for all n ∈ N. We assume that, for all n ∈ N:

1. The coefficient a n satisfies Assumption (A1) with λ uniformly bounded in n.

2. The function ∇v n is solution, in the distribution sense, to

-div a n (∇ϕ n + ∇v n )|∇ϕ n + ∇v n | p-2 -∇ϕ n |∇ϕ n | p-2 = div(h n ).
(5.7.10)

We also assume the following convergences: By (i), we obtain that ∇v n -→ n→+∞ ∇v in L p (B) up to a subsequence. We easily show that the convergence in fact holds for the whole sequence. We consequently get the L p loc (R d )-convergence since B is arbitrary. We now pass to the limit n → +∞ in (5.7.10). Let Ψ ∈ D(R d ), we test (5.7.10) against Ψ. By (iv), it is clear that

(i) ∇ϕ n -→ n→+∞ ∇ϕ in L ∞ (R d ); (ii) a n -→ n→+∞ a in L ∞ loc (R d ); (iii) ∇v n -⇀ n→+∞ ∇v in L p (R d ) and |∇ϕ n | p-2 2 ∇v n -⇀ n→+∞ |∇ϕ| p-2 2 ∇v in L 2 (R d ) which v ∈ W ∇ϕ ; (iv) h n -→ n→+∞ h in L p ′ loc (R d ) with h ∈ L p ′ (R d ).
¢ R d h n • ∇Ψ -→ n→+∞ ¢ R d h • ∇Ψ. ( 5
a n (∇ϕ n + ∇v n )|∇ϕ n + ∇v n | p-2 -∇ϕ n |∇ϕ n | p-2 -→ n→+∞ a (∇ϕ + ∇v)|∇ϕ + ∇v| p-2 -∇ϕ|∇ϕ| p-2 in L p ′ loc (R d ). This shows that ¢ R d a (∇ϕ + ∇v)|∇ϕ + ∇v| p-2 -∇ϕ|∇ϕ| p-2 • ∇Ψ = - ¢ R d h • ∇Ψ,
and concludes the proof of the Lemma 5.7.2. The required convergences follow from (5.7.21) and Proposition 5.2.1 (ii) and (iv). We get that ∇ w ξn converges when n -→ +∞ to ∇v in L p loc (R d ) and that ∇v solves (5.2.12) in the distribution sense. Thus, ∇v solves (5.2.12) in the weak sense in W ξ+∇w per ξ (see Definition 5. (iii) This is (5.7.28).

Proof of

(iv) By the same argument as in (ii), we have that h n -→ n→+∞ 0 in L p ′ loc (R d ).

We have proved that, up to exacting a subsequence, ∇ w -ξ + ∇w per ξ (• + x) ξ + ∇w per ξ (• + x) p-2 = 0. Thus, for all n ∈ N, ∥∇ w ξn -∇ w ξ ∥ L p unif (R d ) ≥ |B(0, 1)| 1/p δη d/p > 0. which is a contradiction with Theorem 5.2.8. Proof of (i). This point is obvious by the form of (5.2.9) and the uniqueness of w per ξ in W 1,p per (Q)/R.

Introducing V 1 := v 1 (• -x), we get that V 1 ∈ W ξ+∇w per

Appendices

Proof of (ii).

The first estimate follows for example from (5.2.4) and in particular: We apply the Hölder inequality on the right-hand side with exponents p/(p -2), p and p find, using (5.2.16):

c ¢ Q ξ 1 + ∇w per ξ1 -(ξ 2 + ∇w per ξ2 ) p 1-1/p ≤ C |ξ 1 | p-2 + |ξ 1 | p-2 |ξ 1 -ξ 2 |.
This yields (5.2.16) by taking the 1/(p -1)-th power of the above inequality.

Proof of (iv). We argue by contradiction. Suppose that there exist three sequences ( . This is a contradiction with (5.8.4) when taking n -→ +∞. We have proved (5.2.8) for |ξ| = 1 and |ξ -η| ≤ 1 2 . The other cases are treated by homogeneity and with the help of (5.2.16) as in (5.5.30).

Some technical inequalities

We gather in this subsection some useful inequalities. We first have (5.8.7)

In the above inequalities (5.8.5)-(5.8.7), c and C refer to universal constants that only depend on p.

For a proof of these inequalities, we refer to [START_REF] Iwaniec | Projections onto gradient fields and L p -estimates for degenerated elliptic operators[END_REF]. For ξ, x ∈ R d , we introduce the function We fix 0 < ε < 1 independent of ξ such that for all y ∈ B(0, ε), we have that |ξ + y| p-2 ≥ 1 p(p-2) . Let x ∈ B(0, ε), the Taylor Formula for h ξ (or α-convexity) yields on the one hand By compactness, we can assume that the two sequences (ξ n ) n∈N and (x n ) n∈N converge respectively to ξ and x. Since |x n | ≤ A, we have that g ξn (x n ) -→ n→+∞ 0. Thus g ξ (x) = 0 which is a contradiction. This proves that inf |ξ|=1 c ξ = c > 0 and concludes the proof of (5.8.9) when |ξ| = 1.

For ξ ̸ = 0, we note that .8.11) and the conclusion still holds true with C independent of ξ.

g ξ = |ξ| p g ξ |ξ| • |ξ| , ( 5
Lemma 5.8.2. There exists a constant C > 0 such that for all x, h ∈ R, (x + h) 1/(p-1) -x 1/(p-1) ≤ C|h| 1/(p-1) .

(5.8.12) + ∥∇v∥ L p (Q c R/2 ) < ε.

(5.8.13)

Let χ R be a cut-off function such that χ R = 1 in Q R/2 and χ = 0 in Q c 3R/4 . We have that |χ R | + R|∇χ R | ≤ C where C depends only on the dimension d (and in partiuclar not on R). We introduce

w R := v - 2 Q R \Q R/2 v χ R ,
We have immediately that w R is compactly supported in Q R and that w R ∈ W The main lemma concerning the existence of the non-periodic correctors is the following: Lemma 5.8.6. Assume that a satisfies ( 1) and ( 4) in Subsection 5.2.5. Let A be defined by (5.4.17).

Then, the operator A is hemicontinuous, bounded, coercive and strongly monotone. In particular, the equation A(∇v) = F for F ∈ W ξ+∇w per ξ ′ admits a unique solution.

Proof of Lemma 5.8.6. We first show the following properties:

• Hemicontinuity: let u, v, h ∈ W ξ+∇w per (5.8.23)

This consequently implies the strong monotonicity of A.

• A is coercive: let v ∈ W ξ+∇w per We now turn to the proof of Theorem 5.2.4. We only detail the proof of point (iii) since the other assertions are adaptations of the p-Laplacian case to the present setting. Let us first note that the proof of Theorem 5.2.5 still works for the non-variational operators considered in this section. Applying the results of [START_REF] Avellaneda | Lp bounds on singular integrals in homogenization[END_REF][START_REF] Avellaneda | Un théorème de liouville pour des équations elliptiques à coefficients périodiques[END_REF], we deduce that ∇ w ξ ∈ L max(p ′ ,p/2) (R d ). Iterating the argument, we conclude that ∇ w ξ ∈ L p ′ (R d ).

We now detail the proof of Theorem 5.2.4 (iii) in the general setting. We define the homogeneous Sobolev spaces

• W 1,q (R d ) := v ∈ W 1,1 loc (R d ), ∇v ∈ L q (R d ) , 1 < q < +∞ and • H 1 (R d ) := v ∈ W 1,1 loc (R d ), ∇v ∈ L 2 (R d ) .
Proof of Theorem 5.2.4 (iii). Let ξ, η ∈ R d such that |ξ| = 1 and |ξ -η| ≤ 1 2 . We note that, thanks to Assumption (A4), we have that W ξ+∇w per Thus, there exists γ > 0 independent of ξ, η, Z and T such that ∀s ∈ [0, 1], Ψ ′ γ (s) ≥ -λ -1 |ξ -η| γ(p-2) |Z| 2 .

In particular, we get by integration that

Ψ γ (1) -Ψ γ (0) ≥ -λ -1 |ξ -η| γ(p-2) |Z| 2 .
All in all, we have proved that there exists γ > 0 such that a(y, ξ + ∇w per where the constant C does not depend on ξ and η. The estimate (5.2.18) is deduced for any ξ and η by homogeneity. The proof for the case p ≥ 3 follows the same lines.

We close this section by commenting on Theorem 5.2.7 and Theorem 5.2.8 in the two remarks below.

Remark 5.8.7 (About Theorem 5.2.7 for non-variational operators). The main result of Section 5.6 is Lemma 5.6.1 which only uses the continuity of the application ξ -→ ∇ w ξ and its homogeneity. It thus holds true in the present setting, as well as Lemma 5.6.2. The proof of Theorem 5.2.7 relies on the approximation of ∇u * by a piecewise constant function Φ. The analysis remains similar since (5.8.20) still holds true. Remark 5.8.8 (About Theorem 5.2.8 for non-variational operators). The arguments exposed in Section 5.6 use only the PDE and not the minimization problem. They consequently almost directly extend to the present setting with the suitable adaptation of notations and using the regularity conditions imposed in Subsection 5.2.5 on the non-periodic operator a.

Derivation of the weak limit

In this section, we prove Theorem 5.2.7 (i). This proof follows the lines of [START_REF] Fusco | On the homogenization of quasilinear divergence structure operators[END_REF]. We first note that we have the following a priori estimates

∥∇u ε ∥ L p (Ω) ≤ C and a • ε ∇u ε |∇u ε | p-2 L p ′ (Ω) ≤ C.
Up to extracting a subsequence, we have by Rellich's compactness Theorem, that

u ε -→ ε→0 u 0 in L p (Ω), ∇u ε -⇀ ε→0 ∇u 0 in L p (Ω) and a • ε ∇u ε |∇u ε | p-2 -⇀ ε→0 a 0 in L p ′ (Ω).
The main goal is to show that a 0 = a * (∇u 0 ). (5.8.33) This will imply in particular that -div a * (∇u 0 ) = f in the weak sense in W 1,p 0 (Ω). Consequently, this in turn implies that u 0 = u * by strict monotonicity of the operator a * . Finally, by uniqueness of the limit, the whole sequence (u ε ) ε>0 converges in W 1,p (Ω).

Proof of (5.8.33). We introduce a parameter ν > 0 and we define Q ν k := 2 -ν (Q + k) for k ∈ Z d . We also set I ν := {k ∈ Z d , Q ν k ⊂ Ω} and the following discretization of ∇u 0 :

⟨∇u 0 ⟩ ν k := 2 Q ν k ∇u 0 , ∀k ∈ I ν .
We fix k ∈ I ν and we introduce the functions

v ν,ε k := εw ⟨∇u0⟩ ν k • ε + ⟨∇u 0 ⟩ ν k • x, ∇v ν,ε k = ∇w ⟨∇u0⟩ ν k • ε + ⟨∇u 0 ⟩ ν k .
We check that: as ε -→ 0, then η -→ 1 Q in L p (Q) and finally ν -→ +∞. We choose ν large enough so that supp(ϕ) ⊂ k∈Iν Q ν k . We have, using the Hölder inequality together with (5.8.7) that (5.8.37) Let us recall that η ν ∈ C ∞ 0 (Q ν k ). We can pass to the limit ε -→ 0 in the RHS of (5.8.37). This is done by the div-curl Lemma (which is in fact an integration by parts in this case) using the convergences stated at the beginning of the section:

• v ν,ε k -→ ε→0 ⟨∇u 0 ⟩ ν k • x in L p (Ω); • ∇v ν,ε k -⇀ ε→0 ⟨∇u 0 ⟩ ν k in L p (Ω); • div a ε ∇v ν,ε k |∇v ν,ε k | p-2 = 0 in Ω; • a ε ∇v ν,ε k |∇v ν,ε k | p-2 -⇀
|I| ≤ k∈Iν ¢ Q ν k a ε ∇u ε ∇u ε p-2 -∇v ν,ε k ∇v ν,ε k p-2 ϕη ν ≤ λ k∈Iν ¢ Q ν k ∇u ε p + ∇v ν,ε k p p-2 p k∈Iν ¢ Q ν k ∇u ε -∇v ν,ε k p η p
¢ Q ν k ∇u ε -∇v ν,ε k p η ν ≤ cλ ¢ Q ν k a ε ∇u ε ∇u ε p-2 -a ε ∇v ν,ε k ∇v ν,ε k p-2 • (∇u ε -∇v ν,ε k ) η ν
¢ Q ν k a ε ∇u ε ∇u ε p-2 -a ε ∇v ν,ε k ∇v ν,ε k p-2 • (∇u ε -∇v ν,ε k ) η ν -→ ε→0 ¢ Q ν k (a 0 -a * (⟨∇u 0 ⟩ ν k )) • (∇u 0 -⟨∇u 0 ⟩ ν k ) η ν .
(5.8.38) Passing to the limit ε -→ 0 in (5.8.35) and using (5.8.38), we obtain that

¢ Ω a 0 • ϕη ν - k∈Iν ¢ Q ν k a * (⟨∇u 0 ⟩ ν k ) ϕη ν ≤ k∈Iν ¢ Q ν k (a 0 -a * (⟨∇u 0 ⟩ ν k ))
• (∇u 0 -⟨∇u 0 ⟩ ν k ) η ν . (5.8.39) This yields since 0 ≤ η ν ≤ 1,

¢ Ω a 0 • ϕη ν - k∈Iν ¢ Q ν k a * (⟨∇u 0 ⟩ ν k ) ϕη ν ≤ k∈Iν ¢ Q ν k
a 0 -a * (⟨∇u 0 ⟩ ν k ) ∇u 0 -⟨∇u 0 ⟩ ν k . (5.8.40) We want to let η ν -→ 1 in L p (Ω) in the LHS. This is possible for the first term since a 0 ϕ ∈ L p ′ (Ω). For the second term, we recall the bound a * (ξ) ≤ C|ξ| p-1 (which is a consequence of Proposition 5.2.1 (ii) and we deduce that .8.41) This allows to pass to the limit in η η in (5.8.40). We obtain

k∈Iν a * (⟨∇u 0 ⟩ ν k ) 1 Q ν k p ′ L p ′ (Ω) ≤ C k∈Iν |⟨∇u 0 ⟩ ν k | p |Q ν k | ≤ C ¢ Ω |∇u 0 | p . ( 5
¢ Ω a 0 • ϕ - k∈Iν ¢ Q ν k a * (⟨∇u 0 ⟩ ν k ) ϕ ≤ k∈Iν ¢ Q ν k a 0 -a * (⟨∇u 0 ⟩ ν k ) ∇u 0 -⟨∇u 0 ⟩ ν k .
(5.8.42)

We finally pass to the limit ν -→ +∞ in (5.8.42). Using the Hölder inequality, we have that This proves that k∈Iν a * (⟨∇u 0 ⟩ ν k ) ϕ1 Q ν k converges as ν -→ +∞ to a * (∇u 0 )ϕ in L p ′ (Ω). We consequently obtain that ¢ Ω a 0 • ϕ -a * (∇u 0 ) ϕ = 0. Since this is true for all ϕ ∈ D(Ω), we conclude that a 0 = a * (∇u 0 ) a.e. This concludes the proof.

k∈Iν ¢ Q ν k a 0 -a * (⟨∇u 0 ⟩ ν k ) ∇u 0 -⟨∇u 0 ⟩ ν k ≤ k∈Iν ¢ Q ν k a 0 -a * (⟨∇u 0 ⟩ ν k ) p ′ 1 p ′ k∈Iν ¢ Q ν k ∇u 0 -⟨∇u 0 ⟩ ν k p 1 p . ( 5

Figure 1 . 1 :

 11 Figure 1.1 : Procédé d'homogénéisation : la figure de gauche représente une propriété physique d'un matériau (par exemple la conductivité thermique). Celle-ci est fortement hétérogène et l'on note ε sa taille caractéristique d'oscillation. On aimerait remplacer le matériau de gauche par le matériau de droite, homogénéisé.

Figure 1 . 2 :

 12 Figure 1.2 : Représentation graphique de la solution à l'équation -(a(./ε)T′ ε ) ′ = f , T ε (-1 2 ) = T ε ( 1 2 ) = 0 où f (x) =1, a(y) = 2 + cos(2πy), ε = 0.1 et ε = 0.05. La fonction u ε est à gauche, sa dérivée est à droite.

2 1Figure 1 . 3 :

 213 Figure 1.3 : Représentation graphique de l'hypothèse géométrique imposée sur les perforations nonpériodiques.

ε

  considérées dans les chapitres 2 et 3. On va pour cela définir une suite (O k ) k∈Z d de perforations non périodiques s'appuyant sur la distribution périodique (O per k ) k∈Z d de trous. Pour k ∈ Z d , on fixe une perforation O k telle que O k ⊂⊂ Q k et Q k \ O k est connexe. L'hypothèse géométrique principale (voir figure 1.3) que l'on impose est la suivante : il existe une suite(α k ) k∈Z d ∈ ℓ 1 (Z d ) telle que ∀k ∈ Z d , O per,- k (α k ) ⊂ O k ⊂ O per,+ k (α k ), (A) où, pour α > 0 et k ∈ Z d , O per,- k (α) := {x ∈ O per k , d(x, ∂O per k ) > α} et O per,+ k (α) := {x ∈ Q k , d(x, O per k ) < α},et d(x, A) désigne la distance euclidienne d'un point x ∈ R d à l'ensemble A. Nous représentons sur la figure 1.4 un exemple schématique de perturbations considérées dans cette thèse. Dans la suite, on notera O per :=

Figure 1 . 4 :

 14 Figure 1.4 : Perturbations locales d'un domaine périodiquement perforé : la figure de gauche représente un domaine Ω per ε périodiquement perforé. La figure de droite montre un domaine localement perturbé Ω ε . La zone rouge délimite le défaut (ici à support compact) introduit dans le domaine macroscopique. Lorsque ε → 0, cette zone se concentre autour de l'origine.

  |j|, désigne l'opération de dérivation de multi-indice j ∈ N d . Dans (1.2.7), les fonctions w (j) sont les correcteurs d'ordre j, qui se calculent récursivement. Pour |j| = 0 et 1, on a-∆ y w (0) = 1 dans R d \ O w (0) = 0 sur ∂O, -∆ y w (j) = 2∂ j w (0) dans R d \ O w (j) = 0 sur ∂O,

¢Ω(Théorème 1 .

 1 Af -u * )ϕ = 0, A = L 2 -lim faible ε→0 w • ε , pourvu que A existe. Remarquons que la preuve ci-dessus n'utilise en rien la périodicité, mais seulement que (1.2.8) admet une solution dans l'espace H 1 unif (R d ) tel que A existe. C'est en particulier le cas si la fonction w (ou plutôt son prolongement par zéro dans les trous) admet une moyenne sur R d . Dans le cas périodique, on a A = ¢ Q w per (y)dy, où w per est la solution périodique de (1.2.8), et donc u * = ¢ Q w per (y)dy f . Dans le cas non périodique, nous prouvons d'abord l'existence du correcteur w. On pose H 1,per (Q \ O per 0 ) := u ∈ H 1,loc (R d \ O per ), u est périodique où l'exposant loc signifie localement dans R d . Soit (O k ) k∈Z d une configuration non périodique de trous et Ω ε le domaine perforé associé. Soit g ∈ L 2 (R d ) et g := 1 R d \O per + g. Il existe une unique fonction w ∈ H 1 (R d \ O) telle que w := w per + w est solution au sens des distributions de l'équation -∆w = g dans R d \ O w = 0 sur ∂O, (1.2.12) où w per ∈ H 1,per (Q \ O per 0 ) est solution de l'EDP du correcteur périodique -∆w per = 1 dans Q \ O per 0 w per = 0 sur ∂O per 0 .

1. 2 .w

 2 PRÉSENTATION DE LA THÈSE Nous déduisons du Théorème 1 la limite u ε /ε 2 -⇀ ε→0 u * dans L 2 (Ω) avec u * = Af , A := L 2 -lim faible per (y)dy, puisque w ∈ L 2 (R d ) (où w est prolongée par -w per dans les perforations non périodiques). En particulier, la limite faible de u ε /ε 2 est la même que dans le cas périodique : ceci était à prévoir car Ω ε et Ω per ε ne différent qu'à l'échelle microscopique.

j + ∇p per j =

 j e j dans Q \ O per 0

( 1 .Figure 1 . 6 :

 116 Figure 1.6 : Représentation d'un coefficient périodique avec défauts : la figure de gauche représente un coefficient périodique. La figure de droite représente un fond périodique qui est perturbé par un défaut ponctuel.

Theorem 2 . 1 . 3 (

 213 Consequence of Theorem 3.1 of [77]). Assume that O per 0 is an open subset of Q such that O per 0 ⊂⊂ Q. Let Ω ε be defined by (2.1.6). Let f ∈ D(Ω) and u ε be the solution to (2.1.1).

Figure 2 . 2 :

 22 Figure 2.2: On the left, illustration of O per k (red), its widened boundary U per k , its enlargement O per,+ k

  .2 left), we define the reduction and the enlargement of O per k by O per,-

Figure 2 . 3 :

 23 Figure 2.3: Pictures of perforated cells divided into two subdomains (white and light grey) with boundary Γ i , i = 1, 2, 3. Left: O k ∩ O per

  ρ where C(d) depends only on the dimension. Theorem 2.3.1 gives the existence of a constant C 3 = C(d)/ρ chosen greater than or equal to one such that

Lemma 2 . 4 . 8 .

 248 Let (O k ) k∈Z d satisfy Assumption (A1) and Assumption (A2). Then V defined by (2.4.7) is not empty.

Proposition 2 . 4 . 9 .

 249 Under the assumptions (A1) and (A2), the minimization Problem (2.4.6) has a solution.

Lemma 2 . 5 . 1 .

 251 Let (O k ) k∈Z d be a sequence of open sets satisfying Assumption (A2). Then, k∈Z d |O k ∆O per k | < +∞, (2.5.1)

  a) This drawing represents a periodic cell Q k and the related notations.ε ε (b) The set Ωε is filled in light grey. The set Ω ′ ε (see (2.5.9)) is marked in purple. The setting is here periodic.

Figure 2 . 5 :

 25 Figure 2.5: Drawing of the notations used in this section.

  We fix a locally Lipschitz bounded domain Ω ⊂ R 3 and a subset O per 0 such that O per 0 ⊂⊂ Q, O per 0 is of class C 2,α and Q\O per 0 is connected. We define for k ∈ Z 3 , O per k := O per 0 + k. O per will be the set of perforations, that is, O per := k∈Z 3 O per

. 3 .Figure 3 . 1 :

 331 Figure 3.1: Periodic domain Ω per ε

  In (3.1.2), the symmetric and positive definite matrix A is the so-called permeability tensor. Its coefficients are defined by j , 1 ≤ i, j ≤ 3,(3.1.3) where the functions w per j , j = 1, 2, 3 are the cell periodic first correctors and solve the following Stokes problems:

.1. 12 ) 3 . 1 . 1 .

 12311 Remark For each fixed k ∈ Z 3 , the estimates (3.1.9) and (3.1.10) are satisfied with constants C i q,k , i = 0, 1, depending on k, see[START_REF] Galdi | An introduction to the mathematical theory of the Navier-Stokes equations: Steady-state problems[END_REF] Theorem III.3.3]. Similarly, as long as O k is of class C 2 , (3.1.12) is satisfied when k is fixed (see[START_REF] Galdi | An introduction to the mathematical theory of the Navier-Stokes equations: Steady-state problems[END_REF] Theorem IV.5.1]). Assumptions (A4)-(A5) require that the constants appearing in (3.1.9), (3.1.10) and (3.1.12) are uniform with respect to k ∈ Z 3 .

(

  A4)' The functions ζ x , x ∈ ∂O k are Lipschitz functions with Lipschitz constant ∥ζ x ∥ Lip(Ux) satisfying ∥ζ x ∥ Lip(Ux) ≤ M . (A5)' The functions ζ x , x ∈ ∂O k are of class C 2 and satisfy ∇ζ x (0) = 0 with the estimate ∥ζ x ∥ W 2,∞ (Ux) ≤ M .

  (a) Illustration of (A3) ε ε (b) A non periodically perforated grid

Figure 3 . 3 :

 33 Figure 3.3: The non-periodic setting

.3. 3 )

 3 Summing the estimate (3.3.2) over k ∈ Y ε , the estimate (3.3.3) over k ∈ Z ε and using (3.3.1) concludes the proof of Lemma 3.3.1.

  3.7), (3.3.10) and (3.3.11), we get

T

  and w i,per j = w per j • e i ,

Figure 3 . 4 :

 34 Figure 3.4: Illustration of the proof of Lemma 3.3.3

.3. 25 )

 25 Gahtering(3.3.24) and (3.3.25) concludes the proof of(3.3.22).

Figure 3 . 5 :

 35 Figure 3.5: Proof of Theorem 3.2.3

Remark 3 . 4 . 7 .

 347 Lemma 3.4.6.(iv) means that we can relabel the family B 0,per i , i = 1, ..., 2N such that for all i ∈ {1, ..., 2N -1}, we have that Ω 0,peri ∩ Ω 0,per i+1 ∪ • • • ∪ Ω 0,per 2N ̸ = ∅.Proof. The proof of Lemma 3.4.6 relies on the periodic structure and on Assumption (A3). We first fix by compactness N 0 balls B 0,peri = B(x i , ρ/2), i = 1, ..., N 0 such that {x ∈ Q, d(x, ∂O per 0 ) ≤ ρ/4} ⊂ N0 i=1B 0,per i and x i ∈ ∂O per 0 . (3.4.21)

  .4.40) Testing (3.4.38) (resp. (3.4.39)) against v k 1 (resp. v k 2 ) and noticing that

4 . 42 )

 442 Collecting (3.4.41) and (3.4.42) yields

Theorem 4 . 1 . 1 .

 411 .1.1) (in particular the whole range p ∈ [2 * , +∞)), we prove in Theorem 4.1.1 that the two scale expansion (4.1.5) is indeed valid for the PDE (4.1.1). Let 2 ≤ p < +∞ and f ∈ V ′ . The sequence (u ε ) ε>0 converges weakly in V (and strongly in L r for all r ∈ [1, 2 * )) to the solution u 0 of (4.1.6). Besides, assume that D 2 u 0 ∈ L 2 (Ω). Then we have the strong convergence∇R ε -→ ε→0 0 in L 2 (Ω), (4.1.8) where R ε := u ε -u 0 -ε d i=1 w i (./ε)∂ i u 0 and w i := w ei .

  where u 0 solves (4.1.6).(4.2.13)

  (i) We have that u ε -⇀ ε→0 u * weakly in W 1,p (Ω) and u ε -→ ε→0 u * strongly in L p (Ω), where u * solves Problem (5.1.6) and ∀ξ ∈ R d , a * (ξ) = ¢ Q a per (y)(ξ + ∇w per ξ )|ξ + ∇w per ξ | p-2 dy. (5.2.20) Besides, we have the L p ′ (Ω)-weak convergence a(./ε)∇u ε |∇u ε | p-2 -⇀ ε→0 a * (∇u * ).

Theorem 5 . 2 . 8 .

 528 Assume that (A1)-(A2)-(A3)-(A4)' are satisfied. Then the mapping Φ p defined by (5.2.23) is continuous.

Remark 5 . 2 . 11 .

 5211 The space W ξ+∇w per ξ is in general different from the space• W 1,p ∩ • H 1 (R d ), where • W 1,p and • H 1 (R d )are the standard homogeneous Sobolev spaces, unless ξ + ∇w per ξ does not vanish.

  d and (x n ) n∈N ⊂ Q such that |ξ n | = 1 and |ξ n + ∇w per ξn (x n )| -→ n→+∞ 0. By compactness, we have that up to a subsequence, (x n , ξ n ) -→ n→+∞ (x, ξ). By Proposition 5.2.1 (ii) and Proposition 5.2.1 (iv), we obtain that |ξ + ∇w per ξ (x)| = 0 which gives c(ξ) = 0: we thus reach a contradiction.

Lemma 5 . 4 . 1 .

 541 Let ξ ∈ R d and W ξ+∇w per ξ be defined by (5.2.10).(i) The space W ξ+∇w per ξ is a Banach space.

Figure 5 . 1 :

 51 Figure 5.1: Numerical simulation in the particular 1D case.

ξ (∇v 1 ) + tg ξ+∇w per ξ (∇v 2 )Lemma 5 . 4 . 3 .( 5 . 4 . 11 )

 125435411 -g ξ+∇w per ξ ((1 -t)∇v 1 + tv 2 ) = 0 a.e. The strict convexity of g ξ+∇w per ξ implies that ∇v 1 = ∇v 2 a.e. This concludes the proof.Let ξ ∈ R d , h ∈ L p ′ (R d ) d ,F ξ be defined by (5.4.2) over V and the space W ξ+∇w per ξ be defined by (5.2.10). Then the application F ξ is Fréchet-differentiable over W ξ+∇w per ξ . Its differential is given, for v ∈ W ξ+∇w per ξ , by F ′ ξ (v) • u := ¢ R d a (ξ + ∇w per ξ + ∇v) ξ + ∇w per ξ + ∇v p-2 -(ξ + ∇w per ξ ) ξ + ∇w per ξ p-2 + h • ∇u. Proof of Lemma 5.4.3. We fix v ∈ W ξ+∇w per ξ and u ∈ W ξ+∇w per ξ

.5. 35 )

 35 We now use Lemma 5.5.1 applied with δ = c. Taking into account(5.5.32), this gives

Lemma 5 . 7 . 1 . 1 )

 5711 Let ξ ∈ R d , a coefficient a satisfying Assumption (A1) and h ∈ L p ′ (R d ) d . Assume that Assumption (A4)' is satisfied. Let v ∈ W ξ+∇w per ξ be solution in the distribution sense to the following PDE: -div a (ξ + ∇w per ξ + ∇v)|ξ + ∇w per ξ + ∇v| p-2 -(ξ + ∇w per ξ )|ξ + ∇w per ξ | p-2 = div(h). (5.7.Then ∇v solves (5.7.1) in the weak sense of Definition 5.2.2: for all w ∈ W ξ+∇w per ξ , ¢ R d a (ξ +∇w per ξ +∇v)|ξ +∇w per ξ +∇v| p-2 -(ξ +∇w per ξ )|ξ +∇w per ξ | p-2 •∇w = -¢ R d h•∇w. (5.7.2)

  2.2) by Lemma 5.7.1, with(ii) We have a n = a per (• + x n ) + a(• + x n ). Since x n -→ n→+∞ x in T d , we have by Assumption (A2) that a per (• + x n ) -→ n→+∞ a per (• + x) in L ∞ (Q). Let B be a bounded domain, then since a ∈ C 0,α ∩ L p ′ (R d ), we have a -→ |x|→+∞ 0. Thus a(• + x n ) -→ n→+∞ 0 in L ∞ (B)and finally a n converges locally uniformly to a := a per (• + x) when n → +∞.

∇ v 1

 1 in L p loc (R d ) where v 1 ∈ W ξ+∇w per ξ (•+x) solves in the distribution sense the PDE -div a per (• + x) ξ + ∇w per ξ (• + x) + ∇ v 1 ξ + ∇w per ξ (• + x) + ∇ v 1 p-2

ξ 8 . 5 . 7 . 4 .

 8574 and that ∇ V 1 solves in the distribution sense the PDE-diva per (ξ + ∇w per ξ + ∇ V 1 )|ξ + ∇w per ξ + ∇ V 1 | p-2 -(ξ + ∇w per ξ )|ξ + ∇w per ξ | p-2 = 0. (5.7.29) Applying Lemma 5.7.1 to (5.7.29) with w:= V 1 gives now ¢ R d a per (• + x) (ξ + ∇w per ξ + ∇ V 1 )|ξ + ∇w per ξ + ∇ V 1 | p-2 -(ξ + ∇w per ξ )|ξ + ∇w per ξ | p-2 • ∇ V 1 = 0.Applying (5.8.5) allows to conclude that ∇ V 1 = 0 in R d and thus ∇ v 1 = 0. This proves (5.7.26) and concludes the proof of Theorem 5.2.RemarkFrom the above theorem, we can deduce thatR d -→ L ∞ (R d ) ξ -→ ∇ w ξ is continuous.The continuity of ξ → ∇w per ξ is due to Proposition 5.2.1 (iv). We prove that ξ → ∇ w ξ is continuous for the L ∞ (R d ) topology. By contradiction, suppose that there exists ξ ∈ R d , two sequences(ξ n ) n∈N ⊂ R d and (x n ) n∈N ⊂ R d and a δ > 0 such that ξ n -→ n→+∞ ξ and ∀n ∈ N, ∇ w ξn (x n ) -∇ w ξ (x n ) ≥ δ.By Theorem 5.2.4 (ii), there exists η independent of n such that ∀n ∈ N, ∀y ∈ B(x n , η), ∇ w ξn (y) -∇ w ξ (y) ≥ δ 2 .

5. 8 . 1

 81 Proof of Proposition 5.2.1

aQξ 1 ξ 1

 11 per |ξ| p , together with (A1). The proof of the second estimate is exactly the same as the one of Theorem 5.2.4 (ii) (see Subsection 5.5.3) with a replaced by a per and w ξ replaced by w per ξ .Proof of (iii).Let ξ 1 , ξ 2 ∈ R d . Applying (5.2.3) with ϕ = w per ξ1 -w per ξ2 with ξ = ξ i , i = 1, 2 and making the difference between the two expressions gives:¢ Q a per (ξ 1 + ∇w per ξ1 )|ξ 1 + ∇w per ξ1 | p-2 -(ξ 2 + ∇w per ξ2 )|ξ 2 + ∇w per ξ2 | p-2 • ∇w per ξ1 -∇w per ξ2 = 0. (5.8.1)Thus, adding the term¢ Q a per (ξ 1 + ∇w per ξ1 )|ξ 1 + ∇w per ξ1 | p-2 -(ξ 2 + ∇w per ξ2 )|ξ 2 + ∇w per ξ2 | p-2 • ξ 1 -ξ 2 = 0 (5.8.2)in the left and right-hand side of (5.8.1), applying (5.8.5) on the left-hand side and (5.8.7) on the right-hand side provides c ¢ + ∇w per ξ1 -(ξ 2 + ∇w per ξ2 ) + ∇w per ξ1 p-2 + ξ 2 + ∇w per ξ2 p-2 ξ 1 + ∇w per ξ1 -(ξ 2 + ∇w per ξ2 ) ξ 1 -ξ 2 .

(x|x| p- 2 -

 2 y|y| p-2 ) • (x -y) ≥ c|x -y| p , (5.8.5) (x|x| p-2 -y|y| p-2 ) • (x -y) ≥ c |x| p-2 + |y| p-2 |x -y| 2 ,(5.8.6)x|x| p-2 -y|y| p-2 ≤ C |x| p-2 + |y| p-2 |x -y|.

Lemma 5 . 8 . 1 . 9 )-→ |x|→+∞ 1 , 2 |x| p + |x| 2 .

 5819122 g ξ (x) := |ξ + x| p -|ξ| p -pξ|ξ| p-2 • x.(5.8.8)We have the following lemma: There exist two constants c, C > 0 depending only on p such that∀ξ, x ∈ R d , c |x| 2 |ξ| p-2 + |x| p ≤ g ξ (x) ≤ C |x| 2 |ξ| p-2 + |x| p . (5.8.Proof of Lemma 5.8.1. We introduce h ξ (x) := |ξ + x| p . The function h ξ is differentiable, strictly convex andg ξ (x) = h ξ (x) -h ξ (0) -h ′ ξ (0) • x for all x ∈ R d .This proves that g ξ > 0 on R d \ {0}. We fix ξ ∈ R d such that |ξ| = 1 and we notice that for all x ̸ = 0,g ξ (x) |x| p ≥ (|x| -1) p -1 -p|x| |x| pWe can choose A > 1 independently of ξ such that |x| ≥ A implies1 4 |x| p + |x| 2 ≤ g ξ (x) ≤3It remains to study g ξ on B(0, A). We first notice that for y ̸ = -ξ and h ∈ R d ,h ′′ ξ (y)(h, h) = p(p -2) |ξ + y| p-2 I d + (ξ + y) T (ξ + y)|ξ + y| p-4• (h, h) = p(p -2)|ξ + y| p-2 |h| 2 + |h(y + ξ)| 2 ≥ p(p -2)|ξ + y| p-2 |h| 2 .

( 1 -

 1 t) 2 h ′′ ξ (tx) • (x, x)dt ≥ the other hand, ∀x ∈ B(0, A), g ξ (x) ≤ C(A)|x| 2 ≤ C(A) |x| 2 + |x| p .It remains to prove that the left-side inequality of (5.8.9) is true on C ε,A := B(0, A)\B(0, ε). Because x → g ξ (x)/(|x| p + |x| 2 ) is (strictly) positive on C ε,A , we havec ξ := min x∈Cε,A g ξ (x) |x| p + |x| 2 > 0.Suppose by contradiction that inf |ξ|=1 c ξ = 0, then there exist two sequences (ξ n ) n∈N and (x n ) n∈N such that|ξ n | = 1, x n ∈ C ε,A and g ξn (x n ) |x n | p + |x n | 2 -→ n→+∞ 0.

Lemma 5 . 8 . 3 ..Proof of Lemma 5 . 8 . 3 .

 583583 Assume that Hypothesis(A4)' is satisfied. Then C ∞ 0 (R d ) is dense in W ξ+∇w per ξ Let v ∈ W ξ+∇w per ξ and ε > 0. There exists R = R(ε) > 1 such that |ξ + ∇w per ξ |

ξ.•

  We want to show that for s, t ∈ [0, 1],|⟨A(u + tv) -A(u + sv), h⟩| -→ where ⟨•, •⟩ stands for the duality product in W ξ+∇w per ξ . We first notice that due to Assumption (4) in Subsection 5.2.5,a •, ξ + ∇w per ξ + ∇u + t∇v -a •, ξ + ∇w per ξ + ∇u + s∇v ≲ λ|t -s|• ξ + ∇w per ξ p-2 + |∇u| p-2 + |∇v| p-2 |∇v| ≲ λ|t -s| ξ + ∇w per ξ p-2 |∇v| + |∇v| p-1 + |∇u| p-1 ,where we used Young inequality in the last inequality. Using the above inequality (we first integrate on the ball B R and let finally R -→ +∞), we get that⟨A(u + tv) -A(u + sv), h⟩ ≲ λ|t -s| ∥∇v∥ W ξ+∇w per ξ ∥∇h∥ W ξ+∇w per ξ A is bounded: let v, h ∈ W ξ+∇w per ξ. We show immediately due to Assumption (4) in Subsection 5.2.5 that|⟨A(v), h⟩| ≲ ∥∇v∥ W ξ+∇w per ξ ∥∇h∥ W ξ+∇w per ξ + ∥∇v∥ p-1 W ξ+∇w per ξ ∥∇h∥ W ξ+∇w per ξ .• A is strongly monotone: let u, v ∈ W ξ+∇w per ξ . We fix R > 0. For all y ∈ B R , we have thanks to Lemma 5.8.5 that a(y, ξ+∇w per ξ + ∇u) -a(y, ξ + ∇w per ξ ) -a(y, ξ+ ∇w per ξ + ∇v) -a(y, ξ + ∇w per ξ ) • ∇(u -v) = a(y, ξ + ∇w per ξ + ∇u) -a(y, ξ + ∇w per ξ + ∇v) • ∇(u -v) ≥ c ∇u -∇v p .(5.8.22) We integrate (5.8.22) over B R and we let R -→ +∞. By dominated convergence Theorem, this gives ⟨A(∇u) -A(∇v), ∇u -∇v⟩ ≥ c ∇u -∇v p L p (R d ) .

′.

  ∥∇v∥ W ξ+∇w per ξ -→ +∞ as ∥∇v∥ W ξ+∇w per ξ -→ +∞, which gives the coercivity of A. This shows the first part of the Lemma. The surjectivity of A is deduced from [76, Corollary 8.1]. The application of Lemma 5.8.6 with F = div(h) where h is defined by (5.2.34) gives the existence and uniqueness of the non-periodic corrector w ξ ∈ W ξ+∇w per ξ satisfying the conclusion of Theorem 5.2.3. Indeed, it is sufficient to check that F ∈ W ξ+∇w per ξ This follows from Assumption (3) introduced in Subsection 5.2.5:) p ′ ξ + ∇w per ξ p < +∞, since b ∈ L p ′ (R d ) and ∇w per ξ ∈ L ∞ (R d ). Thus, F ∈ • W -1,p ′ (R d ) ⊂ W ξ+∇w per ξ ′. Theorem 5.2.3 is proved in the extended setting of Subsection 5.2.5.

Proof of Theorem 5 . 2 . 5 .p-2 h 2 . 2 .

 52522 The proof consists in a linearization. Assume that |ξ| = 1. Using Assumption (A4) and (5.2.35), we first havea y, ξ + ∇w per ξ + ∇ w ξ -a y, ξ + ∇w per ξ = ∂ ξ a y, ξ + ∇w per ξ • ∇ w ξ + O c |∇ w ξ | 2 , (5.8.24)where the symbol O c is uniform in y ∈ R d but depends, among other, on the constant c appearing in Assumption (A4). We note that, due to (4) in Subsection 5.2.5, we have forh ∈ R d , h T • ∂ ξ a y, ξ + ∇w per ξ • h ≥ λ -1 ξ + ∇w perξ Thus, Assumption (A4) gives that the matrix ∂ ξ a y, ξ + ∇w per ξ is strongly elliptic. It is also bounded and Hölder regular: indeed, we fix y ∈ R d and y ′ ∈ B(y, η), η = η(d, c, p, a per ), such that |∇w per ξ (y) -∇w per ξ (y ′ )| ≤ c By the triangle inequality, the inequality (2) and Proposition 5.2.1, we have that∂ ξ a y, ξ + ∇w per ξ (y) -∂ ξ a y ′ , ξ + ∇w per ξ (y ′ ) ≤ ∂ ξ a y, ξ + ∇w per ξ (y) -∂ ξ a y ′ , ξ + ∇w per ξ (y) + ∂ ξ a y ′ , ξ + ∇w per ξ (y) -∂ ξ a y ′ , ξ + ∇w per ξ (y ′ ) ≤ λC y -y ′ ξ p-2 +λC ∇w per ξ (y) -∇w per ξ (y ′ ) , where we used (5.2.35) in the last inequality. Since ∇w per ξ ∈ C 0,α (R d ), we get that ∂ ξ a y, ξ + ∇w per ξ is Hölder regular, uniformly in y ∈ R d . Besides, we have the decomposition ∂ ξ a per y, ξ + ∇w per ξ + ∂ ξ a y, ξ + ∇w per ξ , ∂ ξ a y, ξ + ∇w per ξ ∈ L p ′ (R d ) d×d .

H 1 ( 2 . ( 5 . 8 . 29 ) 2 ++ T + sZ p- 2 + 2 +

 125829222 R d ). Besides, ∇ w ξ ∈ • W 1,p ∩ • W 1,p ′ (R d )thanks to Theorem 5.2.5 and the same holds true for ∇ w η . Writing the difference between the PDEs satisfied by ∇ w ξ and ∇ w η , we have that -div a y, ξ + ∇w per ξ + ∇ w ξ -a y, ξ + ∇w per ξ -a y, η + ∇w per η + ∇ w η + a y, η + ∇w per η = div a y, ξ + ∇w per ξ -a y, η + ∇w per η =:h ξ,η (5.8.25)in the weak sense in• W 1,p ∩ • H 1 (R d ).We note that, due to Assumption (3) of Subsection 5.2.5 and the L ∞ -bounds on the periodic correctors, we have that|h ξ,η | ≤ b(y) |ξ + ∇w per ξ | p-2 + |η + ∇w per η | p-2 ξ + ∇w per ξ -(η + ∇w per η ) ≤ Cb(y) ξ p-1-γ + η p-1-γ |ξ -η| γ , (5.8.26)where γ is given by Proposition 5.2.1 (iv). Thus, h ξ,η ∈ L p ′ (R d ) and∥h ξ,η ∥ L p ′ (R d ) ≲ ∥b∥ L p ′ (R d ) ξ p-1-γ + η p-1-γ |ξ -η| γ ≤ C|ξ -η| γ . (5.8.27) We now test (5.8.25) against the function ∇ w ξ -∇ w η . This yields ¢ R d a y, ξ + ∇w per ξ + ∇ w ξ -a y, ξ + ∇w per ξ -a y, η + ∇w per η + ∇ w η + a y, η + ∇w per η• ∇ w ξ -∇ w η dy = -¢ R d h ξ,η • ∇ w ξ -∇ w η .(5.8.28)As in the proof of Lemma 5.5.1, we introduceT := ∇ w ξ + ∇ w η 2 and Z := ∇ w ξ -∇ w ηWe also define the functionΨ γ (s) = [Φ γ (sZ) -Φ γ (0)] • Z, s ∈ [0, 1],where, forZ ∈ R d , Φ γ ( Z) := a(y, ξ + ∇w per ξ + T + Z) -a(y, η + ∇w per η + T -Z) -γa(y, Z),with γ > 0 to be chosen later. The function Ψ γ is of class C 1 over [0, 1] andΨ ′ γ (s) = Z T Φ ′ γ (sZ) • Z = Z T ∂ ξ a y, ξ + ∇w per ξ + T + sZ + ∂ ξ a y, η + ∇w per η + T -sZ -γ∂ ξ a(y, sZ) • Z ≥ (4) λ -1 ξ + ∇w per ξ + T + sZ pλ -1 η + ∇w per η + T -sZ p-2 -2γλs p-2 |Z| p-2 |Z| 2 .We note, as in inequality(5.5.6), that sZ p-2 ≤ C p ξ + ∇w per ξ η + ∇w per η + T -sZ p-2 + ξ + ∇w per ξ -(η + ∇w per η ) η + ∇w per η + T -sZ p-2 + |ξ -η| γ(p-2) .

ξ+. 2 ⊂ 2 . 2 +

 222 ∇ w ξ ) -a(y, η + ∇w per η + ∇ w η ) -a(y, ξ + ∇w per ξ + T ) + a(y, η+ ∇w per η + T ) • Z ≥ γa(y, Z) • Z -λ -1 |ξ -η| γ(p-2) |Z| 2 ≥ (4) λ -1 γ|Z| p -λ -1 |ξ -η| γ(p-2) |Z| 2 .(5.8.30) It remains to bounda(y, ξ + ∇w per ξ + T ) -a(y, η + ∇w per η + T ) -a(y, ξ + ∇w per ξ ) -a(y, η + ∇w per η ) . (5.8.31)We apply as in Lemma 5.5.1 the mean-value inequality to the functionχ T (x) := a(y, x + T ) -a(y, x), x ∈ ξ + ∇w per ξ , η + ∇w per η ,which derivative is given byχ ′ T (x) := ∂ ξ a(y, x + T ) -∂ ξ a(y, x), x ∈ ξ + ∇w per ξ , η + ∇w per ηWe choose η sufficiently close to ξ so that for all y ∈ Q,ξ + ∇w per ξ (y), η + ∇w per η (y) ⊂ B ξ + ∇w per ξ (y), c R d \ B 0, cBy separating the cases between |T | ≥ 1 2 |x| and |T | ≤ 1 2 |x| and using (5.2.35) together with homogeneity, we easily show, as for the p-Laplacian case, that, for p < 3,|χ ′ T (x)| ≲ c p-3 |T |, proving that a(y, ξ + ∇w per ξ + T ) -a(y, η + ∇w per η + T ) -a(y, ξ + ∇w per ξ ) + a(y, η + ∇w per η ) ≲ c p-3 |T | ξ + ∇w per ξ -(η + ∇w per η ) ≲ c p-3 |ξ -η| γ |T |,(5.8.32)where we used Proposition 5.2.1 (iv) in the last inequality. The case p ≥ 3 is treated analogously (we obtain a bound analogous to (5.5.14)).Let us go back to(5.8.28). Gathering together (5.8.30) and (5.8.32), we get thata(y, ξ + ∇w per ξ + ∇ w ξ ) -a(y, η + ∇w per η + ∇ w η ) -a(y, ξ + ∇w per ξ ) + a(y, η + ∇w per η ) • Z ≥ γa(y, Z) • Z -Cλ -1 |ξ -η| γ(p-2) |Z| 2 -Cc p-3 |T ||ξ -η| γ |Z|.Noting that, thanks to Theorem 5.2.5, T ∈ L p ′ (R d ) d , we get, for p < 3, :γ ¢ R d ∇ w ξ -∇ w η p ≤ Cλ -1 |ξ -η| γ(p-2) ¢ R d |∇ w ξ -∇ w η | Cc p-3 |ξ -η| γ ¢ R d |∇ w ξ -∇ w η | |∇ w ξ + ∇ w η | + ¢ R d |h ξ,η | |∇ w ξ -∇ w η | ≤ C(d, p, a, λ, c)|ξ -η| γ(p-2) ¢ R d |∇ w ξ -∇ w η | |∇ w ξ + ∇ w η | + ¢ R d |h ξ,η | |∇ w ξ -∇ w η | .Using (5.8.27) and the bound given by Theorem 5.2.5 together with Hölder inequality, we conclude finally that,

a: 2 ≤ a ε ∇v ν,ε k p- 1 +- 2 ∇L

 212 ε→0 a * (⟨∇u 0 ⟩ ν k in L p ′ (Ω).The first point follows from[START_REF] Blanc | Precised approximations in elliptic homogenization beyond the periodic setting[END_REF] Proposition 5.1] which shows that if ∇w ξ ∈ (L ∞,per + L p ) (R d ) then w ξ is sublinear and we have the bound|∇w ξ (x) -∇w ξ (y)| ≤ C|x -y| 1-νp , ν p := min 1, first point.The second point follows from the weak convergence of the periodic part ∇w per the fact that ∇ w ⟨∇u0⟩ ν k ∈ L p (R d ). The third point follows from the corrector equation (5.2.9). The last point follows on the one hand from the convergence (of the periodic function)* (⟨∇u 0 ⟩ ν k ) in L p ′ (Ω)and on the other hand from the estimate (with∇v ν,ε,per k C(d, p, λ) ∇v ν,ε k p-2 + ∇v ν,ε,per k pw ⟨∇u0⟩ k,ν • ε ≤ C(d, p, a) a ε ⟨∇u 0 ⟩ ν k p-1 + C(d, p, a) ⟨∇u 0 ⟩ ν k p-2 ∇ w ⟨∇u0⟩ k,ν • ε ,where we used Proposition 5.2.1 (ii) and Theorem 5.2.4 (ii). Since a ∈ L p ′ (R d ) and ∇ w ⟨∇u0⟩ k,ν ∈ L p (R d ), we get thata ε ∇v ν,ε k ∇v ν,ε k p-2 -a per ε ∇v ν,ε,per k ∇v ν,ε,per k p-2 -→ ε→0 0 in L p ′ (Ω),proving the last point above.Before going further, we estimate in the following lines the quantity k∈Iν ¢ ∞ -bound given by Theorem 5.2.4 (ii), we have that k∈Iν ¢ with the proof of(5.8.33). We fix ϕ ∈ D(Ω) andη ν ∈ C ∞ (R d ) such that for all k ∈ Z d , η ν ∈ C ∞ 0 (Q ν k ) and 0 ≤ η ν ≤ 1.The function η can simply be chosen under the formk∈Z d η 2 ν (• -2 -ν k), where η ∈ C ∞ 0 (R d ) with compact support included in Q and 0 ≤ η ≤ 1. The idea is to estimate¢ Ω a ε ∇u ε ∇u ε p-2 • ϕη ν -

  8.34) and the a priori estimates on ∇u ε , we find that |I| ≤ C∥ϕ∥ L ∞ the constant C depends, among other quantities, on ∥∇u 0 ∥ L p (Ω) . We now estimate the quantityk∈Iν ¢ Q ν k ∇u ε -∇v ν,ε k p η ν .By monotonicity, we have that

.8. 43 )

 43 Using(5.8.41) and that a 0 ∈ L p ′ (Ω), we get thatk∈Iν ¢ Q ν k a 0 -a * (⟨∇u 0 ⟩ ν k ) ∇u 0 -⟨∇u 0 ⟩ ν k ≤ C ∥∇u 0 ∥ L p (Ω), ∥a 0 ∥ L p ′ (Ω) of (5.8.44) tends to zero as ν -→ +∞. Inserting this into (5.8.43), we have that lim sup ⟨∇u 0 ⟩ ν k ) ϕ = 0.We now recall that (see[START_REF] Fusco | On the homogenization of quasilinear divergence structure operators[END_REF] Proposition 2.7])∀ξ, ξ ′ ∈ R d , |a * (ξ) -a * (ξ ′ )| ≤ C |ξ| p-2 + |ξ ′ | p-2 |ξ -ξ ′ |.

  1.2. PRÉSENTATION DE LA THÈSE CHAPITRE 1. INTRODUCTION où les fonctions u i et p i sont supposées régulières et les fonctions u i (x, •) et p i (x, •), x ∈ Ω, traduisent les propriétés microscopiques du matériau (elles sont en particulier périodiques dans le cas périodique et périodique "avec défaut" dans le cadre plus général étudié ici). À cause de la contrainte sur la divergence, l'identification des fonctions u i est plus délicate. Au premier ordre (voir [91, Chapter 7]), on trouve que, formellement,

  ), pourvu que le second membre de (1.2.32) soit dans le bon espace. L'objectif des travaux[START_REF] Blanc | A possible homogenization approach for the numerical simulation of periodic microstructures with defects[END_REF][START_REF] Blanc | Local profiles for elliptic problems at different scales: defects in, and interfaces between periodic structures[END_REF][START_REF] Blanc | On correctors for linear elliptic homogenization in the presence of local defects[END_REF][START_REF] Blanc | Precised approximations in elliptic homogenization beyond the periodic setting[END_REF] est de généraliser ces résultats au cas d'un coefficient périodique et localement perturbé au sens suivant : on se fixe un coefficient A per satisfaisant (1.2.31) que l'on suppose de régularité Hölderienne. Soit 1

  avec les mêmes hypothèses que dans la section 1.2.2.1 sur A, à savoir

  .2.40) La première étape est d'étudier (1.2.39) quand A = A per . Un développement à deux échelles standard suggère que u ε

  1,p ′ (Ω) et a : R d -→ R est un coefficient périodique avec défauts au sens de la section 1.2.2.1. On peut alors décomposer a sous la forme a = a per + a. Le cas a = a per a été étudié dans[START_REF] Fusco | Further results on the homogenization of quasilinear operators[END_REF][START_REF] Fusco | On the homogenization of quasilinear divergence structure operators[END_REF][START_REF] Dal | Correctors for the homogenization of monotone operators[END_REF], tout du moins pour ce qui concerne la limite de la suite (u ε ) ε>0 et son gradient, et la construction des correcteurs. Dans cette section (et dans tout le chapitre 5), on suppose que a ∈ L p ′ ∩ C 0,α (R d ). Nous rappelons d'abord les résultats du cas périodique :

	Théorème 10 ([56, 57, 44] adapté à la forme de l'équation (1.2.42)). Soit a per un coefficient pé-
	riodique

  Soit a = a per + a un coefficient coercif et borné tel que a per soit également coercif et borné. On suppose que a et a per sont lipschitziens. Soit q ≤ p ′ tel que a ∈ L q (R d ). Il existe une unique fonction w ξ ∈ W ξ+∇w per

													,	(1.2.57)
	et l'on montre le théorème suivant.
	Théorème 11 (Theorem 5.2.3 du chapitre 5). ξ	solution au sens faible dans W ξ+∇w per ξ
	de (1.2.54)-(1.2.55), c'est-à-dire : pour tout u ∈ W ξ+∇w per ξ	,
	¢	R d	a ξ + ∇w per ξ	+ ∇ w ξ ξ + ∇w per ξ	+ ∇ w ξ	p-2	-ξ + ∇w per ξ	ξ + ∇w per ξ	p-2	•∇u = -	¢	R d	h•∇u.
	En particulier, w ξ := w per ξ	+ w ξ est solution de (1.2.53) au sens des distributions.

∇w per ξ : Théorème 13 (Extrait du théorème 5.2.4). Sous les hypothèses du théorème 11, supposons que pour tout ξ ∈ R d , on ait ξ + ∇w per ξ > c|ξ|, où la constante c > 0 est indépendante de ξ. On a alors l'existence d'une constante C = C(d, p, a, c) telle que pour tout ξ

	.62)
	où M ε est défini dans (1.2.47).
	Notons que les preuves des points (i) et (iii) du théorème 12 ne requièrent pas l'hypothèse (H),
	comme on peut s'y attendre. Nous souhaitons dans la suite améliorer les résultats sur la continuité de
	l'application ξ -→ ∇w ξ , notamment en vue d'obtenir des taux de convergence. En particulier, nous
	aimerions, comme pour les correcteurs périodiques, obtenir la continuité hölderienne par rapport
	au vecteur ξ. Nous avons obtenu ce résultat, sous une condition de non dégénérescence du poids
	ξ +

  1.2.63) où, au moins formellement, ∇w ξ,h = ∂ ξ ∇w ξ • h. Des bornes sur ∇w ξ,h permettent de retrouver des résultats analogues au théorème 13 (sans hypothèse forte sur ξ + ∇w ξ ). Nous pensons que, en s'appuyant sur les résultats du cadre linéaire, il est possible d'étudier (1.2.63), et donc les opérateurs considérés dans

Assumption (A2) reads : (A2) There exists a sequence (α k ) k∈Z d such that

				.1.12)
	and
	O per,-k	(α) = {x ∈ O per k	s.t. dist(x, ∂O per k ) > α}.	(2.1.13)

  .3.4) Summing (2.3.4) over k, then adding (2.3.3) to the result and defining C := max(|K|C 0 , d/ ρ), we obtain

  .4.24) Let us treat the remaining term. We first recall (see Remark 2.4.7) that the linear form v →

	¢	Γ1	∂w per ∂n ext	v is strongly and thus weakly continous on H 1 0 (R

d \ O). We apply this continuity to v n := w n -ϕ, where ϕ was defined in the proof of Lemma 2.

[START_REF] Allaire | Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes I. Abstract framework, a volume distribution of holes[END_REF]

.8. We deduce ¢ Γ1 ∂w per ∂n ext w n -→ n→+∞ ¢ Γ1 ∂w per ∂n ext w. (2.4.25) Finally, collecting (2.4.23), (2.4.24) and (2.4.25) and letting n → +∞, we conclude that 1

  Ωε) is bounded independently of ε. Next, we multiply (2.4.26) by ϕ ε , integrate by parts and apply the Cauchy-Schwarz inequality:

  For k large enough such that α k < α, one thus has |U per k (α k )| ≤ Cα k where C is a constant. This, together with (2.5.2), proves (4.2.2). The fact that |K| < +∞ is a direct consequence of (4.2.2) and of the fact that for all k ∈ K, |O k ∆O per k | ≥ |O per k \ O k | = |O per 0 |.

		.5.2)
	We now use [53, Theorem 3.2.39] to control the measure of U per k (α k ): there exists α > 0 such that
	∀α < α, |U per 0 (α)| ≤ 2C|∂O per 0 |α.	(2.5.3)
	By translation invariance, the above assertion is true for U per 0 (α) replaced by U per k (α) :
	∀k ∈ Z d , ∀α < α, |U per k (α)| ≤ 2C|∂O per 0 |α.

Lemma 2.5.2. Let (O k ) k∈Z d be a sequence of open sets satisfying Assumption (A2). There exists ρ > 0 such that ∀k ∈ K c , ∃B k s.t |B k | ≥ ρ and B k ⊂ O k ∩ O per k , where B k denotes an open ball and K is defined in Lemma 2.5.1. Proof. Since O per 0 is open, it contains a ball B ⊂ B ⊂ O per 0 . One has δ := dist(B, ∂O per 0 ) > 0. By translation invariance, for all k

  .5.7)Proof of Theorem 2.5.4. Existence and uniqueness of a solution u ε for fixed ε > 0 follow from standard elliptic regularity theory, see[START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF] Theorems 9.15 & 9.19]. We now prove the estimate (2.5.6).

  .2.3)and p 0 is given by the Darcy's law (3.1.2).

	Theorem 3.2.1 (Existence of correctors). Suppose that Assumptions (A1)-(A3) and (A4)

0 are satisfied. For all j ∈ {1, 2, 3}, System (3.2.3) admits in the distribution sense a solution (w j , p j ) of the form w j = w per j + w j and p j = p per j + p j where ( w j , p j

  is improved in some sense when we use w j instead of w per j . We exhibit in Theorem 3.2.3 such a situation (see Remark 3.2.6).Before stating Theorem 3.2.3, we obtain in Theorem 3.2.2 H 2 -estimates for the solution of a Stokes system posed in Ω ε (see[START_REF] Masmoudi | Some uniform elliptic estimates in a porous medium[END_REF] Theorem 4.1] for the periodic case).

	Theorem 3.2.2 (Estimates for a Stokes problem). Suppose that Assumptions (A4) 0 and (A5) are
	satisfied

  2 ) H 1 -convergence rate of R ε obtained in Theorem 3.2.3 is sharper than the O(ε 3/2 ) H 1 -convergence rate obtained in[START_REF] Shen | Sharp convergence rates for Darcy's law[END_REF] Theorem 1.1]. By applying Theorem 3.2.3, we get that R ε ∈ H 2 (Ω ε ). We now note that, in general, one has R per

	Remark 3.2.6.

ε / ∈ H 2 (Ω ε ). This follows from the fact that w per j

  in L p -strong is sufficient to deduce Lemma 4.1.2.

	.1.10)
	Using the a priori estimate (4.1.4) and the L p -strong convergence u ε -→ ε→0 subsequence, we conclude that Lemma 4.1.2 is satisfied when p < 2 * . Note that we have shown, thanks u * after extraction of a
	to (4.1.10), that proving u ε -→
	4.2 Proof of the main result
	4.2.1 Proof of Lemma 4.1.2

ε→0 u * We first state the following Lemma. Lemma 4.2.1. Let f ∈ L ∞ (Ω) and u ε be the solution to (4.0.1). After extraction of a subsequence, we have that u ε |u ε | p-2 converges in the L p ′ (Ω) strong topology to u * |u * | p-2 when ε -→ 0 and that u ε converges in the L p (Ω) strong topology to u * when ε → 0.

Proof of Lemma 4.2.1. We note that it is enough to show that

Table 5 .

 5 1: Numerical errors for different values of ε in L ∞ -norm.

	1	0.156	0.109
	0.05	0.163	0.137
	0.01	0.170	0.0657
	0.005	0.170	0.0288
	0.001	0.170	0.0245
	0.0005	0.171	0.0136
	ε	∥R per ε ∥ L 2	∥R ε ∥ L 2
	0.1	6.39	3.85
	0.05	5.01	3.16
	0.01	2.13	0.740
	0.005	1.47	0.331
	0.001	0.654	0.108
	0.0005	0.46	0.0461

Table 5 .

 5 2: Numerical errors for different values of ε in L 2 -norm.

  Proof ofLemma 5.4.1. We first check that W ξ+∇w per ξ is a Banach space. Let (v n ) n∈N be a Cauchy sequence in W ξ+∇w per ξ . In particular we have that ∇v n n∈N is a Cauchy sequence in L p (R d ) and that |ξ + ∇w per ξ | Cauchy sequence in L 2 (R d ). Thus, there exist ∇v ∈ L p (R d ) and g ∈ L 2 (R d ) such that

	p-2 2 ∇v n is a ∇v n -→ n∈N n→+∞ ∇v in L p (R d ) and |ξ + ∇w per ξ |	p-2 2 ∇v n -→ n→+∞	g in L 2 (R d ).	(5.4.1)
	Up to a subsequence, we have the a.e. convergences in (5.4.1). Thus g = |ξ + ∇w per ξ | p-2 ∇v a.e. This
	gives that v n -→ n→+∞	v in W ξ+∇w per ξ

  This proves (ii) since T (v) = T ′ (∇v, ∇v) for v ∈ W ξ+∇w per ∇v n n∈N is bounded in L 2 (R d )-norm. Thus, after extraction, we have that ∇v n converges weakly in L p (R d )-norm to some ∇v ∈ L p (R d ) and that |ξ + ∇w per ξ | ∇v n converges weakly to some g in L 2 (R d )-norm. To identify g and |ξ + ∇w per ξ | ∇v n converges weakly to ∇v in L 2 (B R ) and thus |ξ+ ∇w per ξ | ∇v in L 2 (B R ) since |ξ + ∇w per ξ | ∈ L ∞ (R d ).By uniqueness of the weak limit, we conclude that g = |ξ + ∇w per ξ | 2 ∇v a.e. (in the Lebesgue measure sense) in B R . Since R is arbitrary, we have that g = |ξ + ∇w per ξ |

			ξ	.
	Let (v n ) n∈N ⊂ W ξ+∇w per ξ	be a bounded sequence. Then ∇v n n∈N is bounded in L p (R d )-norm
	and |ξ + ∇w per ξ |	p-2 2 p-2
			2 p-2 2 ∇v we note that for fixed
	R > 0, p-2 2 ∇v n converges weakly
	to |ξ + ∇w per ξ |	p-2 2 p-2
			p-2 2 ∇v a.e., in particular v ∈ W ξ+∇w per ξ	. It remains to show
	that v n converges weakly to v in W ξ+∇w per ξ

  .4.4) We gather in Lemmas 5.4.2 and 5.4.3 below the key properties satisfied by the functionalF ξ . Let ξ ∈ R d , h ∈ L p ′ (R d ) d , F ξ be defined by (5.4.2) over V and the space W ξ+∇w perIn particular, F ξ (v) is finite if and only if v ∈ W ξ+∇w per The function F ξ is convex over V and strictly convex over W ξ+∇w per The point (i) is a simple application of (5.8.9). Let v ∈ W ξ+∇w per ξ

	Lemma 5.4.2. ξ
	be defined by (5.2.10).
	(i) There exist two constants c, C > 0 such that for all v ∈ W ξ+∇w per ξ	,
	c -1 + ∥v∥ 2 W ξ+∇w per ξ	≤ F ξ (v) ≤ C 1 + ∥v∥ p W ξ+∇w per ξ	.	(5.4.5)
		ξ	.
	(ii) ξ	.
	Proof of Lemma 5.4.2.

  .4.7) and we notice that, since x 2 ≤ x p + 1 for x ∈ R + ,

	∥∇v∥ p L p ≤ ∥v∥ p W ξ+∇w per ξ	and	|ξ + ∇w per ξ |	p-2 2 ∇v	2 L 2 ≤ v	2 W ξ+∇w per ξ	≤ 1 + v	p W ξ+∇w per ξ	. (5.4.8)
	Applying (5.4.7) and (5.4.8), we get that

  ′ in the last inequality. This proves that B ∈ L 1 (R d ) and that

								.4.17)
	Gathering (5.4.16), (5.4.17) and recalling the definition (5.2.11), we have proved that A ∈ L 1 (R d )
	and that	¢
		R d	|A| ≤ C ∥u∥ 2 W ξ+∇w per ξ	+ ∥u∥ p W ξ+∇w per ξ	,	(5.4.18)
	¢
	R d	|B| ≤ C ∥v∥ 2 W ξ+∇w per ξ	+ ∥u∥ 2 W ξ+∇w per ξ	+ ∥v∥ p W ξ+∇w per ξ	+ ∥u∥ p W ξ+∇w per ξ	,	(5.4.20)

where the constant C does not depend on u.

We now turn to estimating B, see

(5.4.15)

. Using (5.8.7) and Cauchy-Schwarz inequality, we have that

|B| ≤ C |ξ + ∇w per ξ + ∇v| p-2 + |ξ + ∇w per ξ | p-2 |∇v||∇u| ≤ C |ξ + ∇w per ξ | p-2 |∇v| 2 + |ξ + ∇w per ξ | p-2 |∇u| 2 + |∇v| p-1 |∇u| ≤ C |ξ + ∇w per ξ | p-

2 |∇v| 2 + |ξ + ∇w per ξ | p-2 |∇u| 2 + |∇v| p + |∇u| p (5.4.19) where we used Young's inequality in form of b 1 ≤ b p 1 /p + b p ′ 2 /p

  Proof of Theorem 5.2.3. We prove below that, for h∈ L p ′ (R d ) d , the PDE -diva ξ + ∇w per it is clear that h ∈ L p ′ (R d ) d . Since(5.4.22) is solvable for this choice of h, Theorem 5.2.3 is proved.We are thus left to study the PDE (5.4.22) for an abstract right-hand side h ∈ L p ′ (R d ) d . With Lemma 5.4.2, Lemma 5.4.3 and Lemma 5.4.1, we prove in a standard way that Problem (5.4.22) admits a unique solution. Indeed, let us consider the minimization Problem: This Problem admits a unique solution. The existence is guaranteed by the following procedure: let (v n ) n∈N ⊂ W ξ+∇w per

	ξ	+ ∇w ξ	p-2 (ξ + ∇w per ξ	+ ∇ w ξ ) -ξ + ∇w per ξ	p-2 (ξ + ∇w per ξ ) = div(h), (5.4.22)
	admits a unique solution w ξ ∈ W ξ+∇w per ξ	in the weak sense (see Definition 5.2.2). Theorem 5.2.3 is
	then proved by defining
			h := a(ξ + ∇w per ξ )|ξ + ∇w per ξ | p-2 .	(5.4.23)
	Because of Proposition 5.2.1 (ii) and Assumptions (A2)-(A3), min ξ v∈W ξ+∇w per F ξ (v).	(5.4.24)
						2 W ξ+∇w per ξ	.
	Lemma 5.4.3 is proved.

ξ be a minimizing sequence. Then, by the left-hand estimate of (5.4.5), we have that the sequence ∥v n ∥ W ξ+∇w per ξ n∈N is bounded (see (5.2.11) for the definition of ∥ • ∥ W ξ+∇w per ξ ). By Lemma 5.4.1 (iii), we get that the sequence (v n ) n∈N weakly converges, up to a subsequence, to some v in W ξ+∇w per ξ when n -→ +∞. Since by Lemma 5.4.2 (ii) and Lemma 5.4.3, F ξ is convex and continuous over W ξ+∇w per ξ

  2.1 (i), to the form of the PDE (5.2.12)-(5.2.13) defining ∇ w ξ and the fact that this PDE is uniquely solvable in the sense of Definition 5.2.2. Note that we use that for t ̸ = 0, W ξ+∇w per This result is proved in [104, Lemma 2.2] but we reproduce the proof here for the sake of completeness. Let ξ ∈ R d . By Definition 5.2.2 with ∇ϕ = ∇ w ξ , the inequality (5.8.5), Hölder inequality together with (5.4.23), we have c

ξ = W tξ+∇w per tξ .

Proof of (ii).

  .5.[START_REF] Brezis | Functional analysis, Sobolev spaces and partial differential equations[END_REF] In particular,(5.5.27) proves that ∇w ξ is bounded and that ∥∇w ξ ∥ L ∞ (R d ) ≤ C(d, p, λ, a Lip )|ξ|. By Assumption (A2), the non-linear operator a(y, z) = a(y)z|z| p-2 falls into the scope of[START_REF] Dibenedetto | C 1+α local regularity of weak solutions of degenerate elliptic equations[END_REF]. Let x ∈ R d , up to subtracting of w ξ (x), we have by(5.5.27) that |w ξ | ≤ C(d, p, λ, a Lip )|ξ| on B(x, 2).

	Thus, applying [48, Theorem 2], there exist α > 0 and C 0 > 0 depending only on λ, a Lip , p, d, p, and
	C(d, p, λ, a Lip )|ξ| such that ∇w ξ ∈ C 0,α (B(x, 1)) and
	[∇w ξ ] C 0,α (B(x,1)) ≤ C 0 .	(5.5.28)

  .6.3) By contradiction, if(5.6.3) does not hold, there exists δ > 0 and two sequences (x n ) n∈N ⊂ R d and (ξ n ) n∈N ⊂ R d such that |x n | -→ n→+∞ +∞ and ∥∇ w ξn ∥ L p (xn+Q) ≥ δ|ξ n |.By Theorem 5.2.4 (i), we can assume that |ξ n | = 1. Thus, up to a subsequence, ξ n -→ However, by (5.6.1), for all n large enough, we have that ∥∇ w ξn -∇ w ξ ∥ L p (xn+Q) ≤ δ/2. Thus, for n large enough, we have that ∥∇ w ξ ∥ L p (xn+Q) ≥ δ/2. Since |x n | -→

	n→+∞	+∞, this contradicts that ∇ w ξ ∈ L p (R d ). Thus (5.6.3) is
	satisfied.

n→+∞ ξ.

  ) • Φ = O(δ β ),(5.6.13) where the O is independent of δ. The term C ε is also treated by replacing M ε ∇u * by Φ and using the div-curl Lemma. Noticing that a(./ε)p(./ε, η j )|p(./ε, η j )| p-2 -⇀ Besides, since |x| p -|y| p ≤ C |x| p-1 + |y| p-1 |x -y| for all x, y ∈ R d , we get

			2, we obtain that
					¢
			lim sup	B ε -
			ε→0
						ε→0	a * (η j ) in L p ′ (Ω), we obtain that
					¢
			lim sup	C ε -	a * (Φ) • ∇u * = O(δ β ).	(5.6.14)
			ε→0		Ω
	We introduce D per ε	:=	¡ Ω a per (./ε) |p per (./ε, M ε ∇u * )|	p . By [44, Step 1, pp.1161-1162], we have that
					¢
			D per ε	-→ ε→0	Ω	a * (∇u * ) • ∇u * .	(5.6.15)

Ω

a * (∇u *

  .7.19) Besides, by (5.8.7), (i), (ii) and the L p loc (R d )-convergence of ∇v n , we have that

  Taking into account Remark 5.7.3, we have that v∈ W ξ+∇w per = a, v n = w ξn ϕ n = ξ n • x + w per ξn , ϕ = ξ • x + w per ξ h n = a|ξ n + ∇w per ξn | p-2 (ξ n + ∇w per ξn ), h = a|ξ + ∇w per ξ | p-2 (ξ + ∇w per ξ ).

					ξ	and
		|ξ n + ∇w per ξn |	p-2 2 ∇ w ξn -⇀ n→+∞	|ξ + ∇w per ξ |	p-2 2 ∇v in L 2 (R d ).
	We apply Lemma 5.7.2 with
	  	a n
	 

Theorem 5.2.8 Let (ξ n ) n∈N ⊂ R d such that ξ n -→ n→+∞ ξ for ξ ∈ R d . We aim at showing that ∇w ξn -→ n→+∞ ∇w ξ in L p unif (R d ). By Proposition 5.2.1 (iii), it is sufficient to show that ∇ w ξn -→ n→+∞ ∇ w ξ in L p unif (R d ). Step 1. We have that ∇ w ξn -⇀ n→+∞ ∇ w ξ in L p (R d ). Indeed, by (5.8.5), (5.8.6) and the form of h, see (5.2.13), we have the following a priori estimate: there exists a constant C = C(d, p, a, a per ) > 0 such that for all n ∈ N, ∥ w ξn ∥ W ξn+∇w per ξn ≤ C |ξ n | + |ξ n | p/2 . (5.7.20) In particular, there exists v ∈ V (see (5.4.3) for the definition of V ) and w ∈ L 2 (R d ) such that ∇ w ξn -⇀ n→+∞ ∇v in L p (R d ) and |ξ n + ∇w per ξn | p-2 2 ∇ w ξn -⇀ n→+∞ w in L 2 (R d ). (5.7.21)

  x n ) n∈N ⊂ Q, (ξ n ) n∈N ⊂ R d and (η n ) n∈N ⊂ R d such that for all n ∈ N, |ξ n | = 1, 0 < |ξ n -η n | ≤ 1 2 and ∇w per ξn (x n ) -∇w per ηn (x n ) ≥ n|ξ n -η n | γ . := |ξ n -η n | γ/α .Integrating (5.8.3) over B(x n , δ n ), we get that -η n | pγ δ d n = Cn p |ξ n -ξ n | γ(p+d/α) = Cn p |ξ n -η n | βp . (5.8.4)

	∇w per ξn -∇w per ηn	p L p (Q) ≥ |B 1 |	n 2
	However, by (5.2.16), ∇w per ξn -∇w per ηn	p L

By Proposition 5.2.1 (ii), we have for n large enough that ∀y ∈ B(x n , δ n ), ∇w per ξn (y) -∇w per ηn (y) ≥ n 2 |ξ n -η n | γ , (5.8.3) where δ n p |ξ n p (Q) ≤ C|ξ n -η n | βp

  1,p 0 (Q R ). Thus there exists a function Φ ∈ C ∞ 0 (Q R ) such that ∥w R -Φ∥ W 1,p (Q R ) ≤ εR -dpWe extend Φ by zero outside Q R . By Hölder inequality, we have that∥w R -Φ∥ H 1 0 (Q R ) ≤ ε. (5.8.15) We next show that ∥v -Φ∥ W ξ +∇w per ξ ≤ C(ξ, d, p, a per , C poinc )ε, (5.8.16)where C poinc denotes the maxmimum between the L p Poincaré-Wirtinger constant on Q \ Q 1/2 and the weighted L 2 Poincaré-Wirtinger constant, given by Assumption (A4)', on Q \ Q 1/2 . By the triangle inequality, we have that ∥v -Φ∥ W ξ +∇w per Assume that the non-linear operator a satisfies the Assumptions of Subsection 5.2.5. Then there exists a constant c > 0 such that ∀ξ, ξ ′ ∈ R d , ∀y ∈ R d , {a(y, ξ) -a(y, ξ ′ )} • {ξ -ξ ′ } ≥ c|ξ -ξ ′ | p .(5.8.20)

	p-2	(≤ ε) .	(5.8.14)

ξ ≤ ∥v -w R ∥ W ξ +∇w per ξ + ∥w R -Φ∥ W ξ +∇w per ξ .

(5.8.17)

Lemma 5.8.5.

où la convergence a lieu en un certain sens que nous ne précisons pas.

On note e 1 , ..., e d la base canonique de R d .

Nous avons appris plus tard qu'elle avait déjà été démontrée dans un cadre différent (mais contenant le nôtre) dans[START_REF] Donato | Convergence of Dirichlet problems for monotone operators in a class of porous media[END_REF].

On teste d'abord (1.2.51) contre une fonction de D(R d ) puis on raisonne par densité pour approcher ∇y u 0 .

Notons que, dans le cas périodique, on peut se limiter à intégrer sur Q.

Enfin, une dernière piste concernerait des taux de convergence dans le cas p > 2 pour l'équation de p-Laplace. Comme indiqué plus haut, des résultats sont déjà disponibles dans la littérature pour des équations non dégénérées[START_REF] Clozeau | Quantitative nonlinear homogenization: control of oscillations[END_REF][START_REF] Fischer | Optimal homogenization rates in stochastic homogenization of nonlinear uniformly elliptic equations and systems[END_REF]. Ces résultats s'appuient, entre autres, sur la continuité Hölder de l'application ξ -→ ∇w ξ . Dans le cas périodique, cette continuité est simple à établir. Dans le cas non périodique, nous y sommes parvenus sous hypothèse de non dégénérescence du correcteur périodique. Il serait intéressant, sous cette hypothèse -certes assez forte -, de regarder dans quelle mesure la méthode présentée dans[START_REF] Clozeau | Quantitative nonlinear homogenization: control of oscillations[END_REF] peut être exploitée dans le cadre du chapitre 5.

i.e. a function u belongs toH 1 loc (R d \ O per ) if for all bounded open set U of R d , we have that u ∈ H 1 U ∩ (R d \ O per ) .

Rigorously, we should first take ϕ ∈ D(R d \ O), multiply and integrate by parts, and then approximate u 0 (x, •) in the H 1 -norm by a sequence (ϕn) n∈N of such functions.

The proof of this Theorem may be adapted to the new definition (2.5.5) of Ωε introduced in this section.

See Figure2.5 right. We have that χ = 1 in the complement of the purple boxes.

see[START_REF] Nirenberg | On elliptic partial differential equations[END_REF] Remark 5. p. 126] for the inequality in Ω. We deduce this estimate in the perforated domain using the Stein extension operators (see[1, Chapter 5]) of each perforation. These operators are bounded uniformly in k since the perforations are uniformly in k of class C 2 , see[START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF] Theorem

7.25].

The following proof of(3.3.22) differs from[START_REF] Wolf | Homogenization of the Stokes system in a non periodically perforated domain[END_REF] since in the proof there, we did not pay sufficiently attention to the perforations that intersect the boundary of Ω.

Acknowledgments

We thank C. Le Bris for comments and suggestions that greatly improved the manuscript. We also thank Patrizia Donato for pointing out the reference [51].

Acknowledgments

I am very grateful to my PhD advisor Xavier Blanc for many fruitful discussions and for careful reading of the manuscript. I also thank Claude le Bris for suggesting this subject to me and supporting this project.

Acknowledgments

The author warmly thanks his PhD advisor Xavier Blanc for fruitful discussions and for reading many versions of this manuscript.

Appendices

Technical Lemmas

We recall that if R > 0, we define

(3.4.1) Lemma 3.4.1 (Divergence Lemma on Ω R ). Suppose that Assumption (A4) 0 is satisfied. Let 1 < q < +∞ and R > 0. Let f ∈ L q (Ω R ) be such that

admits a solution v ∈ W 1,q (Ω R ) 3 such that

where C > 0 is a constant independent of f and R.

Deuxième partie

Homogénéisation d'équations non linéaires avec défauts locaux

CHAPTER 4

HOMOGENIZATION OF SEMILINEAR EQUATIONS IN A PERIODIC SETTING WITH A LOCAL DEFECT

In this chapter, we consider the homogenization of the semi-linear PDE

where ε ≪ 1 and the coefficient A ε is of the form A ε = A(•/ε) where A is periodic (possibly perturbed by a local defect). We prove that, when ε tends to zero, u ε converges to a homogenized limit u * that is solution to an equation of the same type as (4.0.1). We also derive the coefficients of the homogenized equation and we compare it to the well-known corresponding linear problem:

(4.0.2)

In particular, we show that the correctors used for the homogenization of the semi-linear PDE (4.0.1) are the same as the ones used in the homogenization of (4.0.2). We then discuss the extension of the results beyond the periodic setting and, in particular, to the setting of [START_REF] Blanc | On correctors for linear elliptic homogenization in the presence of local defects[END_REF][START_REF] Blanc | A possible homogenization approach for the numerical simulation of periodic microstructures with defects[END_REF][START_REF] Blanc | Local profiles for elliptic problems at different scales: defects in, and interfaces between periodic structures[END_REF].

Introduction

Setting of the Problem

In the sequel, we consider d ≥ 1 and we denote by Q := (-1 2 , 1 2 ) d the unit cube of R d . We fix 2 ≤ p < +∞ and we set p ′ = p/(p -1). Let Ω be a Lipschitz bounded domain of R d . We set V := H 1 0 ∩ L p (Ω) endowed with the norm ∥v∥ V := ∥v∥ L p (Ω) + ∥v∥ H 1 0 (Ω) , where ∥u∥ H 1 0 (Ω) := ∥∇u∥ L 2 (Ω) . It is clear that (V, ∥ • ∥ V ) is a Banach reflexive space. The topological dual space of V is denoted by V ′ and the pairing between V ′ and V will be written ⟨•, •⟩. We have V ′ = H -1 (Ω) + L p ′ (Ω). We fix f ∈ V ′ and we consider the following semi-linear equation

Results in the non-periodic case

The first result of this contribution concerns the corrector equation (5.1.8) in the setting (5.1.1)-(5.1.2). For a fixed direction ξ ∈ R d , this equation, posed on the whole space R d , is

where the coefficient a is of the form a := a per + a and a per is a periodic coefficient. We assume that a and a per satisfy the following assumptions:

(A1) there exists λ > 0 such that (5.1.3) is satisfied;

(A2) the coefficients a and a per are Lipschitz-continuous;

(A3) the perturbation a vanishes at infinity in the sense that a ∈ L p ′ (R d ).

A few comments are in order. First, if a satisfies a ∈ C 0,1 (R d ) and a ∈ L q (R d ) for some q ≤ p ′ then a satisfies (A3) by interpolation. Second, Assumption (A2) allows to ensure local regularity (see Proposition 5.2.1 above) of the periodic and non-periodic correctors. Finally, the assumptions of [START_REF] Blanc | A possible homogenization approach for the numerical simulation of periodic microstructures with defects[END_REF] in the linear setting correspond to the case p = 2 in the assumptions (A1)-(A2)-(A3) above.

We now consider the equation (5.2.9) when the coefficient a has the non-periodic structure (5.1.2). For u ∈ L ∞ (R d ), we define the spaces

The space W u is endowed with the norm

(5.2.11)

In the sequel, we denote undifferently functions and equivalence classes for the relation: f ∼ g if and only if f -g is almost everywhere constant. Lemma 5.4.1 below gathers some properties satisfied by spaces of the form (5.2.10). In order to solve (5.2.9), we seek for w ξ of the form w ξ = w per ξ + w ξ where w per ξ is the solution to (5.2.3) such that 1 Q w per ξ = 0. We transform the equation (5.2.9) into

We easily check using Appendix 5.8.2 that each integral appearing in Definition 5.2.2 is convergent. Note that if w ξ is a solution to (5.2.12) in the sense of Definition 5.2.2, then it is a solution to (5.2.12) in the distribution sense but it is not clear that the converse holds true. This is true if the weight ξ + ∇w per ξ satisfies Assumption (A4)' below (see also Remark 5.2.13).

Theorem 5.2.3 (Existence of the non-periodic correctors). Assume that the coefficient a = a per + a satisfies Assumptions (A1)-(A2)-(A3). Then, for all ξ ∈ R d , there exists a unique solution w ξ to equation (5.2.9) such that w ξ ∈ W (i) The map ξ -→ ∇w ξ is homogeneous in the sense that for all ξ ∈ R d and t ∈ R,

(5.2.15)

) and, moreover, we have the estimates

(5.2.17

where γ is given by (5.2.8).

(iv) Assume that Assumption (A4) is satisfied. Then there exists a constant C = C(d, p, a, c) > 0 and an exponent γ > 0 both independent of ξ and η such that, for all ξ, η ∈ R d ,

An important tool to obtain Theorem 5.2.4(iii) is the following Theorem:

holds true where

Remark 5.2.6. Note that, under Assumption (A4), the non-periodic part ∇ w ξ of the corrector has the same integrability as the defect a at infinity. This is reminiscent of the linear case p = 2, see [START_REF] Blanc | Local profiles for elliptic problems at different scales: defects in, and interfaces between periodic structures[END_REF].

Using Theorem 5.2.4, we can prove qualitative results concerning the homogenization of (5.1.1) in the non-periodic setting.

Theorem 5.2.7.

Let Ω be a bounded smooth domain, f ∈ L p ′ (Ω), a := a per + a be a

scalar-valued coefficient satisfying Assumptions (A1)-(A2)-(A3)-(A4).

For ε > 0, let u ε ∈ W 1,p 0 (Ω) be the solution to (5.1.1).

However, because of (5.8.9), we have that

Thus F ξ is not finite over V and, consequently, cannot be continuous.

Properties of the non-periodic correctors: proof of

Theorem 5.2.4

A useful Lemma

We begin by introducing the following function: for all ξ, η ∈ R d , the function G ξ,η is defined over

(5.5.1) The following Lemma gives a lower bound for G ξ,η that will allow to prove Theorem 5.2.5 (iii). Lemma 5.5.1. Suppose that 2 ≤ p < 3. For all δ > 0, there exist constants γ p = γ(p) > 0 and c p = c(p) > 0 such that for all X, Y ∈ R d , for all ξ ∈ R d \ B(0, δ) and η ∈ B(ξ, δ/2), we have that

(5.5.2)

Suppose that p ≥ 3. There exist constants γ p = γ(p) > 0 and c p = c(p) > 0 such that for all

Proof of Lemma 5.5.1. We first give the proof of Estimate (5.5.2). We have that ξ ̸ = 0 and η ̸ = 0. For all X, Y ∈ R d , we define Z := X-Y 2 and T := X+Y 2 . Inequality (5.5.2) is equivalent to the following inequality: for any

(5.5.4)

We prove (5.5.4) for any Z, T ∈ R d . We fix T ∈ R d and we introduce the function

where γ p > 0 is to be chosen later. Since p ≥ 2, the function Φ γp is of class C 2 . Besides, denoting by I the identity matrix, we have that

We now treat the case |T | ≤ 1 2 |x|. In particular T |x| ≤ 1 2 and thus

since the function y → x |x| + y p-2 is regular on B(0, 3 4 ) with derivative uniformly bounded in x. Estimate (5.5.9) is proved the same way. We have concluded the proof.

Proof of (5.5.3). We assume that p ≥ 3. With the above variables T and Z, (5.5.3) is equivalent to proving that for all Z, T, ξ and η ∈ R d , the following inequality holds true:

(5.5.12)

Applying the same method as for the proof of (5.5.2), we only have to prove that

(5.5.13)

We once again appeal to the mean-value inequality on Ψ T , noticing that, in this case, see (5.5.10) and (5.5.11), we have for all x ∈ R d ,

Note that, contrary to the case p < 3, estimate (5.5.14) does not depend on δ. This gives (5.5.13) and finally (5.5.12).

Proof of Theorem 5.2.5

We start this section with a Remark: Proof of Theorem 5.2.5. By homogeneity, we can prove Theorem 5.2.5 for all ξ ∈ R d such that |ξ| = 1. We fix such a ξ ∈ R d . By (A4), there exists a constant c > 0 independent of ξ such that |ξ + ∇w per ξ | ≥ c. In the proof, we introduce the notations

where these quantities are well-defined owing to Proposition 5.2.1 (ii) and Theorem 5.2.4 (ii). We use the following Taylor inequality (5.5. We may write that A ξ = A per ξ + A ξ , where

and

The matrix A per ξ is symmetric, periodic, Hölder continuous, bounded and coercive while the ma- We have that h ∈ L p ′ ∩ L ∞ (R d ) and, thanks to the estimate (5.5.18) and the fact that

Applying [13, Theorem p. 247] and [12, Theorem A] to (5.5.22) gives

and we iterate the argument. We have, thanks to (5.5.18), that

If all other assumptions are satisfied, the assumption |∇ϕ

2 ∇v n is L 2 -weakly convergent. Indeed, following the proof of Lemma 5. [START_REF] Allaire | Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes I. Abstract framework, a volume distribution of holes[END_REF]

.2(iii), we can prove that if ∇v

In particular, we have that v ∈ W ∇ϕ .

Proof of Lemma 5.7.2. We fix two bounded smooth domains B, B ′ such that B ⊂⊂ B ′ . Let χ ∈ D(B ′ ) such that χ = 1 on B and 0 ≤ χ ≤ 1 in B ′ . We introduce the function

We immediately check that, up to extracting a subsequence, we have by the Rellich compactness Theorem and (i), (iii) that

(5.7.12)

(5.7.13)

Since Ψ n ∈ W 1,p 0 (B ′ ), we can test Ψ n against (5.7.10). We re-organize the terms and get that

(5.7.14) We study each term separetely. The term A n vanishes when n -→ +∞ since

which is bounded in L p ′ (B ′ ), uniformly with respect to n by (i) and (iii) and (5.7.12). The term B n vanishes as n -→ +∞ by (5.7.13) and since, by (i), (ii) and (iii):

The term C n vanishes by (5.7.13) and the L p loc -strong convergence of the sequence (h n ) n∈N . The term D n vanishes by (5.7.13) and the convergence of

(5.7.15)

However, since (∇ϕ + ∇v)|∇ϕ + ∇v| p-2 ∈ L p ′ (B ′ ) and because of (5.7.12) and (i), we also have that .7.16) The difference between (5.7.15) and (5.7.16) gives that

(5.7.17) h given by (5.2.13). In addition, by Theorem 5.2.3, the solution of this PDE is unique in W ξ+∇w per ξ . Thus ∇v = ∇ w ξ and this concludes Step 1 since the sequence (∇ w ξn ) n∈N has one possible limit.

Step 2. Suppose by contradiction that ∇ w ξn does not converge to ∇ w ξ in L p unif (R d ) when n -→ +∞. Then there exists δ > 0, a subsequence of (ξ n ) n∈N that we de not relabel and a sequence (x n ) n∈N such that ∀n ∈ N, ∥∇ w ξn -∇ w ξ ∥ L p (B(xn,1)) ≥ δ.

(5.

)

In particular, (5.7.22) gives

We show in the sequel that, up to a subsequence, for i = 1, 2,

(5.7.26)

In particular, passing to the limit n -→ +∞ in (5.7.25) will lead to a contradiction.

Step 3. Proof of (5.7.26). We prove (5.7.26) for i = 1, the proof is standard for i = 2. We have that ∇ w 1 n solves in the distribution sense the PDE

(5.7.27) and that w

we get because of (5.7.20) that the sequences ∥∇ w

n∈N are uniformly bounded in n. Thus, up to extracting a subsequence,

(5.7.28)

We may apply Lemma 5.7.2 with

We check the required convergences. We study separately each term of (5.8.17). By Proposition 5.2.1 (iv), (5.8.14) and (5.8.15), we have that

As for the first term of (5.8.17), we write that

Thus, applying the L p Poincaré-Wirtinger inequality, we have that

(5.8.19) Gathering together (5.8.17), (5.8.18) and (5.8.19), we get (5.8.16) and conclude the proof of the Lemma.

Extension to other non-linear operators

This section does not appear in the paper [START_REF] Wolf | Homogenization of p-laplace type equations in a periodic setting with defects[END_REF]. This is a complement to Subsection 5.2.5 above.

We fix an operator a : R d × R d -→ R d satisfying the properties (1)-( 2

where the space

As a product of two separables spaces, X is separable. By [

Étude de problèmes d'homogénéisation dans un cadre périodique avec défauts

Résumé

Cette thèse s'intéresse à des problèmes d'homogénéisation dans un cadre périodique avec défauts. Elle se divise en deux parties. Dans une première partie, nous étudions le cas de l'homogénéisation en milieu perforé. En s'appuyant sur l'homogénéisation périodique des problèmes de Poisson et de Stokes, nous construisons une notion de perturbation locale du domaine perforé périodique. Cela permet d'obtenir les mêmes types de résultats, à la fois pour l'équation de Poisson et pour le système de Stokes, que dans le cas périodique. Nous construisons en particulier les correcteurs associés aux problèmes et nous obtenons des taux de convergence vers la solution homogénéisée. Dans une seconde partie, nous regardons des équations dont les coefficients sont oscillants à l'échelle ε ≪ 1. Partant à nouveau du cas d'un coefficient périodique, nous étudions des perturbations locales de ce coefficient. Ce cadre a été introduit par X. Blanc, C. Le Bris et P.-L. Lions dans le cas linéaire. Nous étudions ici des cas non-linéaires. Nous construisons les correcteurs associés et nous obtenons des théorèmes de convergence sous certaines hypothèses.

Mots-clés: homogénéisation périodique, homogénéisation non-linéaire, milieu perforé, équations elliptiques.

Some Homogenization problems in a periodic setting with defects Abstract

This PhD thesis focuses on some homogenization problems in a periodic setting with defects. Our study is divided in two parts. In the first part, we study the homogenization in porous medium. Relying on the case of periodically perforated domains, we build a framework that allows to address local perturbation of the periodic setting for the Poisson problem and the Stokes system. We obtain the same type of results as in the periodic case for both equations. We build in particular the correctors and we obtain convergence rates to the homogenized solution. In a second part, we consider equations with oscillating coefficients at scale ε ≪ 1. Given a periodic background, we introduce some local perturbation of the coefficient. This setting has been introduced by X. Blanc, C. Le Bris and P. L. Lions for the linear case. We study here nonlinear cases. We build the associated correctors and we obtain convergence theorems under some assumptions.