In this thesis, we study a class of quasilinear elliptic equations posed in a two-component domain with an L 1 data and its asymptotic analysis. More precisely, we consider a two-component domain, denoted by Ω, which can be written as the disjoint union

where the open sets Ω 1 and Ω 2 are the two components of Ω,a n dΓ is the interface between these components. We study the following quasilinear elliptic problem posed in Ω:

est une réunion périodique d'ensembles déconnectés, qui est présentée dans le chapitre 5. Dans ce chapitre, nous identifions d'abord, en utilisant les estimations a priori obtenues dans la première partie, le problème éclaté (théorème 12). Nous obtenons ensuite le problème homogénéisé dans Ω (théorème 13).

 pour des domaines fixes et dans [29] pour des domaines perforées. Elle a été étendue successivement au cas de domaines àd e u xc o m p o s a n t e sd a n s[ 4 6 ]e t[ 4 5 ]( p o u ru n ep r é s e n t a t i o ng é n é r a l en o u s renvoyons au livre récent [32]).

)) garde cette propriété. Les résultats obtenus dans cette thèse sont présentés ci-dessous en détails.

where ⌫ 1 is the unit outward normal to Ω 1 , f is an L 1 function, and B is a coercive matrix field which has a restricted growth assumption (B(x, r) is bounded on any compact set of R).

The first part of this thesis is dedicated to existence and uniqueness results for this problem in the framework of renormalized solutions, which was introduced by R.J. DiPerna and P.L. Lions. In the second part, we study the corresponding homogenization problem for a two-component domain with a (disconnected) periodic second component by combining the notion of renormalized solutions and the periodic unfolding method, introduced D. Cioranescu, A. Damlamian and G. Griso. It has been successively adapted to two-component domains by P. Donato, K.H. Le Nguyen, and R. Tardieu.

In order to obtain a uniqueness result for the homogenized problem, we study the properties of the corresponding cell problem. In particular, we show that if the matrix field in the cell problem, denoted A(y, t),islocalLipsc hitzcontinuous with respect to t,thentheresultinghomogenizedmatrixA 0 keeps this property. This uniqueness result ensures that the convergences obtained in the homogenization process hold for the whole sequence of the periodicity parameter (and not only a subsequence). i

Résumé

Dans cette thèse, nous étudions une classe de problèmes elliptiques quasilinéaires posés dans un domaine à deux composantes avec une donnée L 1 et son analyse asymptotique. Plus précisement, on considère un domaine Ω, que l'on écrit comme une réunion disjointe Ω = Ω 1 [ Ω 2 [ Γ, où les ensembles ouverts Ω 1 et Ω 2 sont les deux composantes de Ω,e tΓ est l'interface entre les composantes. Nous étudions le problème elliptique quasi-linéaire suivant posé dans Ω :

8 > > > > > > < > > > > > > : div(B(x, u 1 )ru 1 )=f dans Ω 1 , div(B(x, u 2 )ru 2 )=f dans Ω 2 ,
(B(x, u 1 )ru 1 )⌫ 1 =(B(x, u 2 )ru 2 )⌫ 1 sur Γ, (B(x, u 1 )ru 1 )⌫ 1 = h(x)(u 1 u 2 ) sur Γ,

u 1 =0 sur @Ω,
où ⌫ 1 est le vecteur normal unitaire extérieur à Ω 1 , f 2 L 1 (Ω) et B est une matrice coercitive qui vérifie une hypothèse assez générale (B(x, r) n'est pas uniformément borné mais borné sur tout ensemble compact de R).

La première partie de cette thèse est donc dédiée à des résultats d'existence et d'unicité de ce problème dans le cadre des solutions renormalisées, qui a été introduit par R.J. DiPerna et P.L. Lions. Dans la deuxième partie, nous étudions l'homogénéisation d'un problème du même type, posé dans un domaine à deux composantes dont la deuxième est une réunion périodique d'ensembles déconnectés, en mélangeant le notion des solution renormalisées et la méthode de l'éclatement périodique. Cette méthode a été introduite par D. Cioranescu, A. Damlamian and G. Griso et adaptée aux domaines à deux composantes par P. Donato, K.H. Le Nguyen, et R. Tardieu.

Pour obtenir un résultat d'unicité pour le problème homogénéisé qui puisse assurer que les convergences obtenues sont valables pour toute la suite du paramètre de périodicité (et non pas à une sous-suite près), nous étudions les propriétés du problème périodique correpondant, posé dans la cellule de référence. En particulier, nous démontrons que si la matrice A(y, t) du problème dans la cellule de référence est localement lipschitzienne par rapport à t,a l o r sl am a t r i c eh o m o g é n é i s é er é s u l t a n t eA 0 (t) garde cette propriété.

iii Dans cette thèse, nous étudions une classe de problèmes elliptiques quasilinéaires posés dans un domaine à deux composantes avec une donnée L 1 et nous en faisons l'analyse asymptotique dans un domaine périodique à deux composantes. Plus précisement, on considère un domaine Ω,q u el ' o né c r i t comme une réunion disjointe

Ω = Ω 1 [ Ω 2 [ Γ,
où les ensembles ouverts Ω 1 et Ω 2 sont les deux composantes de Ω,e tΓ est l'interface entre les composantes. Nous étudions le problème elliptique quasi-linéaire suivant posé dans Ω : 8 > > > > > > < > > > > > > : div(B(x, u 1 )ru 1 )=f dans Ω 1 , div(B(x, u 2 )ru 2 )=f dans Ω 2 , (B(x, u 1 )ru 1 )⌫ 1 =(B(x, u 2 )ru 2 )⌫ 1 sur Γ, (B(x, u 1 )ru 1 )⌫ 1 = h(x)(u 1 u 2 ) sur Γ,

u 1 =0 sur @Ω, (1)
où ⌫ 1 est le vecteur normal unitaire extérieur à Ω 1 , f 2 L 1 (Ω) et B est une matrice coercitive qui vérifie une hypothèse assez générale (B(x, r) n'est pas uniformément borné mais borné sur tout ensemble compact de R).

Observons que sous les hypothèses précédentes sur f et B nous ne pouvons pas, en général, obtenir l'existence d'une solution faible. Même si f 2 L 2 (Ω) sans hypothèse de bornitude sur B(x, r) par rapport à r,o nn es a i tp a s démontrer en général l'existence d'une solution faible (de même si B est borné et f 2 L 1 (Ω)).

Rappelons que le problème div(A(x, u)ru)=f avec des conditions de Dirichlet sur le bord, si A(x, r) est bornée, elliptique et f appartient à L 1 (Ω) (ou même est une mesure bornée de Radon), il existe d'apres Boccardo-Gallouët [START_REF] Boccardo | Nonlinear elliptic and parabolic equations involving measure data[END_REF] une solution au sens des distributions (et ces résultats sont valable pour une classe plus général d'opérateurs non linéaires àc r o i s s a n c ep). Les auteurs démontrent que u 2 W 1,q 0 (Ω), 81 <q<

N N 1 et verifie Z Ω A(x)rur' dx = Z Ω f ' dx, 8' 2 C 1 0 (Ω).
Cependant, cette solution au sens des distributions ne peut pas avoir, en général, une énergie finie, au sens où u/ 2 H 1 0 (Ω).D e p l u s m ê m e d a n s l e cas linéaire, c'est-à-dire A(x, r)=A(x),l as o l u t i o na us e n sd e sd i s t r i b u t i o n s n'est pas unique en général d'apres le contre exemple de Serrin [START_REF] Serrin | Pathological solutions of elliptic equations,A n n .S c u o l a Norm[END_REF] (voir aussi [START_REF] Prignet | Remarks on existence and uniqueness of solutions of el liptic problems with right hand side measures[END_REF]).

Pour pallier cet inconvénient, plusieurs notions de solutions ont été développées : solutions entropiques (voir [START_REF] Bénilan | An L 1 theory of existence and uniqueness of nonlinear elliptic equations[END_REF]), SOLA (solutions obtenues comme limite d'approximation, voir [START_REF] Dall | Approximated solutions of equations with L 1 data. Application to the h-convergence of quasi-linear parabolic equations[END_REF]) et solutions renormalisées.

Pour mener à bien notre étude sur le problème (1) (existence, unicité, analyse asymptotique), nous avons besoin d'une notion de solution qui permet des résultats d'unicité et de stabilité. Nous utiliserons dans cette thèse la notion de solution renormalisée.

La notion de solution renormalisée a été introduite dans [START_REF] Diperna | On the Cauchy problem for Boltzmann equations: Global existence and weak stability[END_REF] par R.J. DiPerna et P.L. Lions pour des équations du premier ordre. Elle a été développée ensuite par F. Murat dans [START_REF] Murat | Soluciones renormalizadas de EDP elipticas no lineales[END_REF], par P.L. Lions et F. Murat dans [START_REF] Lions | Sur les solutions renormalisées d'équations elliptiques[END_REF] pour des équations elliptiques avec conditions de Dirichlet et données L 1 , puis par G. Dal Maso et al. dans [START_REF] Maso | Renormalized solutions of elliptic equations with general measure data[END_REF] pour des équations elliptiques avec données mesures. La plupart des travaux concernant le développement des solutions renormalisées traite de problèmes elliptiques (ou paraboliques) àd o n n é e sL 1 et avec des conditions de Dirichlet, mais plus rarement le cas d'autres conditions sur le bord (citons [START_REF] Betta | Neumann problems for nonlinear elliptic equations with L 1 data[END_REF] pour des conditions de Neumann, [START_REF] Guibé | Renormalized solutions of el liptic equations with Robin boundary conditions[END_REF] pour un domaine perforé). L'existence et l'unicité d'une solution renormalisée ont été étudiées pour des domaines perforés dans [START_REF] Guibé | Renormalized solutions of el liptic equations with Robin boundary conditions[END_REF], avec une condition de Fourier sur le bord des trous, mais, à notre connaissance les équations de type (1) avec donnée L 1 et saut à l'interface n'ont pas été abordées dans la littérature.

La première partie de cette thèse est donc dédiée à des résultats d'existence et d'unicité pour la solution du problème [START_REF] Allaire | Homogenization and two-scale convergence[END_REF]. Nous avons donné d'abord une définition appropriée de solution renormalisée du problème. Cette définition, ainsi que le résultat d'existence, sont présentés dans le chapitre 2. L'unicité de cette solution est démontrée dans le chapitre 3, où une hypothése supplémentaire de lipschitzianité locale pour la matrice B est nécessaire.

Dans la deuxième partie, nous étudions l'homogénéisation d'un problème du même type, posé dans un domaine à deux composantes dont la deuxième

Partie I

Dans cette partie, nous étudions l'existence et l'unicité de la solution renormalisée de [START_REF] Allaire | Homogenization and two-scale convergence[END_REF]. On définit d'abord le domaine à deux composantes Ω,q u i est un ensemble ouvert borné connexe de R N de frontière @Ω.N o u sd é c o mposons le domaine comme la réunion disjointe Ω = Ω 1 [ Ω 2 [ Γ,o ùΩ 2 est un ensemble ouvert tel que Ω 2 ⇢ Ω de frontière lipschitzienne Γ,e tΩ 1 = Ω \ Ω 2 (voir figure 1).

Ω Ω 2 Γ Ω 2 Γ Ω 1 ∂Ω Figure 1: Le domaine à deux composantes Ω lim n!1 1 n Z Γ (u 1 u 2 )(T n (u 1 ) T n (u 2 )) d =0;
(3b) et pour tout S 1 ,S 2 2 C 1 (R) (ou S 1 ,S 2 2 W 1,1 (R))às u p p o r tc o m p a c t ,u satisfait Z

Ω 1 S 1 (u 1 )B(x, u 1 )ru 1 • rv 1 dx + Z Ω 1 S 0 1 (u 1 )B(x, u 1 )ru 1 • ru 1 v 1 dx + Z Ω 2 S 2 (u 2 )B(x, u 2 )ru 2 • rv 2 dx + Z Ω 2 S 0 2 (u 2 )B(x, u 2 )ru 2 • ru 2 v 2 dx + Z Γ h(x)(u 1 u 2 )(v 1 S 1 (u 1 ) v 2 S 2 (u 2 )) d = Z Ω 1 fv 1 S 1 (u 1 ) dx + Z Ω 2 fv 2 S 2 (u 2 ) dx, (4)
pour tout v 2 V \ (L 1 (Ω 1 ) ⇥ L 1 (Ω 2 )).
Notons que, dans le cadre des solutions renormalisées, une solution u peut ne pas avoir assez de régularité pour avoir un gradient et une trace dans le sens classique des espaces de Sobolev. Nous devons donc d'abord donner une définition appropriée du gradient et de la trace d'une solution renormalisée. Dans ce but, nous démontrons la proposition suivante (qui est une généralisation de [10, Lemma 2.1] et [START_REF] Guibé | Renormalized solutions of el liptic equations with Robin boundary conditions[END_REF]Proposition 2.3]) : Proposition 2. Soit u =(u 1 ,u 2 ):Ω \ Γ ! R une fonction mesurable telle que T k (u) 2 V pour tout k>0.

1. Pour i =1 , 2,i le x i s t eu n ef o n c t i o nm e s u r a b l eu n i q u eG i : Ω i ! R N telle que pour tout k>0, rT k (u i )=G i {|u i |<k} p.p. dans Ω i , où {|u i |<k} dénote la fonction caractéristique de l'ensemble

{x 2 Ω i : |u i (x)| <k}.
On définit G i comme le gradient de u i et on écrit G i = ru i .

Si sup

k 1 1 k kT k (u)k 2 V < 1,
alors il existe une fonction mesurable unique

w i : Γ ! R, for i =1, 2,
telle que pour tout k>0, i (T k (u i )) = T k (w i ) p.p. sur Γ, où i : H 1 (Ω i ) ! L 2 (Γ) est l'opérateur de trace. On définit la fonction w i comme la trace de u i sur Γ et on écrit

i (u i )=w i ,i =1, 2.
L'originalité de cette définition réside dans la régularité (2b), la décroissance d'une énergie sur le bord (3b) ainsi que la présence du terme sur le bord

Z Γ h(x)(u 1 u 2 )(v 1 S 1 (u 1 ) v 2 S 2 (u 2 )) d .
La régularité (2a), la décroissance de l'énergie (3a) sont classiques et permettent via notamment 1 de la proposition 2 de donner un sens à tous les termes de (4) excepté le terme sur le bord.

En effet, soit S i 2 C 1 (R), i =1 , 2 às u p p o r tc o m p a c t . P o u rt o u tv 2

V \ (L 1 (Ω 1 ) ⇥ L 1 (Ω 2 )),s isupp S i ⇢ [ k, k] (i =1 , 2), alors pour i =1 , 2, on a S i (u i )B(x, u i )ru i • rv i = S i (u i )B(x, T k (u i ))rT k (u i ) • rv i 2 L 1 (Ω i ), S 0 i (u i )B(x, u i )ru i • ru i v i = S 0 i (u i )B(x, T k (u i ))rT k (u i ) • rT k (u i ) v i 2 L 1 (Ω i ), et fv i S i (u i ) 2 L 1 (Ω i ).
Corcernant le terme sur le bord Γ,ilestimportan tderemarquerque(2a) et (3a) ne suffisent pas à donner un sens à h(x)(u 1 u 2 )(v 1 S 1 (u 1 ) v 2 S 2 (u 2 )). En général, il n'y a aucune raison d'avoir

h(x)(u 1 u 2 )(v 1 S 1 (u 1 ) v 2 S 2 (u 2 )) = h(x)(u 1 u 2 )(v 1 S 1 (u 1 ) v 2 S 2 (u 2 )) {|u 1 |n} {|u 2 |n}
pour n grand. Pour traiter l'intégrale sur Γ,n o u sa l l o n su t i l i s e r( 2 b ) . P o u rt o u tn 2 N, on définit ✓ n : R ! R (voir figure 3) par Alors, il existe une solution renormalisée de (1) dans le sens de définition 1.

✓ n (s)= 8 > > > > > > > < > > > > > > > : 0, si s  2n, s n +2, si 2n  s  n, 1, si n  s  n, s n +2, si n  s  2n, 0, si s 2n.
La preuve du theorème 3 se fait par passage à la limite dans un problème approché. La première étape, si {f ε } ⇢ L 2 (Ω) telle que

f ε ! f fortement dans L 1 (Ω), et B ε (x, t)=B(x, T 1/ε (t)),n o u sc o n s i d é r o n su n es o l u t i o nu ε 2 V vérifiant 8 > > > > > > < > > > > > > : div(B ε (x, u ε 1 )ru ε 1 )=f ε dans Ω 1 , div(B ε (x, u ε 2 )ru ε 2 )=f ε dans Ω 2 , (B ε (x, u ε 1 )ru ε 1 )⌫ 1 =(B ε (x, u ε 2 )ru ε 2 )⌫ 1 sur Γ, (B ε (x, u ε 1 )ru ε 1 )⌫ 1 = h(x)(u ε 1 u ε 2 ) sur Γ, u ε 1 =0 sur @Ω.
La deuxième étape consiste à établir des estimations a priori, puis de construire (à l'aide de résultats de compacité de Rellich-Kondrachov) u telle que, àu n es o u s -s u i t ep r è s , 8 > > > < > > > :

u ε i ! u i p.p. dans Ω, T k (u ε i ) * T k (u i ) faiblement dans H 1 (Ω i ), i (u ε i ) ! i (u i ) p.p. dans Γ, i (T k (u ε i )) ! i (T k (u i ))
fortement dans L 2 (Γ), p.p. sur Γ. La nouveauté ici vient une fois encore des termes sur le bord et d'exploiter efficacement, par exemple, dans la troisième étape, l'estimation a priori 8k>0, (u ε

1 u ε 2 )(T k (u ε 1 ) T k (u ε 2 )) borné dans L 1 (Γ), et la limite lim n!1 lim sup ε!0 1 n ✓ Z Ω\Γ B(x, T n (u ε ))rT n (u ε )rT n (u ε ) dx + Z Γ (u ε 1 u ε 2 )(T n (u ε 1 ) T n (u ε 2 )) d ◆ =0.
Dans la quatrième étape, on passe à la limite avec un choix judicieux de fonction test et on démontre que u est une solution renormalisée de [START_REF] Allaire | Homogenization and two-scale convergence[END_REF].

Remarque 4. Pour définir le terme sur le bord, il est possible de remplacer (2b) et (3b) par une condition de régularité, donnée par

u 1 u 2 2 W 1 1
q ,q (Γ), avec q>1.

Ce type de régularité découle des estimations du type Boccardo-Gallouët, mais dépend fortement des constantes de Sobolev. Comme ces constantes de Sobolev peuvent exploser dans l'analyse asymptotique, il n'est pas possible d'utiliser [START_REF] Auriault | Macroscopic modelling of heat transfer in composites with interfacial thermal barrier[END_REF].A j o u t o n sq u e(6) sera mise en défaut pour des problèmes non linéaires plus généraux. C'est pour ces raisons que nous avons choisi (2b) et (3b) dans la définition de solution renormalisée.

Chapitre 3 : L'unicité de la solution renormalisée

Dans ce chapitre, nous démontrons l'unicité d'une solution renormalisée de [START_REF] Allaire | Homogenization and two-scale convergence[END_REF]. Pour cela, nous rajoutons aux hypothèses (A1)-(A3) du théorème 3 la condition de lipschitzianité locale suivante sur la matrice B :

(A4) B(x, r) est localement lipschitzienne par rapport à r,c ' e s t -à -d i r e ,po u r tout compact K de R il existe M K > 0 tel que |B(x, r) B(x, s)|  M K |r s|, 8r, s 2 K, p.p. x 2 Ω.

Pour démontrer le résultat d'unicité, nous appliquons la méthode dévelopée dans [START_REF] Blanchard | Quasi-linear degenerate elliptic problems with L 1 data[END_REF][START_REF] Di Nardo | Uniqueness of renormalized solutions to nonlinear parabolic problems with lower-order terms[END_REF]. Cette méthode utilise l'existence d'une fonction auxiliaire ' 2 C 1 (R) qui vérifie des propriétés intéressantes. Plus précisément, nous utilisons la proposition suivante de [START_REF] Di Nardo | Uniqueness of renormalized solutions to nonlinear parabolic problems with lower-order terms[END_REF] : Proposition 5 ( [START_REF] Di Nardo | Uniqueness of renormalized solutions to nonlinear parabolic problems with lower-order terms[END_REF]). On suppose (A4). Alors, il existe une fonction ' 2 C 1 (R) telle que '(0) = 0 et ' 0 1.

De plus, il existe des constantes > 1/2, 0 <k 0 < 1,e tL>0 telles que

' 0 (1 + |'|) 2δ 2 L 1 (R).
En outre, pour tout r, s 2 R tels que |'(r) '(s)|  k, pour 0 <k<k 0 , on a B(x, r) ' 0 (r) B(x, s) ' 0 (s)  1 ' 0 (s)

Lk (1 + |'(r)| + |'(s)|) δ et 1 L  ' 0 (s) ' 0 (r)  L.
Une des difficultés vient de l'intégrale sur le bord Γ qui est liée au saut de la solution. La proposition suivante est un outil important pour la démonstration du théorème d'unicité, liée aux estimations de Boccardo-Gallouët : Proposition 6. Pour i =1, 2,s o i t i l'opérateur de trace défini sur H 1 (Ω i ). Sous les hypothèses de théorème 3, si u est une solution renormalisée de (1), alors i (u i ) 2 L 1 (Γ), i =1, 2.

La méthode développée dans [START_REF] Blanchard | Quasi-linear degenerate elliptic problems with L 1 data[END_REF][START_REF] Di Nardo | Uniqueness of renormalized solutions to nonlinear parabolic problems with lower-order terms[END_REF] consiste formellement à utiliser T k ('(u) '(v)) comme fonction test. La justification est très technique et se fait par passage à la limite à l'aide de la fonction ✓ n (u)T k ('(u) '(v)) autorisée dans la définition de solution renormalisée.

Cependant, la non linéarité du terme T k ('(u) '(v)) n'est pas compatible avec le terme du bord dans le sens où le terme obtenu Z Γ h(x)((u 1 v 1 ) (u 2 v 2 ))(T k ('(u 1 ) '(v 1 ) T k ('(u 2 ) '(v 2 ))) d n'est pas nécessairement de signe positif.

Pour surmonter cette difficulté, nous démontrons une propriété de signe sur Γ : Lemme 7. On suppose (A1)-(A4). Si u et v sont deux solutions renormalisées de (1),a l o r ssgn(u 1 v 1 )=sgn(u 2 v 2 ) p.p. sur Γ.

Ce lemme est essentiel pour la démonstration du théorème d'unicité suivant, qui est le résultat principal de ce chapitre : Théorème 8. Sous les hypothéses (A1)-(A4), la solution renormalisée de (1) est unique.

Une des étapes principales de la démonstration du théorème 8 est de montrer que

lim k!0 1 k 2 Z Ω i ✓ 1 ' 0 (u i ) + 1 ' 0 (v i ) ◆ |rT k ('(u i ) '(v i ))| 2 dx =0,i =1, 2, (7)
où u =( u 1 ,u 2 ) et v =( v 1 ,v 2 ) sont deux solutions renormalisées de (1) et ' est la fonction introduite dans proposition 5.

Pour le démontrer, nous obtenons d'abord, aprés de long calculs, l'inégalité suivante :

lim sup k!0 ✓ 1 k 2 Z U k 1 ✓ 1 ' 0 (u 1 ) + 1 ' 0 (v 1 ) ◆ |r'(u 1 ) r'(v 1 )| 2 dx + 1 k 2 Z U k 2 ✓ 1 ' 0 (u 2 ) + 1 ' 0 (v 2 ) ◆ |r'(u 2 ) r'(v 2 )| 2 dx + 1 k 2 C k ◆  0, où U k i = {x 2 Ω i :0< |'(u i ) '(v i )| <k},i =1, 2, et C k = Z Γ h(x)[(u 1 v 1 ) (u 2 v 2 )][T k ('(u 1 ) '(v 1 )) T k ('(u 2 ) '(v 2 ))] d .
Alors, pour montrer [START_REF] Beltran | Homogenization of a quasilinear elliptic problem in a twocomponent domain with an imperfect interface[END_REF], il suffitd em o n t r e rq u e lim sup k!0

1 k 2 C k 0. (8)
Comme écrit précédemment à cause de la présence de la fonction ',n o u s ne connaissons pas en général le signe de C k .L ec o n t r o l ed e 1 k 2 C k quand k ! 0 est l'une des difficultés principale. Pour ce faire, on utile la propriété de signe sur Γ (lemme 7) et on divise l'ensemble {x 2 Γ ; u 1 (x) v 1 (x) > 0} (défini à un ensemble de mesure nulle près) en 4 sous-ensembles disjoints,

{u 1 v 1 > 0} = P 1 [ P 2 [ P 3 [ P 4 , où P 1 := {'(u 1 ) '(v 1 ) k} \ {'(u 2 ) '(v 2 ) k}, P 2 := {0 < '(u 1 ) '(v 1 ) <k} \ {0 < '(u 2 ) '(v 2 ) <k}, P 3 = {'(u 1 ) '(v 1 ) k} \ {0 < '(u 2 ) '(v 2 ) <k}, P 4 := {0 < '(u 1 ) '(v 1 ) <k} \ {'(u 2 ) '(v 2 ) k}.
Grâce au lemme 7 et à ce découpage, on démontre [START_REF] Ben Cheikh | Homogénéisation des solutions renormalisées dans des domaines perforés[END_REF]. Une fois que [START_REF] Beltran | Homogenization of a quasilinear elliptic problem in a twocomponent domain with an imperfect interface[END_REF] est démontrée, on s'inspire de [START_REF] Blanchard | Quasi-linear degenerate elliptic problems with L 1 data[END_REF][START_REF] Di Nardo | Uniqueness of renormalized solutions to nonlinear parabolic problems with lower-order terms[END_REF] pour obtenir dans un premier temps que u 1 = v 1 dans Ω 1 (car u 1 et v 1 ont une trace nulle sur @Ω,c eq u ip e r m e td ' u t i l i s e rl ' i n é g a l i t éd eP o i n c a r é ) . A i n s iu 1 = v 1 sur Γ,c e qui, combiné au lemme 7 et [START_REF] Beltran | Homogenization of a quasilinear elliptic problem in a twocomponent domain with an imperfect interface[END_REF] donnent u 2 = v 2 dans Ω 2 . quasi-linéaire suivant :

8 > > > > > > > > > > > < > > > > > > > > > > > : div ⇣ A ⇣ x " ,u ε 1 ⌘ ru ε 1 ⌘ = f dans Ω ε 1 , div ⇣ A ⇣ x " ,u ε 2 ⌘ ru ε 2 ⌘ = f dans Ω ε 2 , ⇣ A ⇣ x " ,u ε 1 ⌘ ru ε 1 ⌘ ⌫ ε 1 = ⇣ A ⇣ x " ,u ε 2 ⌘ ru ε 2 ⌘ ⌫ ε 1 sur Γ ε , ⇣ A ⇣ x " ,u ε 1 ⌘ ru ε 1 ⌘ ⌫ ε 1 = " 1 h ⇣ x " ⌘ (u ε 1 u ε 2 ) sur Γ ε , u ε =0 sur @Ω, (9)
sous les mêmes hypothèses considérées dans la première partie, pour " fixé. On suppose de plus ici la condition de périodicité usuelle pour A et h. Concernant l'étude de l'homogénéisation de (9) avec f 2 L 2 (Ω) (voir par exemple [START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF][START_REF] Donato | Homogenization of diffusion problems with a nonlinear interfacial resistance[END_REF]), l'hypothèse de proportionnalité du saut de la solution et du flux sur Γ ε dépend de " γ (au lieu de " 1 ), où  1 est un paramètre. L'homogénéisation est étudiée alors dans les trois cas :

2 ( 1, 1], = 1,e t 2 ( 1, 1).L ad i fférence principale entre ces cas réside dans le problème périodique posé dans la cellule de réference, qui permet de décrire la matrice homogénéisée. Nous nous bornons ici au cas = 1,d o n tl a particularité est la présence dans le problème elliptique dans la cellule de référence, du saut de la solution sur l'interface de référence Γ.

Chapitre 4 : Propriétés du problème dans la cellule de référence

Dans ce chapitre, nous étudions le problème elliptique suivant, posé dans la cellule de référence qui est lié au problème homogénéisé de [START_REF] Bendib | Homogénéisation d'une classe de problèmes non linéaires avec des conditions de Fourier dans des ouverts perforés[END_REF] :

8 > > > > > > > > < > > > > > > > > : div(Ar λ 1 )=G λ 1 dans Y 1 , div(Ar λ 2 )=G λ 2 dans Y 2 , Ar λ 1 • n 1 = Ar λ 2 • n 2 sur Γ, Ar λ 1 • n 1 = h(y)( λ 1 λ 2 ) sur Γ, λ 1 Y périodique, M Γ ( λ 1 )=0, (10)
où 2 R N et G λ i est défini par hG λ i ,vi = Z Y i A rvdy, 8v 2 H 1 (Y i ),i =1, 2, (11)
qui appartient à (H 1 (Y i )) 0 . Nous nous sommes intéressés aux propriétés de la solution de [START_REF] Bénilan | An L 1 theory of existence and uniqueness of nonlinear elliptic equations[END_REF], qui n'étaient pas étudiées dans la littérature. Ceci est motivé par le fait que la matrice homogénéisée A 0 qui correspond à (9) est définie en fonction de la solution λ de [START_REF] Bénilan | An L 1 theory of existence and uniqueness of nonlinear elliptic equations[END_REF], et pas seulement de A. Plus précisément, la matrice A 0 est définie par

A 0 (t)=A 0 1 (t)+A 0 2 (t), (12)
où

A 0 i (t) = 1 |Y | Z Y i A(y, t)r y w λ i (y, t) dy, i =1, 2, 8 2 R N , avec w λ i (y, t)= • y λ i (y, t), et λ =( λ 1 , λ 2
) solution de [START_REF] Bénilan | An L 1 theory of existence and uniqueness of nonlinear elliptic equations[END_REF]. Les propriétés démontrées dans ce chapitre sont, à notre avis, intéressantes en elles-mêmes.

Nous démontrons, en particulier, que si la matrice A est lipschitzienne par rapport à la deuxième variable, alors A 0 garde cette propriété. Grâce á cette propriété, nous pouvons obtenir un résultat d'unicité pour le problème homogénéisé posé dans Ω (voir théorème 13) correspondant à [START_REF] Bendib | Homogénéisation d'une classe de problèmes non linéaires avec des conditions de Fourier dans des ouverts perforés[END_REF].

Rappelons que dans [START_REF] Cabarrubias | Homogenization of a quasilinear elliptic problem with nonlinear Robin boundary conditions[END_REF] les auteurs ont démontré un résultat similaire pour l'homogénéisation des problèmes elliptiques dans un domaine perforé, en utilisant une estimation de type Meyers bien connue.

Dans notre cas, où la solution de [START_REF] Bénilan | An L 1 theory of existence and uniqueness of nonlinear elliptic equations[END_REF] présente un saut sur l'interface Γ, nous avons dû d'abord démontrer le résultat suivant, qui établit une estimation du type Meyers, adaptée à notre problème périodique.

Théorème 9. Soit 2 R N et soit λ =( λ 1 , λ 2 ) 2 H la solution de (10). Alors, pour tout 2 R N ,i le x i s t ep i > 2, i =1, 2,t e lq u e λ i 2 W 1,p i (Y i ).
De plus, pour tout q i tel que 2  q i  p i , i =1 , 2,i le x i s t eu n ec o n s t a n t e positive c i ,q u id e p e n dd e↵, , q i et Y i ,t e ll eq u e

kr λ i k L q i (Y i )  c i | |.
Nous montrons ce théorème en utilisant les estimations prouvées par T. Gallouët et A. Monier dans [START_REF] Gallouët | On the regularity of solutions to elliptic equations[END_REF] pour des equations elliptiques avec des conditions de Neumann non homogènes.

Ceci nous a permis de démontrer le résultat principal de ce chapitre, énoncé ci-dessous :

Théorème 10. Soit A :(y, t) 2 Y ⇥ R 7 ! A(y, t) 2 R N ⇥N une matrice reélle qui vérifie : (P1) A(•,t) appartient à M (↵, ,Y) pour tout t 2 R; (P2) A(•,t)={a ij } i,
|A 0 (s) A 0 (t)|  C r |s t| 8s, t 2 ( r, r).
Ce théorème est ce dont nous avions besoin pour démontrer un résultat d'unicité pour le problème homogénéisé, qui est présenté dans la section suivante.

Chapitre 5 : Résultats d'homogénéisation

Dans ce chapitre, nous étudions le comportement asymptotique du problème [START_REF] Bendib | Homogénéisation d'une classe de problèmes non linéaires avec des conditions de Fourier dans des ouverts perforés[END_REF]. Nous utilisons une adaptation de la méthode de l'éclatement périodique aux domaines à deux composantes introduite dans [START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF]. Cette méthode utilise l'opérateur d'éclatement périodique T ε i , i =1 , 2,a g i s s a n tp o u r toute fonction mesurable u i ,d é fi n i ed a n sΩ ε i .S o ni n t é r ê tp r i n c i p a le s tq u ' i l transforme les intégrales sur les ensembles variables Ω ε i en des intégrales sur les ensembles Ω ⇥ Y i qui sont indépendants de ".

Ànotreconnaissance,lapremièreétudequicombinelecadredessolutions renormalisées et la méthode de l'éclatement périodique a été faite dans [START_REF] Donato | Homogenization of quasilinear elliptic problems with nonlinear Robin conditions and L 1 data[END_REF], pour des problèmes elliptiques dans des domaines périodiquement perforés avec des conditions de Robin sur le bord des trous. Nous avons adopté ici une approche similaire.

L'homogénéisation dans le cadre des solutions renormalisées est encore plus difficile que celle dans le cas de données L 2 . Ceci est dû au fait que la fonction u ε i ,q u ie s tl ar e s t r i c t i o nd el as o l u t i o nu ε de (9) à Ω ε i ,n ' a p p a r t i e n t pas nécessairement à H 1 (Ω ε i ), i =1, 2. On rappelle que quand la donnée f dans (9) appartient á L 2 (Ω),o n peut obtenir des estimations a priori sur u ε i dans H 1 (Ω ε i ) (voir [START_REF] Beltran | Homogenization of a quasilinear elliptic problem in a twocomponent domain with an imperfect interface[END_REF]). Donc, en utilisant les résultats montrés dans [START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF], on en déduit les convergences suivantes : 8 < :

T ε i (u ε i ) ! u 1 fortement dans L 2 (Ω,H 1 (Y i )),i =1, 2 T ε i (ru ε i ) * ru 1 + r y b u i faible dans L 2 (Ω ⇥ Y i ),i =1, 2, (13)
avec

u 1 2 H 1 0 (Ω) et b u i 2 L 2 (Ω,H 1 (Y i )), i =1 , 2.
A i n s i ,g r â c eàc e sc o n v e rgences, on obtient le problème homogénéisé dans Ω,q u ie s ts a t i s f a i tp a rl a fonction u 1 .

Par contre, dans notre cas, u ε i n'appartient pas à H 1 (Ω ε i ), i =1, 2 et par conséquent nous ne pouvons pas procéder de cette façon. Nous considérons alors les troncatures de

u ε i (i.e. T k (u ε i )), puisque dans le cadre des solutions renormalisées, T k (u ε i ) 2 H 1 (Ω ε i ) (voir définition 1), i =1, 2
,po u rt o u tk>0. Donc, au lieu à (13), en mélangeant les techniques des solutions renormalisées et celles de l'éclatement périodique (en particulier les résultats de compacité), nous démontrons qu'il existe u 1 et une suite {b

u n i } n2N ⇢ L 2 (Ω,H 1 (Y i )), i = 1, 2,v é r i fi a n tpo u rt o u tn 2 N, i =1, 2, 8 > > > < > > > : T n (u 1 ) 2 H 1 0 (Ω), T ε i (T n (u ε i )) ! T n (u 1 ) fortement dans L 2 (Ω,H 1 (Y i )), T ε i (rT n (u ε i )) * rT n (u 1 )+r y b u n i faiblement dans L 2 (Ω ⇥ Y i ). (14)
Même s'il y a des similitudes entre l'homogénéisation de [START_REF] Bendib | Homogénéisation d'une classe de problèmes non linéaires avec des conditions de Fourier dans des ouverts perforés[END_REF] et celle étudiée dans [START_REF] Donato | Homogenization of quasilinear elliptic problems with nonlinear Robin conditions and L 1 data[END_REF], il y a des difficultés supplémentaires dans notre cas, à cause de la présence du saut de la solution à l'interface. La première différence peut être vue dans la définition d'une solution renormalisée de (9) (voir définition 1), qui contient des hypothèses supplémentaires, comme discutées dans le chapitre 2. De plus, la démonstration du théorème suivant, qui est la construction de la partie oscillante b u i , i =1 , 2, àp a r t i rd el as u i t ed ef o n c t i o n s{b u n i } n2N et d'un résultat d'identification, est encore plus délicate que celle du théorème analogue dans [START_REF] Donato | Homogenization of quasilinear elliptic problems with nonlinear Robin conditions and L 1 data[END_REF] :

Théorème 11. Soit b u n 1 2 L 2 (Ω,H 1 per (Y 1 )) et b u n 2 2 L 2 (Ω,H 1 (Y 2 )), n 2 N, les fonctions introduites dans (14),a v e cM Γ (b u n 1 )=0 .A l o r s ,i le x i s t eu n e unique fonction b u i : Ω ⇥ Y i ! R,i =1, 2, telle que pour tout R 2 C 1 (R) àsupportcompactvérifiantsupp R ⇢ [ m, m], pour un m 2 N,o na R(u 1 )b u n i = R(u 1 )b u i p.p. dans Ω ⇥ Y i ,
pour tout n m,o ùu 1 est la fonction donnée par [START_REF] Betta | Uniqueness for Neumann problems for nonlinear elliptic equations[END_REF]. De plus,

b u i (x, •) 2 H 1 (Y i ),i =1, 2, avec M Γ (b u 1 )=0, p.p. x 2 Ω.
La partie la plus originale de la démonstration est liée au fait que la moyenne de b u n 2 ,po u rn 2 N,n ' e s tp a sn é c e s s a i r e m e n tz é r o . Grâce à ce théorème et aux convergences dans [START_REF] Betta | Uniqueness for Neumann problems for nonlinear elliptic equations[END_REF], on peut alors démontrer le théorème suivant, qui décrit le problème homogénéisé éclaté satisfait par

(u 1 , b u 1 , b u 2 ) : Théorème 12 (Le problème homogénéisé éclaté). Soit u 1 , b u 1 et b u 2 les fonc- tions définies par (14).S o i e n t 1 , 2 des fonctions appartenant à C 1 (R) (ou 1 , 2 2 W 1,1 (R))às u p p o r tc o m p a c t . A l o r s ,(u 1 , b u 1 , b u 2 ) satisfait 8 > > > > > > > < > > > > > > > : 2 X i=1 1 |Y | Z Ω⇥Y i A(y, u 1 )(ru 1 + r y b u i )(r( 1 (u 1 )')+ 2 (u 1 )r y Φ i ) dx dy + 1 |Y | Z Ω⇥Γ h(y) 2 (u 1 )(b u 1 b u 2 )(Φ 1 Φ 2 ) dx d y = Z Ω f (x) 1 (u 1 )'(x) dx 8' 2 H 1 0 (Ω) \ L 1 (Ω), Φ i 2 L 2 (Ω,H 1 per (Y i )),i =1, 2.
De plus, pour k>0,o nal e sl i m i t e ss u i v a n t e s:

lim k!1 1 k Z {|u 1 |<k}⇥Y i A(y, u 1 )(rT k (u 1 )+r y b u i )(rT k (u 1 )+r y b u i ) dx dy =0, pour i =1, 2,e t lim k!1 1 k Z {|u 1 |<k}⇥Γ (b u 1 b u 2 ) 2 dx d y =0.
Enfin, nous obtenons le problème homogénéisé dans Ω,c eq u ic o m p l è t e le chapitre : Théorème 13 (Le problème homogénéisé dans Ω). Soit u 1 une valeur d'adhérence de la suite

{T ε i (u ε i )}, i =1, 2.A l o r su 1 est une solution renormalisée du prob- lème ( div(A 0 (u 1 )ru 1 )=f dans Ω u 1 =0 sur @Ω, (15)
i.e.,

T k (u 1 ) 2 H 1 0 (Ω), pour tout k>0, (16)
lim k!1 1 k Z {|u 1 |<k} A 0 (u 1 )ru 1 ru 1 dx =0, ( 17
)
et pour tout 2 C 1 (R) (ou 2 W 1,1 (R))às u p p o r tc o m p a c t , Z Ω (u 1 )A 0 (u 1 )ru 1 r' dx + Z Ω 0 (u 1 )A 0 (u 1 )ru 1 ru 1 ' dx = Z Ω f (u 1 )' dx, (18)
pour tout ' 2 H 1 0 (Ω) \ L 1 (Ω),o ùA 0 est la matrice homogénéisée définie au dessus (voir (12)).

Si de plus (A4) est vérifiée, alors u 1 est l'unique solution renormalisée de [START_REF] Bendib | Homogénéisation d'une classe de problèmes non linéaires avec des conditions de Fourier dans des ouverts perforés[END_REF],l e sf o n c t i o n sb u 1 , b u 2 sont définies de manière unique et toutes les suites dans (14) convergent (et pas seulement les sous-suites).

La démonstration de la dernière affirmation de ce théorème utilise le théorème 10 du chapitre précédent. Soulignons ici que la preuve de la condition de décroissance de l'énergie (condition [START_REF] Boccardo | Existence of a solution for a weaker form of a nonlinear elliptic equation. Recent advances in nonlinear elliptic and parabolic problems[END_REF]) n'est pas classique.

En conclusion, soulignons que, comme on peut voir tout au long de cette thèse, gérer l'intégrale sur le bord qui provient du saut de la solution sur l'interface est délicat. Cette difficulté ne se limite pas aux résultats d'homogénéisation. Elle peut aussi être observée dans l'étude de l'existence et de l'unicité de la solution renormalisée de [START_REF] Allaire | Homogenization and two-scale convergence[END_REF], ainsi que dans l'étude des propriétés de la solution du problème dans la cellule de référence [START_REF] Bénilan | An L 1 theory of existence and uniqueness of nonlinear elliptic equations[END_REF].

Chapter 1 Introduction

The aim of this thesis is to study a class of elliptic partial differential equations (PDE) with weak data. More precisely, we study a quasilinear elliptic problem posed in a domain with an imperfect interface, where the data is an L 1 function and the matrix field of the quasilinear term has a restricted growth assumption (it is only bounded with respect to the solution on the compact sets of R)a n dw ec o n s i d e rt h ec o r r e s p o n d i n ga s y m p t o t i ca n a l y s i s in a periodic two-component domain.

Let us point out that since we have these weak assumptions, a weak solution may not exist (even in the presence of just one of them). Hence, the convenient framework of renormalized solution needs to be introduced for our problem.

This notion is related to partial differential equations (PDE) with data less summable than L 2 ,e . g . d a t ai nL 1 ,o rm e a s u r ed a t a( m o r eo nr e n o r m a lized solutions is presented in the next section).

The existence and uniqueness results in the framework of renormalized solution for a domain with imperfect interface has previously not been studied in the literature. The first part of this study is dedicated to obtaining such results.

The second part of this thesis is concerned with the corresponding homogenization for a two-component domain with a (disconnected) periodic second component. We use the periodic unfolding method, originally introduced in [START_REF] Cioranescu | Periodic unfolding and homogenization[END_REF] (see also the recent book [START_REF] Cioranescu | The periodic unfolding method[END_REF]), and adapted to a domain with imperfect interface in [START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF] (more on the periodic unfolding method in Section 1.2.4).

In the study of the homogenization, the framework usually considered is the variational setting, where the data are L 2 (or H 1 ) functions. The framework considered in this thesis presents non-trivial additional difficulties, which need to be treated specifically. However, combining homogenization with the notion of renormalized solution is not completely new, one may refer to the pioneer paper [START_REF] Murat | Homogenization of renormalized solutions of elliptic equations[END_REF], see also [START_REF] Ben Cheikh | Homogénéisation des solutions renormalisées dans des domaines perforés[END_REF][START_REF] Briane | Ac l a s so fs e c o n d -o r d e rl i n e a re ll i ptic equations with drift: renormalized solutions, uniqueness and homogenization[END_REF] for some cases. More recently, in [START_REF] Gaudiello | Homogenization of the brush problem with a source term in L 1[END_REF], the authors studied the homogenization of a linear elliptic problem with Neumann boundary conditions, highly oscillating boundary and L 1 data.

In addition, P. Donato, O. Guibé and A. Oropeza studied in [START_REF] Donato | Homogenization of quasilinear elliptic problems with nonlinear Robin conditions and L 1 data[END_REF] the homogenization of a quasilinear elliptic problem with nonlinear boundary conditions and L 1 data. In their study, as far as we know, the notion of renormalized solution is combined for the first time with the period unfolding method. We follow a similar approach for this thesis.

In this chapter, we present in details the framework of renormalized solutions and the homogenization theory. In the next section, dedicated to the notion of renormalized solution, we present the existence and uniqueness results for a model case. In addition, we give a summary of the results obtained in the first part of this thesis.

In Section 1.2, devoted to the homogenization theory, we discuss several methods that were developed for periodic homogenization. We also present there a summary of the second part of this thesis.

Renormalized solutions

In this section, we discuss the framework of renormalized solutions.

When considering a weak data (e.g., L 1 data and measure data), we cannot, in general, show the existence of a weak solution. Recall that the problem div(A(x, u)ru)=f

with Dirichlet boundary conditions, if the matrix field A(x, r) is bounded and coercive, and f belongs to L 1 (Ω) (similarly, when f is a bounded Radon measure), from Boccardo-Gallouët [START_REF] Boccardo | Nonlinear elliptic and parabolic equations involving measure data[END_REF], there exists a solution in the sense of distributions (this result is also true for a class of more general nonlinear operators with increasing p). The authors show that u 2 W 1,q 0 (Ω), 81 <q< N N 1 and verifies

Z Ω A(x)rur' dx = Z Ω f ' dx, 8' 2 C 1 0 (Ω).
However, this solution in the sense of distributions can not have, in general, afi n i t ee n e r g y ,i nt h es e n s et h a tu/ 2 H 1 0 (Ω). Moreover, even in the linear case, that is, A(x, r)=A(x),t h es o l u t i o ni nt h es e n s eo fd i s t r i b u t i o n si sn o t unique in general, one can see the counter example by Serrin in [START_REF] Serrin | Pathological solutions of elliptic equations,A n n .S c u o l a Norm[END_REF] (see also [START_REF] Prignet | Remarks on existence and uniqueness of solutions of el liptic problems with right hand side measures[END_REF]).

To overcome this inconvenience, a number of notions of solutions were developed: entropy solutions (see [START_REF] Bénilan | An L 1 theory of existence and uniqueness of nonlinear elliptic equations[END_REF]), SOLA (solutions obtained as limit approximations, see [START_REF] Dall | Approximated solutions of equations with L 1 data. Application to the h-convergence of quasi-linear parabolic equations[END_REF]) and renormalized solutions.

The notion of renormalized solutions was originally introduced in [39] by R.J. Di Perna and P.L. Lions for first order equations. It was then further developed by F. Murat in [START_REF] Murat | Soluciones renormalizadas de EDP elipticas no lineales[END_REF], by P.L. Lions and F. Murat in [START_REF] Lions | Sur les solutions renormalisées d'équations elliptiques[END_REF] for elliptic equations with Dirichlet boundary conditions and L 1 data, and by G. Dal Maso et al. in [START_REF] Maso | Renormalized solutions of elliptic equations with general measure data[END_REF] for elliptic equations with general measure data. Most of the works concerning the development of renormalized solutions consider elliptic (or parabolic) problems with L 1 data and Dirichlet boundary conditions, but rarely other boundary conditions (see for example [START_REF] Betta | Neumann problems for nonlinear elliptic equations with L 1 data[END_REF] for Neumann conditions and [START_REF] Guibé | Renormalized solutions of el liptic equations with Robin boundary conditions[END_REF] for perforated domains).

There are some physical motivations in considering a weaker data (e.g. L 1 or measure data). As an example, in [START_REF] Liu | Renormalized solutions to a reactiondiffusion system applied to image denoising[END_REF] the authors considered a reactiondiffusion system with L 1 data, which is then applied to image processing. In addition, some engineering problems can require the source to be a mass concentration in a point, which is represented by a measure.

To have more understanding of the notion of renormalized solutions, we present in the sequel the existence and uniqueness results of a model case (see (1.1)).

A model case

Let Ω be an open bounded set in R N with Lipschitz continuous boundary @Ω, where N 2.

We are interested to study the following quasilinear Dirichlet problem:

8 < : div(A(x, u)ru)+ u = f in Ω u =0 on @Ω, (1.1)
where 0 and the matrix A : Ω⇥R ! R N ⇥N is a Carathéodory function, that is, 1. the map r 7 ! A(x, r) is continuous for a.e. x 2 Ω;a n d 2. the map x 7 ! A(x, r) is measurable for a.e. r 2 R, which satisfies the following properties for some ↵, 2 R with 0 < ↵ < ,

( A(x, r)⇠ • ⇠ ↵|⇠| 2 , 8⇠ 2 R N , 8r 2 R, |A(x, r)|  , 8r 2 R. (1.2)
Before we consider the case when the function f belongs to L 1 (Ω),l e tu s first look at the variational case, where f 2 L 2 (Ω).U n d e rt h i sa s s u m p t i o n , the variational formulation of problem (1.1) is

8 > > > > < > > > > : Find u 2 H 1 0 (Ω) such that Z Ω A(x, u)rurvdx+ Z Ω uv dx = Z Ω fvdx, for any v 2 H 1 0 (Ω). (1.3)
Note that all three of the integrals in this formulation are well-defined. Moreover, using Lax-Milgram Theorem and Schauder's Fixed Point Theorem, we can show the existence of a solution to (1.3).

As for showing the uniqueness of said solution, we need to separate the cases for > 0 and =0 .W h e n > 0,a na d d i t i o n a la s s u m p t i o nt h a t A(x, r) is locally Lipschitz continuous with respect to the second variable r must be made to show uniqueness. The case =0is even more difficult and in order to show the uniqueness of solution of (1.3), the additional condition of global Lipschitz continuity with respect to r of A(x, r),hastobedone(see e.g. [START_REF] Artola | Sur une classe de problèmes paraboliques quasi-linéaires[END_REF][START_REF] Boccardo | Nonlinear elliptic and parabolic equations involving measure data[END_REF]). Now, let us consider the case when the function f belongs to L 1 (Ω).A s mentioned before, the weak formulation can not be used, since the integrals involved may not make sense. We then consider the notion of renormalized solution. We present here the definition of a renormalized solution of (1.1).

For simplicity, we only prove in this section the existence and uniqueness results for the linear case (that is, A(x, r)=A(x)) with > 0. The proof of existence and uniqueness for the case where =0requires more arguments (see Remark 1.6).

Before giving the definition of renormalized solution, we need to make sure that the integrals that will be in the renormalized formulation make sense. In particular, it is not clear if a solution u of (1.1) (with f 2 L 1 (Ω)) belongs to any Sobolev space. This means that a solution u may not have enough regularity to have a gradient in the usual sense of Sobolev spaces. Hence, one must first give a proper definition for the gradient of any measurable function using its truncate, where the truncation operator is defined by T k (t) = min{k, max{t, k}} (see Figure 1.1). Proposition 1.1 ([10, Lemma 2.1]). Let u be a measurable function defined from Ω to R.I fT k (u) 2 H 1 0 (Ω) for any k>0,t h e nt h e r ee x i s t sau n i q u e measurable function v defined from Ω to R N such that

rT k (u)=v {|u|<k} , 8k>0, Let S 2 C 1 (R) with supp S ⇢ [ k, k],f o rs o m ek>0.T h e n ,w eh a v e S(u)u = S(u)T k (u) S 0 (u)ruru = S 0 (u)ruru |u|<k = S 0 (u)rT k (u)rT k (u). Let v 2 H 1 0 (Ω) \ L 1 (Ω).T h ef u n c t i o n sS(u), T k (u)
,a n dv are bounded in Ω,w h i c hi m p l i e st h a tS(u)u and S(u)v both belong to L 1 (Ω).I tf o l l o w st h a t the third and fourth integral of (1.6) are well-defined. Furthermore, since

T k (u) 2 H 1 0 (Ω),w eh a v eb y(1.29), S(u)A(x, u)ru 2 (L 2 (Ω)) N S 0 (u)A(x, u)ruru = S 0 (u)A(x, u)rT k (u)rT k (u) 2 L 1 (Ω).
Hence, the first two integrals of (1.6) make sense.

On the other hand, condition (1.5) is important in showing the stability and uniqueness of the renormalized solution.

We now consider, for simplicity, the linear case and present the pro of of the following theorem from [START_REF] Murat | Soluciones renormalizadas de EDP elipticas no lineales[END_REF]: [START_REF] Murat | Soluciones renormalizadas de EDP elipticas no lineales[END_REF]). Suppose that A(x, r)=A(x) for a.e. (x, r) 2 Ω ⇥ R (that is, the problem is linear) and > 0.T h e nt h e r ee x i s t sau n i q u e renormalized solution to (1.1) in the sense of Definition 1.2.

Theorem 1.4 ([
Proof. The proof is divided into 2 steps. The first step is dedicated to show the existence of a renormalized solution. This will be done by approximating f by a sequence {f ε } in L 2 (Ω),a n dc o n s i d e r i n gas e q u e n c eo fa p p r o x i m a t e solutions {u ε } in H 1 0 (Ω). The limit of {u ε } will be a candidate for a renormalized solution, and we show that this is the case. In the second step, we show the uniqueness of the obtained solution.

Step 1. Existence of a renormalized solution.

Let {f ε } be a sequence in L 2 (Ω) such that as " tends to 0,

f ε ! f strongly in L 1 (Ω). (1.7)
For a fixed " > 0, we consider the following variational formulation:

Z Ω A(x)ru ε rvdx+ Z Ω u ε vdx= Z Ω f ε vdx, 8v 2 H 1 0 (Ω). (1.8)
Since > 0 and A satisfies (1.29), the Lax-Milgram Theorem gives the existence and uniqueness of the solution u ε 2 H 1 0 (Ω) of (1.8). Now, we want to show that u ε is Cauchy in L 1 (Ω).F o r ", " 0 > 0,b y linearity, we obtain that u ε u ε 0 satisfies

Z Ω A(x)r(u ε u ε 0 )rvdx+ Z Ω (u ε u ε 0 )vdx= Z Ω (f ε f ε 0 )vdx, 8v 2 H 1 0 (Ω). Note that T k (u ε u ε 0 ) 2 H 1 0 (Ω)
,a n ds o ,w ec a nu s ei ta sat e s tf u n c t i o ni n this last formulation. We then obtain

Z Ω A(x)r(u ε u ε 0 )rT k (u ε u ε 0 ) dx + Z Ω (u ε u ε 0 )T k (u ε u ε 0 ) dx = Z Ω (f ε f ε 0 )T k (u ε u ε 0 ) dx.
(1.9)

Note that from Proposition 1.1, we have

A(x)r(u ε u ε 0 )rT k (u ε u ε 0 ) = {|uε u ε 0 |<k} A(x)r(u ε u ε 0 )r(u ε u ε 0 ) 0,
a.e. in Ω. Moreover, by Hölder's inequality and the fact that

|T k (t)|  k for any t 2 R,w eh a v e Z Ω (f ε f ε 0 )T k (u ε u ε 0 ) dx  kkf ε f ε 0 k L 1 (Ω) .
It then follows from (1.9) that

1 k Z Ω (u ε u ε 0 )T k (u ε u ε 0 ) dx  1 kf ε f ε 0 k L 1 (Ω) .
Taking the limit of both sides of this inequality as k tends to 0, and noting that

(u ε u ε 0 ) T k (u ε u ε 0 ) k  |u ε u ε 0 | 2 L 1 (Ω), and
T k (u ε u ε 0 ) k ! sgn(u ε u ε 0 ) as k ! 0 a.e. in Ω,
where sgn is the usual sign function (i.e., sgn(r)=r/|r| if r 6 =0and sgn(0) = 0), we obtain by Lebesgue dominated convergence theorem,

ku ε u ε 0 k L 1 (Ω)  1 kf ε f ε 0 k. (1.10)
We then conclude from (1.7) that {u ε } is a Cauchy sequence in L 1 (Ω).H e n c e , there exists u 2 L 1 (Ω) (and so u is finite a.e. in Ω)s u c ht h a t( u pt oa subsequence)

( u ε ! u strongly in L 1 (Ω), u ε ! u a.e. in Ω. (1.11)
We now claim that u is a renormalized solution of (1.1). To show this, we need to show that u satisfies (1.4), (1.5), and (1.6).

We first obtain some estimates for T k (u ε ),f o ra n yk>0,t h e ns h o ws o m e convergences for the sequence {T k (u ε )} as " approaches 0. We start by using

T k (u ε ) 2 H 1 0 (Ω) as a test function in (1.8), which gives Z Ω A(x)ru ε rT k (u ε ) dx + Z Ω u ε T k (u ε ) dx = Z Ω f ε T k (u ε ) dx.
(1.12)

Since T k is an increasing function, we know that u ε T k (u ε ) 0 a.e. in Ω. Then, we deduce from (1.12) and Hölder's inequality that

Z Ω A(x)rT k (u ε )rT k (u ε ) dx  Z Ω f ε T k (u ε ) dx  kkf ε k L 1 (Ω) . (1.13)
By coercivity of A and (1.7), we have

krT k (u ε )k (L 2 (Ω)) N  kM,
for any k>0,f o ra n y" > 0,a n df o rs o m eM>0 independent of k and ". This implies that for any k>0,thesequence{T k (u ε )} ε is uniformly bounded in H 1 0 (Ω). By a diagonal process and Rellich Theorem, we can extract a subsequence (which will still be denoted by ")s u c ht h a tf o ra n yk>0 (taken from a countable set), there exists

v k 2 H 1 0 (Ω) such that ( T k (u ε ) * v k weakly in H 1 0 (Ω), T k (u ε ) ! v k
strongly in L 2 (Ω) and a.e. in Ω.

From (1.11), the continuity of T k ,andtheuniquenessofthelimit,w ededuce that for any k>0, T k (u)=v k 2 H 1 0 (Ω) (this shows that u satisfies (1.4)), and 8 > < > :

T k (u ε ) * T k (u) weakly in H 1 0 (Ω), T k (u ε ) ! T k (u)
strongly in L 2 (Ω) and a.e. in Ω,

rT k (u ε ) * rT k (u) weakly in (L 2 (Ω)) N .
(1.14)

We now show that u satisfies (1.5). By (1.13), (1.14), and (1.7), for all k>0,w eh a v e

1 k Z Ω A(x)rT k (u)rT k (u) dx  lim sup ε!0 1 k Z Ω A(x)rT k (u ε )rT k (u ε ) dx  lim sup ε!0 1 k Z Ω f ε T k (u ε ) dx = 1 k Z Ω fT k (u) dx.
Since u is finite a.e. in Ω,w eh a v e

T k (u) k ! 0 a.e. in Ω as k !1.
Moreover, from the definition of T k ,w eh a v ef o ra n yk>0,

f T k (u) k  |f | 2 L 1 (Ω).
Then the Lebesgue dominated convergence theorem implies

lim k!1 1 k Z Ω A(x)rT k (u)rT k (u) dx =0
(which gives that u satisfies (1.5)), and

lim k!1 lim sup ε!0 1 k Z Ω A(x)rT k (u ε )rT k (u ε ) dx =0. (1.15)
To end this step, we show that u satisfies (1.6). For n>0,l e tu sd e fi n et h e function h n as (see Figure 1.2)

h n (s)= 8 > > > > > > > > > < > > > > > > > > > : 0, if s  2n s n +2, if 2n  s  n 1, if n  s  n s n +2, if n  s  n 0, if s 2n.
Moreover, we have

h n (u ε )ru ε = h n (u ε )rT 2n (u ε ), which implies that Z Ω A(x)ru ε rvS(u)h n (u ε ) dx = Z Ω A(x)rT 2n (u ε )rvS(u)h n (u ε ) dx,
and

Z Ω S 0 (u)A(x)ru ε ruh n (u ε )vdx= Z Ω S 0 (u)A(x)rT 2n (u ε )ruh n (u ε )vdx.
Since h n is continuous, we know from (1.11) that h n (u ε ) ! h n (u) a.e. in Ω as " tends to 0.W ea l s oh a v e|h n |  1 from the definition of h n , which implies that

h n (u ε ) ! h n (u) in L 1 (Ω) weak-* as " ! 0.
Combining this with (1.14), we obtain

lim ε!0 Z Ω A(x)ru ε rvS(u)h n (u ε ) dx = Z Ω A(x)rurvS(u)h n (u) dx, (1.20)
and

lim ε!0 Z Ω S 0 (u)A(x)ru ε ruh n (u ε )vdx= Z Ω S 0 (u)A(x)ruruh n (u)vdx.
(1.21) If we choose n sufficiently large such that supp S ⇢ [ n, n],t h e nw eh a v e S(u)h n (u)=S(u) a.e. in Ω.

Applying this identity to (1.17), (1.18), (1.20), and (1.21), we can pass to the limit as n tends to infinity. Together with (1.19) and (1.16), we have the desired result.

Step 2. Uniqueness of the renormalized solution.

From the previous step, we know that there exists at least one solution. Suppose that u and v are two renormalized solutions of (1.1). Let n 2 N and k>0.F r o mt h ed e fi n i t i o n so fT k and h n , it follows that

h n (u)h n (v)T k (T 2n (u) T 2n (v)) 2 H 1 0 (Ω) \ L 1 (Ω).
Hence, this can be used as a test function for the renormalized formulation (1.6) for u (with

S = h n and v = h n (v)T k (T 2n (u) T 2n (v)))a n da l s of o rv (with S = h n and v = h n (u)T k (T 2n (u) T 2n (v))
). Subtracting the resulting equations and the fact that

h n (u)h n (v)T k (T 2n (u) T 2n (v)) = h n (u)h n (v)T k (u v) a.e. in Ω, yield I 1 + I 2 + I 3 I 4 I 5 + I 6 =0
, where

I 1 = Z Ω A(x)(ru rv)(ru rv) {|u v|<k} h n (u)h n (v) dx I 2 = Z Ω h 0 n (u)A(x)ruruh n (v)T k (u v) dx I 3 = Z Ω h 0 n (v)A(x)rurvh n (u)T k (u v) dx I 4 = Z Ω h 0 n (u)A(x)rvruh n (v)T k (u v) dx I 5 = Z Ω h 0 n (v)A(x)rvrvh n (u)T k (u v) dx I 6 = Z Ω (u v)h n (u)h n (v)T k (u v) dx.
We want to evaluate the limit of each term as n approaches +1.

By the decay of the truncate energy (1.5), the properties of the matrix A in (1.29), the fact that supp h n ⇢ [ 2n, 2n],a n dY o u n g ' si n e q u a l i t y( f o rI 3 and I 4 ), it follows that

lim n!1 I 2 =l i m n!1 I 3 =l i m n!1 I 4 =l i m n!1 I 5 =0. Note that I 1 0,a n dt h u s ,f r o mF a t o u ' sL e m m a Z Ω (u v)T k (u v) dx  lim inf n!1 I 6  0.
Since T k is an increasing function, it follows that u v =0a.e. in Ω. This concludes this step and this proof.

We can also have the following stability result. This can be proved by using the same method used in the proof above.

Remark 1.5 (Stability). Let {f ε } be a sequence that strongly converges to a function f in L 1 (Ω).S u p p o s eu ε is the renormalized solution corresponding to the data f ε .T h e n{u ε } converges to u in L 1 (Ω),w h e r eu is the renormalized solution corresponding to the function f .I na d d i t i o n ,T k (u ε ) converges strongly to T k (u) in H 1 0 (Ω),f o ra n yk>0.

Remark 1.6 (The case =0). Note that if we consider the case =0,w e clearly do not have the estimate (1.10) from the proof of the existence result of Theorem 1.4. To show existence for this case, we argue first by showing that the approximate solution u ε of (1.8) is Cauchy in measure then use the Rel lich-Kondrachov compactness result to obtain convergences (1.14).

For the uniqueness result, if u and v are two renormalized solutions, notice that from the proof of the uniqueness part of Theorem 1.4, the argument to show that u = v involves mainly integral I 6 ,whichistheinte gr alr elate dtothe term with .A s =0 ,w ed on o th a v et h i st e r mw h e ns h o w i n gu n i q u e n e s s for this case. The proof of uniqueness of the renormalized solution for this case requires more delicate arguments.

These delicate arguments can be observed in the first part of this thesis, as we consider there the case =0.

Our renormalized solution results

In the first part of this thesis, we study the existence and uniqueness of renormalized solution of a class of quasilinear elliptic equations posed in a two-component domain with an L 1 data. More precisely, the domain Ω is a connected bounded open set in R N with boundary @Ω. We write Ω as the disjoint union

Ω = Ω 1 [ Ω 2 [ Γ,
where Ω 2 is an open set such that Ω 2 ⇢ Ω with a Lipschitz boundary Γ,a n dΩ 1 = Ω \ Ω 2 (see Figure 1.3). We study the following quasilinear elliptic problem posed in Ω:

8 > > > > > > < > > > > > > : div(B(x, u 1 )ru 1 )=f in Ω 1 , div(B(x, u 2 )ru 2 )=f in Ω 2 , (B(x, u 1 )ru 1 )⌫ 1 =(B(x, u 2 )ru 2 )⌫ 1 on Γ, (B(x, u 1 )ru 1 )⌫ 1 = h(x)(u 1 u 2 ) on Γ, u 1 =0 on @Ω, (1.22)
where ⌫ 1 is the unit outward normal to Ω 1 , f is an L 1 function, and B is a coercive matrix field which has a restricted growth assumption (B(x, r) is bounded on any compact set of R).

To properly define a renormalized solution of (1.22), we first introduce the space that we will be working with, that is well adapted to our problem.

We define the space V as

V := {v ⌘ (v 1 ,v 2 ):v 1 2 V 1 and v 2 2 H 1 (Ω 2 )},
equipped with the norm

kvk 2 V := krv 1 k 2 L 2 (Ω 1 ) + krv 2 k 2 L 2 (Ω 2 ) + kv 1 v 2 k 2 L 2 (Γ) , Ω Ω 2 Γ Ω 2 Γ Ω 1 ∂Ω Figure 1.3: The two-component domain Ω where V 1 is the space defined by V 1 = {v 2 H 1 (Ω 1 ):v =0on @Ω} with kvk V 1 := krvk L 2 (Ω 1 ) .
We follow a similar definition from [START_REF] Maso | Renormalized solutions of elliptic equations with general measure data[END_REF] with the additional term on the interface Γ and condition for the jump. More precisely:

Definition 1.7. Let u =( u 1 ,u 2 ):Ω \ Γ ! R be a measurable function. Then u is a renormalized solution of (1) if T k (u) 2 V, 8k>0;
(1.23a)

(u 1 u 2 )(T k (u 1 ) T k (u 2 )) 2 L 1 (Γ), 8k>0; (1.23b) lim n!1 1 n Z {|u|<n} B(x, u)ru • rudx =0; (1.24a) lim n!1 1 n Z Γ (u 1 u 2 )(T n (u 1 ) T n (u 2 )) d =0; (1.24b)
and for any S 1 ,S 2 2 C 1 (R) (or equivalently for any

S 1 ,S 2 2 W 1,1 (R))w i t h compact support, u satisfies Z Ω 1 S 1 (u 1 )B(x, u 1 )ru 1 • rv 1 dx + Z Ω 1 S 0 1 (u 1 )B(x, u 1 )ru 1 • ru 1 v 1 dx + Z Ω 2 S 2 (u 2 )B(x, u 2 )ru 2 • rv 2 dx + Z Ω 2 S 0 2 (u 2 )B(x, u 2 )ru 2 • ru 2 v 2 dx + Z Γ h(x)(u 1 u 2 )(v 1 S 1 (u 1 ) v 2 S 2 (u 2 )) d = Z Ω 1 fv 1 S 1 (u 1 ) dx + Z Ω 2 fv 2 S 2 (u 2 ) dx, (1.25
)

for all v 2 V \ (L 1 (Ω 1 ) ⇥ L 1 (Ω 2 )).
Note that since we are in the renormalized framework, a solution u of (1) may not have enough regularity to have a gradient and trace in the usual sense of Sobolev spaces. Hence, we first have to make sure that the gradient and the trace of a solution are properly defined. To this aim, we prove the following proposition (which is a generalization of [10, Lemma 2.1] and [59, Proposition 2.3]):

Proposition 1.8. Let u =( u 1 ,u 2 ):Ω \ Γ ! R be a measurable function such that T k (u) 2 V for every k>0.
1. For i =1, 2,t h e r ee x i s t sau n i q u em e a s u r a b l ef u n c t i o nG i :

Ω i ! R N
such that for all k>0,

rT k (u i )=G i {|u i |<k} a.e. in Ω i ,
where {|u i |<k} denotes the characteristic function of

{x 2 Ω i : |u i (x)| <k}.
We define G i as the gradient of u i and write G i = ru i .

If

sup k 1 1 k kT k (u)k 2 V < 1,
then there exists a unique measurable function

w i : Γ ! R, for i =1, 2,
such that for all k>0,

i (T k (u i )) = T k (w i ) a.e. in Γ,
where i :

H 1 (Ω i ) ! L 2 (Γ)
is the trace operator. We define the function w i as the trace of u i on Γ and set

i (u i )=w i ,i =1, 2.
The originality of this definition is found in the regularity (1.23b) and the decay of the energy on the interface (1.24b), together with the presence of the term on Γ,

Z Γ h(x)(u 1 u 2 )(v 1 S 1 (u 1 ) v 2 S 2 (u 2 )) d .
The regularity (1.23a) and the decay of the truncated energy (1.24a) are classical and it allows us (with Proposition 1.8) to give a sense to all the integrals in (1.25) except for the boundary integral on Γ.

Indeed, let

S i 2 C 1 (R), i =1 , 2, with compact support. Then for all v 2 V \ (L 1 (Ω 1 ) ⇥ L 1 (Ω 2 )),i fsupp S i ⇢ [ k, k] (i =1, 2), then for i =1, 2, we have S i (u i )B(x, u i )ru i • rv i = S i (u i )B(x, T k (u i ))rT k (u i ) • rv i 2 L 1 (Ω i ), S 0 i (u i )B(x, u i )ru i • ru i v i = S 0 i (u i )B(x, T k (u i ))rT k (u i ) • rT k (u i ) v i 2 L 1 (Ω i ), fv i S i (u i ) 2 L 1 (Ω i ).
Concerning the boundary integral on Γ,n o t et h a t( 1 . 2 3 a )a n d( 1 . 2 4 a )a r e not enough to give a sense to h(x)(u 1 u 2 )(v 1 S 1 (u 1 ) v 2 S 2 (u 2 )).I ng e n e r a l , there is no reason to have

h(x)(u 1 u 2 )(v 1 S 1 (u 1 ) v 2 S 2 (u 2 )) = h(x)(u 1 u 2 )(v 1 S 1 (u 1 ) v 2 S 2 (u 2 )) {|u 1 |n} {|u 2 |n}
for large n.

For the b oundary term, for any n 2 N,l e tu sd e fi n e✓ n : R ! R by (see Figure 1.4)

✓ n (s)= 8 > > > > > > > < > > > > > > > : 0, if s  2n, s n +2, if 2n  s  n, 1, if n  s  n, s n +2, if n  s  2n, 0, if s 2n.
Then since S 1 has a compact support, we have for some large enough n,

h(u 1 u 2 )v 1 S 1 (u 1 )=hv 1 (u 1 u 2 )(S 1 (u 1 ) S 1 (u 2 ))✓ n (u 1 ) + hv 1 (u 1 u 2 )S 1 (u 2 )✓ n (u 1 ).
Since ✓ n also has a compact support, then hv 1 (u 1 u 2 )S 1 (u 2 )✓ n (u 1 ) is bounded, and is therefore in L 1 (Γ). Moreover, since

S 1 (u 1 ) S 1 (u 2 )=S 1 (T 2n (u 1 )) S 1 (T 2n (u 2 ))
and S 1 are Lipschitz-continuous, we have

|hv 1 (u 1 u 2 )(S 1 (u 1 ) S 1 (u 2 ))✓ n (u 1 )| khv 1 k L 1 (Γ) kS 0 1 k L 1 (R) |u 1 u 2 ||T 2n (u 1 ) T 2n (u 2 )|,
and we let

B ε (x, t)=B(x, T 1/ε (t)).W ec o n s i d e ras o l u t i o nu ε 2 V of 8 > > > > > > < > > > > > > : div(B ε (x, u ε 1 )ru ε 1 )=f ε in Ω 1 , div(B ε (x, u ε 2 )ru ε 2 )=f ε in Ω 2 , (B ε (x, u ε 1 )ru ε 1 )⌫ 1 =(B ε (x, u ε 2 )ru ε 2 )⌫ 1 on Γ, (B ε (x, u ε 1 )ru ε 1 )⌫ 1 = h(x)(u ε 1 u ε 2 ) on Γ, u ε 1 =0 on @Ω.
The second step is to obtain a priori estimates, then construct (with the help of Rellich-Kondrachov compactness results) a function u such that (up to a subsequence) 8 > > > < > > > :

u ε i ! u i a.e. in Ω, T k (u ε i ) * T k (u i ) weakly in H 1 (Ω i ), i (u ε i ) ! i (u i ) a.e. in Γ, i (T k (u ε i )) ! i (T k (u i )) strongly in L 2 (Γ)
, a.e. on Γ. The originality here comes again from the boundary integral on Γ,a n d using, for example, in the third step, the a priori estimate 8k>0, (u ε

1 u ε 2 )(T k (u ε 1 ) T k (u ε 2
)) bounded in L 1 (Γ), and the limit

lim n!1 lim sup ε!0 1 n ✓ Z Ω\Γ B(x, T n (u ε ))rT n (u ε )rT n (u ε ) dx + Z Γ (u ε 1 u ε 2 )(T n (u ε 1 ) T n (u ε 2 )) d ◆ =0.
In the final step of the proof, we are able then to pass to the limit with an appropriate choice of test function and show that u is a renormalized solution of (1).

Remark 1.10. To give a sense to the integral on Γ,i ti sp o s s i b l et or e p l a c e (1.23b) and (1.24b) by a regularity condition given by To prove the uniqueness result, we adopt the method developed in [START_REF] Blanchard | Quasi-linear degenerate elliptic problems with L 1 data[END_REF][START_REF] Di Nardo | Uniqueness of renormalized solutions to nonlinear parabolic problems with lower-order terms[END_REF]. This method makes use of an auxiliary function ' 2 C 1 (R) with interesting properties. To be more precise, we use the following proposition from [START_REF] Di Nardo | Uniqueness of renormalized solutions to nonlinear parabolic problems with lower-order terms[END_REF]: Proposition 1.11 ([38]). Suppose that (A4) holds. Then there exists a function ' 2 C 1 (R) that satisfies the following properties: '(0) = 0 and ' 0 1.

u 1 u 2 2 W 1 1 q ,q (Γ),
In addition, there are constants > 1/2, 0 <k 0 < 1,a n dL>0 such that

' 0 (1 + |'|) 2δ 2 L 1 (R). Moreover, for any r, s 2 R satisfying |'(r) '(s)|  k,f o r0 <k<k 0 , B(x, r) ' 0 (r) B(x, s) ' 0 (s)  1 ' 0 (s) Lk (1 + |'(r)| + |'(s)|) δ and 1 L  ' 0 (s) ' 0 (r)  L.
One of the difficulties in the proof of uniqueness comes from the boundary integral on Γ, which is related to the jump of the solution. The following proposition is an important tool in the proof of the uniqueness theorem, which is related to the Boccardo-Gallouët estimates: Proposition 1.12. For i =1 , 2,l e t i be the trace function defined on H 1 (Ω i ).U n d e rt h ea s s u m p t i o n so fT h e o r e m1 . 9 ,i fu is a renormalized solution of (1),t h e n i (u i ) 2 L 1 (Γ), i =1, 2.

The method developed in [START_REF] Blanchard | Quasi-linear degenerate elliptic problems with L 1 data[END_REF][START_REF] Di Nardo | Uniqueness of renormalized solutions to nonlinear parabolic problems with lower-order terms[END_REF] consists of using T k ('(u) '(v)) as test function. The proof is very technical and is done by passing to the limit with the help of the function ✓ n (u)T k ('(u) '(v)) in the renormalized formulation in the definition of a renormalized solution.

However, the nonlinearity of the term T k ('(u) '(v)) is not compatible with the boundary integral on Γ,s i n c et h et e r mo b t a i n e d

Z Γ h(x)((u 1 v 1 ) (u 2 v 2 ))(T k ('(u 1 ) '(v 1 ) T k ('(u 2 ) '(v 2 ))) d
is not necessarily positive.

To overcome this difficulty, we show a sign property on Γ:

Lemma 1.13. Suppose assumptions (A1)-(A4) hold. If u and v are two renormalized solutions of (1),t h e nsgn(u 1 v 1 )=sgn(u 2 v 2 ) a.e. on Γ.

This lemma is essential in the proof of the following uniqueness theorem, which is the main result of this chapter: Theorem 1.14. If assumptions (A1)-(A4) hold, then the renormalized solution of (1) is unique.

One of the major steps of the proof of Theorem 1.14 consists of showing that

lim k!0 1 k 2 Z Ω i ✓ 1 ' 0 (u i ) + 1 ' 0 (v i ) ◆ |rT k ('(u i ) '(v i ))| 2 dx =0,i =1, 2,
(1.27) where u =( u 1 ,u 2 ) and v =( v 1 ,v 2 ) are two renormalized solutions of (1), and ' is the function in Proposition 1.11.

To prove this, after some long computations, we obtain

lim sup k!0 ✓ 1 k 2 Z U k 1 ✓ 1 ' 0 (u 1 ) + 1 ' 0 (v 1 ) ◆ |r'(u 1 ) r'(v 1 )| 2 dx + 1 k 2 Z U k 2 ✓ 1 ' 0 (u 2 ) + 1 ' 0 (v 2 ) ◆ |r'(u 2 ) r'(v 2 )| 2 dx + 1 k 2 C k ◆  0,
where

U k i = {x 2 Ω i :0< |'(u i ) '(v i )| <k},i =1, 2
, and

C k = Z Γ h(x)[(u 1 v 1 ) (u 2 v 2 )][T k ('(u 1 ) '(v 1 )) T k ('(u 2 ) '(v 2 ))] d .
Hence, to show (1.27), it is enough to show that

lim sup k!0 1 k 2 C k 0.
As already mentioned above, due to the presence of the function ',w e don't know in general the sign of C k . The limit of 1 k 2 C k as k ! 0 is one of the main difficulties. To proceed, we use the sign property on Γ (Lemma 1.13) and we divide the set {x 2 Γ ; u 1 (x) v 1 (x) > 0} (up to a zero measure subset) into 4 disjoint subsets,

{u 1 v 1 > 0} = P 1 [ P 2 [ P 3 [ P 4 ,
where

P 1 := {'(u 1 ) '(v 1 ) k} \ {'(u 2 ) '(v 2 ) k}, P 2 := {0 < '(u 1 ) '(v 1 ) <k} \ {0 < '(u 2 ) '(v 2 ) <k}, P 3 = {'(u 1 ) '(v 1 ) k} \ {0 < '(u 2 ) '(v 2 ) <k}, P 4 := {0 < '(u 1 ) '(v 1 ) <k} \ {'(u 2 ) '(v 2 ) k}.
This division is possible due to Lemma 1.13. Then we are able to compute the limit of the boundary integral over each subset. Once (1.27) is shown, we follow the same approach as in [START_REF] Blanchard | Quasi-linear degenerate elliptic problems with L 1 data[END_REF][START_REF] Di Nardo | Uniqueness of renormalized solutions to nonlinear parabolic problems with lower-order terms[END_REF] to show first that u 1 = v 1 in Ω 1 (since u 1 and v 1 have zero trace on @Ω, which allows us to use the Poincaré inequality). As a consequence, we have u 1 = v 1 on Γ. This, combined with Lemma 1.13 and (1.27), gives

u 2 = v 2 in Ω 2 .

Homogenization theory

Homogenization theory is motivated from the study of the macroscopic behaviour of microscopic composite materials. Composite materials are composed of 2 or more finely mixed components and their main physical characteristics (e.g. thermal or electric conductivity) can be modelled by PDEs with oscillating coefficients, describing the heterogeneities at the micro-scale. Then, the mathematical homogenization theory allows to give a macroscopic description of these materials, considered as homogeneous, at the macroscale.

There are different types of composite materials that can be studied by homogenization, some examples are plywood (a layered material) and concrete (a periodic material). Also, additional oscillations can come, in some case, from the domain. Let us mention the ones with oscillating boundaries, such as heat radiators and engines, or perforated materials like sponge (a porous medium), or trusses.

In this thesis, we are concerned in the homogenization of finely periodically mixed materials, that is, periodic homogenization. Indeed, when the components of these materials are finely mixed, a reasonable assumption is that the distribution of the heterogeneities is periodic.

To explain intuitively how periodic homogenization works, imagine a composite material with two components, Material 1 and Material 2. We suppose that an "-sized Material 1 is periodically scattered throughout Material 2, with "-periodicity in each axis-direction. This is obtained by a change of scale from a fixed unit cell (see Figure 1.2).

Let " be a positive parameter taken from a positive sequence that goes to 0. This parameter characterizes the periodicity of the domain.

We now intro duce the reference cell Y ,d e fi n e db y

Y = N Y j=1 [0,l j ),
for some l j > 0, j =1,...,N.

The heat conductivity in Ω, can be modelled by the following problem:

( div(A ε ru ε )=f in Ω, u ε =0 on @Ω, (1.28)
where f 2 L 2 (Ω) is the heat source, u ε is the temperature distribution, and A ε represents the conductivity of the material. We assume that

A ε (x)=A ⇣ x " ⌘ ,
where A(y) is a Y -periodic measurable matrix field, that satisfies the following properties, for some fixed

↵, 2 R + , 0 < ↵ < : ( A(y)⇠ • ⇠ ↵|⇠| 2 , |A(y)⇠|  |⇠|, (1.29)
for all ⇠ 2 R N ,a n df o ra l ly 2 Y . Note that the conductivity A ε is, in general, discontinuous due to the nature of the domain Ω. This means that the heat flux A ε ru ε may not be differentiable in the classical sense. Thus, we need to consider derivatives in the weak sense, and we look for a weak solution u ε in an appropriate Sobolev space.

In order to find a weak solution, we consider the variational formulation of problem (1.28), which is the following:

8 < : Find u ε 2 H 1 0 (Ω) such that Z Ω A ε ru ε rvdx= Z Ω fvdx, 8v 2 H 1 0 (Ω). (1.30) Since f 2 L 2 (Ω),f o rafi x e d",

it is easily established by the Lax-Milgram

Theorem that this problem has a unique solution with the following a priori estimate:

ku ε k H 1 0 (Ω)  1 ↵ kf k L 2 (Ω) . (1.31)
Our goal is to pass to the limit in the integrals in (1.30) as " goes to 0. Actually, we are only concerned with the integral

Z Ω A ε ru ε rvdx, (1.32)
since the other integral in (1.30) is independent of ".L e tu se x a m i n et h e sequences {A ε } and {u ε }.

From (1.31), we know that the sequence {u ε } is uniformly bounded in H 1 0 (Ω). Thus, we can extract a subsequence from {u ε } such that

u ε * u 0 weakly in H 1 0 (Ω), (1.33)
which then implies that

ru ε * ru 0 weakly in L 2 (Ω) N .
Now, to easily pass to the limit of (1.32), we need a strong convergence for A ε . However, this is not the case. In fact, from classical results (see Theorem 2.6 of [START_REF] Cioranescu | An introduction to homogenization[END_REF]), we only have a weak convergence for functions with the form similar to A ε . In general, we cannot pass to the limit of an integral of a product of two weakly convergent sequences. This difficulty justifies the interest and development of the homogenization theory. The various methods for periodic homogenization were developed specifically to solve this problem.

It is worth noting that the weak limit u 0 of u ε describes the thermal distribution in the homogenized material. However, except for this u 0 ,w e have no information about this homogenized material. By passing to the limit of (1.30), we will obtain an elliptic problem satisfied by this u 0 . This homogenized problem gives the conductivity matrix A 0 (see (1.35)) for the homogenized material.

We now discuss the different methods for periodic homogenization by proving the following homogenization result for problem (1.28):

Theorem 1.15. Let A ε be the matrix A ε (x)=A( x ε ),w h e r eA is Y -periodic and satisfies (1.29),a n df be a function in L 2 (Ω).I fu ε is the weak solution of (1.30),t h e nt h e r ee x i s t su

0 2 H 1 0 (Ω) such that ( u ε * u 0 weakly in H 1 0 (Ω), A ε ru ε * A 0 ru 0 weakly in (L 2 (Ω)) N , (1.34)
where A 0 is the constant matrix defined by (see [START_REF] Bensoussan | Asymptotic analysis for periodic structures[END_REF])

A 0 = 1 |Y | Z Y A(y, t)r y c w λ (y) dy, 8 2 R N , (1.35)
or equivalently,

t A 0 = 1 |Y | Z Y t A(y, t)r y w λ (y) dy, 8 2 R N .
(1.36)

The functions c w λ and w λ are respectively the unique solutions of the following cel l problems:

8 > > < > > : div(A(y)rc w λ )=0 in Y, c w λ • yY -periodic M Y (c w λ • y)=0, (1.37
)

and 8 > > < > > : div( t A(y)rw λ )=0 in Y, w λ • yY -periodic M Y (w λ • y)=0, (1.38)
for every 2 R N . The function u 0 is the unique solution in

H 1 0 (Ω) of the homogenized prob- lem ( div(A 0 ru 0 )=f in Ω, u 0 =0 on @Ω. (1.39)
Remark 1.16. The uniqueness of the function u 0 implies that the convergences in (1.34) applies to the whole sequence and not just for a subsequence. This is very important in homogenization since this means that the function u 0 approximates the thermal distribution in Ω for whatever value of " we choose.

In the sequel, we discuss briefly some methods, originally developed to prove this last theorem. For more in-depth discussion of these methods, one can see [START_REF] Cioranescu | An introduction to homogenization[END_REF].

Multiple-scales method

The multiple-scales method, which is also known as asymptotic expansion method, is a non-rigorous method of homogenization introduced in [START_REF] Bensoussan | Asymptotic analysis for periodic structures[END_REF][START_REF]Asymptotic analysis for periodic structures[END_REF] (see also [START_REF] Sanchez-Palencia | Non-homogeneous media and vibration theory[END_REF]). It features the formal asymptotic expansion of the solution u ε of (1.30) (see (1.40)). It must be mentioned first that the problem can be characterized in two scales: the macroscopic scale, which gives the position of a point on the domain Ω,a n dt h em i c r o s c o p i cs c a l ey = x ε , which locates apo i n ti nt h ec e l lY and describes the micro-oscillations. This leads to the consideration of the following formal expansion (which is also called ansatz) for u ε :

u ε (x)=u 0 ⇣ x, x " ⌘ +"u 1 ⇣ x, x " ⌘ +" 2 u 2 ⇣ x, x " ⌘ +••• = 1 X i=0 " i u i ⇣ x, x " ⌘ , (1.40)
where the functions u i = u i (x, y) are functions from Ω ⇥ Y to R and are Yperiodic with respect to the second variable y.I ti sc l e a rt h a tu 0 is the limit of u ε as " tends to 0. The idea of this method is to substitute this formal expansion of u ε in (1.28) and then solve for the functions u j , j 2 N [ {0}.

We need a few preliminaries b efore we can further discuss this metho d. Let Ψ = Ψ(x, y) be a function defined on R N ⇥ R N .L e tΨ ε be defined by

Ψ ε (x)=Ψ ⇣ x, x " ⌘ .
Define the operator A ε by

A ε = div(A ε r)= X 1i,jN @ @x i ✓ a ε ij @ @x j ◆ . Note that @Ψ ε @x i (x)= 1 " @Ψ @y i ⇣ x, x " ⌘ + @Ψ @x i ⇣ x, x "
⌘ .

Then, we can write

A ε Ψ ε (x)=[(" 2 A 0 + " 1 A 1 + A 2 )Ψ] ⇣ x, x " ⌘ ,
where

A 0 = X 1i,jN @ @y i ✓ a ij (y) @ @y j ◆ , A 1 = X 1i,jN @ @x i ✓ a ij (y) @ @y j ◆ X 1i,jN @ @y i ✓ a ij (y) @ @x j ◆ , A 2 = X 1i,jN @ @x i ✓ a ij @ @x j ◆ .
It then follows, using (1.40), that we have the following infinite systems of equations:

( A 0 u 0 =0 in Y u 0 Y -periodic in y, ( A 0 u 1 = A 1 u 0 in Y u 1 Y -periodic in y, ( A 0 u 2 = f A 1 u 1 A 2 u 0 in Y u 2 Y -periodic in y,

and

(

A 0 u s+2 = A 1 u s+1 A 2 u s in Y u s+2 Y -periodic in y,
for s 1. The following theorem is the result of solving these systems. One can refer to [START_REF] Cioranescu | An introduction to homogenization[END_REF] for details.

Theorem 1.17. Let f 2 L 2 (Ω) and u ε be the unique solution of (1.28).

Then u ε can be written as

u ε = u 0 " N X k=1 b k ⇣ x " ⌘ @u 0 @x k + " 2 N X k,l=1 b ✓ kl ⇣ x " ⌘ @ 2 u 0 @x k @x k + ••• , where u 0 is the solution to (1.39), b k = y k b w e k ,w i t h b w e k the unique solution of the cell problem (1.37), {e k } is the canonical basis of R N ,a n d b ✓ kl is the solution of 8 > > < > > : div(A(y)r b ✓ kl )= a kl N X i,j=1 @(a ij ki b l ) @y i N X j=1 a kj @(b l y l ) @y j in Y, b ✓ kl Y -periodic, M Y ( b ✓ kl )=0.
As mentioned earlier, this is a non-rigorous method. In particular, there is no justification as to why we can write u ε as the formal expansion (1.40). However, the results of this method is justified by the subsequent methods.

Tartar's method of oscillating test functions

Tartar's method of oscillating test functions is introduced by L. Tartar in [START_REF] Tartar | [END_REF][START_REF]Quelques remarques sur l'homogénéisation,inF unctionalAnalysis and Numerical Analysis[END_REF]. As the name suggests, this method makes use of a suitable oscillating test function to remove the problem of passing to the limit of an integral with a product of two weakly convergent sequences. This is a more mathematically rigorous method than the multiple-scales method and it also confirms the results obtained in the previous method.

We now present a sketch of the proof of Theorem 1.15 using this method (the details of which can be found in Section 8 of [START_REF] Cioranescu | An introduction to homogenization[END_REF]).

We first recall that we have convergence (1.33), and it follows that we have for a subsequence 8 > < > :

u ε * u 0 weakly in H 1 0 (Ω) u ε ! u 0 strongly in L 2 (Ω) ru ε * ru 0 weakly in (L 2 (Ω)) N .
(1.41)

From (1.29) and (1.31), we deduce that

kA ε ru ε k L 2 (Ω)  ↵ kf k L 2 (Ω) .
This implies that there exists ⇠ 0 2 (L 2 (Ω)) N such that

A ε ru ε *⇠ 0 weakly in (L 2 (Ω)) N . (1.42)
This allows us to pass to the limit of the integral in (1.30) as " tends to 0, which gives

Z Ω ⇠ 0 rvdx= Z Ω fvdx,
for any v 2 H 1 0 (Ω). Hence, to prove Theorem 1.15, it is enough to prove that

⇠ 0 = A 0 ru 0 ,
where A 0 is defined in (1.35). Define

w ε λ (x)="w λ ⇣ x " ⌘ = • x " λ ⇣ x " ⌘ ,
where w λ is the solution of (1.38) and λ is the solution of 8 > < > :

Find λ 2 H 1 per (Y )with M Y ( λ )=0such that Z Y t A(y)r λ rvdy = Z Y t A(y) rvdy, 8v 2 H 1 per (Y ).
(1.43)

Functions taking the form similar to w ε λ are the main feature of this method. By classical results, it can be shown that

r x w ε λ * weakly in L 2 (Ω).
By compact embedding of

H 1 (Ω) in L 2 (Ω),w eh a v e ( w ε λ * • x weakly in H 1 (Ω) w ε λ ! • x strongly in L 2 (Ω). (1.44)
Now, let ⌘ ε λ be the vector function

⌘ ε λ = t A ε rw ε λ =( t Ar y w λ ) ⇣ x " ⌘ .
Note that t Ar y w λ is Y -periodic, and it follows from classical results on functions of the form ⌘ ε λ that

⌘ ε λ * M Y ( t Arw λ )= t A 0 weakly in (L 2 (Ω)) N . (1.45)
Using (1.43), it can be shown that

Z Ω ⌘ ε λ rvdx=0, 8v 2 H 1 0 (Ω). (1.46)
Note also that [START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF]. We then obtain, by subtracting the resulting equations,

A ε ru ε rw ε λ = t A ε rw ε λ ru ε = ⌘ ε λ ru ε . Let ' 2 D(Ω). Choose 'w ε λ as test function in (1.38) and 'u ε in (1.
Z Ω A ε ru ε r'w ε λ dx Z Ω ⌘ ε λ r'u ε dx = Z Ω f 'w ε λ dx, 8' 2 D(Ω).
Note that since the integrals with a product of two weakly convergent sequences cancelt, we don't have any problem passing to the limit of this equation. Using (1.41), (1.42), (1.44), and (1.45), and doing some calculations, we have

Z Ω ⇠ 0 ' dx = Z Ω t A 0 ru 0 ' dx, 8' 2 D(Ω).
By the arbitrary nature of ,w eh a v et h ed e s i r e dr e s u l t ,t h a ti s

⇠ 0 = A 0 ru 0 .

Two-scale convergence method

The two-scale convergence method was introduced by Nguetseng in [START_REF] Nguetseng | Agener alc onver genc er esultforafunctionalr elate dto the theory of homogenization[END_REF], and further developed by Allaire in [START_REF] Allaire | Homogenization and two-scale convergence[END_REF][START_REF] Allaire | Two-scale convergence: A new method in periodic homogenization.,i nN o n l i n e a rp a r t i a ld i fferential equations and their applications[END_REF]. This method introduced a new type of convergence which is called the two-scale convergence. This kind of convergence is concerned with the limit of the integrals of the form

Z Ω v ε (x) ⇣ x, x " ⌘ dx,
where

{v ε } is a sequence in L 2 (Ω) and (x, y) is a smooth Y -periodic func- tion.
Before we present the definition of this convergence, we need to define the appropriate space for the function . The space D(Ω;

C 1 per (Y )) is the space of measurable functions defined from Ω ⇥ R N to R such that for any fixed x 2 Ω, u(x, •) 2 C 1 per (Y ),a n dt h em a px 2 Ω 7 ! u(x, •) 2 C 1 per (Y ) is indefinitely differentiable with a compact support inside Ω.
We now have the following definition of the two-scale convergence:

Definition 1.18. Let {v ε } be a sequence of functions in L 2 (Ω).T h es e q u e n c e {v ε } is said to be two-scale convergent to a function

v 0 = v 0 (x, y) 2 L 2 (Ω⇥Y ) if lim ε!0 Z Ω v ε (x) ⇣ x, x " ⌘ dx = 1 |Y | Z Ω Z Y v 0 (x, y) (x, y) dy dx, (1.47)
for any = (x, y) 2 D(Ω; C 1 per (Y )). We remark that the two-scale convergence justifies the formal asymptotic expansion that was introduced in the multiple-scale method (see Remark 9.6 of [START_REF] Cioranescu | An introduction to homogenization[END_REF] for more details).

The following theorems are two of the main results of this method. These compactness results allow us to pass to the limit when proving the homogenization results.

Theorem 1.19. Let {v ε } be a bounded sequence in L 2 (Ω).T h e n ,w ec a ne xtract a subsequence {v ε 0 } such that {v ε 0 } two-scale convergences to a function

v 0 2 L 2 (Ω ⇥ Y ). Theorem 1.20. Let {v ε } be a sequence of functions in H 1 (Ω) such that v ε * v 0 weakly in H 1 (Ω).
Then {v ε } two-scale converges to v 0 ,a n dt h e r ee x i s tas u b s e q u e n c e" 0 and

v 1 = v 1 (x, y) 2 L 2 (Ω ⇥ Y ) such that rv ε 0 two-scale convergences to r x v 0 + r y v 1 .
We now give the idea of the proof of Theorem 1.15 using the two-scale convergence method.

Note that we still have the convergences (1.41), specifically the first two. Then, from Theorem 1.20, u ε two-scale converges to u 0 .

In addition, we can find

u 1 = u 1 (x, y) 2 L 2 (Ω ⇥ Y ) such that ru ε two-scale converges to r x u 0 + r y u 1 . We cho ose v 0 (x)+"v ⇣ x,
x " ⌘ as test function in (1.30), where v 0 2 D(Ω) and

v 1 = v 1 (x, y) 2 D(Ω; C 1 per (Y )).W et h e nh a v e Z Ω A ε ru ε h rv 0 (x)+"r x v 1 ⇣ x, x " ⌘ + r y v 1 ⇣ x, x " ⌘i dx = Z Ω f ⇣ v 0 (x)+v 1 ⇣ x, x " ⌘⌘ dx.
By using the two-scale convergence of ru ε and other classical results, we can pass to the limit of both of these integrals to obtain

1 |Y | Z Ω Z Y A(y)(ru 0 + r y u 1 (x, y))(rv 0 (x)+r y v 1 (x, y)) dxdy = Z Ω fv 0 dx, (1.48) for any (v 0 ,v 1 ) 2 H 1 0 (Ω) ⇥ L 2 (Ω ⇥ Y ).
This variational formulation can be shown to have a unique solution

(u 0 ,u 1 ) 2 H 1 0 ⇥ L 2 (Ω ⇥ Y ) by the Lax- Milgram Theorem.
Choosing first (0,v 1 ) and then (v 0 , 0) as test functions in (1.48), it can be shown that (1.48) is equivalent to the problem

8 > > > > > > < > > > > > > : div y (A(y)r y u 1 (x, y)) = div y (A(y)ru 0 ) in Ω ⇥ Y div x Z Y A(y)(ru 0 (x)+r y u 1 (x, y)) dy = |Y |f in Ω u 0 =0 on @Ω u 1 (x, •) Y -periodic.
By performing some calculations, it follows that u 1 can be written as

u 1 (x, y)= N X j=1 b j (y) @u 0 @x j , (1.49)
where, for j =1,...,n, b j satisfies 8 <

:

Find b j 2 H 1 per (Y )with M Y ( b j )=0such that Z Y A(y)rb j rvdy = Z Y A(y)e j rvdy, 8v 2 H 1 per (Y ),
with {e j } N j=1 the canonical basis of R N . Replacing u 1 in (1.48) with its rewritten form in (1.49), and performing some calculations, we can deduce that u 0 satisfies (1.39).

The periodic unfolding method

The periodic unfolding method was introduced by D. Cioranescu, A. Damlamian and G. Griso in [START_REF] Cioranescu | Periodic unfolding and homogenization[END_REF] and further developed in [START_REF] Cioranescu | The periodic unfolding method in homogenization[END_REF] for the study of periodic homogenization in classical domains (that is, with no holes or interfacial resistance). It was then later developed for perforated domains in [START_REF] Cioranescu | The periodic unfolding method in domains with holes[END_REF] and for two-component domains in [START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF]. This method relies heavily on an operator which is called the unfolding operator and usually denoted by T ε . This operator, which sends an L p (Ω) function to an L p (Ω ⇥ Y ) function, has a lot of interesting properties to help in passing to the limit. One of which is it transforms an integral over Ω to an integral over Ω ⇥ Y . The resulting integral over Ω ⇥ Y is then relatively easier to manage than the original integral. One can refer to [START_REF] Cioranescu | The periodic unfolding method[END_REF] for a very detailed discussion on the periodic unfolding method.

In the sequel, we will discuss the proof of Theorem 1.15 using the periodic unfolding method on classical domains. As we will be using this method for this thesis, we will give more details in the proof using this method. In addition, since this thesis focuses on two-component domains, we will also say some few words about the Periodic Unfolding Method on two-component domains.

The periodic unfolding method for classical domains

Y = ( `2 R N : `= N X i=1 `ib i , (l 1 ,...,l N ) 2 (0, 1) N )
.

(1.50)

We now denote for almost every z 2 R N ,

[z] Y = N X j=1 l j b j ,l j 2 Z,j =1,...,N, the unique integer combination such that z [z] Y 2 Y .W e s e t {z} Y = z [z] Y 2 Y ,f o ra . e . z 2 R N . Then we can write z = {z} Y +[z] Y , for a.e. z 2 R N ,
that is, we can decompose z as the sum of {z} Y , which is its local position in the cell Y ,a n d[z] Y , which gives the location of the cell where z is. Then, writing this decomposition for z

= x " ,f o ra n yx 2 R N ,w eh a v e x " = n x " o Y + h x " i Y .
Now, we define the following sets:

• G = ( ⇠ 2 R N : ⇠ = N X i=1 k i b i , (k 1 ,...,k N ) 2 Z N ) , • Ξ ε = {⇠ 2 G : "(⇠ + Y ) ⇢ Ω}, • b Ω ε = interior ( [ ξ2Ξε "(⇠ + Y ) ⇢ Ω ) , • Λ ε = Ω \ b Ω ε .
Clearly, from the definition of Ξ ε ,t h es e t b Ω ε is the interior of the largest union of cells "(⇠ + Y ) such that the cells "(⇠ + Y ) are entirely inside Ω.W e can also see that its complement Λ ε is the subset of Ω that contains the parts of the cells "(⇠ + Y ) that intersects the boundary @Ω.

We now present the following definition of the unfolding operator T ε for classical domains. Definition 1.21. Let ' be a measurable function. The unfolding operator T ε is given by

T ε (')(x, y)= 8 < : ' ⇣ " h x " i Y + "y ⌘ for a.e. (x, y) 2 b Ω ε ⇥ Y, 0 for a.e. (x, y) 2 Λ ε ⇥ Y.
(1.51)

It must be noted that the unfolding operator T ε doubles the dimension of the space, and all the oscillations are in the second variable y.

We also need the following definition for the mean value operator.

Definition 1.22. For p 2 [1, 1],t h em e a nv a l u eo p e r a t o rM Y is defined from L p (Ω ⇥ Y ) to L p (Ω),a n di sg i v e nb y M Y (')(x)= 1 |Y | Z Y '(x, y) dy for a.e. x 2 Ω.
(1.52)

We now present the following proposition which states the interesting properties of the unfolding operator T ε . These properties are very helpful in passing to the limit during the homogenization process. The proof of this proposition can be found in [START_REF] Cioranescu | The periodic unfolding method in homogenization[END_REF][START_REF] Cioranescu | The periodic unfolding method[END_REF]. Proposition 1.23 ([31,[START_REF] Cioranescu | The periodic unfolding method[END_REF]). Let p 2 [1, 1).

1. T ε is a linear and continuous operator from L p (Ω) to L p (Ω ⇥ Y ).

For any measurable functions ' and ,w eh a v e

T ε (' )=T ε (')T ε ( ).

For any

' 2 L p (Ω), T ε (') ! ' strongly in L p (Ω ⇥ Y ). 4. For any ' 2 L 1 (Ω),w eh a v e Z b Ωε '(x) dx = Z Ω '(x) dx Z Λε '(x) dx = 1 |Y | Z Ω⇥Y T ε (')(x, y) dx dy. 5. For any ' 2 L p (Ω), kT ε (')k L p (Ω⇥Y )  |Y | 1/p kwk L p (Ω) . For the case p = 1, kT ε (')k L 1 (Ω⇥Y ) kwk L 1 (Ω) . 6. Let {w ε } be a bounded sequence in L p (Ω).I tf o l l o w sf r o mt h ep r e v i o u s item that the sequence {T ε (w ε )} is bounded in L p (Ω ⇥ Y ) (which also holds when p =+1). Suppose further that T ε (w ε ) * b w weakly in L p (Ω ⇥ Y ), then w ε * M Y ( b w) weakly in L p (Ω). If p =+1 and T ε (w ε ) converges weakly-* to b w in L 1 (Ω),t h e n w ε * M Y ( b w) weakly-* in L 1 (Ω).
7. Let {w ε } be a sequence in L p (Ω) such that for some w 2 L p (Ω),

w ε ! w strongly in L p (Ω). Then T ε (w ε ) ! w strongly in L p (Ω ⇥ Y ).
The following proposition gives an interesting result on the action of the unfolding operator to a highly oscillating function. This is very helpful in periodic homogenization as we deal with highly oscillating coefficients. Proposition 1.24. Let f be a function defined on Y ,t h e ne x t e n d e db yYperiodicity to the whole of R N .D e fi n et h es e q u e n c eo ff u n c t i o n s{f ε } by

f ε (x)=f ⇣ x " ⌘ for a.e. x 2 R N . Then T ε (f ε | Ω )(x, y)= ( f (y) for a.e. (x, y) 2 b Ω ε ⇥ Y, 0 for a.e. (x, y) 2 Λ ε ⇥ Y. Let p 2 [1, 1).I ff 2 L p (Y ),t h e n T ε (f ε | Ω ) ! f strongly in L p (Ω ⇥ Y ).
It is worth noting that there is a relationship between the two-scale convergence of a sequence and the weak convergence of the unfolded sequence. This is stated in the following proposition, which is proved in [START_REF] Cioranescu | The periodic unfolding method in homogenization[END_REF].

Proposition 1.25 ([31]). Let {w ε } be a bounded sequence in L p (Ω),w i t h p 2 (1, 1),a n dw 2 L p (Ω).T h ef o l l o w i n gs t a t e m e n t sa r ee q u i v a l e n t :

1. {T ε (w ε )} converges weakly to w in L p (Ω).
2. {w ε } two-scale converges to w. Now, we present a proposition that gives the convergence results for unfolded sequence of functions and unfolded sequence of gradients of these functions. These convergences are essential to the homogenization process when using this method. Proposition 1.26 ([31]). Let p 2 (1, +1).L e t{w ε } be a bounded sequence sequence in W 1,p (Ω).T h e n t h e r e e x i s t a s u b s e q u e n c e ( w h i c h w i l l s t i l l b e denoted by "),

w 2 W 1,p (Ω),a n d b w 2 L p (Ω; W 1,p per (Y )) such that ( T ε (w ε ) * w weakly in L p (Ω; W 1,p (Y )) T ε (rw ε ) * rw + r y b w weakly in L p (Ω ⇥ Y ).
Moreover, if p =+ 1,t h e nw eh a v et h es a m ec o n v e r g e n c e si nt h ew e a k -* topology.

We are now ready to prove Theorem 1.15 using the periodic unfolding method.

Note that we still have estimate (1.31) and convergence (1.33). Then from Proposition 1.23 and Proposition 1.26, we can extract a subsequence (which we will still denote by "), such that for some u 0 2 H 1 0 (Ω),a n d u 1 2 L 2 (Ω; H 1 per (Y )), we have the following convergences: 8 > > < > > :

u ε * u 0 weakly in H 1 0 (Ω), T ε (u ε ) * u 0 weakly in L 2 (Ω; H 1 (Y )), T ε (ru ε ) * ru 0 + r y u 1 weakly in L 2 (Ω ⇥ Y ).
(1.53)

Now, we choose a test function ' 2 H 1 0 (Ω) in the variational formulation (1.30) and we obtain

Z Ω A ε (x)ru ε r' dx = Z Ω f ' dx.
Using the properties of the unfolding operator in Proposition 1.23 and the convergences in (1.53), we have

lim ε!0 Z Ω A ⇣ x " ⌘ ru ε r' dx =lim ε!0 1 |Y | Z Ω⇥Y T ε ⇣ A ⇣ x " ⌘ ru ε r' ⌘ dx dy =lim ε!0 1 |Y | Z Ω⇥Y A(y)T ε (ru ε )T ε (r') dx dy = 1 |Y | Z Ω⇥Y A(y)(ru 0 + r y u 1 )r' dx dy.
Moreover, by Proposition 1.23 and the fact that both functions f and ' are independent of y,

lim ε!0 Z Ω f ' dx =lim ε!0 1 |Y | Z Ω⇥Y T ε (f ') dx dy =lim ε!0 1 |Y | Z Ω⇥Y T ε (f )T ε (') dx dy = 1 |Y | Z Ω⇥Y f ' dx dy = Z Ω f ' dx. It follows that 1 |Y | Z Ω⇥Y A(y)(ru 0 + r y u 1 )r' dx dy = Z Ω f ' dx, (1.54)
for all ' 2 H 1 0 (Ω). We now consider the function v ε (x)="'(x) ( x ε ), where ' 2 D(Ω) and 2 H 1 per (Y ). By the properties of the unfolding operator in Proposition 1.23, we have

T ε (v ε )="T ε (') (y), T ε (rv ε )="T ε (r') + T ε (')r y .
Then, as " tends to 0, we obtain

( T ε (v ε ) ! 0 strongly in L 2 (Ω ⇥ Y ), T ε (rv ε ) ! 'r y strongly in L 2 (Ω ⇥ Y ). (1.55)
Choosing v ε as a test function in (1.30) gives

Z Ω A ε (x)ru ε rv ε dx = Z Ω fv ε dx.
Similar to what was done above, we again use Proposition 1.23 to pass to the limit of the left-hand side of this equation. Together with the convergences in (1.55), we will obtain

lim ε!0 Z Ω A ⇣ x " ⌘ ru ε rv ε dx = 1 |Y | Z Ω⇥Y A(y)(ru 0 + r y u 1 )'r y dx dy and lim ε!0 Z Ω fv ε dx =0. This gives 1 |Y | Z Ω⇥Y A(y)(ru 0 + ru 1 )'r y dx dy =0.
Adding this last equation to (1.54), and using the density of the tensor product

D(Ω) ⌦ H 1 per (Y ) in L 2 (Ω; H 1 per (Y )),w eo b t a i n 1 |Y | Z Ω⇥Y A(y)(ru 0 + r y u 1 )(r' + r y Φ) dx dy = Z Ω f ' dx,
for any ' 2 H 1 0 (Ω) and for all Φ 2 L 2 (Ω; H 1 per (Y )). This is equivalent to the formulation (1.48) obtained in the previous section by the the two-scale convergence method. To obtain the homogenized problem (1.39) that is satisfied by u 0 , we argue as in the proof above. This concludes the proof.

The periodic unfolding method for an imperfect interface We briefly discuss here the method for a two-component domain, which is the domain considered in this thesis. The main difference of a classical domain and a two-component domain is the jump of the solution on the interface of the two materials. This jump may be caused by an imperfect contact in the interface.

The method will be presented in more detail in Chapter 5, which is dedicated to the homogenization. One can also see [START_REF] Donato | Homogenization of diffusion problems with a nonlinear interfacial resistance[END_REF][START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF] for a full discussion of this method.

We study the asymptotic behavior of the following quasilinear elliptic problem in Ω:

8 > > > > > > > > > > > < > > > > > > > > > > > : div ⇣ A ⇣ x " ,u ε 1 ⌘ ru ε 1 ⌘ = f in Ω ε 1 , div ⇣ A ⇣ x " ,u ε 2 ⌘ ru ε 2 ⌘ = f in Ω ε 2 , ⇣ A ⇣ x " ,u ε 1 ⌘ ru ε 1 ⌘ ⌫ ε 1 = ⇣ A ⇣ x " ,u ε 2 ⌘ ru ε 2 ⌘ ⌫ ε 1 on Γ ε , ⇣ A ⇣ x " ,u ε 1 ⌘ ru ε 1 ⌘ ⌫ ε 1 = " 1 h ⇣ x " ⌘ (u ε 1 u ε 2 ) on Γ ε , u ε =0 on @Ω, (1.56)
where A(y,t) is a coercive matrix field with restricted growth assumption which is also Y -periodic.

⌫ ε 1 is the unit outward normal to Ω ε 1 , f 2 L 1 (Ω), h is a Y -periodic L 1 (Γ) function and
In the literature, in the study of homogenization of (1.56) for f 2 L 2 (Ω), the proportionality assumption on Γ ε depends on " γ (instead of " 1 ), where  1. The homogenization then is done in three cases: 2 ( 1, 1], = 1, and 2 ( 1, 1). The major difference between these cases is the corresponding cell problem. We only consider here the case = 1, which has the particularity that also the cell problem presents a jump on the reference interface.

Before we obtain homogenization results, we first study the following periodic elliptic cell problem, related to the homogenized problem of (1.56):

8 > > > > > > > > < > > > > > > > > : div(Ar λ 1 )=G λ 1 in Y 1 , div(Ar λ 2 )=G λ 2 in Y 2 , Ar λ 1 • n 1 = Ar λ 2 • n 2 on Γ, Ar λ 1 • n 1 = h(y)( λ 1 λ 2 ) on Γ, λ 1 Y periodic, M Γ ( λ 1 )=0, (1.57)
where 2 R N and G λ i is defined by

hG λ i ,vi = Z Y i A rvdy, 8v 2 H 1 (Y i ),i =1, 2, (1.58)
which belongs to (H 1 (Y i )) 0 . We are interested in the properties of the solution of (1.57), which were not studied in the literature. This is motivated by the fact that the corresponding homogenized matrix A 0 corresponding to (1.56) is defined with through the solution λ of (1.57). More precisely, the definition of A 0 is as follows:

A

0 (t)=A 0 1 (t)+A 0 2 (t), (1.59)
where

A 0 i (t) = 1 |Y | Z Y i A(y, t)r y w λ i (y, t) dy, i =1, 2, 8 2 R N , with w λ i (y, t)= • y λ i (y, t), and λ =( λ 1 , λ 2
) the solution of (1.57). The properties that we obtained in this thesis are interesting in itself. In particular, we prove that if the matrix field A is Lipschitz-continuous with respect to the second variable, then also A 0 retains this property. This allows us to obtain a uniqueness result for the homogenized problem in Ω (see Theorem 1.31) corresponding to (1.56).

Let us recall, in [START_REF] Cabarrubias | Homogenization of a quasilinear elliptic problem with nonlinear Robin boundary conditions[END_REF], the authors proved a similar result for the homogenization of elliptic problems in a perforated domains, which involves the use of a Meyers-type estimate.

We then proceed by proving first the following theorem, which states that a suitable Meyers-type estimate holds for the periodic solution of (1.57): Theorem 1.27. Let 2 R N and let λ =( λ 1 , λ 2 ) 2 H be the solution of (1.57).T h e nf o ra n y 2 R N ,t h e r ee x i s t sp i > 2, i =1, 2,s u c ht h a t

λ i 2 W 1,p i (Y i ).
Moreover, for i =1, 2,f o re v e r y2  q i  p i ,t h e r ee x i s t sap o s i t i v ec o n s t a n t c i ,d e p e n d e n to n↵, , q i ,a n dY i ,s u c ht h a t

kr λ i k L q i (Y i )  c i | |.
This theorem is proved using the estimates given by Gallouët and Monier in [START_REF] Gallouët | On the regularity of solutions to elliptic equations[END_REF] for elliptic equations with nonhomogeneous Neumann boundary conditions.

We are then able to prove the following main result:

Theorem 1.28. Let A :( y, t) 2 Y ⇥ R 7 ! A(y, t) 2 R N
(s) A 0 (t)|  C r |s t| 8s, t 2 ( r, r).
This theorem is used to show uniqueness results for the homogenized problem in Ω presented below (see Theorem 1.31).

We now proceed with the homogenization of (1.56). We use the periodic unfolding method adapted to the domain with imperfect interface introduced in [START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF]. This method makes use of the periodic unfolding operators T ε i , i =1, 2, which is defined for any measurable function u i defined in

Ω ε i .
O n e of the main interesting properties of this operator is that it transforms the integrals over the varying domain

Ω ε i to integrals over the set Ω ⇥ Y i , i =1, 2, which is independent of ".
As far as we know, a study that combines the framework of renormalized solution and the periodic unfolding method has been first done in [START_REF] Donato | Homogenization of quasilinear elliptic problems with nonlinear Robin conditions and L 1 data[END_REF], and we adopt a similar approach in this thesis.

The homogenization in the framework of renormalized solution is not as straightforward as in the case with L 2 data. This is because the restrictions to Ω ε i of the solution u ε of (9) do not necessarily belong to H 1 (Ω ε i ), i =1, 2. In particular, when f in (9) belongs to L 2 (Ω),o n ec a no b t a i ns o m ea priori estimates on u ε i in H 1 (Ω ε i ) (see [START_REF] Beltran | Homogenization of a quasilinear elliptic problem in a twocomponent domain with an imperfect interface[END_REF]). Then, using the results in [START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF], these estimates lead to the following convergences: 8 < :

T ε i (u ε i ) ! u 1 strongly in L 2 (Ω,H 1 (Y i )),i =1, 2 T ε i (ru ε i ) * ru 1 + r y b u i weakly in L 2 (Ω ⇥ Y i ),i =1, 2, (1.60)
for some

u 1 2 H 1 0 (Ω) and b u i 2 L 2 (Ω,H 1 (Y i )), i =1 , 2.
Then using these convergences, one obtains the homogenized problem in Ω satisfied by u 1 .

However, in our case, u ε i does not belong to H 1 (Ω ε i ), i =1 , 2 and hence, this is not how we will proceed. We instead work on the truncates of

u ε i (i.e., T k (u ε i )), since in the framework of renormalized solutions, T k (u ε i ) 2 H 1 (Ω ε i ) (see Definition 1.7), i =1, 2,forallk>0.
Thus, in place of (1.60), combining the techniques of the framework of renormalized solutions and that of the periodic unfolding method (in particular, the compactness results), we show that there exists u 1 and a sequence {b

u n i } n2N ⇢ L 2 (Ω,H 1 (Y i )), i =1 , 2,s u c h that for any n 2 N, i =1, 2, 8 > > < > > : T n (u 1 ) 2 H 1 0 (Ω) T ε i (T n (u ε i )) ! T n (u 1 ) strongly in L 2 (Ω,H 1 (Y i )),i =1, 2, T ε i (rT n (u ε i )) * rT n (u 1 )+r y b u n i weakly in L 2 (Ω ⇥ Y i ),i =1, 2.
(1.61) Even if there are similarities between the homogenization of ( 9) and that studied in [START_REF] Donato | Homogenization of quasilinear elliptic problems with nonlinear Robin conditions and L 1 data[END_REF], there are additional difficulties that arise here, due to the presence of the jump of the solution on the interface Γ.

The first main difference can be observed in the definition of a renormalized solution of (9) (see Definition 1.7), containing additional conditions, as presented in Chapter 2. In addition, the proof of the following theorem, which is the construction of the oscillating part b u i , i =1, 2,fromthesequence of functions {b u n i } n2N and an identification result, is also more delicate than the corresponding theorem in [START_REF] Donato | Homogenization of quasilinear elliptic problems with nonlinear Robin conditions and L 1 data[END_REF]:

Theorem 1.29. Let b u n 1 2 L 2 (Ω,H 1 per (Y 1 )) and b u n 2 2 L 2 (Ω,H 1 (Y 2 )), n 2 N, be the functions given in (1.61) with M Γ (b u n 1 )=0.
T h e nt h e r ee x i s t sau n i q u e measurable function

b u i : Ω ⇥ Y i ! R,i =1, 2, such that for every R 2 C 1 (R) with compact support, that is, supp R ⇢ [ m, m],f o rs o m em 2 N,w eh a v e R(u 1 )b u n i = R(u 1 )b u i a.e. in Ω ⇥ Y i ,
for all n m,w h e r eu 1 is the function given above. Moreover, we have

b u i (x, •) 2 H 1 (Y i ),i =1, 2, with M Γ (b u 1 )=0, for a.e. x 2 Ω.
The delicate part of the proof comes from the fact that the average value on Γ of b u n 2 , n 2 N,i sn o tn e c e s s a r i l yz e r o . Using this theorem and the convergences in (1.61), we are able to show the following theorem, describing the unfolded homogenized problem satisfied by the triple

(u 1 , b u 1 , b u 2 ): Theorem 1.30 (The unfolded homogenized problem). Let u 1 , b u 1 ,a n db u 2 be functions introduced in (1.61).L e t 1 , 2 be functions in C 1 (R) (or equiva- lently, 1 , 2 2 W 1,1 (R))w i t hc o m p a c ts u p p o r t s . Then the triple (u 1 , b u 1 , b u 2 ) satisfies 8 > > > > > > > < > > > > > > > : 2 X i=1 1 |Y | Z Ω⇥Y i A(y, u 1 )(ru 1 + r y b u i )(r( 1 (u 1 )')+ 2 (u 1 )r y Φ i ) dx dy + 1 |Y | Z Ω⇥Γ h(y) 2 (u 1 )(b u 1 b u 2 )(Φ 1 Φ 2 ) dx d y = Z Ω f (x) 1 (u 1 )'(x) dx 8' 2 H 1 0 (Ω) \ L 1 (Ω), Φ i 2 L 2 (Ω,H 1 per (Y i )),i =1, 2.
In addition, for k>0,t h ef o ll o w i n gl i m i t sh o l d :

lim k!1 1 k Z {|u 1 |<k}⇥Y i A(y, u 1 )(rT k (u 1 )+r y b u i )(rT k (u 1 )+r y b u i ) dx dy =0, for i =1, 2,a n d lim k!1 1 k Z {|u 1 |<k}⇥Γ (b u 1 b u 2 ) 2 dx d y =0.
From this unfolded problem, we obtain the homogenized problem in Ω, which completes the chapter.

Theorem 1.31 (The homogenized problem in Ω). Let u 1 be a cluster point of the sequence

{T ε i (u ε i )}, i =1, 2.T h e nu 1 is a renormalized solution of ( div(A 0 (u 1 )ru 1 )=f in Ω u 1 =0 on @Ω, (1.62)
that is,

T k (u 1 ) 2 H 1 0 (Ω), for any k>0, (1.63)
lim k!1 1 k Z {|u 1 |<k} A 0 (u 1 )ru 1 ru 1 dx =0, (1.64)
and for every

2 C 1 (R) (or equivalently, 2 W 1,1 (R))w i t hc o m p a c t support, Z Ω (u 1 )A 0 (u 1 )ru 1 r' dx + Z Ω 0 (u 1 )A 0 (u 1 )ru 1 ru 1 ' dx = Z Ω f (u 1 )' dx, (1.65)
for every ' 2 H 1 0 (Ω)\L 1 (Ω),w h e r eA 0 is the homogenized matrix as defined above.

If in addition, (A4) holds, then u 1 is the unique renormalized solution of (9) and all of the sequences in (1.61) converge (not just a subsequence).

The proof of the last assertion of this theorem makes use of Theorem 10 of the previous chapter. It is worth noting that proving of the decay of the "truncated" energy (condition (1.64)) is not standard and also delicate.

Finally, let us point out that, as observed along this thesis, managing the boundary integral that arise from the jump of the solution on the interface is not an easy task. This difficulty is not limited to the homogenization result. This can also be seen in the existence and uniqueness of the renormalized solution of (1) and also in studying the properties of the solution of the cell problem [START_REF] Bénilan | An L 1 theory of existence and uniqueness of nonlinear elliptic equations[END_REF].

Part I Existence and uniqueness results

Chapter 2

Quasilinear elliptic problems in a two-component domain with L 1 data

Introduction

In the present chapter, we study the existence of a solution u := (u 1 ,u 2 ) of the following class of quasi-linear equations:

8 > > > > > > < > > > > > > : div(B(x, u 1 )ru 1 )=f in Ω 1 , div(B(x, u 2 )ru 2 )=f in Ω 2 u 1 =0 on @Ω, (B(x, u 1 )ru 1 )⌫ 1 =(B(x, u 2 )ru 2 )⌫ 1 on Γ, (B(x, u 1 )ru 1 )⌫ 1 = h(x)(u 1 u 2 ) on Γ. (2.1)
Here, Ω is our two-component domain and @Ω is its boundary. The open sets Ω 1 and Ω 2 are the two disjoint components of Ω, Γ is the interface between them (see Figure 1), and the vector ⌫ i is the unit outward normal to Ω i . The matrix field B(x, r) is coercive and is not restricted by any growth condition with respect to r (B(x, r) is bounded on any compact set of R), and the data f is an L 1 -function. On the boundary @Ω,w eh a v eaD i r i c h l e tb o u n d a r y condition, while on the interface Γ,w eh a v eac o n t i n u o u sfl u xa n dt h ej u m p of the solution is proportional to the flux. We refer to [START_REF] Carslaw | Conduction of heat in solids[END_REF] for a justification of the model in the case of the conduction of heats in solids.

The existence and uniqueness of solution of (2.1) when f 2 L 2 (Ω) was studied in [START_REF] Beltran | Homogenization of a quasilinear elliptic problem in a twocomponent domain with an imperfect interface[END_REF][START_REF] Donato | Homogenization of diffusion problems with a nonlinear interfacial resistance[END_REF][START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF]. In [START_REF] Donato | Homogenization of diffusion problems with a nonlinear interfacial resistance[END_REF][START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF] the equations are linear, that is, the matrix field B does not depend on the solution u, while in [START_REF] Beltran | Homogenization of a quasilinear elliptic problem in a twocomponent domain with an imperfect interface[END_REF], the equations are quasilinear, which is also the case in this study. The above mentioned papers are all motivated by homogenization, which is also our main goal (see [START_REF] Donato | Homogenization results for quasilinear elliptic problems in a two-component domain with L 1 data[END_REF]).

Since we consider in this chapter an L 1 data, we need an appropriate notion of solution. Let us recall that, for the elliptic equation div(A(x, u)ru)=f with Dirichlet boundary condition, if the matrix A is bounded, a solution in the sense of distribution exists (see [START_REF] Boccardo | Nonlinear elliptic and parabolic equations involving measure data[END_REF]) but it is not unique in general (see the counterexamples in [START_REF] Prignet | Remarks on existence and uniqueness of solutions of el liptic problems with right hand side measures[END_REF][START_REF] Serrin | Pathological solutions of elliptic equations,A n n .S c u o l a Norm[END_REF]). If the matrix field is not bounded, then we cannot expect to have a solution in the sense of distribution since there is no reason to have A(x, u) 2 L 1 loc .I nt h i sc h a p t e r ,w eu s et h en o t i o no f renormalized solution, which was first discussed in [START_REF] Diperna | On the Cauchy problem for Boltzmann equations: Global existence and weak stability[END_REF] by R.J. DiPerna and P.L. Lions for first order equations. This notion was then further developed by F. Murat in [START_REF] Murat | Soluciones renormalizadas de EDP elipticas no lineales[END_REF], by P.L. Lions and F. Murat in [START_REF] Lions | Sur les solutions renormalisées d'équations elliptiques[END_REF] for elliptic equations with Dirichlet boundary conditions and L 1 data, and by G. Dal Maso et al. in [START_REF] Maso | Renormalized solutions of elliptic equations with general measure data[END_REF] for elliptic equations with general measure data. There is a wide range of literature for elliptic equations with Dirichlet boundary condition and L 1 data, among them are [START_REF] Bénilan | An L 1 theory of existence and uniqueness of nonlinear elliptic equations[END_REF][START_REF] Boccardo | Nonlinear elliptic and parabolic equations involving measure data[END_REF][START_REF] Maso | Renormalized solutions of elliptic equations with general measure data[END_REF][START_REF] Dall | Approximated solutions of equations with L 1 data. Application to the h-convergence of quasi-linear parabolic equations[END_REF][START_REF] Lions | Sur les solutions renormalisées d'équations elliptiques[END_REF][START_REF] Murat | Soluciones renormalizadas de EDP elipticas no lineales[END_REF]. Considering elliptic equations with Neumann or Robin boundary conditions and L 1 data, which are connected to our problem, gives, in general, additional difficulties due the lack of Poincaré inequality or the low regularity of the solution (definition of the trace for e.g.). In the case of one-component domain, L 1 data and Neumann or Robin boundary conditions, let us mention [START_REF] Andreu | Quasi-linear elliptic and parabolic equations in L 1 with nonlinear boundary conditions[END_REF][START_REF] Andreu | Existence and uniqueness for a degenerate parabolic equation with L 1data[END_REF][START_REF] Ouaro | Entropy solution to an elliptic problem with nonlinear boundary conditions[END_REF] using the framework of entropy solutions, [START_REF] Droniou | Solving convection-diffusion equations with mixed, Neumann and Fourier boundary conditions and measures as data, by a duality method[END_REF] using a duality method and [START_REF] Betta | Neumann problems for nonlinear elliptic equations with L 1 data[END_REF][START_REF] Guibé | Renormalized solutions of el liptic equations with Robin boundary conditions[END_REF] using the framework of renormalized solutions.

The main originality here is the jump of the solution which produces in the formulation a term in the interface Γ.R e c a l l i n gt h a tt h er e g u l a r i t yo f the renormalized solution is given through the truncate, the first difficulty is to give a sense on the interface for functions (u 1 ,u 2 ) whose truncates belong to H 1 only in each component. Following the ideas of [START_REF] Andreu | Quasi-linear elliptic and parabolic equations in L 1 with nonlinear boundary conditions[END_REF][START_REF] Guibé | Renormalized solutions of el liptic equations with Robin boundary conditions[END_REF] (but in the case of one-component domain), we define an appropriate notion of trace (see Proposition 2.2). The second difficulty is the regularity of 1 (u 1 )

2 (u 2 ) (where 1 is the trace function for H 1 (Ω 1 ) functions and 2 is the trace function for H 1 (Ω 2 ) functions), since we have to deal, in the renormalized formulation, with terms on the boundary like ( 1 (u 1 ) 2 (u 2 ))S( (u 1 )), where S is a C 1 function with compact support. To have ( 1 (u 1 )

2 (u 2 ))S( (u 1 )) belonging to L 1 (Γ) is then equivalent to have 2 (u 2 )S( (u 1 )) 2 L 1 (Γ), which is unusual and is in some sense a coupled regularity on the boundary. It is worth noting that it is not a direct consequence of T k (u 1 ) 2 H 1 (Ω 1 ) and T k (u 2 ) 2 H 1 (Ω 2 ) (T k is the usual truncation function at height ±k,s e e (2.4)). Using the structure of the equation, we impose an extra regularity (see (2.12b)), namely

( 1 (u 1 ) 2 (u 2 ))[T k ( (u 1 )) T k ( (u 2 ))] 2 L 1 (Γ), for any k>0,
which allows one to prove that 2 (u 2 )S( (u 1 )) 2 L 1 (Γ) and then ( 1 (u 1 )

2 (u 2 ))S( (u 1 )) 2 L 1 (Γ) (see Remark 2.5
). We also impose a decay of the energy of the trace (see (2.13b)) in addition to the usual decay of the energy which are crucial to obtain stability results (see Remark 2.10). Consequently, we are able to give a definition of renormalized solution for problem (2.1) for which we prove the existence (see Theorem 2.8).

This chapter is organized as follows. The next section discusses the assumptions on our problem and some definitions, including the definition of ar e n o r m a l i z e ds o l u t i o no f( 2 . 1 )( s e eD e fi n i t i o n2 . 4 ) . S e c t i o n2 . 3i sd e v o t e d to the proof of the existence of a renormalized solution for (2.1). We also remark here that by using the Boccardo-Gallouët estimates, we can actually replace conditions (2.12b) and (2.13b) of Definition 2.4 by another regularity condition on the interface. However, we prefer not to use these estimates because we have the homogenization process in mind (see Remark 2.9).

Assumptions and Definitions

In this section, we present the assumptions and definitions necessary for our problem. We begin by introducing the two-component domain Ω. The domain Ω is a connected bounded open set in R N with boundary @Ω.W e can write Ω as the disjoint union

Ω = Ω 1 [ Ω 2 [ Γ, where Ω 2 is an open set such that Ω 2 ⇢ Ω with a Lipschitz boundary Γ,a n dΩ 1 = Ω \ Ω 2 .
W ed e n o t e by ⌫ i the unit outward normal to Ω i .

If we have a function u defined on Ω \ Γ,t h e nw ed e n o t eu i = u Ω i the restriction of u in Ω i . Furthermore, we have the following assumptions:

(A1) The data f belongs to L 1 (Ω). and it has the following properties:

(A2) The function h satisfies

(A3.1) B(x, r)⇠ • ⇠ ↵|⇠| 2 ,f o rs o m e↵ > 0,f o ra . e . x 2 Ω, 8r 2 R, 8⇠ 2 R N ; (A3.2) for any k>0, B(x, r) 2 L 1 (Ω ⇥ ( k, k)) N ⇥N .
The space for this class of equations is not a usual L p space or a Sobolev space due to the jump on the interface. We need the normed space V defined as follows. Let V 1 be the space defined by

V 1 = {v 2 H 1 (Ω 1 ):v =0on @Ω} with kvk V 1 := krvk L 2 (Ω 1 ) . Define V := {v ⌘ (v 1 ,v 2 ):v 1 2 V 1 and v 2 2 H 1 (Ω 2 )}, equipped with the norm kvk 2 V := krv 1 k 2 L 2 (Ω 1 ) + krv 2 k 2 L 2 (Ω 2 ) + kv 1 v 2 k 2 L 2 (Γ) . (2.3)
Identifying rv := g rv 1 + g rv 2 ,w eh a v et h a tkvk 2 V = krvk 2 L 2 (Ω\Γ) + kv 1 v 2 k 2 L 2 (Γ) . Proposition 2.1 ([68]). The norm given in (2.3) is equivalent to the norm of V 1 ⇥ H 1 (Ω 2 )
,t h a ti s ,t h e r ee x i s tt w op o s i t i v ec o n s t a n t sc 1 ,c 2 such that

c 1 kvk V kvk V 1 ⇥H 1 (Ω 2 )  c 2 kvk V , 8v 2 V.
We now define the function T k , which is known as the truncation function at height ±k. The function T k : R ! R is given by

T k (t)= 8 > < > : k, if t  k, t, if k  t  k, k, if t k.
(2.4) This function will be crucial in the definition of a renormalized solution of (2.1).

In the case of L 1 data, we cannot expect to have the solution u belonging to V .I ng e n e r a l ,i nt h ef r a m e w o r ko fr e n o r m a l i z e ds o l u t i o n ,t h er e g u l a r i t yo f the solution is given through the regularity of any truncate. So it is necessary in our case to define the gradient and the trace of the solution u.F o rt h e gradient, we follow the definition given in [START_REF] Bénilan | An L 1 theory of existence and uniqueness of nonlinear elliptic equations[END_REF]. For the trace, we have to precise the trace of u 1 on Γ and the one of u 2 on Γ.W i t hr e s p e c tt o[ 3 ,5 9 ] , we have the additional difficulty for u 2 since we do not have the Poincaré inequality.

Proposition 2.2. Let u =( u 1 ,u 2 ):Ω \ Γ ! R be a measurable function such that T k (u) 2 V for every k>0.
1. For i =1, 2,t h e r ee x i s t sau n i q u em e a s u r a b l ef u n c t i o nG i :

Ω i ! R N
such that for all k>0,

rT k (u i )=G i {|u i |<k} a.e. in Ω i , (2.5)
where {|u i |<k} denotes the characteristic function of

{x 2 Ω i : |u i (x)| <k}.
We define G i as the gradient of u i and write G i = ru i .

If

sup k 1 1 k kT k (u)k 2 V < 1, (2.6)
then there exists a unique measurable function

w i : Γ ! R, for i =1, 2,
such that for all k>0,

i (T k (u i )) = T k (w i ) a.e. in Γ, (2.7
)

where i : H 1 (Ω i ) ! L 2 (Γ)
is the trace operator. We define the function w i as the trace of u i on Γ and set

i (u i )=w i ,i =1, 2.
Proof.

1. This is proved in [START_REF] Bénilan | An L 1 theory of existence and uniqueness of nonlinear elliptic equations[END_REF] (see Lemma 2.1).

2. The case i =1,o rm o r eg e n e r a l l yt h et r u n c a t e sh a v eaz e r ot r a c eo na part of the boundary (which allows one to use Poincaré-kind inequality) is presented in [START_REF] Guibé | Renormalized solutions of el liptic equations with Robin boundary conditions[END_REF]. We just have to prove the result for i =2.

The uniqueness is in the almost everywhere sense. Note that if we find two functions that satisfies (2.7), then the uniqueness of w 2 is assured by the monotonicity of T k and the fact that w 2 is finite a.e. in Γ.

By Proposition 2.1, we know that

kT k (u 2 )k H 1 (Ω 2 )  c 1 kT k (u)k V ,
for some positive constant c 1 ,i n d e p e n d e n to fk. It follows from (2.6)

that kT k (u 2 )k 2 H 1 (Ω 2 )  Mk, (2.8)
with M 2 R + independent of k.D u et ot h er e g u l a r i t yo fΓ, 2 (T n (u 2 )) is well-defined and

k 2 meas Γ {| 2 (T k (u 2 ))| k} = Z Γ\{|T k (u 2 )| k} ( 2 (T k (u 2 ))) 2 d k 2 (T k (u 2 ))k 2 L 2 (Γ) .
Hence, by Trace Theorem and (2.8), we have

k 2 meas Γ {| 2 (T k (u 2 ))| k} k 2 (T k (u 2 ))k 2 L 2 (Γ) kT k (u 2 )k 2 L 2 (Ω 2 ) + krT k (u 2 )k 2 L 2 (Ω 2 )  Mk.
As a result,

meas Γ {| 2 (T k (u 2 ))| k} ! 0 as k ! 0. (2.9) Define Γ n = {x 2 Γ : | 2 (T n (u 2 ))| <n} for n 2 N.
From (2.9), it follows that

Γ = [ n 1 Γ n [ A, (2.10)
where A is a subset of Γ with zero measure.

Note that for k<n ,w eh a v eT k (T n (u 2 )) = T k (u 2 ).F i xk>0. Then for every n 2 N such that n>k, we have the following equality

T k ( 2 (T n (u 2 ))) = 2 (T k (T n (u 2 ))) = 2 (T k (u 2 )) a.e. on Γ,
and then

2 (T k (u 2 )) = 2 (T n (u 2 )) a.e. on Γ k . (2.11)
Since for every n 1  n,w eh a v eΓ n 1 ✓ Γ n ,i nv i e wo f( 2 . 1 0 )a n d( 2 . 1 1 ) , we can define w 2 in the following way:

w 2 = 2 (T n (u 2 )) on Γ n ,
and noting that Γ = S n 1

Γ n (up to measure zero set), we have for any

k>0 2 (T k (u 2 )) = T k (w 2 )
a.e. on Γ. This concludes the proof.

Remark 2.3. In the following, we give an example of a measurable function u where T k (u) 2 V but u 2 is not defined on a part of the interface. We consider [START_REF] Allaire | Two-scale convergence: A new method in periodic homogenization.,i nN o n l i n e a rp a r t i a ld i fferential equations and their applications[END_REF] and Ω 2 =( 0 , 1) (so Γ = {0, 1}), and u =(u 1 ,u 2 ) is defined as

Ω =( 1, 2) with Ω 1 =( 1, 0) [ (1,
u(x)= ( u 1 (x)=(x +1)(x 2) if x 2 Ω 1 u 2 (x)=x 2 if x 2 Ω 2 .
We have for some positive constants

C 1 ,C 2 , krT k (u 1 )k 2 L 2 (Ω 1 ) = Z {|u 1 |<k} (2x 1) 2 dx  Z Ω 1 (2x 1) 2 dx  C 1 ,
and

krT k (u 2 )k 2 L 2 (Ω 2 ) = Z 1 k 1/2 ( 2x 3 ) 2 dx =4  x 7 7 1 x=k 1/2 = 4 7 (k 7/2 1).
Thus, we can see that

k 7/2 C kT k (u)k 2 V  Ck 7/2 ,
for some C>0 but clearly u 2 does not have a trace on {0} ⇢ Γ.

We are now in a p osition to give the definition of renormalized solution.

Definition 2.4. Let u =( u 1 ,u 2 ):Ω \ Γ ! R be a measurable function.

Then u is a renormalized solution of (2.1) if

T k (u) 2 V, 8k>0; (2.12a) (u 1 u 2 )(T k (u 1 ) T k (u 2 )) 2 L 1 (Γ), 8k>0; (2.12b) lim n!1 1 n Z {|u|<n} B(x, u)ru • rudx =0; (2.13a) lim n!1 1 n Z Γ (u 1 u 2 )(T n (u 1 ) T n (u 2 )) d =0; (2.13b)
and for any S 1 ,S 2 2 C 1 (R) (or equivalently for any

S 1 ,S 2 2 W 1,1 (R))w i t h compact support, u satisfies Z Ω 1 S 1 (u 1 )B(x, u 1 )ru 1 • rv 1 dx + Z Ω 1 S 0 1 (u 1 )B(x, u 1 )ru 1 • ru 1 v 1 dx + Z Ω 2 S 2 (u 2 )B(x, u 2 )ru 2 • rv 2 dx + Z Ω 2 S 0 2 (u 2 )B(x, u 2 )ru 2 • ru 2 v 2 dx + Z Γ h(x)(u 1 u 2 )(v 1 S 1 (u 1 ) v 2 S 2 (u 2 )) d = Z Ω 1 fv 1 S 1 (u 1 ) dx + Z Ω 2 fv 2 S 2 (u 2 ) dx, (2.14
)

for all v 2 V \ (L 1 (Ω 1 ) ⇥ L 1 (Ω 2 )).
Remark 2.5. Conditions (2.12a) (the regularity of the truncate) and (2.13a) (the decay of the "truncated energy") are standard in the framework of renormalized solutions. As mentioned in the introduction, the main originality here is the presence of the traces in conditions (2.12b) and (2.13b).

In view of Proposition 2.2, (u 1 ) and (u 2 ) are well-defined. Condition (2.12b) is an extra regularity of (u

1 u 2 )(T k (u 1 ) T k (u 2 )). Indeed, (u 1 u 2 )(T k (u 1 ) T k (u 2 )
) cannot be written as

(u 1 u 2 )(T k (u 1 ) T k (u 2 )) {|u 1 |<n} {|u 2 |<n} , for any n 2 N,s ot h a th a v i n g(u 1 u 2 )(T k (u 1 ) T k (u 2 )) belonging to L 1 (Γ)
is not a consequence of (2.12a). Conditions (2.12a) and (2.12b) allow one to give a sense of all the terms in (2.14).L e tS i 2 C 1 (R), i =1, 2,w i t hc o m p a c ts u p p o r t .

Then for all

v 2 V \ (L 1 (Ω 1 ) ⇥ L 1 (Ω 2 )),w eh a v ei fsupp S i ⇢ [ k, k] (i =1, 2), then for i =1, 2, S i (u i )B(x, u i )ru i • rv i = S i (u i )B(x, T k (u i ))rT k (u i ) • rv i 2 L 1 (Ω i ), S 0 i (u i )B(x, u i )ru i • ru i v i = S 0 i (u i )B(x, T k (u i ))rT k (u i ) • rT k (u i ) v i 2 L 1 (Ω i ), fv i S i (u i ) 2 L 1 (Ω i ).
For the boundary term, for any n 2 N,l e tu sd e fi n e✓ n : R ! R by

✓ n (s)= 8 > > > > > > > < > > > > > > > : 0, if s  2n, s n +2, if 2n  s  n, 1, if n  s  n, s n +2, if n  s  2n, 0, if s 2n.
(2.15)

Then since S 1 has a compact support, for some large enough n,w eh a v e

h(u 1 u 2 )v 1 S 1 (u 1 )=hv 1 (u 1 u 2 )(S 1 (u 1 ) S 1 (u 2 ))✓ n (u 1 ) + hv 1 (u 1 u 2 )S 1 (u 2 )✓ n (u 1 ).
Since both S 1 and ✓ n have compact support, we have that hv 1 (u

1 u 2 )S 1 (u 2 )✓ n (u 1 )
is bounded and is therefore in L 1 (Γ).M o r e o v e r ,s i n c e

S 1 (u 1 ) S 1 (u 2 )=S 1 (T 2n (u 1 )) S 1 (T 2n (u 2 ))
and S 1 is Lipschitz, we have

|hv 1 (u 1 u 2 )(S 1 (u 1 ) S 1 (u 2 ))✓ n (u 1 )| khv 1 k L 1 (Γ) kS 0 1 k L 1 (R) ⇥ |u 1 u 2 ||T 2n (u 1 ) T 2n (u 2 )|, a.e. in Γ.T h u s ,i nv i e wo f(2.12b), h(u 1 u 2 )v 1 S 1 (u 1 ) 2 L 1 (Γ).S i m i l a r l y , h(u 1 u 2 )v 2 S 2 (u 2 ) 2 L 1 (Γ).
It is worth noting that condition (2.12b) is equivalent to have

u 2 {|u 1 |<k} 2 L 1 (Γ) and u 1 {|u 2 |<k} 2 L 1 (Γ), (2.16)
for any k>0.I n d e e d ,

u 2 {|u 1 |<k} =(u 2 u 1 ) {|u 1 |<k} (✓ n (u 1 ) ✓ n (u 2 )) + u 2 ✓ n (u 2 ) {|u 1 |<k} + u 1 ✓ n (u 1 ) {|u 1 |<k} u 1 ✓ n (u 2 ) {|u 1 |<k} ,
and by condition (2.12b),t h efi r s tt e r mo nt h er i g h t -h a n ds i d eb e l o n g st o L 1 (Γ) while the next 3 terms are bounded and thus also belong to L 1 (Γ).

Finally, let us comment that conditions (2.13a) and (2.13b) are crucial to recover that formally, for any k>0, T k (u) is an admissible function in (2.1),t h a ti s , Z

Ω 1 B(x, u 1 )ru 1 rT k (u 1 ) dx + Z Ω 2 B(x, u 2 )ru 2 rT k (u 2 ) dx + Z Γ h(x)(u 1 u 2 )(T k (u 1 ) T k (u 2 )) d = Z Ω fT k (u 1 ) dx.
To prove this, fix k>0.F o rn 2 N,u s i n gS

1 = S 2 = ✓ n and v = T k (u) as a test function in (2.14),w eh a v e Z Ω 1 ✓ n (u 1 )B(x, u 1 )ru 1 • rT k (u 1 ) dx + Z Ω 1 ✓ 0 n (u 1 )B(x, u 1 )ru 1 • ru 1 T k (u 1 ) dx + Z Ω 2 ✓ n (u 2 )B(x, u 2 )ru 2 • rT k (u 2 ) dx + Z Ω 2 ✓ 0 n (u 2 )B(x, u 2 )ru 2 • ru 2 T k (u 2 ) dx + Z Γ h(x)(u 1 u 2 )(✓ n (u 1 )T k (u 1 ) ✓ n (u 2 )T k (u 2 )) d = Z Ω fT k (u)✓ n (u) dx. (2.17)
Condition (2.13a) allows one to pass to the limit of the second and fourth integral in (2.17) while condition (2.13b) is useful for passing to the limit of the integral on the boundary in (2.17).

Remark 2.6. As observed in the previous remark, the main purpose of introducing condition (2.12b) is to allow us to make sense of the integral on the interface. We can avoid introducing this extra regularity condition on Γ by using the Boccardo-Gallouët estimates presented in [START_REF] Boccardo | Nonlinear elliptic and parabolic equations involving measure data[END_REF]. However, these estimates are heavily dependent on the Sobolev constants. With the final aim of doing the homogenization process, we try as much as possible to refrain from using these estimates (see Remark 2.9).

Remark 2.7. In the variational case (i.e. B is a bounded matrix field and f 2 L q with q (N +2)/(2N )), if B(x, r) is global Lipschitz continuous with respect to r or if its modulus of continuity is strongly controlled, the (variational) solution is unique (see [START_REF] Donato | Uniqueness result for a class of singular elliptic problems in two-component domains[END_REF]). In the L 1 case, the uniqueness question is addressed in [START_REF] Fulgencio | Uniqueness results for quasilinear elliptic problems in a two-component domain with L 1 data[END_REF]: under assumptions (A1)-(A3) and a local Lipschitz condition on B(x, r) with respect to r,w ep r o v et h a tt h er e n o r m a l i z e d solution is unique.

Existence Results

In this section, we present the proof for the existence of a renormalized solution of (2.1).

Theorem 2.8. Suppose assumptions (A1)-(A3) hold. Then there exists a renormalized solution to (2.1) in the sense of Definition 2.4.

Proof. The proof is divided into 4 steps. In Step 1, we consider an approximate problem (see (2.18) below) in which B is approximated by a bounded function and f ε is an L 2 data. Using Schauder's fixed point theorem, the existence of at least a variational solution of (2.18) can be shown. Step 2 is devoted to prove some a priori estimates and then extracting a convergent subsequence. In Step 3, we prove that conditions (2.12a), (2.12b), (2.13a) and (2.13b) are satisfied by the limit. Finally, in Step 4, we pass to the limit and we show that the constructed function is a renormalized solution.

From this p oint until the end of the pro of, we let i 2 {1, 2}.

Step 1: Introducing the approximate problem and showing the existence of solution of the approximate problem Let " > 0.S u p p o s e{f ε } ⇢ L 2 (Ω) such that

f ε ! f strongly in L 1 (Ω),
as " ! 0.D e fi n eB ε (x, t)=B(x, T 1/ε (t)). We now consider the following approximate problem

8 > > > > > > < > > > > > > : div(B ε (x, u ε 1 )ru ε 1 )=f ε in Ω 1 , div(B ε (x, u ε 2 )ru ε 2 )=f ε in Ω 2 , u ε 1 =0 on @Ω, (B ε (x, u ε 1 )ru ε 1 )⌫ 1 =(B ε (x, u ε 2 )ru ε 2 )⌫ 1 on Γ, (B ε (x, u ε 1 )ru ε 1 )⌫ 1 = h(x)(u ε 1 u ε 2 ) on Γ.
(2.18)

The variational formulation of problem (2.18) is the following

8 > > > > < > > > > : Find u ε 2 V such that 8' 2 V Z Ω 1 B ε (x, u ε 1 )ru ε 1 • r' 1 dx + Z Ω 2 B ε (x, u ε 2 )ru ε 2 • r' 2 dx + Z Γ h(x)(u ε 1 u ε 2 )(' 1 ' 2 ) d = Z Ω f ε ' dx. (2.19)
Using Proposition 2.1 and Schauder's Fixed Point Theorem, the proof of the existence of solution for (2. [START_REF] Boccardo | Unicité de la solution de certaines équations elliptiques non linéaires. (Uniqueness of the solution of some nonlinear elliptic equations)[END_REF]) is quite standard (see e.g. [START_REF] Beltran | Homogenization of a quasilinear elliptic problem in a twocomponent domain with an imperfect interface[END_REF]).

Step 2: Extracting subsequences and examining convergences Let

u ε =( u ε 1 ,u ε
2 ) be a solution to the approximate problem (2.18). By Stampacchia's theorem, for k>0, T k (u ε ) 2 V since u ε 2 V .

Using T k (u ε ) as a test function in the variational formulation (2.19), we have Z

Ω 1 B ε (x, u ε 1 )ru ε 1 rT k (u ε 1 ) dx + Z Ω 2 B ε (x, u ε 2 )ru ε 2 rT k (u ε 2 ) dx + Z Γ h(x)(u ε 1 u ε 2 )(T k (u ε 1 ) T k (u ε 2 )) d = Z Ω f ε T k (u ε ) dx. (2.20)
By the definition of T k ,t h ec oe r c i v i t yo fB,a n dt h ea s s u m p t i o no nh,w e have Z

Ω 1 B ε (x, u ε 1 )ru ε 1 rT k (u ε 1 ) dx + Z Ω 2 B ε (x, u ε 2 )ru ε 2 rT k (u ε 2 ) dx + Z Γ h(x)(u ε 1 u ε 2 )(T k (u ε 1 ) T k (u ε 2 )) d ↵krT k (u ε 1 )k 2 L 2 (Ω 1 ) + ↵krT k (u ε 2 )k 2 L 2 (Ω 2 ) + h 0 kT k (u ε 1 ) T k (u ε 2 )k 2 L 2 (Γ) C 1 kT k (u ε )k 2 V ,
for some positive constant C 1 .O nt h eo t h e rh a n d ,b yH o l d ë ri n e q u a l i t y ,

Z Ω f ε T k (u ε ) dx = Z Ω 1 f ε T k (u ε 1 ) dx + Z Ω 2 f ε T k (u ε 2 ) dx kf ε k L 1 (Ω) k  Mk,
for some positive constant M , which is independent of ". Thus,

kT k (u ε )k 2 V  Mk C 1 , (2.21)
that is, the sequence {T k (u ε )} is bounded in V for every k>0.

By the Rellich theorem, the inclusions V ,! L 2 (Ω 1 )⇥L 2 (Ω 2 ) and H 1/2 (Γ) ,! L 2 (Γ) are compact. Consequently, since {T k (u ε )} is bounded in V for every k>0 (countable), by a diagonal process, we can extract a subsequence of {T k (u ε )} such that for any k>0 (k being a rational number), there is a v k 2 V such that 8 > < > :

T k (u ε 0 i ) ! v k,i strongly in L 2 (Ω i ), a.e. in Ω i , T k (u ε 0 i ) * v k,i weakly in V, i (T k (u ε 0 i )) ! i (v k,i
) strongly in L 2 (Γ), a.e. on Γ.

(2.22)

Now, we show that {u ε 0 i } and { i (u ε 0 i )} are Cauchy sequences in measure. For u ε 0 i ,w ef o l l o wt h ea r g u m e n t sd e v e l o p e di n[ 1 0 ] . F o r i (u ε 0 i ),w eh a v e additional difficulties which are overcome by using Proposition 2.1.

Note that we have

kT k (u ε 0 i )k 2 L 2 (Ω i ) = Z {|u ε 0 i | k} |T k (u ε 0 i )| 2 dx + Z {|u ε 0 i |<k} |T k (u ε 0 i )| 2 dx = Z {|u ε 0 i | k} k 2 dx + Z {|u ε 0 i |<k} |u ε 0 i | 2 dx.
It follows by Poincaré inequality, Proposition 2.1, and (2.21), we have

k 2 meas{|u ε 0 | k} = Z {|u ε 0 1 | k} k 2 dx + Z {|u ε 0 2 | k} k 2 dx kT k (u ε 0 1 )k 2 L 2 (Ω 1 ) + kT k (u ε 0 2 )k 2 L 2 (Ω 2 )  C 3 krT k (u ε 0 1 )k 2 L 2 (Ω 1 ) + kT k (u ε 0 2 )k 2 H 1 (Ω 2 )  C 4 kT k (u ε 0 )k 2 V  C 4 Mk C 1 .
for some C 3 ,C 4 2 R + . Thus, we can find a positive constant C independent of " such that meas{|u

ε 0 i | k}  C k . (2.23)
For 1 (u ε 0 1 ),o b s e r v et h a tb yP o i n c a r éi n e q u a l i t ya n d( 2 . 2 1 ) ,

k 2 meas Γ {| 1 (u ε 0 1 )| k} = Z {|γ 1 (u ε 0 1 )| k} k 2 d = Z {|γ 1 (u ε 0 1 )| k} 1 (T k (u ε 0 1 )) 2 d k 1 (T k (u ε 0 1 ))k 2 L 2 (Γ)  C 5 krT k (u ε 0 1 )k 2 L 2 (Ω 2 )  C 6 k.
Consequently,

meas Γ {| 1 (u ε 0 1 )| k}  C 6 k ! 0 as k ! 1 . (2.24)
For 2 (u ε 0 2 ), by the Trace Theorem, Proposition 2.1, and (2.21), we have

k 2 meas Γ {| 2 (u ε 0 2 )| k} = Z {|γ 2 (u ε 0 2 )| k} k 2 d = Z {|γ 2 (u ε 0 2 )| k} 2 (T k (u ε 0 2 )) 2 d k 2 (T k (u ε 0 2 ))k 2 L 2 (Γ)  C 7 kT k (u ε 0 2 )k 2 H 1 (Ω 2 )  C 8 k. It follows that meas Γ {| 2 (u ε 0 2 )| k}  C 8 k ! 0 as k ! 1 . (2.25)
By (2.24) and (2.25), for every ⌘ > 0,t h e r ee x i s t sk 0 such that for every

k k 0 , meas Γ {x 2 Γ; | i (u ε 0 i )| k} < ⌘. (2.26)
Let !, ⌘ > 0.B y( 2 . 2 3 )a n d( 2 . 2 6 ) ,w ec a nfi n dk large enough such that

meas{|u ε 0 i | k}  ⌘ 3 , (2.27
)

meas Γ {x 2 Γ; | i (u ε 0 i )| k}  ⌘ 3 , (2.28)
for every " 0 > 0.N o t et h a tf r o m( 2 . 2 2 ) ,w ec a nd e d u c et h a tt h es e q u e n c e s

{T k (u ε i )}, { i (T k (u ε i ))} are Cauchy in measure. Hence, there exists " 0 > 0 such that meas{|T k (u ε 0 i ) T k (u ε 00 i )| !} < ⌘ 3 , (2.29)
meas

Γ {| i (T k (u ε 0 i )) i (T k (u ε 00 i ))| !} < ⌘ 3 , (2.30)
for every 0 < " 0 , " 00 < " 0 . Observe that

{|u ε 0 i u ε 00 i | !} ⇢ {|u ε 0 i | k} [ {|u ε 00 i | k} [ {|T k (u ε 0 i ) T k (u ε 00 i )| !}, and thus, meas{|u ε 0 i u ε 00 i | !}  meas{|u ε 0 i | k} + meas{|u ε 00 i | k} + meas{|T k (u ε 0 i ) T k (u ε 00 i )| !}.
It follows from (2.27) and (2.29) that

meas{|u ε 0 i u ε 00 i | !} < ⌘,
that is, {u ε 0 i } is actually Cauchy in measure. Using the inequalities (2.28) and (2.30), and similar arguments, it can be shown that

{ i (u ε 0 i )} is Cauchy in measure.
Consequently, there is a subsequence of {u ε 0 i } that is convergent a.e. to some measurable function u i :

Ω i ! R,t h a ti s u ε 0 i ! u i a.e. in Ω i .
(2.31)

It follows from (2.23) that u i is finite a.e. in Ω i . This u := (u 1 ,u 2 ) is our candidate for a renormalized solution for problem (2.1). We now prove that u satisfies the conditions (2.12). Indeed, by the continuity of T k ,w eh a v e

T k (u ε 0 ) ! T k (u)=v k 2 V a.e. in Ω \ Γ. (2.32)
Moreover, we can deduce that { i (u ε 0 i )} is convergent a.e. on Γ up to a subsequence. Hence, there exists

! i : Γ ! R such that i (u ε 0 i ) ! ! i a.e. on Γ, (2.33)
with ! i finite a.e. on Γ by (2.26). We now identify w i and i (u i ).U s i n g (2.21) and (2.22), we obtain

1 k kT k (u)k 2 V  M C 1 ,
for any k>0. By Proposition 2.2, i (u i ) (the trace in the truncate sense) is well defined. From (2.22), (2.32) and (2.33), we obtain that for any k>0,

T k (! i )= i (v k,i )= i (T k (u i )) = T k ( i (u i )) a.e. on Γ.
Then we have ! i = i (u i ) a.e. on Γ.B y F a t o u ' s L e m m a ,T k being nondecreasing, we have for all k>0,

Z Γ (u 1 u 2 )(T k (u 1 ) T k (u 2 )) d  lim inf ε 0 !0 Z Γ (u ε 0 1 u ε 0 2 )(T k (u ε 0 1 ) T k (u ε 0 2 )) d  kM, which is (2.12b).
From this p oint, we just denote our sequence by ". Rewriting all the results we got in terms of ", we have the following: for all k>0, 8 > > > < > > > :

u ε i ! u i a.e. in Ω, T k (u ε i ) ! T k (u i ) strongly in L 2 (Ω i ), a.e. in Ω i , i (u ε i ) ! i (u i ) a.e. on Γ, i (T k (u ε i )) ! i (T k (u i )) strongly in L 2 (Γ), a.e. in Γ.
(2.34)

In addition, we have

rT k (u ε i ) * rT k (u i ) weakly in (L 2 (Ω i )) N . (2.

35)

Step 3: Showing conditions (2.13) of Definition 2.4.

From the continuity of B and (2.34), we have that for any fixed n>0, B(x, T n (u ε )) ! B(x, T n (u)) a.e. in Ω and weakly* in L 1 (Ω). (2.36)

Due to assumption (A3.1) and the lower semi-continuity of the weak convergence,

1 n Z {|u|<n} B(x, u)ru•rudx = 1 n Z Ω\Γ B(x, T n (u))rT n (u) • rT n (u) dx  lim inf ε!0 1 n Z Ω\Γ B(x, T n (u ε ))rT n (u ε )rT n (u ε ) dx,
and by Fatou's Lemma,

1 n Z Γ (u 1 u 2 )(T n (u 1 ) T n (u)) d  lim inf ε!0 1 n Z Γ (u ε 1 u ε 2 )(T n (u ε 1 ) T n (u ε 2 )) d . Since Z Ω\Γ B(x, T n (u ε ))rT n (u ε )rT n (u ε ) dx and Z Γ (u ε 1 u ε 2 )(T n (u ε 1 ) T n (u ε 2 )) d are nonnegative, it is sufficient to show that lim n!1 lim sup ε!0 1 n ✓ Z Ω\Γ B(x, T n (u ε ))rT n (u ε )rT n (u ε ) dx + Z Γ (u ε 1 u ε 2 )(T n (u ε 1 ) T n (u ε 2 )) d ◆ =0.
(2.37)

We use

1 n T n (u ε ) as a test function in (2.19) to obtain 1 n Z Ω 1 B ε (x, u ε 1 )ru ε 1 rT n (u ε 1 ) dx + 1 n Z Ω 2 B ε (x, u ε 2 )ru ε 2 rT n (u ε 2 ) dx + 1 n Z Γ h(x)(u ε 1 u ε 2 )(T n (u ε 1 ) T n (u ε 2 )) d = 1 n Z Ω f ε T n (u ε ) dx.
Consequently, for " small enough, we have

1 n Z Ω 1 B(x,T n (u ε 1 ))ru ε 1 rT n (u ε 1 ) dx + 1 n Z Ω 2 B(x, T n (u ε 2 ))ru ε 2 rT n (u ε 2 ) dx + 1 n Z Γ (u ε 1 u ε 2 )(T n (u ε 1 ) T n (u ε 2 )) d = 1 n Z Ω f ε T n (u ε ) dx.
Furthermore, since T n (u ε ) converges to T n (u) weakly⇤ in L 1 (Ω) and f ε converges to f in L 1 (Ω),w eh a v e

1 n Z Ω f ε T n (u ε ) dx ! 1 n Z Ω fT n (u) dx as " ! 0.
It follows that

lim sup ε!0 1 n ✓ Z Ω\Γ B(x,T n (u ε ))rT n (u ε )rT n (u ε ) dx + Z Γ (u ε 1 u ε 2 )(T n (u ε 1 ) T n (u ε 2 )) d ◆ = 1 n Z Ω fT n (u) dx.
Observe that since u is finite a.e.,

1 n T n (u) ! 0 a.e. in Ω.
In addition, for any n>0, |T n (u)|  n a.e. and thus,

1 n fT n (u)  |f | 2 L 1 (Ω).
By the Lebesgue Dominated Convergence Theorem, we obtain

lim n!1 1 n Z Ω fT n (u) dx =0,
which gives (2.37).

Step 4. Show that u satisfies (2.14) of Definition 2.4.

Let S 1 ,S 2 2 C 1 (R) with compact support and let k>0 such that

supp S i ⇢ [ k, k]. (2.38)
We need to show that for any v 2 V \ (L 1 (Ω 1 ) ⇥ L 1 (Ω 2 )), u satisfies (2.14). We use the function ✓ n defined in (2.15). Note that

✓ n (u ε i )=✓ n (T 2n (u ε i )) 2 H 1 (Ω i ) \ L 1 (Ω i ),
and thus, if we define

i = v i S i (u i )✓ n (u ε i ), for v 2 V \ (L 1 (Ω 1 ) ⇥ L 1 (Ω 2 )),w eh a v et h a t =( 1 , 2 ) 2 V \ (L 1 (Ω 1 ) ⇥ L 1 (Ω 2 )).
Using as a test function in (2.19), we have

I 11 + I 12 + I 21 + I 22 + I 31 + I 32 + I 4 = I 51 + I 52 , (2.39)
where

I 1i = Z Ω i B ε (x, u ε i )ru ε i • rv i S i (u i )✓ n (u ε i ) dx I 2i = Z Ω i B ε (x, u ε i )ru ε i • ru i S 0 i (u i )✓ n (u ε i ) dx I 3i = Z Ω i B ε (x, u ε i )ru ε i • ru ε i S i (u i )S 0 n (u ε i ) dx I 4 = Z Γ h(x)(u ε 1 u ε 2 )(v 1 S 1 (u 1 )✓ n (u ε 1 ) v 2 S 2 (u 2 )✓ n (u ε 2 )) d I 5i = Z Ω i fv i S i (u i )✓ n (u ε i ) dx.
We look at the behavior of each integral. In particular, we will pass to the limit as " ! 0 and then as n ! 1 . Note that for n k,s i n c esupp

S i ⇢ [ k, k],w eh a v e ✓ n (s)S i (s)=S i (s) and ✓ n (s)S 0 i (s)=S 0 i (s), for a.e. s 2 R. (2.40)
We first lo ok at I 1i .O b s e r v et h a ti f" is small enough, we have

B ε (x, u ε i )ru ε i rv i S i (u i )✓ n (u ε i )=B(x, T 1/ε (u ε i ))rT 2n (u ε i )S i (u i )✓ n (u ε i ). 82
Choosing " small enough, we have

B(x, T 1/ε (u ε i ))✓ n (u ε i )=B(x, T 2n (u ε i ))✓ n (u ε i ) ! ✓ n (u i )B(x, T n (u i )),
a.e. in Ω i . Moreover, by the assumptions on B,w eh a v e

|B(x, T 1/ε (u ε i ))✓ n (u ε i )|  sup Ω i ⇥[ 2n,2n] |B(x, r)|.
It follows from the Lebesgue Dominated Convergence Theorem that

B(x, T 1/ε (u ε i ))✓ n (u ε i )=B(x, T 2n (u ε i ))✓ n (u ε i ) ! ✓ n (u i )B(x, T 2n (u i )).
a.e. in Ω i and in L 1 (Ω i ) weak-⇤. This and (2.35) imply as " ! 0,

I 1i ! Z Ω i B(x, T 2n (u i ))rT 2n (u i )rv i S i (u i )✓ n (u i ) dx = Z Ω i B(x, u i )ru i rv i S i (u i )✓ n (u i ) dx.
By (2.40) we have,

lim n!1 lim ε!0 I 1i = Z Ω i B(x, u i )ru i rv i S i (u i ) dx. (2.41)
We now observe the b ehavior of I 2i .F o rs m a l le n o u g h",w eh a v e

B ε (x, u ε i )ru ε i ru i v i S 0 i (u i )✓ n (u ε i )=B(x, T 2n (u ε i ))rT 2n (u ε i )ru i v i S 0 i (u i )✓ n (u ε i ), a.e. in Ω i . Since ru i v i S 0 i (u i )=rT 2n (u i )vS 0 i (u i ) 2 (L 2 (Ω i )) N ,b y( 2 . 3 5 )w eo b t a i n as " ! 0, I 2i ! Z Ω i B(x, T 2n (u i ))rT 2n (u i )ru i v i S 0 i (u i )✓ n (u i ) dx = Z Ω i B(x, u i )ru i ru i v i S 0 i (u i )✓ n (u i ) dx.
By (2.40),

lim n!1 lim ε!0 I 2i = Z Ω i B(x, u i )ru i • ru i v i S 0 i (u i ) dx. (2.42)
For the b ehavior of I 3i ,w eo b s e r v et h a t

|✓ 0 n (s)|  1 n , for |s|  2n. |I 3i |  kv i k L 1 (Ω i ) kS i k L 1 (R) n Z {|u ε i |<2n} B(x, u ε i )rT 2n (u ε i )rT 2n (u ε i ) dx.
By (2.37), we have lim

n!1 lim sup ε!0 I 3i =0. (2.43)
For I 4 ,w en o t et h a t

h(x)(u ε 1 u ε 2 )v i S i (u i )✓ n (u ε i )=h(x)(u ε 1 u ε 2 )v i S i (u i )✓ n (u ε i )✓ 2n (u ε i ).
Then we can write I 4 as

I 4 = I 41 + I 42 + I 43 I 44 ,
where

I 41 = Z Γ h(x)(u ε 1 u ε 2 )v 1 S 1 (u 1 )✓ 2n (u ε 1 )(✓ n (u ε 1 ) ✓ n (u ε 2 )) d , I 42 = Z Γ h(x)(u ε 1 u ε 2 )v 1 S 1 (u 1 )✓ n (u ε 2 )✓ 2n (u ε 1 ) d , I 43 = Z Γ h(x)(u ε 1 u ε 2 )v 2 S 2 (u 2 )✓ 2n (u ε 2 )(✓ n (u ε 1 ) ✓ n (u ε 2 )) d , I 44 = Z Γ h(x)(u ε 1 u ε 2 )v 2 S 2 (u 2 )✓ n (u ε 1 )✓ 2n (u ε 2 ) d .
Observe that ✓ n is Lipschitz and ✓ n (u ε i )=✓ n (T 2n (u ε i )). This gives

|✓ n (u ε 1 ) ✓ n (u ε 2 )| = |✓ n (T 2n (u ε 1 )) ✓ n (T 2n(u ε 2 ) )|  1 n |T 2n (u ε 1 ) T 2n (u ε 2 )|.
Consequently,

|I 41 |  khk L 1 (Γ) kv 1 k L 1 (Γ) kS 1 k L 1 (R) k✓ n k L 1 (R) n Z Γ |u ε 1 u ε 2 ||T 2n (u ε 1 ) T 2n (u ε 2 )| d ,
u ε 2 )v 1 S 1 (u 1 )✓ 2n (u 1 )✓ n (u ε 2 )|  M,
where the constant M depends only on the L 1 norms of h, S 1 , ✓ n and ✓ 2n , and n.A l s o ,

h(x)(u ε 1 u ε 2 )v 1 S 1 (u 1 )✓ 2n (u ε 1 )✓ n (u ε 2 ) ! h(x)(u 1 u 2 )v 1 S 1 (u 1 )✓ 2n (u 1 )✓ n (u 2 )
, a.e. on Γ as " ! 0. By Lebesgue Dominated Convergence Theorem, as " ! 0,

I 42 ! Z Γ h(x)(u 1 u 2 )v 1 S 1 (u 1 )✓ 2n (u 1 )✓ n (u 2 ) d ,
and similarly,

I 44 ! Z Γ h(x)(u 1 u 2 )v 2 S 2 (u 2 )✓ 2n (u 2 )✓ n (u 1 ) d . For large enough n,f o rj =1 , 2, i 6 = j,w eh a v eS i (u i )✓ 2n (u i )✓ n (u j )= S i (u i )✓ n (u j ).I n v i e w o f ( 2 . 1 6 ) i n R e m a r k 2 . 5 , (u 1 u 2 )S i (u i ) 2 L 1 (Γ), for i =1,
I 42 = Z Γ h(u 1 u 2 )v 1 S 1 (u 1 ) d , (2.46)
lim n!1 lim ε!0 I 44 = Z Γ h(u 1 u 2 )v 2 S 2 (u 2 ) d . ( 2
I 4 = Z Γ h(u 1 u 2 )(v 1 S 1 (u 1 ) v 2 S 2 (u 2 )) d . (2.48)
Finally for I 5 ,o b s e r v i n gt h a t✓ n (u ε i ) weakly converges to ✓ n (u i ) in L 1 (Ω i ) weakly⇤ and a.e. in Ω i , f ε converges strongly to f in L 1 (Ω),w eh a v e

I 5i = Z Ω i f ε v i S i (u i )✓ n (u ε i ) dx ! Z Ω i fv i S i (u i )✓ n (u i ) dx.
From (2.40), we have

lim n!1 lim ε!0 I 5i = Z Ω i fv i S i (u i ) dx.
(2.49)

Passing through the limit of (2.39) and using (2.41), (2.42), (2.43), (2.48), and (2.49), we have the desired conclusion.

This concludes the proof for the existence of a renormalized solution.

Remark 2.9. As explained in the introduction, it is possible to use the Boccardo-Gallouët estimates, that is, to show that

u ε =(u ε 1 ,u ε 2 ) is bounded in W 1,q (Ω 1 ) ⇥ W 1,q (Ω 2 ),a n dt h a tl e a d st ou =(u 1 ,u 2 ) 2 W 1,q (Ω 1 ) ⇥ W 1,q (Ω 2 ),
for all q< N N 1

.S u c h a r e s u l t m a y s i m p l i f y t h e p r o o f s i n c e i t i m p l i e s that 1 (u ε 1 ), 2 (u ε 2 ) 2 W 1 1 q ,q (Γ) and in particular, 1 (u ε 1 ), 2 (u ε 2 ) are bounded in L 1+η (Γ),f o rs o m es m a l le n o u g h⌘.I tf o l l o w st h a tw ec a ng i v ea n o t h e r definition of renormalized solution including u 1 u 2 2 W 1 1 q ,q (Γ) instead of (2.12b) and then (2.13b) is not necessary since it is a direct consequence of the regularity u 1 u 2 2 W 1 1 q ,q (Γ). However, since we plan to deal with the periodic homogenization of this problem (see [START_REF] Donato | Homogenization results for quasilinear elliptic problems in a two-component domain with L 1 data[END_REF]), we cannot use the Boccardo-Gallouët estimates since they are strongly related to the Sobolev constant which may blow up in a varying domain. Moreover, our techniques allow us to consider more general equations (with a nonlinear boundary terms) for which the Boccardo-Gallouët estimates are not useful.

div(B ε (x, u ε 1 )ru ε 1 )=f ε in Ω 1 , div(B ε (x, u ε 2 )ru ε 2 )=f ε in Ω 2 , u ε 1 =0 on @Ω, (B ε (x, u ε 1 )ru ε 1 )⌫ 1 =(B ε (x, u ε 2 )ru ε 2 )⌫ 1 on Γ, (B ε (x, u ε 1 )ru ε 1 )⌫ 1 = h ε (x)(u ε 1 u ε 2 ) on Γ, (2.50)
where

1. f ε 2 L 1 (Ω); 2. B ε (x, t) is a Carathéodory matrix verifying (a) B ε (x, r)⇠ • ⇠ ↵|⇠| 2 ,a . e . x 2 Ω,f o ra l lr 2 R,f o ra n y⇠ 2 R N , and
(b) for any k>0, B ε (x, r) 2 L 1 (Ω ⇥ ( k, k)) N ⇥N ;
3. h ε 2 L 1 (Γ) with 0 <h 0 <h ε (y) a.e. on Γ and h ε (y) <M (uniform), for some M>0.

Let f 2 L 1 (Ω), B : Ω ⇥ R ! R N ⇥N is a Carathéodory function, and h : Γ ! R with h 0.I f f ε ! f strongly in L 1 (Ω); > < > : B ε (x, r ε ) ! B(x, r)
for every sequence r ε 2 R such that r ε ! r a.e. on R; h ε ! h a.e. in Γ, then u ε converges to u a.e. where u is a renormalized solution of

8 > > > > > > < > > > > > > : div(B(x, u 1 )ru 1 )=f in Ω 1 , div(B(x, u 2 )ru 2 )=f in Ω 2 u 1 =0 on @Ω, (B(x, u 1 )ru 1 )⌫ 1 =(B(x, u 2 )ru 2 )⌫ 1 on Γ, (B(x, u 1 )ru 1 )⌫ 1 = h(x)(u 1 u 2 ) on Γ.
The main point is to obtain the a priori estimates of Step 2. In view of (2.17) in Remark 2.5, T k (u ε ) is an "admissible" test function, so that Z

Ω 1 B ε (x, u ε 1 )ru ε 1 • rT k (u ε 1 ) dx + Z Ω 2 B ε (x, u ε 2 )ru ε 2 • rT k (u ε 2 ) dx + Z Γ h ε (x)(u ε 1 u ε 2 )(T k (u ε 1 ) T k (u ε 2 )) d = Z Ω f ε T k (u ε ) dx,
which gives all the necessary estimates. Then we can extract subsequences so that (2.34) hold true. In view of conditions on f ε , B ε ,a n dh ε ,w ec a n perform Step 3 of the proof of Theorem 2.8.

Chapter 3

Uniqueness for quasilinear elliptic problems in a two-component domain with L 1 data

Introduction

In this chapter, we study the uniqueness of the renormalized solution of the following class of quasilinear elliptic problems:

8 > > > > > > < > > > > > > : div(B(x, u 1 )ru 1 )=f in Ω 1 , div(B(x, u 2 )ru 2 )=f in Ω 2 , u 1 =0 on @Ω, (B(x, u 1 )ru 1 )⌫ 1 =(B(x, u 2 )ru 2 )⌫ 1 on Γ, (B(x, u 1 )ru 1 )⌫ 1 = h(x)(u 1 u 2 ) on Γ. (3.1)
The domain Ω can be written as the disjoint union of Ω 1 , Ω 2 and Γ, where Ω 1 and Ω 2 are the two open components of Ω,andΓ is the interface between them. The matrix field B is a Carathéodory function that is uniformly elliptic (see assumption (A3)). The function h is in L 1 (Γ) while f belongs to L 1 (Ω).

When f belongs to L 2 (Ω) and the domain is composed of only one component, that is, div(B(x, u)ru)=f in Ω with Dirichlet boundary conditions, the uniqueness of the solution was obtained in [START_REF] Artola | Sur une classe de problèmes paraboliques quasi-linéaires[END_REF] and [START_REF] Carrillo | On some nonlinear elliptic equations involving derivatives of the nonlinearity[END_REF] under ag l o b a lL i p s c h i t z -k i n dc o n d i t i o no nB with respect to the second variable. Some generalizations to nonlinear elliptic problems were addressed in [START_REF] Boccardo | Unicité de la solution de certaines équations elliptiques non linéaires. (Uniqueness of the solution of some nonlinear elliptic equations)[END_REF], [START_REF] Chipot | Elliptic equations. An introductory course[END_REF] and [START_REF] Chipot | Uniqueness of monotonicity properties for strongly nonlinear elliptic variational inequalities[END_REF].

As far as the two-component domain is concerned, additional difficulties arise due to the jump at the interface. When f 2 L 2 (Ω),t h eu n i q u e n e s so f the variational solution was proved in [START_REF] Beltran | Homogenization of a quasilinear elliptic problem in a twocomponent domain with an imperfect interface[END_REF] by adapting the method of Chipot introduced in [START_REF] Chipot | Elliptic equations. An introductory course[END_REF] (see also [START_REF] Donato | Uniqueness result for a class of singular elliptic problems in two-component domains[END_REF] for (3.1) with a singular term).

Elliptic problems in the usual one component domain and L 1 data, that is, div(B(x, u)ru)=f in Ω with Dirichlet boundary conditions, are not in the standard variational setting. Furthermore, in the sense of distribution, we do not have uniqueness of the solution (see [START_REF] Serrin | Pathological solutions of elliptic equations,A n n .S c u o l a Norm[END_REF]). Thus, we need a convenient framework to prove the uniqueness of the solution.

Uniqueness results were proved by using the notion of entropy solutions (see [START_REF] Porretta | Uniqueness of solutions for some nonlinear Dirichlet problems[END_REF]) or by using the (equivalent) notion of renormalized solutions (see [START_REF] Blanchard | Quasi-linear degenerate elliptic problems with L 1 data[END_REF], [START_REF] Di Nardo | Uniqueness of renormalized solutions to nonlinear parabolic problems with lower-order terms[END_REF], and [START_REF] Feo | Uniqueness for elliptic problems with locally Lipschitz continuous dependence on the solution[END_REF]).

Since we consider L 1 data, we choose the appropriate framework of renormalized solutions (see [START_REF] Maso | Renormalized solutions of elliptic equations with general measure data[END_REF][START_REF] Murat | Soluciones renormalizadas de EDP elipticas no lineales[END_REF]). The existence of a renormalized solution (which is motivated by homogenization, see [START_REF] Donato | Homogenization results for quasilinear elliptic problems in a two-component domain with L 1 data[END_REF]) has been obtained in [START_REF] Fulgencio | Uniqueness results for quasilinear elliptic problems in a two-component domain with L 1 data[END_REF] (see Definition 3.2).

The main novelty of this chapter is the uniqueness of the renormalized solution under a fairly used assumption on the matrix field B(x, s) in s (similar to [START_REF] Di Nardo | Uniqueness of renormalized solutions to nonlinear parabolic problems with lower-order terms[END_REF], see assumption (A4)). With respect to the already mentioned references, let us point out that mixing technical test functions developed in [START_REF] Blanchard | Quasi-linear degenerate elliptic problems with L 1 data[END_REF] for L 1 problem and the jump give additional difficulties.

In particular, we cannot expect to control the sign of the contribution of the interface terms. To overcome this, we first prove in Lemma 3.6 that if u and v are two renormalized solutions of (3.1), then u 1 v 1 and u 2 v 2 have the same sign on the interface Γ. This sign property is crucial to prove the uniqueness result, Theorem 3.8, which we accomplish by adapting the method of [START_REF] Di Nardo | Uniqueness of renormalized solutions to nonlinear parabolic problems with lower-order terms[END_REF].

This chapter is organized as follows. The next section is devoted to the assumptions and the definitions that are necessary to achieve our aim. Here, we present the definition of a renormalized solution of (3.1) (see Definition 3.2). Section 3.3 is devoted to prove some properties of the renormalized solution to (3.1), in particular, the sign property (see Lemma 3.6) mentioned above. Our uniqueness result (see Theorem 3.8) is proved in Section 3.4.

Assumptions and Definitions

Due to the jump of a solution on the interface Γ,theusualSobolevspaces are not suitable to work with for our problem. Hence, we need to define a special normed space V .

Let V 1 be the normed space defined as

V 1 = {v 2 H 1 (Ω 1 ):v =0on @Ω} with kvk V 1 := krvk L 2 (Ω 1 ) .
The space V is defined as

V := {v ⌘ (v 1 ,v 2 ):v 1 2 V 1 and v 2 2 H 1 (Ω 2 )},
equipped with the norm

kvk 2 V := krv 1 k 2 L 2 (Ω 1 ) + krv 2 k 2 L 2 (Ω 2 ) + kv 1 v 2 k 2 L 2 (Γ) . (3.4)
As presented in [START_REF] Fulgencio | Uniqueness results for quasilinear elliptic problems in a two-component domain with L 1 data[END_REF], since the data f is in L 1 (Ω),w edonotexpectasolution u of (3.1) to be in any L p -space. Moreover, it is also not expected to have the regularity required to have a gradient and trace in the usual sense. The following proposition was proved in [START_REF] Fulgencio | Uniqueness results for quasilinear elliptic problems in a two-component domain with L 1 data[END_REF] to give a definition for the gradient and trace of any measurable function. This proposition made use of the truncation function T k : R ! R,g i v e nb y

T k (t)= 8 > < > : k, if t  k, t, if k  t  k, k, if t k.
(3.5)

Proposition 3.1 ([54]

). Let u =( u 1 ,u 2 ):Ω \ Γ ! R be a measurable function such that T k (u) 2 V for every k>0.

1. For i =1, 2,t h e r ee x i s t sau n i q u em e a s u r a b l ef u n c t i o nG i :

Ω i ! R N such that for all k>0, rT k (u i )=G i {|u i |<k} a.e. in Ω i , (3.6)
where {|u i |<k} denotes the characteristic function of

{x 2 Ω i : |u i (x)| <k}.
We define G i as the gradient of u i and write G i = ru i .

If

sup k 1 1 k kT k (u)k 2 V < 1, (3.7)
then there exists a unique measurable function

w i : Γ ! R,f o ri = 1, 2,s u c ht h a tf o ra llk>0 i (T k (u i )) = T k (w i ) a.e. in Γ, (3.8)
where i : H 1 (Ω i ) ! L 2 (Γ) is the trace operator. We define the function w i as the trace of u i on Γ and set

i (u i )=w i .
With this proposition, we can now present the definition of a renormalized solution of (3.1) given in [START_REF] Fulgencio | Uniqueness results for quasilinear elliptic problems in a two-component domain with L 1 data[END_REF].

Definition 3.2. Let u =( u 1 ,u 2 ):Ω \ Γ ! R be a measurable function. Then u is a renormalized solution of (3.1) if T k (u) 2 V, 8k>0; (3.9a) (u 1 u 2 )(T k (u 1 ) T k (u 2 )) 2 L 1 (Γ), 8k>0; (3.9b) lim n!1 1 n Z {|u|<n} B(x, u)ru • rudx =0; (3.10a) lim n!1 1 n Z Γ (u 1 u 2 )(T n (u 1 ) T n (u 2 )) d =0; (3.10b)
and for any S 1 ,S 2 2 C 1 (R) (or equivalently for any S 1 ,S 2 2 W 1,1 (R))w i t h compact support, u satisfies Z

Ω 1 S 1 (u 1 )B(x, u 1 )ru 1 • r 1 dx + Z Ω 1 S 0 1 (u 1 )B(x, u 1 )ru 1 • ru 1 1 dx + Z Ω 2 S 2 (u 2 )B(x, u 2 )ru 2 • r 2 dx + Z Ω 2 S 0 2 (u 2 )B(x, u 2 )ru 2 • ru 2 2 dx + Z Γ h(x)(u 1 u 2 )( 1 S 1 (u 1 ) 2 S 2 (u 2 )) d = Z Ω 1 f 1 S 1 (u 1 ) dx + Z Ω 2 f 2 S 2 (u 2 ) dx, (3.11
) As presented in [START_REF] Fulgencio | Uniqueness results for quasilinear elliptic problems in a two-component domain with L 1 data[END_REF], assumptions (A1)-(A3) are enough to show the existence of a renormalized solution of (3.1) in the sense of this previous definition. However, to have uniqueness of the solution, an additional assumption on matrix B must be added (see (A4)), as will be seen in the next section.

for all 2 V \ (L 1 (Ω 1 ) ⇥ L 1 (Ω 2 )).

Preliminary Results

In this section, we prove some properties on renormalized solutions of (3.1) (see Lemma 3.4 and Proposition 3.5), which are standard in the L 1 framework. Moreover, we prove Lemma 3.6, which states that if u and v are two renormalized solutions of (3.1) for the same data f ,t h e nw eh a v et h es i g n condition on the interface Γ,t h a ti ssgn(u 1 v 1 )=sgn(u 2 v 2 ) on Γ. This result is crucial for the proof of our uniqueness result (see Theorem 3.8).

Lemma 3.4. Let u be a renormalized solution of (3.1).I f' is a bounded and increasing function belonging in C 1 (R) such that '(0) = 0,t h e n

' 0 (u i )B(x, u i )ru i • ru i 2 L 1 (Ω i ),i =1, 2, (3.12)
(u 1 u 2 ) '(u 1 ) '(u 2 ) 2 L 1 (Γ). (3.13)
Proof. Let ' be a bounded increasing function that belongs in C 1 (R) such that '(0) = 0.L e tn>0.D e fi n et h ef u n c t i o n✓ n : R ! R by ✓ n (u 1 )' 0 (u 1 )B(x, u 1 )ru

✓ n (s)= 8 > > > > > > > > > < > > > > > > > > > : 0, if s  2n s n +2, if 2n  s  n 1, if n  s  n s n +2, if n  s  2n 0, if s 2n.
1 ru 1 dx + Z Ω 2 ✓ n (u 2 )' 0 (u 2 )B(x, u 2 )ru 2 ru 2 dx + Z Ω 1 ✓ 0 n (u 1 )'(u 1 )B(x, u 1 )ru 1 ru 1 dx + Z Ω 2 ✓ 0 n (u 2 )'(u 2 )B(x, u 2 )ru 2 ru 2 dx + Z Γ h(x)(u 1 u 2 )(✓ n (u 1 )'(u 1 ) ✓ n (u 2 )'(u 2 )) d = Z Ω f ✓ n (u)'(u) dx.
(3.16)

We now study the terms in (3.16) to pass to the limit as n goes to infinity.

Regarding the third and fourth terms we have, for i =1, 2, Z

Ω i ✓ 0 n (u i )'(u i )B(x, u i )ru i ru i dx  k'k L 1 (R) n Z {n<|u i |<2n} B(x, u i )ru i ru i dx,
so that the decay of the energy of the truncates (3.10a) implies that

lim n!+1 Z Ω i ✓ 0 n (u i )'(u i )B(x, u i )ru i ru i dx =0, for i =1, 2.( 3 . 1 7 )
As far as the fifth term of (3.16) is concerned, we have in view of the definition of ✓ n ,

h(x)(u 1 u 2 )(✓ n (u 1 )'(u 1 ) ✓ n (u 2 )'(u 2 )) = h(x)(u 1 u 2 )✓ n (u 1 ) ✓ 2n (u 1 ) ✓ 2n (u 2 ) '(u 1 ) + h(x)(u 1 u 2 )✓ n (u 1 )✓ 2n (u 2 ) '(u 1 ) '(u 2 ) h(x)(u 1 u 2 )✓ 2n (u 2 ) ✓ n (u 1 ) ✓ n (u 2 ) '(u 2 ). (3.18)
Since the functions ✓ n and ✓ 2n are Lipschitz continuous and recalling that ' is bounded, we deduce that

h(x)(u 1 u 2 )✓ n (u 1 ) ✓ 2n (u 1 ) ✓ 2n (u 2 ) '(u 1 )  khk L 1 (Γ) 2n (u 1 u 2 )(T 4n (u 1 ) T 4n (u 2 ))k'k L 1 (R) , h(x)(u 1 u 2 )✓ 2n (u 2 ) ✓ n (u 1 ) ✓ n (u 2 ) '(u 2 )  khk L 1 (Γ) 2n (u 1 u 2 )(T 2n (u 1 ) T 2n (u 2 ))k'k L 1 (R) ,
so that condition (3.10b) leads to

lim n!+1 Z Ω h(x)(u 1 u 2 )✓ n (u 1 ) ✓ 2n (u 1 ) ✓ 2n (u 2 ) '(u 1 ) dx =0, (3.19) lim n!+1 Z Ω h(x)(u 1 u 2 )✓ 2n (u 2 ) ✓ n (u 1 ) ✓ n (u 2 ) '(u 2 ) dx =0. (3.20)
At last, the integral on the right-hand side of (3.16), by (3.15), is bounded by

Z Ω f ✓ n (u)'(u) dx kf k L 1 (Ω) k'k L 1 (R) . ( 3
Z Ω 1 ✓ n (u 1 )' 0 (u 1 )B(x, u 1 )ru 1 ru 1 dx + Z Ω 2 ✓ n (u 2 )' 0 (u 2 )B(x, u 2 )ru 2 ru 2 dx + Z Γ h(x)(u 1 u 2 )✓ n (u 1 )✓ 2n (u 2 ) '(u 1 ) '(u 2 ) d  !(n)+kf k L 1 (Ω) k'k L 1 (R) ,
where w(n) ! 0 as n goes to infinity. Since u 1 (resp. u 2 ) is finite almost everywhere in Ω 1 (resp. Ω 2 ), the definition of ✓ n and Fatou's Lemma allow one to deduce that Z

Ω 1 B(x, u 1 )' 0 (u 1 )ru 1 ru 1 dx + Z Ω 2 B(x, u 2 )' 0 (u 2 )ru 2 ru 2 dx + Z Γ h(x)(u 1 u 2 ) '(u 1 ) '(u 2 ) d kf k L 1 (Ω) k'k L 1 (R) . (3.22)
This gives (3.12) and (3.13).

As mentioned in Remark 3.3, we do not impose in Definition 3.2 that i (u i ) (i =1 , 2)b e l o n g st oL 1 (Γ).H o w e v e r ,h a v i n gn or e g u l a r i t yo n i (u i ), for i =1, 2, seems to be an obstacle to prove Theorem 3.8. By adapting the estimates of Boccardo-Gallouët (see [START_REF] Boccardo | Nonlinear elliptic and parabolic equations involving measure data[END_REF]) to our two-component domain, we are able to prove in Proposition 3.5 that i (u i ) belongs to L 1 (Γ),fori =1, 2. Proposition 3.5. For i =1, 2,let i be the trace function defined on

H 1 (Ω i ). If u is a renormalized solution of (3.1),t h e n i (u i ) 2 L 1 (Γ), i =1, 2.
Proof. By taking ' = T k in Lemma 3.4, and by observing the precise estimate (3.22) at the end of the proof of Lemma 3.4, we have

kT k (u)k 2 V  kkf k L 1 (Ω) , 8k>0.
Since on V ,thenormk•k V is equivalent to the norm of H 1 (Ω 1 )⇥H 1 (Ω 2 ),the Boccardo-Gallouët estimates hold true, so that

u i 2 W 1,p (Ω i ), i =1 , 2,f o r any p< N N 1 .W ea r et h e na b l et oc o n c l u d et h a t i (u i ) 2 L 1 (Γ), i =1, 2.
In proving the uniqueness result, one of the main difficulties we encountered is managing the integral on the interface Γ with test functions which are nonlinear with respect to the unknown. The very first step to overcome this difficulty is the following lemma which establishes a sign property of the difference of any two renormalized solutions of (3.1) on the interface. We will denote by sgn the usual sign function (sgn(r)=r/|r| if r 6 =0and sgn(0) = 0). Lemma 3.6. Suppose (A1)-(A4) hold. If u and v are two renormalized solutions of (3.1),t h e nsgn(u 1 v 1 ) = sgn(u 2 v 2 ) a.e. on Γ.

Proof. Let u and v be renormalized solutions of (3.1). Writing (3.11)

of Definition 3.2 for S 1 = S 2 = ✓ n and = 1 k T k (u v)
, where 0 <k<1,f o ru and v,a n ds u b t r a c t i n gt h er e s u l t i n ge q u a t i o n s ,w eh a v e

I k,n 1 + I k,n 2 + J k,n 1 + J k,n 2 + L k,n = M k,n , (3.23)
where for i =1, 2,

I k,n i = 1 k Z Ω i (✓ n (u i )B(x, u i )ru i ✓ n (v i )B(x, v i )rv i )rT k (u i v i ) dx, J k,n i = 1 k Z Ω i (✓ 0 n (u i )B(x, u i )ru i ru i ✓ 0 n (v i )B(x, v i )rv i rv i )T k (u i v i ) dx, L k,n = 1 k Z Γ h(x)(u 1 u 2 )(✓ n (u 1 )T k (u 1 v 1 ) ✓ n (u 2 )T k (u 2 v 2 )) d 1 k Z Γ h(x)(v 1 v 2 )(✓ n (v 1 )T k (u 1 v 1 ) ✓ n (v 2 )T k (u 2 v 2 )) d , M k,n = 1 k Z Ω f (✓ n (u) ✓ n (v))T k (u v) dx.
We study the b ehavior of each term first as k ! 0 and then as n ! 1 . We can write I k,n i , i =1, 2,a s

I k,n i = I k,n i,1 + I k,n i,2 + I k,n i,3
, where

I k,n i,1 = 1 k Z Ω i ✓ n (u i )B(x, u i )rT k (u i v i )rT k (u i v i ) dx, i =1, 2, I k,n i,2 = 1 k Z Ω i ✓ n (u i )(B(x, u i ) B(x, v i ))rv i rT k (u i v i ) dx, i =1, 2, I k,n i,3 = 1 k Z Ω i (✓ n (u i ) ✓ n (v i ))B(x, v i )rv i rT k (u i v i ) dx, i =1, 2.
Clearly,

I k,n i,1 0, i =1, 2.F o rI k,n i,2 , i =1, 2,w euse(3.3)and(3.15)toobtain |I k,n i,2 | = 1 k Z Ω i ✓ n (u i )(B(x, u i ) B(x, v i ))rv i rT k (u i v i ) dx  1 k Z {0<|u i v i |<k} \{|u i |2n} \{|v i |2n+1} |✓ n (u i )||B(x, u i ) B(x, v i )||rv i ||rT k (u i v i )| dx  1 k Z {0<|u i v i |<k} \{|u i |2n} \{|v i |2n+1} C|u i v i ||rv i rT k (u i v i )| dx  C Z {0<|u i v i |<k} |rT 2n+1 (v i )rT k (u i v i )| dx.
From (3.9a) of Definition 3.2, we know that for any 0 <k<1, i =1, 2,

|rT 2n+1 (v i )rT k (u i v i ) {0<|u i v i |<k} |  |rT 2n+1 (v i )rT 1 (u i v i )| 2 L 1 (Ω i ).
In addition, we have,

rT k (u i v i ) {0<|u i v i |<k} ! 0 as k ! 0, a.e. in Ω i ,i =1, 2.
By Lebesgue Dominated Convergence Theorem, we conclude that

C Z {0<|u i v i |<k} |rT 2n+1 (v i )rT k (u i v i )| dx ! 0 as k ! 0,i =1, 2, which gives lim n!1 lim k!0 I k,n i,2 =0,i =1, 2. ( 3
|I k,n i,3 | = 1 k Z Ω i (✓ n (u i ) ✓ n (v i ))B(x, v i )rv i rT k (u i v i ) dx  1 k Z {0<|u i v i |<k} \{|u i |<2n+1} \{|v i |<2n+1} |✓ n (u i ) ✓ n (v i )||B(x, u i )rv i r(u i v i )| dx  1 n Z {0<|u i v i |<k} \{|u i |<2n+1} \{|v i |<2n+1} |B(x, u i )rv i r(u i v i )| dx  1 n Z {0<|u i v i |<k} \{|u|<2n+1} \{|v i |<2n+1} |B(x, v i )ru i rv i | dx + 1 n Z {0<|u i v i |<k} \{|u i |<2n+1} \{|v i |<2n+1} B(x, v i )rv i rv i dx  1 n Z {0<|u i v i |<k} |B(x, v i )rT 2n+1 (u i )rT 2n+1 (v i )| dx + 1 n Z {0<|u i v i |<k} B(x, v i )rT 2n+1 (v i )rT 2n+1 (v i ) dx.
Again, from (3.9a) of Definition 3.2, we deduce that for i =1, 2,

|rT 2n+1 (u i )rT 2n+1 (v i )| {0<|u i v i |<k}  |rT 2n+1 (u i )rT 2n+1 (v i )| 2 L 1 (Ω i ) and |rT 2n+1 (v i )| 2 {0<|u i v i |<k}  |rT 2n+1 (v i )| 2 2 L 1 (Ω i ). Furthermore, rT 2n+1 (u i )rT 2n+1 (v i ) {0<|u i v i |<k} ! 0 as k ! 0, a.e. in Ω i ,i =1, 2,
and
|rT 2n+1 (v i )| 2 {0<|u i v i |<k} ! 0 as k ! 0, a.e. in Ω i ,i =1, 2.
Using the Lebesgue Dominated Convergence Theorem, we have

1 n Z {0<|u i v i |<k} |B(x, v i )rT 2n+1 (u i )rT 2n+1 (v i )| dx + 1 n Z {0<|u i v i |<k} B(x, v i )|rT 2n+1 (v i )| 2 dx ! 0 as k ! 0.
Hence,

lim n!1 lim k!0 I k,n i,3 =0,i =1, 2. (3.25) For J k,n i , i =1, 2,s i n c ew eh a v e( 3 . 1 5 )a n d |T k (r)|  k, 8r 2 R, 8k>0, (3.26)
we obtain for i =1, 2,

|J k,n i | = 1 k Z Ω i (✓ 0 n (u i )B(x, u i )ru i ru i ✓ 0 n (v i )B(x, v i )rv i rv i )T k (u i v i ) dx  Z Ω i |✓ 0 n (u i )B(x, u i )ru i ru i | dx + Z Ω i |✓ 0 n (v i )B(x, v i )rv i rv i | dx  1 n Z {|u i |<2n} B(x, u i )ru i ru i dx + 1 n Z {|v i |<2n} B(x, v i )rv i rv i dx.
By (3.10a) of Definition 3.2, it follows that

lim n!1 lim k!0 J k,n i =0,i =1, 2. (3.27)
For the integral on the b oundary, we use Prop osition 3.5 to pass to the limit. Note that by (3.15) and (3.26), we have

1 k h(x)(u 1 u 2 )✓ n (u 1 )T k (u 1 v 1 ) khk L 1 (Γ) |u 1 u 2 | 2 L 1 (Γ).
Furthermore,

1 k h(x)(u 1 u 2 )✓ n (u 1 )T k (u 1 v 1 ) ! k!0 h(x)(u 1 u 2 )✓ n (u 1 )sgn(u 1 v 1 )
a.e. on Γ,a n d

h(x)(u 1 u 2 )✓ n (u 1 )sgn(u 1 v 1 ) ! n!1 h(x)(u 1 u 2 )sgn(u 1 v 1 ),
a.e. on Γ. The Lebesgue Dominated Convergence Theorem implies

lim n!1 lim k!0 1 k Z Γ h(x)(u 1 u 2 )✓ n (u 1 )T k (u 1 v 1 ) d = Z Γ h(x)(u 1 u 2 )sgn(u 1 v 1 ) d , 100
Similarly, we obtain

lim n!1 lim k!0 1 k Z Γ h(x)(u 1 u 2 )✓ n (u 2 )T k (u 2 v 2 ) d = Z Γ h(x)(u 1 u 2 )sgn(u 2 v 2 ) d , lim n!1 lim k!0 1 k Z Γ h(x)(u 1 u 2 )✓ n (v i )T k (u 2 v 2 ) d = Z Γ h(x)(u 1 u 2 )sgn(u 2 v 2 ) d ,i =1, 2.
Thus,

lim n!1 lim k!0 L k,n = Z Γ h(x)[(u 1 v 1 ) (u 2 v 2 )](sgn(u 1 v 1 ) sgn(u 2 v 2 )) d .
(3.28)

For the integral on the right-hand side of (3.23),

|M k,n | = 1 k Z Ω f (✓ n (u) ✓ n (v))T k (u v) dx  Z Ω |f ||✓ n (u) ✓ n (v)| dx. Note that |f ||✓ n (u) ✓ n (v)|  2|f | 2 L 1 (Ω) with ✓ n (u) ✓ n (v) ! 0 as n ! 1 , a.e. in Ω.
Thus, by the Lebesgue Dominated Convergence Theorem,

Z Ω |f ||✓ n (u) ✓ n (v)| dx ! 0 as n !1, which gives lim n!1 lim k!0 M k,n =0. (3.29)
From (3.23) and the fact that I k,n i,1 0,w eo b t a i n

I k,n i,2 + I k,n i,3 + X k,n + L k,n  M k,n .
Taking the limit of both sides of the last inequality first as k ! 0 then as

n ! 1 ,w eg e t Z Γ h(x)((u 1 v 1 ) (u 2 v 2 ))(sgn(u 1 v 1 ) sgn(u 2 v 2 )) d =0.
That is, sgn(u 1 v 1 )=sgn(u 2 v 2 ) on Γ.

Main Result

This section is devoted to our main result, Theorem 3.8, namely the uniqueness of the renormalized solution under assumptions (A1)-(A4). The proof of this uniqueness result makes use of the results of the previous section and the method developed in [START_REF] Blanchard | Quasi-linear degenerate elliptic problems with L 1 data[END_REF][START_REF] Di Nardo | Uniqueness of renormalized solutions to nonlinear parabolic problems with lower-order terms[END_REF]. The following proposition, proved in [START_REF] Di Nardo | Uniqueness of renormalized solutions to nonlinear parabolic problems with lower-order terms[END_REF], states that assuming a very local Lipschitz control of B(x, s) with respect to s,w eh a v et h ee x i s t e n c eo faf u n c t i o n' which controls the Lipschitz continuous character of the matrix field B through very technical conditions.

Proposition 3.7 ( [START_REF] Di Nardo | Uniqueness of renormalized solutions to nonlinear parabolic problems with lower-order terms[END_REF]). Suppose that (3.3) holds. Then there exists a function ' 2 C 1 (R) that satisfies the following properties:

'(0) = 0 and ' 0 1. (3.30)
In addition, there are constants > 1/2, 0 <k 0 < 1,a n dL>0 such that

' 0 (1 + |'|) 2δ 2 L 1 (R), (3.31)
and for any r,

s 2 R satisfying |'(r) '(s)|  k,f o r0 <k<k 0 , B(x, r) ' 0 (r) B(x, s) ' 0 (s)  1 ' 0 (s) Lk (1 + |'(r)| + |'(s)|) δ (3.32) and 1 L  ' 0 (s) ' 0 (r)  L. (3.33)
We now state and prove the main theorem.

Theorem 3.8. If assumptions (A1)-(A4) hold, then the renormalized solution of (3.1) is unique.

Proof. In view of Theorem 1 in [START_REF] Fulgencio | Uniqueness results for quasilinear elliptic problems in a two-component domain with L 1 data[END_REF], assumptions (A1)-(A3) are sufficient to give the existence of at least one solution to (3.1). Let u and v be two renormalized solutions of (3.1). Since (3.3) holds, by Proposition 3.7, we can find a function ' 2 C 1 (R), such that for some constants > 1/2, 0 <k 0 < 1,a n dL>0, ' satisfies (3.30)- (3.33).

The proof is then decomposed into two steps.

Step 1 is devoted to show the very technical result (3.34). Roughly speaking, (3.34) is an extension of the method developed by Artola in [START_REF] Artola | Sur une classe de problèmes paraboliques quasi-linéaires[END_REF] (see also [START_REF] Boccardo | Unicité de la solution de certaines équations elliptiques non linéaires. (Uniqueness of the solution of some nonlinear elliptic equations)[END_REF]), and allows one to consider very general dependency of B(x, s) with respect to s and L 1 data. Limit (3.34) was also derived in [START_REF] Blanchard | Quasi-linear degenerate elliptic problems with L 1 data[END_REF] (see also [START_REF] Di Nardo | Uniqueness of renormalized solutions to nonlinear parabolic problems with lower-order terms[END_REF]) for elliptic equations with Dirichlet boundary condition. Since we have to deal with the boundary term, we give here a complete proof of (3.34). In Step 2, we are able to conclude that u = v a.e. in Ω.

Step 1. In this step we prove that

lim k!0 1 k 2 Z Ω i ✓ 1 ' 0 (u i ) + 1 ' 0 (v i ) ◆ |rT k ('(u i ) '(v i ))| 2 dx =0,i =1, 2.
(3.34) Writing (3.11) of Definition 3.2 for u and v, with

S 1 = S 2 = ✓ n and = W k := T k ('(T 3n (u)) '(T 3n (v)))
, where n 2 N, n 1 and 0 <k<1, and subtracting the resulting equations, we have

A k,n 1 + A k,n 2 + B k,n 1 + B k,n 2 + C k,n = D k,n , (3.35)
where

A k,n i = Z Ω i (✓ n (u i )B(x, u i )ru i ✓ n (v i )B(x, v i )rv i )rW k dx, i =1, 2, B k,n i = Z Ω i (✓ 0 n (u i )B(x, u i )ru i ru i ✓ 0 n (v i )B(x, v i )rv i rv i )W k dx, i =1, 2, C k,n = Z Γ h(x)(u 1 u 2 )(W k,1 ✓ n (u 1 ) W k,2 ✓ n (u 2 )) d Z Γ h(x)(v 1 v 2 )(W k,1 ✓ n (v 1 ) W k,2 ✓ n (v 2 )) d , D k,n = Z Ω fW k (✓ n (u) ✓ n (v)) dx.
We will first look at the limit of each term as n ! 1 . We can write A k,n i as

A k,n i = Z Ω i ✓ n (u i ) B(x, u i ) ' 0 (u i ) (r'(u i ) r'(v i ))rW k dx + Z Ω i ✓ n (u i ) ✓ B(x, u i ) ' 0 (u i ) B(x, v i ) ' 0 (v i ) ◆ ' 0 (v i )rv i rW k dx + Z Ω i (✓ n (u i ) ✓ n (v i ))B(x, v i )rv i rW k dx, i =1, 2.
Using the symmetry with respect to v i ,w eo b t a i n

A k,n i = A k,n i,1 + A k,n i,2 + A k,n i,3 ,i =1, 2,
where

A k,n i,1 = 1 2 Z Ω i ✓ ✓ n (u i ) B(x, u i ) ' 0 (u i ) + ✓ n (v i ) B(x, v i ) ' 0 (v i ) ◆ (r'(u i ) r'(v i ))rW k dx, A k,n i,2 = 1 2 Z Ω i ✓ B(x, u i ) ' 0 (u i ) B(x, v i ) ' 0 (v i ) ◆ ⇥ (✓ n (u i )' 0 (v i )rv i + ✓ n (v i )' 0 (u i )ru i )rW k dx, A k,n i,3 = 1 2 Z Ω i (✓ n (u i ) ✓ n (v i ))(B(x, u i )ru i + B(x, v i )rv i )rW k dx.
For i =1 , 2,l e tu sd e fi n eU

k i = {x 2 Ω i :0< |'(u i ) '(v i )| <k }.F o r any k>0 small enough, since supp ✓ n =[ 2n, 2n],w eh a v ea . e . i nU k i , ✓ n (u i )rT k ('(u i ) '(v i )) = ✓ n (u i )rT k ('(T 3n (u i )) '(T 3n (v i ))) = ✓ n (u i )rW k ,i =1, 2. (3.36)
As a consequence of (3.36), for any k>0 small enough, we get for i =1, 2,

A k,n i,1 = 1 2 Z Ω i ✓ n (u i ) B(x, u i ) ' 0 (u i ) (r'(u i ) r'(v i ))rW k dx + 1 2 Z Ω i ✓ n (v i ) B(x, v i ) ' 0 (v i ) (r'(u i ) r'(v i ))rW k dx = 1 2 Z U k i ✓ n (u i ) B(x, u i ) ' 0 (u i ) (r'(u i ) r'(v i ))(r'(u i ) r'(v i )) dx + 1 2 Z U k i ✓ n (v i ) B(x, v i ) ' 0 (v i ) (r'(u i ) r'(v i ))(r'(u i ) r'(v i )) dx.
Using the coercivity of B we obtain that

↵ 2 Z U k i ✓ ✓ n (u i ) ' 0 (u i ) + ✓ n (v i ) ' 0 (v i ) ◆ |r('(u i ) '(v i ))| 2 dx  A k,n i,1 ,i =1, 2. (3.37)
As far as A k,n i,2 are concerned, by (3.32), we have for i =1 , 2,a n da n yk>0 small enough

|A k,n i,2 |  1 2 Z U k i B(x, u i ) ' 0 (u i ) B(x, v i ) ' 0 (v i ) |✓ n (u i )r'(v i )+✓ n (v i )r'(u i )| ⇥ |r('(u i ) '(v i ))| dx  1 2 Z U k i 1 ' 0 (u i ) Lk (1 + |'(u i )| + |'(v i )|) δ ⇥ |✓ n (u i )r'(v i )+✓ n (v i )r'(u i )||r('(u i ) '(v i ))| dx  1 2 Z U k i 1 ' 0 (u i ) Lk (1 + |'(u i )| + |'(v i )|) δ ⇥ ✓ n (u i )|r'(v i )||r('(u i ) '(v i ))| dx + 1 2 Z U k i 1 ' 0 (u i ) Lk (1 + |'(u i )| + |'(v i )|) δ ⇥ ✓ n (v i )|r'(u i )||r('(u i ) '(v i ))| dx.
For " > 0 (which will be chosen later), Young's inequality leads to, for i =1, 2,

|A k,n i,2 |  Z U k i ✓ n (u i )' 0 (v i ) 2 " 1 " ✓ Lk ' 0 (u i ) |r'(v i )| (1 + |'(u i )| + |'(v i )|) δ ◆ 2 + " ✓ 1 ' 0 (v i ) |r'(u i ) r'(v i )| ◆ 2 # dx + Z U k i ✓ n (v i )' 0 (u i ) 2 " 1 " ✓ Lk ' 0 (u i ) |r'(u i )| (1 + |'(u i )| + |'(v i )|) δ ◆ 2 + " ✓ 1 ' 0 (u i ) |r'(u i ) r'(v i )| ◆ 2 # dx  C 1 k 2 Z U k i (✓ n (u i )' 0 (v i )|r'(v i )| 2 + ✓ n (v i )' 0 (u i )|r'(u i )| 2 ) (' 0 (u i )) 2 (1 + |'(u i )| + |'(v i )|) 2δ dx + C 2 " Z U k i ✓ ✓ n (u i ) ' 0 (u i ) + ✓ n (v i ) ' 0 (v i ) ◆ |r'(u i ) r'(v i )| 2 dx.
In view of assumption (3.33), we deduce that, for i =1, 2,

C 1 k 2 Z U k i [✓ n (u i )' 0 (v i )|r'(v i )| 2 + ✓ n (v i )' 0 (u i )|r'(u i )| 2 ] (' 0 (u i )) 2 (1 + |'(u i )| + |'(v i )|) 2δ dx  C 1 k 2 Z U k i (L 2 ✓ n (u i )' 0 (v i )|rv i | 2 + ✓ n (v i )' 0 (u i )|ru i | 2 ) (1 + |'(u i )| + |'(v i )|) 2δ dx  C 3 k 2 Z U k i (✓ n (u i )+✓ n (v i )) ' 0 (u i )|ru i | 2 + ' 0 (v i )|rv i | 2 (1 + |'(u i )| + |'(v i )|) 2δ dx.
It follows that

|A k,n i,2 |  C 3 k 2 Z U k i (✓ n (u i )+✓ n (v i )) ' 0 (u i )|ru i | 2 + ' 0 (v i )|rv i | 2 (1 + |'(u i )| + |'(v i )|) 2δ dx + C 2 " Z U k i ✓ ✓ n (u i ) ' 0 (u i ) + ✓ n (v i ) ' 0 (v i ) ◆ |r'(u i ) r'(v i )| 2 dx, i =1, 2, (3.38)
where C 2 and C 3 are positive constants independent of k and n (with C 2 also independent of ").

We now turn to the term A k,n i,3 .B y( 3 . 3 0 )a n d( 3 . 3 3 ) ,w eh a v e

|u i v i |  L ' 0 (u i ) |'(u i ) '(v i )| a.e. in U k i ,i =1, 2, (3.39)
and since ✓ n is a Lipschitz continuous function verifying

|✓ 0 n (r)| = 1 n {n<|r|<2n}
a.e. in R,w eo b t a i n

|✓ n (u i ) ✓ n (v i )|  1 n |u i v i |  Lk n' 0 (u i ) a.e. in U k i ,i =1, 2. (3.40)
Observe that this inequality still holds if the roles of u i and v i are interchanged.

Therefore using (3.40), we obtain, for i =1, 2,

|A k,n i,3 |  1 2 Z U k i \ {|u i |<2n+1} \{|v i |<2n+1} |✓ n (u i ) ✓ n (v i )| B(x, u i )' 0 (u i )ru i ru i + B(x, u i )' 0 (u i )ru i rv i B(x, v i )' 0 (v i )ru i rv i B(x, v i )' 0 (v i )rv i rv i dx  Lk 2n Z U k i \ {|u i |<2n+1} \{|v i |<2n+1} (B(x, u i )ru i ru i + B(x, v i )rv i rv i ) dx + Lk 2n Z U k i \ {|u i |<2n+1} \{|v i |<2n+1} (B(x, u i )|ru i rv i | + B(x, v i )|ru i rv i |) dx
Applying Young's Inequality on the second term of the previous inequality, we get, for i =1, 2,

Lk 2n Z U k i \ {|u i |<2n+1} \{|v i |<2n+1} (B(x, u i )|ru i rv i | + B(x, v i )|ru i rv i |) dx  Lk 4n Z U k i \ {|u i |<2n+1} \{|v i |<2n+1} h B(x, u i )(ru i ru i + rv i rv i ) +[B(x, v i )(ru i ru i + rv i rv i ) i dx.
It follows that, for i =1, 2,

|A k,n i,3 |  C 4 k n Z U k i \ {|u i |<2n+1} \{|v i |<2n+1} (B(x, u i )ru i ru i + B(x, v i )rv i rv i ) dx + C 5 k n Z U k i \ {|u i |<2n+1} \{|v i |<2n+1} (B(x, u i )rv i rv i + B(x, v i )ru i ru i ) dx.
(3.41) By (3.10a) of Definition 3.2, the first term of the right-hand side of (3.41) goes to zero as n goes to infinity. It is worth noting that the second term of the right-hand side of (3.41) contains non symmetric terms in u i and v i , so that without any bound on B,t h eb e h a v i o ro ft h i st e r mi sn o tad i r e c t consequence of the decay of the truncate energy (3.10a). Using (3.10a) and condition (3.32), we claim that the second term also goes to zero as n goes to infinity. Indeed, writing for i =1, 2, Z

U k i \ {|u i |<2n+1} \{|v i |<2n+1} B(x, u i )rv i rv i dx = Z U k i \ {|u i |<2n+1} \{|v i |<2n+1} ✓ B(x, u i ) ' 0 (u i ) B(x, v i ) ' 0 (v i ) ◆ ' 0 (u i )rv i rv i dx + Z U k i \ {|u i |<2n+1} \{|v i |<2n+1} B(x, v i ) ' 0 (v i ) ' 0 (u i )rv i rv i dx,
and using (3.32) and (3.33), we have, for i =1, 2,

C 5 k n Z U k i \ {|u i |<2n+1} \{|v i |<2n+1} B(x, u i )rv i rv i dx  C 5 Lk 2 n Z {|v i |<2n+1} |rv i | 2 (1 + |'(u i )| + |'(v i )|) δ dx + C 5 Lk n Z {|v i |<2n+1} B(x, v i )rv i rv i dx.
It follows that by (3.10a) of Definition 3.2, we have

lim n!1 C n Z U k i \{|u i |<2n+1} \{|v i |<2n+1} B(x, u i )rv i rv i dx =0,i =1, 2.
By similar computations, it can be shown that

lim n!1 C n Z U k i \{|u i |<2n+1} \{|v i |<2n+1} B(x, v i )ru i ru i dx =0,i =1, 2.

Consequently, lim

n!1 A k,n i,3 =0,i =1, 2. (3.42) Regarding the term B k,n i , i =1, 2,w eh a v e |B k,n i | = Z Ω i (✓ 0 n (u i )B(x, u i )ru i ru i ✓ 0 n (v i )B(x, v i )rv i rv i )W k dx  Z Ω i |✓ 0 n (u i )|B(x, u i )ru i ru i |W k | dx + Z Ω i |✓ 0 n (v i )|B(x, v i )rv i rv i |W k | dx  k n Z {|u i |<2n} B(x, u i )ru i ru i dx + k n Z {|v i |<2n} B(x, v i )rv i rv i dx.
These last two integrals go to zero as n goes to infinity by (3.10a) of Definition 3.2. Thus,

lim n!1 B k,n i =0,i =1, 2.
(3.43)

To pass to the limit of C k,n as n goes to 1, we use Proposition 3.5. Note that |h(x)(u

1 u 2 )W k,1 ✓ n (u 1 )| khk L 1 (Γ) |u 1 u 2 |k 2 L 1 (Γ), and
h(x)(u 1 u 2 )W k,1 ✓ n (u 1 ) ! n!1 h(x)(u 1 u 2 )T k ('(u 1 ) '(v 1 )
) a.e. on Γ.

By the Lebesgue Dominated Convergence Theorem, we have

lim n!1 Z Γ h(x)(u 1 u 2 )W k,1 ✓ n (u 1 ) d = Z Γ h(x)(u 1 u 2 )T k ('(u 1 ) '(v 1 )) d .
Using similar arguments, we obtain that

lim n!1 Z Γ h(x)(u 1 u 2 )W k,2 ✓ n (u 2 ) d = Z Γ h(x)(u 1 u 2 )T k ('(u 2 ) '(v 2 )) d and for i =1, 2, lim n!1 Z Γ h(x)(v 1 v 2 )W k,i ✓ n (v i ) d = Z Γ h(x)(v 1 v 2 )T k ('(u i ) '(v i )) d .
Therefore, we conclude that

lim n!1 C k,n = Z Γ h(x)[(u 1 u 2 ) (v 1 v 2 )] ⇥ [T k ('(u 1 ) '(v 1 )) T k ('(u 2 ) '(v 2 ))] d . (3.44)
Finally, concerning D k,n ,s i n c e

|fW k (✓ n (u) ✓ n (v))|  2k|f | 2 L 1 (Ω), while ✓ n (u) ✓ n (v) ! 0 a.e. in Ω as n ! 1 ,
the Lebesgue Dominated Convergence Theorem leads to

lim n!1 D k,n =0. (3.45)
Combining (3.37), (3.38), (3.42), (3.43), and (3.45), and choosing " small enough, we obtain Z

U k 1 ✓ ✓ n (u 1 ) ' 0 (u 1 ) + ✓ n (v 1 ) ' 0 (v 1 ) ◆ |r'(u 1 ) r'(v 1 )| 2 dx + Z U k 2 ✓ ✓ n (u 2 ) ' 0 (u 2 ) + ✓ n (v 2 ) ' 0 (v 2 ) ◆ |r'(u 2 ) r'(v 2 )| 2 dx + C k,n  Ck 2 Z U k 1 (✓ n (u 1 )+✓ n (v 1 )) ' 0 (u 1 )|ru 1 | 2 + ' 0 (v 1 )|rv 1 | 2 (1 + |'(u 1 )| + |'(v 1 )|) 2δ dx + Ck 2 Z U k 2 (✓ n (u 2 )+✓ n (v 2 )) ' 0 (u 2 )|ru 2 | 2 + ' 0 (v 2 )|rv 2 | 2 (1 + |'(u 2 )| + |'(v 2 )|) 2δ dx + ⇢(n), (3.46)
where C is a positive constant independent of k and n, and where ⇢(n) goes to zero as n goes to 1.

Let ⌧ : R ! R such that

⌧ (r)= Z r 0 ' 0 (t) (1 + |'(t)|) 2δ dt.
Clearly, ⌧ is an increasing C 1 (R)-function and since 2 > 1, ⌧ is bounded. Then, by Lemma 3.4 and (3.30) of Proposition 3.7, we deduce that

' 0 (u i )|ru i | 2 (1 + |'(u i )|) 2δ and ' 0 (v i )|rv i | 2 (1 + |'(v i )|) 2δ 2 L 1 (Ω i ). (3.47)
Hence, we can pass to the limit on the right-hand side of (3.46). Furthermore,

✓ n (u i ) ! 1 as n ! 1 , a.e. in Ω i ,i =1, 2.
By Fatou's Lemma and (3.44), we have

Z U k 1 ✓ 1 ' 0 (u 1 ) + 1 ' 0 (v 1 ) ◆ |r'(u 1 ) r'(v 1 )| 2 dx + Z U k 2 ✓ 1 ' 0 (u 2 ) + 1 ' 0 (v 2 ) ◆ |r'(u 2 ) r'(v 2 )| 2 dx + C k  Ck 2 Z U k 1 ' 0 (u 1 )|ru 1 | 2 + ' 0 (v 1 )|rv 1 | 2 (1 + |'(u 1 )| + |'(v 1 )|) 2δ dx + Ck 2 Z U k 2 ' 0 (u 2 )|ru 2 | 2 + ' 0 (v 2 )|rv 2 | 2 (1 + |'(u 2 )| + |'(v 2 )|) 2δ dx, (3.48)
where

C k = Z Γ h(x)[(u 1 v 1 ) (u 2 v 2 )][T k ('(u 1 ) '(v 1 )) T k ('(u 2 ) '(v 2 ))] d .
Dividing both sides of (3.48) by k 2 and noting that U k i ! 0 a.e. in Ω i as k ! 0, (3.47) and the Lebesgue Dominated Convergence Theorem allow one to conclude that lim sup

k!0 ✓ 1 k 2 Z U k 1 ✓ 1 ' 0 (u 1 ) + 1 ' 0 (v 1 ) ◆ |r'(u 1 ) r'(v 1 )| 2 dx + 1 k 2 Z U k 2 ✓ 1 ' 0 (u 2 ) + 1 ' 0 (v 2 ) ◆ |r'(u 2 ) r'(v 2 )| 2 dx + 1 k 2 C k ◆  0. (3.49)
As a consequence, proving (3.34) is equivalent to showing that

lim sup k!0 1 k 2 C k 0. (3.50)
We now study the b ehavior of C k /k 2 as k goes to zero. To shorten the notation, we will denote by g k the function given by

g k = h[(u 1 v 1 ) (u 2 v 2 )] ⇥ [T k ('(u 1 ) '(v 1 )) T k ('(u 2 ) '(v 2 ))].
The main difficulty in managing this term is its non-linearity. Indeed, even if sgn(u 1 v 1 )=s g n ( u 2 v 2 ) a.e. on Γ,t h e r ei sn or e a s o nt oh a v eg k 0 nor to give a bound of g k /k 2 .I no r d e rt os t u d yt h eb e h a v i o ro fC k /k 2 ,w e decompose the integral on Γ into the integral on different subsets. Since from Lemma 3.6, sgn(u 1 v 1 )=sgn(u 2 v 2 ) a.e. on Γ,i nv i e wo ft h es y m m e t r y of g k with respect to u i and v i (i =1, 2), proving (3.50) is equivalent to prove

lim sup k!0 1 k 2 Z {u 1 v 1 >0} g k d 0.
We now split the set {x 2 Γ ; u 1 (x) v 1 (x) > 0} (up to a zero measure subset) into 4 subsets,

{u 1 v 1 > 0} = P 1 [ P 2 [ P 3 [ P 4 ,
where

P 1 := {'(u 1 ) '(v 1 ) k} \ {'(u 2 ) '(v 2 ) k}, P 2 := {0 < '(u 1 ) '(v 1 ) <k} \ {0 < '(u 2 ) '(v 2 ) <k}, P 3 = {'(u 1 ) '(v 1 ) k} \ {0 < '(u 2 ) '(v 2 ) <k}, P 4 := {0 < '(u 1 ) '(v 1 ) <k} \ {'(u 2 ) '(v 2 ) k}.
Since we have

T k ('(u 1 ) '(v 1 )) T k ('(u 2 ) '(v 2 )) = 0 a.e. on P 1 ,
we obtain that

1 k 2 Z P 1 g k d =0. (3.51)
As far as

R P 2 g k d is concerned, recalling that ' 2 C 1 (R) with ' 0 (t) 1 for any t 2 R gives 0 <u 1 v 1  '(u 1 ) '(v 1 ) <ka.e. on P 2 and 0 <u 2 v 2  '(u 2 ) '(v 2 ) <ka.e. on P 2 .
As a consequence, we deduce that

1 k 2 |g k | = 1 k 2 |h||(u 1 v 1 ) (u 2 v 2 )||('(u 1 ) '(v 1 )) ('(u 2 ) '(v 2 ))| khk L 1 (Γ) a.e. on P 2 .
Since P 2 ! 0 as k ! 0 a.e. on Γ, the Lebesgue Dominated Convergence Theorem leads to

lim k!0 1 k 2 Z P 2 g k d =0. (3.52)
We now study R P 3 g k d by splitting P 3 into P 3 \ {u 1 v 1 k} and P 3 \ {u 1 v 1 <k}.W i t ha l r e a d yu s e da r g u m e n t s ,w eh a v e 0 <u 2 v 2 <ka.e. on P 3 .

It follows that

[(u 1 v 1 ) (u 2 v 2 )] 0 a.e. on P 3 \ {u 1 v 1 k}, so that g k 0 a.e. on P 3 \ {u 1 v 1 k}.
On the other hand, we have

1 k 2 |g k | = 1 k 2 |h||(u 1 v 1 ) (u 2 v 2 )|(k ('(u 2 ) '(v 2 ))) khk L 1 (Γ) a.e. on P 3 \ {0 <u 1 v 1 <k}.
Since P 3 ! 0 as k ! 0, a.e. on Γ, the Lebesgue Dominated Convergence Theorem leads to

lim k!0 1 k 2 Z P 3 \{0<u 1 v 1 <k} g k d =0.
Noting that we can write Z

P 3 g k d = Z P 3 \{u 1 v 1 k} g k d + Z P 3 \{0<u 1 v 1 <k} g k d ,
we deduce that

lim sup k!0 1 k 2 Z P 3 g k d 0. (3.53)
At last, by writing

Z P 4 g k d = Z P 4 \{u 2 v 2 k} g k d + Z P 4 \{0<u 2 v 2 <k} g k d ,
and by proving with similar arguments that Z

P 4 \{u 2 v 2 k} g k d 0 and lim k!0 1 k 2 Z P 4 \{0<u 2 v 2 <k} g k d =0, yield lim sup k!0 1 k 2 Z P 4 g k d 0. (3.54)
The results (3.51)-(3.54) give

lim sup k!0 1 k 2 C k 0. (3.55)
Therefore, (3.34) holds true.

Step 2. In this step, we prove that u 1 = v 1 a.e. in Ω 1 and u 2 = v 2 a.e. in Ω 2 . We first show that u 1 = v 1 a.e. in Ω 1 .T od ot h i s ,w ec o n s i d e rt h ef u n c t i o n

✓ n (u 1 )T k ('(u 1 ) '(v 1 )) 2 L 1 (Ω 1 ) \ V 1 .
Since u 1 = v 1 =0on @Ω, we can apply Poincaré inequality which leads to

Z Ω 1 (✓ n (u 1 )) 2 T k ('(u 1 ) '(v 1 )) k 2 dx  C Z Ω 1 (✓ n (u 1 )) 2 ✓ rT k ('(u 1 ) '(v 1 )) k ◆ 2 dx + C Z Ω 1 (✓ 0 n (u 1 )) 2 |ru 1 | 2 T k ('(u 1 ) '(v 1 )) k 2 dx, (3.56)
where C>0 does not depend on k and n. The second integral on the right-hand side of (3.56) can be bounded by

Z Ω 1 (✓ 0 n (u 1 )) 2 |ru 1 | 2 T k ('(u 1 ) '(v 1 )) k 2 dx  1 n 2 Z {|u 1 |<2n} |ru 1 | 2 dx.
The integral on the right-hand side goes to zero as n goes to 1 by (3.10a) of Definition 3.2. This implies

lim n!1 lim k!0 Z Ω 1 (✓ 0 n (u 1 )) 2 |ru 1 | 2 T k ('(u 1 ) '(v 1 )) k 2 dx =0. (3.57)
For the first integral on the right-hand side of (3.56), we have

Z Ω 1 (✓ n (u 1 )) 2 ✓ rT k ('(u 1 ) '(v 1 )) k ◆ 2 dx  max s2[ 2n,2n] ' 0 (s) k 2 Z U k 1 ' 0 (u 1 ) |r'(u 1 ) r'(u 2 )| 2 dx.
The integral on the right-hand side of this inequality goes to 0 as k goes to 0 by (3.34). Thus, Z

Ω 1 {ϕ(u 1 )6 =ϕ(v 1 )} dx =l i m n!1 lim k!0 Z Ω 1 (✓ n (u 1 )) 2 T k ('(u 1 ) '(v 1 )) k 2 dx =0, (3.58)
that is, '(u 1 )='(v 1 ) a.e. in Ω 1 .S i n c e' 0 1,w eh a v eu 1 = v 1 a.e. in Ω 1 . As a consequence, 1 (u 1 )= 1 (v 1 ),t h a ti s ,u 1 = v 1 a.e. on Γ. From Lemma 3.6, sgn(u 1 v 1 )=s g n ( u 2 v 2 ) on Γ.S i n c eu 1 v 1 =0 a.e. on Γ,w ea l s oh a v eu 2 v 2 =0a.e. on Γ. Thus, u 2 = v 2 a.e. on Γ.

It only remains to prove that

u 2 = v 2 in Ω 2 . Consider the function ✓ n (u 2 )T k ('(u 2 ) '(v 2 )) 2 L 1 (Ω 2 ) \ H 1 (Ω 2 ),
which has a zero value a.e. on Γ since u 2 = v 2 a.e. on Γ.W ec a nt h e na p p l y Poincaré inequality which implies that

Z Ω 2 (✓ n (u 2 )) 2 T k ('(u 2 ) '(v 2 )) k 2 dx  C Z Ω 2 (✓ n (u 2 )) 2 ✓ rT k ('(u 2 ) '(v 2 )) k ◆ 2 dx + C Z Ω 2 (✓ 0 n (u 2 )) 2 |ru 2 | 2 T k ('(u 2 ) '(v 2 )) k 2 dx.
Using the same arguments to show (3.58), we conclude that

Z Ω 2 {ϕ(u 2 )6 =ϕ(v 2 )} dx =l i m n!1 lim k!0 Z Ω 2 (✓ n (u 2 )) 2 T k ('(u 2 ) '(v 2 )) k 2 dx =0. This implies '(u 2 )='(v 2 ) a.e. in Ω 2 . Therefore, since ' 0 1, u 2 = v 2 a.e. in Ω 2 .
This concludes the proof of the uniqueness of the renormalized solution of (3.1).

Part II Homogenization results

Chapter 4

Some properties of an elliptic periodic problem with an interfacial resistance

Introduction

In several models of partial differential equations, an imperfect contact on the interface between two materials gives rise to an interfacial resistance, often modelled by a jump of the solution on the interface, which is proportional to the normal derivative. For a heat diffusion model, this means that the jump of the temperature is proportional to the flux (see [START_REF] Carslaw | Conduction of heat in solids[END_REF] for a justification of the model).

When the coefficients are periodically oscillating, and the interface is "periodic, the asymptotic behaviour of the problem as " ! 0 has been widely studied in literature. In this case, the proportionality factor is of order " γ , and the homogenized problem depends on the value of .L e t u s b r i e fl y describe the domain, more precisely defined in Section 4, and the problem we are concerned about in this chapter (see Figure 4 1 is connected and defined by Ω ε 1 = Ω\Ω ε 2 , while the interface is given by Γ ε = @Ω ε 2 .

Let us recall the results for the case = 1 (see [START_REF] Donato | Homogenization of two heat conductors with an interfacial contact resistance.,A n a l .A p p l[END_REF], [START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF]), which motivate our study. For i =1, 2,s e t✓ i =

|Y i | |Y | , with Y 1 = Y \ Y 2 . Let u ε =( u ε 1 ,u ε 2 )
be the solution of problem (4.1) for = 1. Then, there exists

u 1 2 H 1 0 (Ω) such that 8 > > < > > : e u ε i *✓ i u 1 weakly in L 2 (Ω) , i =1, 2, A ε g ru ε 1 * A 0 1 ru 1 weakly in (L 2 (Ω)) N , A ε g ru ε 2 * A 0 2 ru 1 weakly in (L 2 (Ω)) N , (4.2)
where e • denotes the zero extension to the whole of Ω,a n du 1 is the unique solution of the problem ( div (A 0 ru 1 )=f in Ω,

u 1 =0 on @Ω. (4.3)
The homogenized matrix A 0 is defined as

A 0 = A 0 1 + A 0 2 ,A 0 i = 1 |Y | Z Y i A(y)( r λ i (y)) dy, i =1, 2, 8 2 R N ,
where λ i = λ | Y i , λ is the solution of 8 > > > > > > < > > > > > > : div(Ar λ )= div(A ) in Y 1 [ Y 2 , [Ar λ • n 1 ]=0 on Γ, Ar λ 1 • n 1 = h(y)[ λ ] on Γ, λ 1 Y periodic, M Γ ( λ 1 )=0, (4.4)
and [•] denotes the jump through Γ,t h et e r m div(A ) being intended as in (4.13)-(4.14) below.

For similar elliptic homogenization problems we refer to [START_REF] Canon | Homogénéisation d'un problème de diffusion en milieu composite avec barrière à l'interface[END_REF], [START_REF] Ene | On the microstructure models of porous media[END_REF], [START_REF] Ene | Model of diffusion in partially fissured media[END_REF], [START_REF] Lipton | Heat conduction in fine scale mixtures with interfacial contact resistance[END_REF], [START_REF] Lipton | Composites with imperfect interface[END_REF] and [START_REF] Pernin | Homogénéisation d'un problème de diffusion en milieu composite à deux composantes[END_REF]. More recently, a similar boundary condition has been considered in [START_REF] Lochner | Homogenization of linearized elasticity in a two-component medium with slip displacement conditions[END_REF], where the authors study the homogenization of linearized elasticity in a two-component medium with slip displacement conditions.

We also refer to [START_REF] Cabarrubias | Homogenization of a quasilinear elliptic problem with nonlinear Robin boundary conditions[END_REF] for the homogenization, by unfolding, of a quasilinear elliptic problem in perforated domains with Robin conditions on the interface, which motivates this study.

Indeed, in the homogenization of quasilinear elliptic problems, it is crucial for the homogenized matrix to have some kind of Lipschitz continuity to prove the uniqueness of the solution of the limit problem. For example, in [START_REF] Cabarrubias | Homogenization of a quasilinear elliptic problem with nonlinear Robin boundary conditions[END_REF]Theorem 3.3], it is proved that if a sequence of periodic matrices satisfies the assumption introduced by Chipot in [START_REF] Chipot | Elliptic equations. An introductory course[END_REF] (see Remark 4.12), then this assumption still holds true for the homogenized matrix field. To prove this result, a Meyers type estimate for the solution of the periodic cell problem is needed.

In this chapter, we present some results similar to Theorem 3.3 of [START_REF] Cabarrubias | Homogenization of a quasilinear elliptic problem with nonlinear Robin boundary conditions[END_REF], but in the case where the solution of the cell problem describing the homogenized matrix is the solution of problem (4.4), which present a jump, which corresponds to the homogenization of problem (4.1) in the case = 1.

To do that, we need first to prove a Meyers type estimate for the solution of the cell problem in this case, which makes use of a Meyers type result proved by T. Gallouët and A. Monier in [START_REF] Gallouët | On the regularity of solutions to elliptic equations[END_REF] for elliptic equations with nonhomogeneous Neumann boundary conditions. We also prove, for completeness, abo u n d e d n e s sr e s u l tf o rt h a ts o l u t i o n .

The main difficulty here comes from the fact that a weak solution of this problem is not an H 1 (Y )-function, since only the restriction of the solution to each component Y i belongs to H 1 (Y i ). Hence, we have to deal with a jump on the interface Γ, which gives rise to a boundary term in the variational formulation (see (4.15)). This implies that one cannot merely repeat the arguments of the classical case without jump in the proofs, but most of the steps are more delicate and need suitable arguments.

The present chapter is organised as follows:

In the next section, we introduce some definitions and we state the periodic problem in a two-component cell.

Section 4.3 is devoted to the Meyers type result. In Section 4.4, we prove some results for a rescaled version of (4.15), which will be needed for the main result of the last section.

Section 4.5 contains some additional properties adapted to the quasilinear case, where A = B(y, t) is a Carathéodory matrix field depending also on a real parameter t. More precisely, we prove an estimate of the difference of two homogenized matrices in terms of the difference of the oscillating ones. As a consequence, in Corollary 4.11 (see also Remark 4.12) we deduce a generalization of Theorem 3.3 of [START_REF] Cabarrubias | Homogenization of a quasilinear elliptic problem with nonlinear Robin boundary conditions[END_REF] to the case with jump.

Finally, in Section 4.6, we prove the boundedness of the solution to (4.15).

We b elieve that these results can b e very helpful not only to extend the homogenization results proved for problem (4.1) with = 1 to the quasilinear case, but also in the homogenization of other elliptic problems in progress (e.g. quasilinear problems with singularity or with L 1 data in two-component domains).

Recall that (4.8) defines a norm since a Poincaré-Wirtinger inequality holds on W per (Y 1 ).

From now on, for any measurable function u defined on Y ,f o ri =1 , 2, we denote by u i = u Y i ,t h er e s t r i c t i o no fu on Y i and by e • the zero extension to the whole of Y .

Let us introduce the space

H = {u 2 L 2 (Y ) | u 1 2 W per (Y 1 ),u 2 2 H 1 (Y 2 )}
equipped with the norm

kuk 2 H = kru 1 k 2 L 2 (Y 1 ) + kru 2 k 2 L 2 (Y 2 ) + ku 1 u 2 k 2 L 2 (Γ) . (4.9)
N.B. In the sequel, we identify the gradient ru of a function u in H with its absolutely continuous part g ru 1 + g ru 2 ,a n dw es e t

kruk 2 L 2 (Y \Γ) = kru 1 k 2 L 2 (Y 1 ) + kru 2 k 2 L 2 (Y 2 ) , (4.10)
so that, (4.9) reads

kuk 2 H = kruk 2 L 2 (Y \Γ) + ku 1 u 2 k 2 L 2 (Γ) . Remark 4.1. The norm given in (4.9) is equivalent to the norm kuk 2 Wper(Y 1 )⇥H 1 (Y 2 ) = ku 1 k 2 Wper(Y 1 ) + ku 2 k 2 H 1 (Y 2 ) , (4.11)
that is, there exist 2 positive constants c 1 ,c 2 ,d e p e n d i n go nY 1 ,Y 2 ,s u c ht h a t

c 1 kuk Wper(Y 1 )⇥H 1 (Y 2 ) kuk H  c 2 kuk Wper(Y 1 )⇥H 1 (Y 2 ) . (4.12)
The proof of the equivalence is similar to that of Proposition 2.6 of [START_REF] Monsurrò | Homogenization of a two-component composite with interfacial thermal barrier[END_REF], where the result is presented for the space

V 1 ⇥ H 1 (Y 2 ),w i t h V 1 = {v 2 H 1 (Y 1 ):v =0on @Y } with kvk V 1 = krvk L 2 (Y 1 )
.

In [START_REF] Monsurrò | Homogenization of a two-component composite with interfacial thermal barrier[END_REF], the estimates for

V 1 ⇥ H 1 (Y 2
) is obtained by using the Poincaré inequality in V 1 .T h eo n l yd i fference here is that one has to use the Poincaré-Wirtinger inequality on W per (Y 1 ) instead.

Our aim is to study some properties of the solution of the following problem:

8 > > > > > > > > < > > > > > > > > : div(Ar λ 1 )=G λ 1 in Y 1 , div(Ar λ 2 )=G λ 2 in Y 2 , Ar λ 1 • n 1 = Ar λ 2 • n 2 on Γ, Ar λ 1 • n 1 = h(y)( λ 1 λ 2 ) on Γ, λ 1 Y periodic, M Γ ( λ 1 )=0, (4.13)
where 2 R N and G λ i is defined by

hG λ i ,vi = Z Y i A rvdy, 8v 2 H 1 (Y i ),i =1, 2, (4.14)
which belongs to (H 1 (Y i )) 0 .

Remark 4.2. Observe that a Poincaré-Wirtinger inequality still holds in

W per (Y 1 ) if in (4.7) we replace the condition M Γ (u)=0by M Y 1 (u)=0 , where M Y 1 (u)= 1 |Y 1 | Z Y 1 udy.
Consequently, if we set

b λ i = λ i M Y 1 ( λ 1 ),i =1, 2, 8 2 R N , then b λ =(b λ 1 , b λ 2 ) is the unique solution of 8 > > > > > > > > < > > > > > > > > : div(Arb λ 1 )=G λ 1 in Y 1 , div(Arb λ 2 )=G λ 2 in Y 2 , Arb λ 1 • n 1 = Arb λ 2 • n 2 on Γ, Arb λ 1 • n 1 = h(y)(b λ 1 b λ 2 ) on Γ, b λ 1 Y periodic, M Y 1 (b λ 1 )=0
, which is a problem of the same type as (4.13).

Hence, all the results stated here for λ are still true for b λ .W ep r e s e n t here the problem under the form (4.13),s i n c ei ti st h ep r o b l e mw h i c hi s used for describing the homogenization of elliptic problems with a jump when = 1(see [START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF]- [START_REF] Donato | Homogenization of two heat conductors with an interfacial contact resistance.,A n a l .A p p l[END_REF]).

We make the following assumptions on A and h:

(A1) The matrix field A belongs to M (↵, ,Y), where 0 < ↵ < .

(A2) The function h is in L 1 (Γ) with 0 <h 0 <h(y) for a.e. y 2 Γ,f ors ome h 0 2 R.

Using (4.14), the variational formulation of (4.13) is 8 > > > > < > > > > :

Find λ =( λ 1 , λ 2 ) 2 H such that Z Y \Γ Ar λ rvdy+ Z Γ h(y)( λ 1 λ 2 )(v 1 v 2 ) d = Z Y \Γ A rvdy, for any v =(v 1 ,v 2 ) 2 H. (4.15)
The existence and uniqueness of the solution of (4.15) can be easily established by the Lax-Milgram theorem. In the next sections, we investigate some properties of the solution of (4.15).

A Meyers type estimate

We prove here that the gradient of the solution, which is in L 2 (Y \ Γ),h a sa better summability.

Theorem 4.3. Let 2 R N and let λ =( λ 1 , λ 2 )
2 H be the solution of (4.15).T h e nf o ra n y 2 R N ,t h e r ee x i s t sp i > 2, i =1, 2,s u c ht h a t

λ i 2 W 1,p i (Y i ). (4.16)
Moreover, for i =1, 2,f o re v e r y2  q i  p i ,t h e r ee x i s t sap o s i t i v ec o n s t a n t c i ,d e p e n d e n to n↵, , q i ,a n dY i ,s u c ht h a t

kr λ i k L q i (Y i )  c i | |. (4.17)
To prove this theorem, we apply the following Meyers type result, which has been proved by Gallouët and Monier in [START_REF] Gallouët | On the regularity of solutions to elliptic equations[END_REF].

( div(Aru)=F in Ω Aru • n + hu =0 on @Ω, (4.18)
where A 2 M (↵, , Ω),f o rs o m e0 < ↵ < ,a n dh 2 L 1 (@Ω) with 0 <h 0  h(y),f o ra . e .y 2 @Ω,f o rs o m er e a ln u m b e rh 0 . Define

2 ⇤ = 8 < : 2N N 2 , if N 3 any number in (2, +1), if N =2. (4.19)
Then there exists p 0 > 2 (with p 0 < 2 ⇤ for N 3)s u c ht h a tf o re v e r y 2 <q<p 0 ,i fF 2 (W 1,q 0 (Ω)) 0 then u 2 W 1,q (Ω),a n df o rs o m eC>0,

kuk W 1,q (Ω)  CkF k (W 1,q 0 (Ω)) 0 , (4.20)
where C depends on ↵, , Ω,a n dq.

Remark 4.5.

1. It is easy to check that Theorem 4.4 still holds if Ω = Y 1 and we prescribe the boundary condition only on Γ,andtheY periodicity of u.

2. Theorem 4.4 also holds if Ω = Y 2 as defined in (4.5) since we can apply it to every connected component Y j 2 of Y 2 . We establish the following lemma which is needed to prove Theorem 4.3.

Lemma 4.6. Let 2 R N and let λ =( λ 1 , λ 2 ) be the solution to (4.15).

For i =1, 2,s e t t = 8 > < > : 2(N 1) N 2 if N 3,
any number in (2, +1) if N =2, (4.21)
and let µ λ 1 and µ λ 2 be the linear functionals defined by

µ λ 1 (v 1 )= Z Γ h λ 2 v 1 d , 8v 1 2 W 1,t 0 (Y 1 ), (4.22)
and

µ λ 2 (v 2 )= Z Γ h λ 1 v 2 d , 8v 2 2 W 1,t 0 (Y 2 ), (4.23)
respectively. Then, for i =1, 2,f o re v e r y1  r i  t we have

µ λ i 2 (W 1,r 0 i (Y i )) 0 , (4.24)
and for some b i > 0,

kµ λ i k (W 1,r 0 i (Y i )) 0  b i | |, 8 2 R N . (4.25)
Proof. We first observe that

k λ k Wper(Y 1 )⇥H 1 (Y 2 )  c 1 1 k λ k H  c| |, (4.26)
for some positive constant c that depends on ↵, , h 0 , Y 1 ,andY 2 .I n d e e d ,t h e first inequality is a consequence of the equivalence of the norms in (4.12). To prove the second inequality, take λ =( λ 1 , λ 2 ) as a test function in (4.15), which gives

Z Y \Γ Ar λ r λ dy + Z Γ h(y)( λ 1 λ 2 )( λ 1 λ 2 ) d = Z Y \Γ A r λ dy.
Then, by the ellipticity of A,t h ea s s u m p t i o n so nh,a n dH ö l d e r ' si n e q u a l i t y , we obtain

min{↵,h 0 }k λ k 2 H  ↵kr λ k 2 L 2 (Y \Γ) + h 0 k λ 1 λ 2 k 2 L 2 (Γ)  Z Y \Γ Ar λ r λ dy + Z Γ h(y)( λ 1 λ 2 )( λ 1 λ 2 ) d = Z Y \Γ A r λ dy  | |kr λ k L 2 (Y \Γ)  | |k λ k H .
Hence, (4.26) holds true. Note that for i =1, 2, r i  t,w eh a v er 0 i t 0 and hence v i 2 W 1,r 0 i (Y i ) ✓ W 1,t 0 (Y i ). Moreover, we have the following continuous embeddings,

H 1 (Y 1 ) ⇢ L t (Γ) ✓ L r 2 (Γ), H 1 (Y 2 ) ⇢ L t (Γ) ✓ L r 1 (Γ).
Then, since λ i 2 H 1 (Y i ) for i =1 , 2, it follows that for some positive constants a 1 and a 2 ,

λ 1 2 L r 2 (Γ) with k λ 1 k L r 2 (Γ)  a 1 k λ 1 k H 1 (Y 1 ) , (4.27)
λ 2 2 L r 1 (Γ) with k λ 2 k L r 1 (Γ)  a 2 k λ 2 k H 1 (Y 2 ) . (4.28)
Note that for i =1 , 2, µ λ i is linear and using the Trace Theorem, Hölder's inequality, (4.27), (4.28), and (4.26), we have

|hµ λ 1 ,v 1 i| khk L 1 (Γ) k λ 2 k L r 1 (Γ) kv 1 k L r 0 1 (Γ)  a 2 khk L 1 (Γ) k λ 2 k H 1 (Y 2 ) kv 1 k W 1,r 0 1 (Y 1 )  b 1 | |kv 1 k W 1,r 0 1 (Y 1 ) , and
|hµ λ 2 ,v 2 i| khk L 1 (Γ) k λ 1 k L r 2 (Γ) kv 2 k L r 0 2 (Γ)  a 1 khk L 1 (Γ) k λ 1 k H 1 (Y 1 ) kv 2 k W 1,r 0 2 (Y 2 )  b 2 | |kv 2 k W 1,r 0 2 (Y 2 ) ,
for some positive constants b 1 ,b 2 which depend on ↵, , h, Y 1 , Y 2 , r 1 ,andr 2 . These imply (4.24) and (4.25).

Proof of Theorem 4.3: Fix 2 R N . From (4.13)-(4.14), we can write the following system for λ 1 :

8 > > > < > > > : div(Ar λ 1 )=G λ 1 in Y 1 , Ar λ 1 • n 1 + h λ 1 = h λ 2 on Γ, λ 1 Y periodic, M Γ ( λ 1 )=0, (4.29)
while for λ 2 we have

( div(Ar λ 2 )=G λ 2 in Y 2 , Ar λ 2 • n 2 + h λ 2 = h λ 1 on Γ. (4.30)
System (4.29) can be rewritten as

8 > > > < > > > : div(Ar λ 1 )=G λ 1 + µ λ 1 in Y 1 , Ar λ 1 • n 1 + h λ 1 =0 on Γ, λ 1 Y periodic, M Γ ( λ 1 )=0, (4.31
8q i 2 (2,p i 0 ), if F i 2 (W 1,q 0 i (Y i )) 0 , then the solution λ i 2 W 1,q i (Y i ), (4.34
) and due to (4.20), for some c i > 0,

8q i 2 (2,p i 0 ), k λ i k W 1,q i (Y i )  c i kF i k (W 1,q 0 i (Y i )) 0 . (4.35)
Let now t be as in (4.21). Set

s i =min{t, p i 0 },i =1, 2.
From Lemma 4.6 (written for t = s i and r i = p i ), we deduce that

µ λ i 2 (W 1,p 0 i (Y i )) 0 , 8p i 2 (2,s i ),i =1, 2. (4.36)
Moreover, for i =1 , 2,s i n c eA 2 L 1 (Y i ),f o re v e r yv i 2 W 1,p 0 i (Y i ),w eh a v e by Hölder's inequality,

|hG λ i ,v i i| = Z Y i A rv i dy kA k L p i (Y i ) krv i k L p 0 i (Y i )  d i | |kv i k W 1,p 0 i (Y i ) ,
where d i depends on and Y i . This implies that for i =1, 2,foreveryp i <s i ,

G λ i 2 (W 1,p 0 i (Y i )) 0 with kG λ i k (W 1,p 0 i (Y i )) 0  d i | |. (4.37)
This, together with (4.33) and (4.36), implies

F i 2 (W 1,p 0 i (Y i )) 0 , 8p i 2 (2,s i ),
which, due to (4.34), proves the first statement of the theorem for every p i in (2,s i ),s i n c es i <p i 0 . Fix now, for i =1 , 2,s u c hap i and let us prove the second statement. For every q i 2 (2,p i ), i =1, 2,f r o m( 4 . 3 5 ) ,

kr λ i k L q i (Y i )  c i kF i k (W 1,q 0 i (Y i )) 0 .
Hence, to prove (4.17), it suffices to show that for some positive constant c i ,

kF i k (W 1,q 0 i (Y i )) 0  c i | |, 8 2 R N ,i=1, 2. (4.38)
Since q i <p i <s i , from (4.33), (4.25) of Lemma 4.6 (written for r i = q i ), and (4.37) (written for p i = q i ), we have

kF i k (W 1,q 0 i (Y i )) 0 kG λ i k (W 1,q 0 i (Y i )) 0 + kµ λ i k (W 1,q 0 i (Y i )) 0  c i | |.
This shows (4.38) and ends the proof.

The rescaled problem

In this section, we rescale the function λ and prove that it satisfies a rescaled version of (4.15) in R N .

Let Ω be a connected open set of R N with a Lipschitz continuous boundary @Ω.U n d e rt h en o t a t i o n so fS e c t i o n2 ,f o rk 2 Z N ,letk l =(k 1 l 1 ,k 2 l 2 ,...,k N l N ) and define

Y k = k l + Y and Y k i = k l + Y i ,i =1, 2, and Y # 2 = [ k2Z N Y k 2 , Γ # = @Y # 2 ,Y # 1 = R N \ Y # 2 . (4.39)
Let {"} be a sequence of positive real numbers that converges to zero.

Set G ε = {k 2 Z N | "Y k 2 ⇢ Ω}. We define the sets Ω ε 1 , Ω ε 2 ,a n dΓ ε as Ω ε 2 = [ k2Gε "Y k 2 with Γ ε = @Ω ε 2 , and Ω ε 1 = Ω \ Ω ε 2 . (4.40)
In what follows, for any measurable function u defined in Ω,w es e t

u i = u Ω ε i , the restriction of u in Ω ε i ,i =1, 2,
and we denote by e u i ,t h ez e r oe x t e n s i o nt oΩ of any function u i defined in Ω ε i . N.B. In the sequel, we identify a function in H 1 (Y 2 ) with its extension by periodicity to Y # 2 .M o r e o v e ra si n(4.10),f o ra n yf u n c t i o nu such that u i 2 H 1 (Ω ε i ), i =1, 2 we identify the gradient with its absolutely continuous part, and we set

kruk 2 L 2 (Ω\Γ ε ) = kru 1 k 2 L 2 (Ω ε 1 ) + kru 2 k 2 L 2 (Ω ε 2 )
. Let now λ the solution of (4.15) and set Proof. We adapt here the proof of Theorem 4.28 of [START_REF] Cioranescu | An introduction to homogenization[END_REF] to the case with a jump on Γ.F i r s t ,f o ri =1 , 2,l e t i 2 C 1 per (Y i ),a n ds e tv i = i M Γ ( 1 ). Then v =(v 1 ,v 2 ) 2 H and choosing v as test function in (4.15), we have

Z Y \Γ Ar λ r dy + Z Γ h(y)( λ 1 λ 2 )( 1 2 ) d = Z Y \Γ A r dy. (4.44)
Let ' be a function in L 2 (R N ) with a compact support K and

' i 2 H 1 (Y i ), i =1, 2
.L e t{Y j } m j=1 be a finite set of translated cells of Y such that

K ⇢ m [ j=1 Y j .
Let Γ j ⇢ Y j be the corresponding translated sets of Γ, j =1,...,m. Let {✓ j } m j=0 be a partition of unity associated to this covering, i.e., a family of functions such that, 8 > < > :

✓ j 2 D(R N ), 0  ✓ j  1, 8j 2 {0, 1,...,m}, m X j=0 ✓ j =1, supp ✓ j ⇢ Y j , 8j 2 {1,...,m}, supp ✓ 0 ⇢ R N \ K. (4.45) Since ' =0in supp ✓ 0 ,w eh a v e ' = ' m X j=1 ✓ j = m X j=1 ('✓ j ) in R N . (4.46)
We denote the extension by p erio dicity of '✓ j by ('✓ j ) # , j =1 ,...,m.B y (4.45), ✓ j =0in a neighborhood of @Y j , j =1,...,m,andhence,fori =1, 2,

(' i ✓ j ) # 2 C 1 per (Y i ).
Writing (4.44) for i =( ' i ✓ j ) # , i =1 , 2,a n dt h e na d d i n gt h er e s u l t i n g equations for j =1,...,m,w eo b t a i n ,

m X j=1 Z Y \Γ Ar λ r('✓ j ) # dy + m X j=1 Z Γ h(y)( λ 1 λ 2 )((' 1 ✓ j ) # (' 2 ✓ j ) # ) d = m X j=1 Z Y \Γ A r('✓ j ) # dy. ( 4
X j=1 Z Y \Γ Ar λ r('✓ j ) # dy = m X j=1 Z Y j \Γ j Ar λ r('✓ j ) dy = Z R N \Γ # Ar λ r' dy. (4.48) Similarly, m X j=1 Z Y \Γ A r('✓ j ) # dy = Z R N \Γ # A r' dy. (4.49)
Also, from (4.45), and (4.46), we have by periodicity, for the integral on the boundary,

m X j=1 Z Γ h(y)( λ 1 λ 2 )((' 1 ✓ j ) # (' 2 ✓ j ) # ) d = m X j=1 Z Γ j h(y)( λ 1 λ 2 )(' 1 ✓ j ' 2 ✓ j ) d = m X j=1 Z Γ # h(y)( λ 1 λ 2 )(' 1 ✓ j ' 2 ✓ j ) d = Z Γ # h(y)( λ 1 λ 2 )(' 1 ' 2 ) d (4.50)
Finally, combining (4.47) with (4.48)-(4.50), we get the desired result.

The main result of this section is the following one:

Theorem 4.8. For =( 1 , 2 ) 2 H 1 (Y 1 ) ⇥ H 1 (Y 2 ),s e t ε (x)=" ⇣ x " ⌘
for a.e. x 2 Ω.T h e n ,f o re v e r y' 2 D(Ω) and for " sufficiently small, the function w λ ε defined by (4.42) satisfies Z

Ω\Γ ε A ε rw λ ε r(' ε ) dx = " 1 Z Γ ε h ε (x)(w λ 1,ε w λ 2,ε )( ε 1 ε 2 )' d , (4.51)
where

A ε (x)=A ⇣ x " ⌘ and h ε (x)=h ⇣ x " ⌘ . (4.52)
Proof. Let ' 2 D(Ω) and " sufficiently small such that supp ' \ "Γ # ⇢ Γ ε . Set ' ε (y)='("y) and

v i (y)=' ε (y) i (y),i =1, 2.
Using v =(v 1 ,v 2 ) as a test function in (4.43), we have Z

R N \Γ # Ar λ r(' ε ) dy + Z Γ # h(y)( λ 1 λ 2 )( 1 2 )' ε d = Z R N \Γ # A r(' ε ) dy.
Using Proposition 4.7 and making the change of scale x = "y,byconstruction we have

" N Z Ω\Γ ε A ⇣ x " ⌘ r y λ ⇣ x " ⌘ r y ⇣ ' ε ⇣ x " ⌘ ⇣ x " ⌘⌘ dx + " N +1 Z Γ ε h ⇣ x " ⌘⇣ λ 1 ⇣ x " ⌘ λ 2 ⇣ x " ⌘⌘ ⇣ 1 ⇣ x " ⌘ 2 ⇣ x " ⌘⌘ ' ε ⇣ x " ⌘ d = " N Z Ω\Γ ε A ⇣ x " ⌘ r y ⇣ ' ε ⇣ x " ⌘ ⇣ x " ⌘⌘ dx. (4.53)
Observe now that

r y λ ⇣ x " ⌘ = r x λ ε (x), r y ⇣ x " ⌘ = r x ε (x), r y ' ε ⇣ x " ⌘ = "r x '(x).
It follows that

r y ⇣ ' ε ⇣ x " ⌘ ⇣ x " ⌘⌘ = ' ε ⇣ x " ⌘ r y ⇣ x " ⌘ + ⇣ x " ⌘ r y ' ε ⇣ x " ⌘ = '(x)r x ε (x)+ ε (x)r x '(x) = r x ('(x) ε (x)).
Then, we can rewrite (4.53) as

" N Z Ω\Γ ε A ε r λ ε r(' ε ) dx + " N 1 Z Γ ε h ε (x)( λ 1,ε λ 2,ε )( ε 1 ε 2 )' d = " N Z Ω\Γ ε A ε r(' ε ) dx.
Multiplying both sides of this equation by " N and using (4.42) we obtain (4.51).

Further properties for a quasilinear case

In this section, we want to prove that some estimates providing the uniqueness for a class of quasilinear problems in a periodic two-component domain, remain valid after the homogenization process.

More precisely, we show that if a sequence of periodic matrices satisfies the assumption introduced by Chipot in [START_REF] Chipot | Elliptic equations. An introductory course[END_REF] (see (4.62)), then this assumption still holds true for the homogenized matrix field. To do that, we combine the Meyer-like estimate obtained in Theorem 4.3 with some ideas from [START_REF] Cabarrubias | Homogenization of a quasilinear elliptic problem with nonlinear Robin boundary conditions[END_REF].

Lemma 4.9. There exists p>2 such that for i =1 , 2, w λ i 2 W 1,p (Y i ). Furthermore, for al l q 2 [2,p],t h e r ee x i s t sc>0,d e p e n d i n go n↵, ,Y 1 ,Y 2 , and q such that

krw λ k L q (Y \Γ)  c| |, for every 2 R N .
(4.54)

In addition,

lim ε!0 krw λ ε k L q (Ω\Γ ε )  c |Ω| 1 q |Y | 1 q | | for every 2 R N . (4.55)
Proof. From Theorem 4.3, for i =1 , 2,w ec a nfi n dp i , 2 <p i < 2 ⇤ (where 2 ⇤ is defined in (4. [START_REF] Boccardo | Unicité de la solution de certaines équations elliptiques non linéaires. (Uniqueness of the solution of some nonlinear elliptic equations)[END_REF])), such that λ i 2 W 1,p i (Y i ). It follows that for p = min{p 1 ,p 2 }, w λ i 2 W 1,p (Y i ), i =1, 2.F u r t h e r m o r e ,f o re v e r y2  q  p,there exists a positive constant b such that

kr λ k L q (Y \Γ)  b| |.
This, together with (4.41), gives (4.54) with c = b +1.

Moreover, since the function | ry w λ i | q i is a Y periodic function in L 1 (Y ), from classical results on functions of the form f (x/") (see for instance [33, Chapter 2]), we get

lim ε!0 k rw λ i,ε k q i L q i (Ω) =lim ε!0 Z Ω ry w λ i ⇣ x " ⌘ q i dx = |Ω| |Y | Z Y i
|r y w λ i (y)| q i dy.

(4.56) This gives, using (4.54),

lim ε!0 krw λ ε k L q (Ω\Γ ε ) = |Ω| 1 q |Y | 1 q krw λ k L q (Y \Γ)  c |Ω| 1 q |Y | 1 q | |,
which completes the proof.

For A 2 M (↵, ,Y),l e tu si n t r o d u c en o wt h eh o m o g e n i z e dm a t r i xA 0 (see [START_REF] Donato | Homogenization of two heat conductors with an interfacial contact resistance.,A n a l .A p p l[END_REF]), defined as

A 0 = A 0 1 + A 0 2 ,A 0 i = 1 |Y | Z Y i A(y)rw λ i (y) dy, i =1, 2, 8 2 R N , (4.57)
where w λ is defined by (4.41).

From classical results (see for instance (see for instance [START_REF] Cioranescu | An introduction to homogenization[END_REF]Chapter 6]) we know that the following convergences hold, for i =1, 2: 8 < :

f λ i,ε ! 0 strongly in L 2 (Ω), A εr w λ i,ε * A 0 i weakly in [L 2 (Ω)] N , (4.58)
where A ε is defined in (4.52).

The following result extends to the periodic case with a jump on Γ a similar result for a fixed domain, proved in the general framework the Gconvergence in Colombini-Spagnolo [START_REF] Colombini | Sur la convergence de solutions d'équations paraboliques[END_REF] and for the H-convergence in [START_REF] Boccardo | Homogénéisation de problèmes quasilinéaires[END_REF]. As i m p l e rp r o o ff o rt h ec a s eo fap e r i o d i cm a t r i xfi e l da n dap e r i o d i c a l l y perforated domain, has been given in [START_REF] Cabarrubias | Homogenization of a quasilinear elliptic problem with nonlinear Robin boundary conditions[END_REF]Proposition 3.2]. We adapt here to the case with jump some arguments used therein. Let us show that for every ,µ 2 R N ,w eh a v e

(B ε D ε )( rw λ,B 1,ε rw µ, t D 1,ε + rw λ,B 2,ε rw µ, t D 2,ε ) * (B 0 D 0 ) µ (4.60)
weakly in L 1 (Ω).T od ot h i s ,w efi r s tp r o v et h a tf o ra n y' 2 D(Ω),

lim ε!0 Z Ω\Γ ε (B ε D ε )rw λ,B ε rw µ, t D ε ' dx = Z Ω (B 0 D 0 ) µ' dx. (4.61)
Let ' 2 D(Ω).W eh a v e

I ε = Z Ω\Γ ε (B ε D ε )rw λ,B ε rw µ, t D ε ' dx = Z Ω\Γ ε B ε rw λ,B ε rw µ, t D ε ' dx Z Ω\Γ ε t D ε rw µ, t D ε rw λ,B ε ' dx.
From Theorem 4.8, written for = w µ, t D ,w eo b t a i n

Z Ω\Γ ε B ε rw λ,B ε rw µ, t D ε ' dx = Z Ω\Γ ε B ε rw λ,B ε r(w µ, t D ε ') dx Z Ω\Γ ε B ε rw λ,B ε w µ, t D ε r' dx = " 1 Z Γ ε h ε (x)(w λ,B 1,ε w λ,B 2,ε )(w µ, t D 1,ε w µ, t D 2,ε )' d Z Ω\Γ ε B ε rw λ,B ε w µ, t D ε r' dx.
Similarly, from Theorem 4.8, written for = w λ,B ,w eh a v e Z

Ω\Γ ε t D ε rw µ, t D ε rw λ,B ε ' dx = Z Ω\Γ ε t D ε rw µ, t D ε w λ,B ε r' dx " 1 Z Γ ε h ε (x)(w µ, t D 1,ε w µ, t D 2,ε )(w λ,B 1,ε w λ,B 2,ε )' d .
Then

I ε = Z Ω\Γ ε B ε rw λ,B ε w µ, t D ε r' dx + Z Ω\Γ ε t D ε rw µ, t D ε w λ,B ε r' dx.
From (4.57), (4.42), and (4.58) (written for A = B), we have

lim ε!0 Z Ω\Γ ε B ε rw λ,B ε w µ, t D ε r' dx =lim ε!0 2 X i=1 Z Ω B ε rw λ,B i,ε (µ • x ] µ, t D i,ε )r' dx = 2 X i=1 Z Ω B 0 i (µ • x)r' dx = Z Ω B 0 µ' dx.
Similarly,

lim ε!0 Z Ω\Γ ε t D ε rw µ, t D ε w λ,B ε r' dx = Z Ω D 0 µ' dx.
It follows that lim ε!0

I ε = Z Ω (B 0 D 0 ) µ' dx.
This proves (4.61).

By Lemma 4.9, there exists ⌘ > 0 such that (B ε D ε ) rw λ,B ε rw µ, t D ε is bounded in L 1+η (Ω). This, together with (4.61), gives us (4.60).

Noting that B 0 and D 0 are constants, and kB ε D ε k L 1 (Ω) = kB Dk L 1 (Y ) , using the Cauchy-Schwarz inequality, from (4.55) written for q =2 and (4.60) we have,

|Ω||(B 0 D 0 ) µ| = k(B 0 D 0 ) µk L 1 (Ω)  lim inf ε!0 (B ε D ε )( rw λ,B 1,ε rw µ, t D 1,ε + rw λ,B 2,ε rw µ, t D 2,ε ) L 1 (Ω) kB Dk L 1 (Y ) lim ε!0 ⇣ kr w λ,B ε k L 2 (Ω\Γ ε ) krw µ, t D ε k L 2 (Ω\Γ ε ) ⌘  c 2 |Ω| |Y | kB Dk L 1 (Y ) | ||µ|.
It follows that for some c 0 depending on ↵, ,Y,Y 1 ,a n dY 2 ,

|(B 0 D 0 ) µ|  c 0 kB Dk L 1 (Y ) | ||µ|.
Writing this inequality with = e j and µ = e k , where {e j } N j=1 is the canonical basis, gives the desired result.

As a consequence, we have the following corollaries: Let A 0 be the corresponding homogenized matrix to A given by (4.57).T h e n He proved that under these assumptions, which are weaker then the Lipschitzcontinuity, the corresponding quasilinear problem with Dirichlet boundary conditions admits a unique solution. Consequently, from Corol lary 4.11 we can derive that under these assumptions on the periodic matrix field, also the homogenized matrix has the same property, so that the homogenized problem has a unique solution. This is crucial in homogenization, since it implies that the whole sequences under consideration converge (and not only subsequences). A similar result has been proved in in [START_REF] Cabarrubias | Homogenization of a quasilinear elliptic problem with nonlinear Robin boundary conditions[END_REF]Theorem 3.3] for the homogenization of a quasilinear problem in perforated domains with some nonlinear Robin conditions. As mentioned in the introduction, we believe that this can be applied to many other interesting homogenization problems. Corollary 4.13. Suppose that the matrix field A(y, t) satisfies (P1) and (P2) from Corollary 4.11 and (P4) A(y, t) is locally Lipschitz with respect to the second variable, that is, for every r>0,t h e r ee x i s t sap o s i t i v ec o n s t a n tM r such that |A(y, s) A(y, t)|  M r |s t| 8s, t 2 ( r, r).

Then the homogenized matrix A 0 is also locally Lipschitz, that is, for every r>0,t h e r ee x i s t sap o s i t i v ec o n s t a n tC r such that |A 0 (s) A 0 (t)|  C r |s t| 8s, t 2 ( r, r).

Proof. It follows again from Theorem 4.10 written for B = A(•,s) and D = A(•,t).

Boundedness of the solution

To complete the study done in the previous sections, adapting classical arguments from G. Stampacchia [START_REF] Stampacchia | Le problème de Dirichlet pour les équations el liptiques du second ordre à coefficients discontinus[END_REF], we prove here that for every ,thesolution λ of (4.15) belongs to L 1 (Y ).

Theorem 4.14. For any 2 R N ,i f λ =( λ 1 , λ 2 ) 2 H is the solution of (4.15),t h e n λ i 2 L 1 (Y i ), i =1, 2. Proof. Fix 2 R N .D e fi n ef o ra n yh 1,

A h = {y 2 Y \ Γ | λ (y) h}.
(4.64)

Let g h (x)=(x h) + , where

(x h) + = 8 < : x h, if x h>0 0, if x h  0.
Using g h ( λ ) as a test function in (4.15), we have

Z Y \Γ Ar λ rg h ( λ ) dy + Z Γ h(y)( λ 1 λ 2 )(g h ( λ 1 ) g h ( λ 2 )) d = Z Y \Γ
A rg h ( λ ) dy. Let us first show that if w and z are two functions defined on Γ,w eh a v e

(w z)[g h (w) g h (z)] [g h (w) g h (z)] 2 . ( 4.66)
If w h and z h,t h e ng h (w)=w h and g h (z)=z h. Thus

(w z)[g h (w) g h (z)] = [(w h) (z h)][g h (w) g h (z)] = [g h (w) g h (z)] 2 .
When w  h and z  h,w eh a v eg h (z)=0=g h (w), and it follows that

(w z)[g h (w) g h (z)] = 0 = [g h (w) g h (z)] 2 .
If w  h and z h,t h e ng h (w)=0and g h (z)=z h.H e n c e ,

(w z)[g h (w) g h (z)] = (w z)[ g h (z)] = (z w)g h (z) (z h)g h (z)=[g h (z)] 2 =[g h (w) g h (z)] 2 .
Finally, by symmetry, we also have the result when w h and z  h. This proves (4.66).

By the ellipticity of A,t h ea s s u m p t i o no nh, (4.66) (written for w = λ 1 and z = λ 2 ), and Hölder's inequality, we obtain from (4.65), min{↵,h 0 }k( λ h)

+ k 2 H  ↵kr( λ h) + k 2 L 2 (Y \Γ) + h 0 k( λ 1 h) + ( λ 2 h) + k 2 L 2 (Γ)  Z Y \Γ A r( λ 1 h) + dy = Z A h A r( λ h) + dy  | ||A h | 1/2 kr( λ h) + k L 2 (Y \Γ)  | ||A h | 1/2 k( λ h) + k H .
Then from the equivalence of the two norms in (4.12), for some positive constant C,

k( λ h) + k 2 Wper(Y 1 )⇥H 1 (Y 2 )  c 2 1 k( λ h) + k 2 H  C|A h |. (4.67)
By the Sobolev embeddings, we have from (4.67),

✓Z Y 1 [( λ 1 h) + ] 2 ⇤ dy ◆ 2/2 ⇤ + ✓Z Y 2 [( λ 2 h) + ] 2 ⇤ dy ◆ 2/2 ⇤  C|A h |, (4.68)
where 2 ⇤ is defined in (4.19). For r h,w eha v efrom(4.64)thatA r ✓ A h . It follows that

( λ i h) + = λ i h r h in A r . Thus, (r h) 2 |A r | 2/2 ⇤  ✓Z Ar [( λ 1 h) + ] 2 ⇤ dy ◆ 2/2 ⇤ + ✓Z Ar [( λ 2 h) + ] 2 ⇤ dy ◆ 2/2 ⇤  C|A h |.
Fix k 0 > 1 and for h k 0 ,d e fi n e'(h)=|A h |. Then ' is decreasing and for all r>h ,

'(r)  C (r h) 2 ⇤ ('(h)) 2 ⇤ /2 . Since 2 ⇤ 2 > 1
k λ i k L 1 (Y i )  k 0 + d.

Chapter 5

Homogenization results for quasilinear elliptic problems in a two-component domain with L 1 data

Introduction

The main goal of this chapter is to study the homogenization of a class of quasilinear elliptic problems in a two-component domain with L 1 data and a matrix field not necessarily bounded with respect to the solution.

To describe the domain, we first introduce the reference cell Y , which has two components Y 1 and Y 2 , with Y 2 ⇢ Y and Y 1 = Y \ Y 2 ,a n dt h ei n t e r f a c e between them denoted by Γ. The two-component domain, denoted Ω,canbe written as the union

Ω ε 1 [ Ω ε 2 [ Γ ε , where Ω ε 2 is a disjoint union of "-periodic translated sets "Y 2 with Ω ε 2 ⇢ Ω, Ω ε 1 = Ω \ Ω ε 2 ,andΓ ε is the interface between Ω ε 1 and Ω ε 2 .
We prescrib e a Dirichlet b oundary condition on @Ω, while on Γ ε we assume that the flux is continuous and proportional to the jump of the solution, with a proportionality function of order " 1 . This models an imperfect contact between the two components Ω ε 1 and Ω ε 2 (see [START_REF] Carslaw | Conduction of heat in solids[END_REF] for the justification of this model).

To be more precise, the elliptic problem has the following form:

8 > > > > > > > < > > > > > > > : div ⇣ A ⇣ x " ,u ε ⌘ ru ε ⌘ = f in Ω ε 1 [ Ω ε 2 , h A ⇣ x " ,u ε ⌘ ru ε i • ⌫ ε =0 on Γ ε , ⇣ A ⇣ x " ,u ε ⌘ ru ε ⌘ ⌫ ε = " 1 h ε (x)[u ε ] on Γ ε , u ε =0 on @Ω, (5.1)
where ⌫ ε is the unit outward normal to Ω ε 1 , [•] denotes the jump on Γ ε ,a n d f 2 L 1 (Ω). Moreover, we do not prescribe a restricted growth assumption on the matrix field A(y, t) with respect to t (see assumption (A3.2) in the next section).

In general, in the study of homogenization of (5.1), the proportionality assumption on Γ ε depends on " γ (instead of " 1 ), where  1. The homogenization then has 3 cases: ). The major difference between these cases is the corresponding cell problem. In this chapter, we consider the case = 1, which has the particularity that also the cell problem presents a jump on the reference interface. The remaining cases will be studied in a forthcoming paper.

2 ( 1, 1], = 1,a n d 2 ( 1, 1
We consider a quasilinear term since for several comp osite materials, the thermal conductivity depends, nonlinearly, on the temperature itself. Some of the dependence are monotone, for example, glass or wood, the conductivity of which nonlinearly increases with the temperature, while for ceramics, the dependence is decreasing. There are also cases where the dependence is not monotone, for instance, aluminium or semi-conductors (see [START_REF] Bendib | Homogénéisation d'une classe de problèmes non linéaires avec des conditions de Fourier dans des ouverts perforés[END_REF]15] for more details).

On the other hand, taking an L 1 data is motivated by thermo-dynamics. Athermo viscoelasticit ymodelthatconsidersthenonlinearmec hanicaldissipation (which depends on the square of the gradient of the displacement), by the methods in [START_REF] Germain | Cours de mécanique des milieux continus[END_REF][START_REF] Suquet | Plasticité et homogénéisation[END_REF] gives a balance energy equation, where the right-hand side is expected to be in L 1 .

Since we are considering an L 1 data, we cannot work in a variational setting. Hence, we choose the appropriate framework of renormalized solution for our problem, which provides existence, stability and, with some additional assumptions, uniqueness results.

The notion of renormalized solution was first introduced by R. J. DiPerna and P. L. Lions in [START_REF] Diperna | On the Cauchy problem for Boltzmann equations: Global existence and weak stability[END_REF] for first order equations. It was then further developed for elliptic equations with L 1 data by F. Murat in [START_REF] Murat | Soluciones renormalizadas de EDP elipticas no lineales[END_REF] and by P. L. Lions and F. Murat in [START_REF] Lions | Sur les solutions renormalisées d'équations elliptiques[END_REF], and for elliptic equations with measure data by G. Dal Maso et al. in [START_REF] Maso | Renormalized solutions of elliptic equations with general measure data[END_REF].

The existence of a renormalized solution to (5.1) for fixed " is proved in [START_REF] Fulgencio | Uniqueness results for quasilinear elliptic problems in a two-component domain with L 1 data[END_REF], while the uniqueness has been studied in [START_REF] Fulgencio | Quasilinear elliptic problems in a twocomponent domain with L 1 data[END_REF], where an additional local Lipschitz continuity assumption must be prescribed on the matrix field A.

As mentioned above, we study in this chapter the homogenization of (5.1). Homogenization theory deals with the study of the macroscopic behaviour of composite materials. Composite materials are composed of two or more finely mixed components and their main physical characteristics (e.g. thermal or electric conductivity) can be modelled by problems with oscillating coefficients, describing the heterogeneities at the micro-scale. Then, the mathematical homogenization theory allows to give a macroscopic description of these materials, considered as homogeneous, at the macro-scale.

There are various literatures concerning the homogenization theory, as there is a wide interest on the subject. One can refer to the following references: [START_REF] Bensoussan | Asymptotic analysis for periodic structures[END_REF][START_REF] Cioranescu | An introduction to homogenization[END_REF][START_REF] Cioranescu | Homogenization of reticulated structures,v o l .1 3 6[END_REF][START_REF]Quelques remarques sur l'homogénéisation,inF unctionalAnalysis and Numerical Analysis[END_REF]. It is worth mentioning that the first results on homogenization was by Spagnolo in [START_REF] Spagnolo | Sul limite delle soluzioni di problemi di Cauchy relativi all'equazione del calore[END_REF].

There are several methods developed for homogenization. Some of which are the multiple-scale method introduced by Bensoussan et al. in [START_REF] Bensoussan | Asymptotic analysis for periodic structures[END_REF] (one can also see [START_REF] Sanchez-Palencia | Non-homogeneous media and vibration theory[END_REF]), the oscillating test function method developed by Tartar in [START_REF]Quelques remarques sur l'homogénéisation,inF unctionalAnalysis and Numerical Analysis[END_REF], and the two-scale convergence method first studied by Nguetseng in [START_REF] Nguetseng | Agener alc onver genc er esultforafunctionalr elate dto the theory of homogenization[END_REF] which is further developed by Allaire in [START_REF] Allaire | Homogenization and two-scale convergence[END_REF].

The most recently developed method, which is what we use in this chapter, is the periodic unfolding method. This method was introduced by D. Cioranescu et al. in [START_REF] Cioranescu | Periodic unfolding and homogenization[END_REF] and further developed in [START_REF] Cioranescu | The periodic unfolding method in homogenization[END_REF] for the study of periodic homogenization in classical domains (that is, with no holes or interfacial resistance). It was then later developed for perforated domains in [START_REF] Cioranescu | The periodic unfolding method in domains with holes[END_REF] and for two-component domains in [START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF]. For a detailed and comprehensive presentation of the method with several applications we refer to the recent book [START_REF] Cioranescu | The periodic unfolding method[END_REF].

The periodic unfolding method for two-component domains makes use of the unfolding operator T ε i , i =1 , 2, which is defined on any measurable functions in Ω ε i (see Definition 5.8 for more details). This operator has some interesting properties that can be used in homogenization, one of which is that it allows us to transform an integral in Ω ε i to an integral on the fixed domain Ω ⇥ Y i .

Moreover, when the data f in (5.1) belongs to L 2 (Ω) and A(y, t) is bounded, we can obtain some a priori estimates on the restriction of the solution u ε to each component Ω ε i , which we denote by [START_REF] Beltran | Homogenization of a quasilinear elliptic problem in a twocomponent domain with an imperfect interface[END_REF]).

u ε i = u ε | Ω ε i ,i nH 1 (Ω ε i ), i =1, 2 (see
Using the results in [START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF], these estimates lead to the following convergences:

8 < : T ε i (u ε i ) ! u 1 strongly in L 2 (Ω,H 1 (Y i )),i =1, 2 T ε i (ru ε i ) * ru 1 + r y b u i weakly in L 2 (Ω ⇥ Y i ),i =1, 2, (5.2)
for some

u 1 2 H 1 0 (Ω) and b u i 2 L 2 (Ω,H 1 (Y i )), i =1, 2.
These convergences allow us to identify first the unfolded limit problem (Theorem 5.22), satisfied by

(u 1 , b u 1 , b u 2 )
. This is the most delicate proof, which is long and quite labourous. Then we obtain, by standard computations, the homogenized problem in Ω,satisfiedb yu 1 (Theorem 5.25). Nevertheless, the proof of the decay of the "truncated" energy (see (5.111)), which proves that u 1 is a renormalized solution, is not standard and also delicate.

In literature, the homogenization results for (5.1) (with f 2 L 2 (Ω))h a v e been studied by different methods. The linear case was studied in [START_REF] Monsurrò | Homogenization of a two-component composite with interfacial thermal barrier[END_REF][START_REF] Monsurrò | Erratum for the paper: "Homogenization of a twocomponent composite with interfacial thermal barrier[END_REF] using the oscillating test function method, and in [START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF][START_REF] Donato | Homogenization of diffusion problems with a nonlinear interfacial resistance[END_REF] using the periodic unfolding method. Some results in the quasilinear case are contained in [START_REF] Borja | On the homogenization and correctors of a quasilinear elliptic problem in a two-component domain with an imperfect interface[END_REF], with the use the periodic unfolding method.

So far, we have only referenced literatures that dealt with homogenization in the variational setting, that is, when the data belongs to L 2 (Ω) and the matrix field A(y, t) is globally bounded. There are various works in homogenization in the framework of renormalized solution, one can see for instance [START_REF] Ben Cheikh | Homogénéisation des solutions renormalisées dans des domaines perforés[END_REF][START_REF] Briane | Ac l a s so fs e c o n d -o r d e rl i n e a re ll i ptic equations with drift: renormalized solutions, uniqueness and homogenization[END_REF][START_REF] Murat | Homogenization of renormalized solutions of elliptic equations[END_REF]. More recent works are [START_REF] Gaudiello | Homogenization of the brush problem with a source term in L 1[END_REF][START_REF] Donato | Homogenization of quasilinear elliptic problems with nonlinear Robin conditions and L 1 data[END_REF]. The authors in [START_REF] Gaudiello | Homogenization of the brush problem with a source term in L 1[END_REF] worked on the homogenization of a linear elliptic problem with Neumann boundary conditions, highly oscillating boundary, and L 1 data.

On the other hand, with the use of periodic unfolding method, the authors in [START_REF] Donato | Homogenization of quasilinear elliptic problems with nonlinear Robin conditions and L 1 data[END_REF] studied the asymptotic behaviour of a quasilinear elliptic problem in perforated domains with nonlinear Robin boundary condition and L 1 data. To our knowledge, [START_REF] Donato | Homogenization of quasilinear elliptic problems with nonlinear Robin conditions and L 1 data[END_REF] is the first study to mix the periodic unfolding method and the notion of renormalized solution, we adopt a similar approach in this chapter.

Let us recall that for a fixed ",thesolutionsu ε i , i =1, 2,doesnotnecessarily belong to H 1 (Ω ε i ),d u et ot h ef a c tt h a tt h ed a t af is an L 1 function. This means that we cannot have the usual convergences as in (5.2). We instead consider the truncates of u ε i , i =1, 2, where the truncation function is defined as T k (r)=m i n ( k, max( k, r)),f o ra n yk>0. This is possible since, in the framework of renormalized solution, the truncated of the solution for any k>0 is an H 1 function. Thus, in place of (5.2), combining the techniques of the framework of renormalized solutions and that of the periodic unfolding method (in particular, the compactness results), we show that there exist a function u 1 : Ω ! R and a sequence {b

u n i } n2N ⇢ L 2 (Ω,H 1 (Y i )), i =1 , 2, such that for any n 2 N, i =1, 2, 8 > > < > > : T n (u 1 ) 2 H 1 0 (Ω) T ε i (T n (u ε i )) ! T n (u 1 ) strongly in L 2 (Ω,H 1 (Y i )),i =1, 2, T ε i (rT n (u ε i )) * rT n (u 1 )+r y b u n i weakly in L 2 (Ω ⇥ Y i ),i =1, 2.
Even if we are using the process developed in [START_REF] Donato | Homogenization of quasilinear elliptic problems with nonlinear Robin conditions and L 1 data[END_REF], important additional differences and difficulties arise here, due to the presence of the jump. We use the definition of a renormalized solution adapted to our problem, introduced in [START_REF] Fulgencio | Quasilinear elliptic problems in a twocomponent domain with L 1 data[END_REF], and which contains an additional condition on the interface integral. In particular, Theorem 5.18 has a more delicate proof in our case compared to the corresponding theorem in [START_REF] Donato | Homogenization of quasilinear elliptic problems with nonlinear Robin conditions and L 1 data[END_REF], since the second component b u n 2 ,forn 2 N, does not necessarily have a zero average on Γ.F u r t h e r m o r e ,t h r o u g h o u tt h i s study, it can be observed that managing the boundary integral resulting from the jump is not a straightforward task and must be handled with care.

This chapter is organized as follows. The next section is dedicated to all the preliminary definitions and assumptions that we need. In particular, we define there the renormalized solution of (5.1) (Definition 5.6) and the periodic unfolding operator T ε i , i =1, 2 (Definition 5.8). We also present the properties of T ε i , i =1, 2 (see Proposition 5.10). In Section 5.3, we obtain some a priori estimates for the renormalized solution of (5.1). We also prove some pointwise convergence of T ε i (u ε i ) on Ω ⇥ Y i , i =1, 2 and on Ω ⇥ Γ (see Theorem 5.17).

Section 5.4 is devoted to the statement and proof of the main homogenization results. In particular, we identify the unfolded problem in Theorem 5.22 and the homogenized problem on Ω in Theorem 5.25.

Preliminaries and Position of the Problem

In this section, we present some definitions and the assumptions relevant to our problem. We also discuss here the unfolding operator that will be used in the homogenization results, as well as its important properties.

We first define our two-comp onent domain. Let Ω be a connected open bounded subset of R N with a Lipschitz continuous boundary @Ω.W et h e n define the reference cell Y as

Y = N Y j=1 [0,l j ),

Lemma 5.4 ([68]

). There exists a constant C>0,i n d e p e n d e n to f",s u c h that

kv i k 2 L 2 (Γ ε )  C(" 1 kv i k 2 L 2 (Ω ε i ) + "krv i k 2 L 2 (Ω ε i ) ), for every v i 2 H 1 (Ω ε i ), i =1, 2
. Note that the above assumptions for problem (5.3) give rise to two main difficulties: the data f belongs to L 1 (Ω),a n da s s u m p t i o n( A 3 . 2 )o nA only supposes local boundedness with respect to t. This means that a solution of (5.3) in the weak sense may not exist (even in the presence of only one of the two difficulties). Hence, we need to introduce a framework that provides existence and uniqueness of a solution of (5.3). Remark that, if so, a solution u ε may not have enough regularity in order to have a gradient and a trace in the usual sense of Sobolev spaces. In the following proposition, which is proved in [START_REF] Fulgencio | Uniqueness results for quasilinear elliptic problems in a two-component domain with L 1 data[END_REF], we give a definition of gradient and trace of any measurable function u,u s i n gt h et r u n c a t i o nf u n c t i o nT k ,d e fi n e db y

T k (t)= 8 > < > : k, if t  k t, if k  t  k k, if t k.
(5.8) Proposition 5.5 ([54]). Let u =( u 1 ,u 2 ):Ω \ Γ ε ! R be a measurable function such that T k (u) 2 H ε for every k>0.T h e n 1. there exists a unique measurable function v i :

Ω ε i ! R N such that for all k>0, rT k (u i )=v i {|u i |<k} a.e. in Ω ε i ,i =1, 2.
(5.9)

We define v i as the gradient of u i and write v i = ru i .

If

sup k 1 1 k kT k (u)k 2 H ε < 1, (5.10)
there exists a unique measurable function w

i : Γ ε ! R,f o ri =1 , 2, such that for all k>0, i (T k (u i )) = T k (w i ) a.e. on Γ ε ,i =1, 2, (5.11
)

where i : H 1 (Ω ε i ) ! L 2 (Γ ε
) is the trace operator. For i =1 , 2,w e define the function w i as the trace of u i on Γ ε and set

i (u i )=w i . while properties of T k imply 0  1 k Z Γ ε (u ε 1 u ε 2 )(T k (u ε 1 ) T k (u ε 2 )) d = 1 k Z Γ ε (u ε 1 u ε 2 )(T k (T n+1 (u ε 1 )) T k (T n+1 (u ε 2 ))) d  1 n Z Γ ε (u ε 1 u ε 2 )(T n+1 (u ε 1 ) T n+1 (u ε 2 )) d .
2. Let us mention that the definition in [START_REF] Fulgencio | Uniqueness results for quasilinear elliptic problems in a two-component domain with L 1 data[END_REF] involves two different C 1 (R) functions, 1 , 2 ,f o rt h et w oc o m p o n e n t sΩ ε 1 and Ω ε 2 as opposed to the one function for both as in (5.14).W eu s eh e r et h el a t t e ro p t i o nf o r simplicity, as there is no difference in the proofs whichever renormalized formulation we use.

3. If we have assumptions (A1)-(A3), then the existence (as well as stability) of a renormalized solution of (5.3) in the sense of the previous definition is assured by [START_REF] Fulgencio | Uniqueness results for quasilinear elliptic problems in a two-component domain with L 1 data[END_REF]. If, in addition, (A4) holds, then the uniqueness of the renormalized solution is proved in [START_REF] Fulgencio | Quasilinear elliptic problems in a twocomponent domain with L 1 data[END_REF].

For the homogenization process, we will use the periodic unfolding operator, which was introduced in [START_REF] Cioranescu | The periodic unfolding method in homogenization[END_REF] (for classical domains) and [START_REF] Cioranescu | The periodic unfolding method in domains with holes[END_REF] (for perforated domains), and extended to the case of two-component domains in [START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF] and [START_REF] Donato | Homogenization of diffusion problems with a nonlinear interfacial resistance[END_REF] (see also the recent book [START_REF] Cioranescu | The periodic unfolding method[END_REF]). We now define the p erio dic unfolding op erator adapted to the two-comp onent domain.

T ε i (' i )(x, y)= ( ' i ⇣ " h x " i Y + "y ⌘ a.e. (x, y) 2 b Ω ε ⇥ Y i 0 a.e. (x, y) 2 Λ ε ⇥ Y i . Remark 5.9. In the sequel, if ' 2 H 1 (Ω ε i ), i =1 , 2,t h et r a c eo fT ε i (') on Ω ⇥ Γ (sometimes denoted T ε b (') in the literature) is still denoted by T ε i ('), for i =1, 2.
We now present the following result from [START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF], completed in [START_REF] Donato | Homogenization of diffusion problems with a nonlinear interfacial resistance[END_REF], which states some very helpful properties of the unfolding operator for two-component domains.

Ω ε i . (P2) For every ' 2 L 1 (Ω ε i ),o n eh a s 1 |Y | Z Ω⇥Y i T ε i (')(x, y) dx dy = Z b Ω ε i '(x) dx = Z Ω ε i '(x) dx Z Λ ε i '(x) dx. (P3) For every ' 2 L p (Ω ε i ), kT ε i (')k L p (Ω⇥Y i )  |Y | 1/p k'k L p (Ω ε i ) . (P4) For ' 2 L p (Ω), T ε i (') ! ' strongly in L p (Ω ⇥ Y i ). (P5) Let {' ε } be a sequence in L p (Ω) such that ' ε ! ' strongly in L p (Ω). Then T ε i (' ε ) ! ' strongly in L p (Ω ⇥ Y i ). (P6) Let ' 2 L p (Y i ) be a Y periodic function and set ' ε (x)=' ⇣ x " ⌘ .T h e n T ε i (' ε ) ! ' strongly in L p (Ω ⇥ Y i ). (P7) Let ' ε 2 L p (Ω ε i ) satisfy k' ε k L p (Ω ε i )  C.I fT ε i (' ε ) * b ' weakly in L p (Ω ⇥ Y i ),t h e n e ' ε *✓ i M Y i ( b ') weakly in L p (Ω). (P8) For any ' 2 W 1,p (Ω ε i ),o n eh a s r y [T ε i (')] = " T ε i (r') and T ε i (') 2 L 2 (Ω,W 1,p (Y i )). (P9) If ' 2 L p (Γ ε ),t h e n kT ε i (')k L p (Ω⇥Γ)  |Y | 1/p " 1/p k'k L p (Γ ε ) .
The following result is crucial for the sequel. It follows from [START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF] and [START_REF] Donato | Homogenization of diffusion problems with a nonlinear interfacial resistance[END_REF], using the notations given in Remark 4.4 of [START_REF] Donato | Homogenization of diffusion problems with a nonlinear interfacial resistance[END_REF] for = 1. Definition 5.6, we obtain

2 X i=1 Z Ω ε i S n (u ε i )A ε (x, u ε i )ru ε i rT k (u ε i ) dx + 2 X i=1 Z Ω ε i S 0 n (u ε i )A ε (x, u ε i )ru ε i ru ε i T k (u ε i ) dx + " 1 Z Γ ε h ε (x)(u ε 1 u ε 2 )(S n (u ε 1 )T k (u ε 1 ) S n (u ε 2 )T k (u ε 2 )) d = Z Ω fS n (u ε )T k (u ε ) dx.
(5.18)

For i =1, 2,n o t et h a t Z Ω ε i S n (u ε i )A ε (x, u ε i )ru ε i rT k (u ε i ) dx = Z Ω ε i S n (u ε i )A ε (x, u ε i )rT k (u ε i )rT k (u ε i ) dx. (5.19)
On the other hand, since

|S 0 n |  1 n , for any n>0, (5.20)
|T k |  k, for any k>0, (5.21) by (5.13a) of Definition 5.6, Z

Ω ε i S 0 n (u ε i )A ε (x, u ε i )ru ε i ru ε i T k (u ε i ) dx  2k • 1 2n Z {|u ε i |2n} A ε (x, u ε i )ru ε i ru ε i dx = k⇢ i (n),i =1, 2, (5.22)
where lim

n!1 ⇢ i (n)=0,i =1, 2. (5.23)
Moreover, since we have (5.21) and

|S n |  1, for any n>0, (5.24)
we obtain

Z Ω fS n (u ε )T k (u ε ) dx  kkf k L 1 (Ω) . (5.25)
for some positive constants C 1 ,C 2 independent of " and k, which gives the desired estimate.

Proposition 5.13. Let u ε =( u ε 1 ,u ε 2 ) be a renormalized solution of (5.3). Then, for any k>0,

kT k (u ε i )k 2 L 2 (Ω ε i )  C 1 k, i =1, 2, (5.35)
for some positive constant C 1 independent of k and ". Moreover, for " < 1,w eh a v ef o ra n yk>0

kT k (u ε i )k 2 L 2 (Γ ε )  C 2 k" 1 ,i =1, 2, (5.36)
where C 2 is a positive constant independent of k and ".

Proof. To prove (5.35), we use the first inequality in (5.7), Remark 5.2, and Proposition 5.12. We then obtain for i =1, 2,

kT k (u ε i )k 2 L 2 (Ω ε i )  C 1 kT k (u ε i )k 2 H 1 (Ω ε i )  C 1 kT k (u ε i )k 2 H ε  C 1 k,
where C 1 is a positive constant independent of k and ".

To prove (5.36), when " < 1 (and hence " < " 1 ), we have from Lemma 5.4 (taking

v i = T k (u ε i ), i =1, 2), kT k (u i )k 2 L 2 (Γ ε )  C" 1 kT k (u i )k 2 H 1 (Ω ε i ) ,i =1, 2.
Arguing as in the proof of (2.21), we obtain (5.36).

Corollary 5.14. Let u ε =( u ε 1 ,u ε 2 ) be a renormalized solution of (5.3).W e can find a subsequence (stil l denoted by ")s u c ht h a tf o ra n yn 2 N,t h e r e exist u n 1 2

H 1 0 (Ω), b u n 1 2 L 2 (Ω,H 1 per (Y 1 )) with M Γ (b u n 1 )=0a.e. in x 2 Ω, and b u n 2 2 L 2 (Ω,H 1 (Y 2 )
) such that the following convergences hold: 8 > > < > > :

T ε i (T n (u ε i )) ! u n 1 strongly in L 2 (Ω,H 1 (Y i )),i =1, 2, T ε i (rT n (u ε i )) * ru n 1 + r y b u n i weakly in L 2 (Ω ⇥ Y i ),i =1, 2, Tn (u ε i ) *✓ i u n 1 weakly in L 2 (Ω),i =1, 2,
(5.37) as " goes to 0.

Furthermore, for any n 2 N,a s" tends to 0,

T ε 1 (T n (u ε 1 )) T ε 2 (T n (u ε 2 )) " * b u n 1 b u n 2 weakly in L 2 (Ω ⇥ Γ). (5.38)
Proof. By Proposition 5.12, for any k>0,t h es e q u e n c e

{T k (u ε )} = {(T k (u ε 1 ),T k (u ε 2 ))} is bounded in H ε . Choosing k = n with n 2 N,t h efi
Tn (u ε i ) *✓ i M Y i (u n 1 )
weakly in L 2 (Ω),i =1, 2.

Since u n 1 is independent of y,w eh a v et h ed e s i r e dr e s u l t .

Remark 5.15. The diagonal process used in the proof of Corollary 5.14 requires that k belongs to some countable set which tends to +1.F o rs i mplicity, we take k 2 N.

Remark 5.16. Let f be a function defined on Ω.S i n c eb yd e fi n i t i o n ,T ε i vanishes in Λ ε ⇥ Y i ,t h ei d e n t i t y

T ε i (f (u ε i )) = f (T ε i (u ε i )),i =1, 2, (5.39)
does not hold in Ω ⇥ Y i ,u n l e s sf (0) = 0 (see also Remark 3.3 of [START_REF] Donato | Homogenization of quasilinear elliptic problems with nonlinear Robin conditions and L 1 data[END_REF]). However, for any Lebesgue measurable function ',w eh a v e

T ε i (f (u ε i ))T ε i ('(x)) = f (T ε i (u ε i ))T ε i ('(x)),i =1, 2 (5.40)
in Ω ⇥ Y i even if f (0) 6 =0,s i n c eT

ε i (')(x, y)=0in Λ ε ⇥ Y i .
The following result, stating a pointwise convergence of T ε i (u ε i ), i =1 , 2, is an important tool for the identification of the homogenized problem in Ω.

Theorem 5.17. Let u ε =(u ε 1 ,u ε 2 ) be a renormalized solution of (5.3).T h e n there exists a measurable function u 1 : Ω ! R,fi n i t ea l m o s te v e r y w h e r e , such that (up to a subsequence) (5.44)

T ε i (u ε i ) ! u 1 a.e
In addition, from the trace theorem,

kT ε i (T n (u ε i )) u n 1 k 2 L 2 (Ω⇥Γ) = Z Ω kT ε i (T n (u ε i )) u n 1 k 2 L 2 (Γ) dx  Z Ω kT ε i (T n (u ε i )) u n 1 k 2 H 1 (Y i ) dx, i =1, 2.
This last quantity goes to 0 in view of Corollary 5.14. It follows from (5.39) that

T n (T ε i (u ε i )) ! u n 1
strongly in L 2 (Ω ⇥ Γ),i =1, 2.

(5.45)

We now claim that T ε i (u ε i ), i =1, 2, is Cauchy in measure and hence, pointwise convergent (up to a subsequence).

Let n>0. By (P3) of Proposition 5.10 and (2.21) of Proposition 5.13, we obtain for i =1, 2,

n 2 meas{|T ε i (u ε i )| n} = Z {|T ε i (u ε i )| n} (T n (T ε i (u ε i ))) 2 dx dy  Z Ω⇥Y i (T ε i (T n (u ε i ))) 2 dx dy  |Y i |kT n (u ε i )k 2 L 2 (Ω ε i )  |Y i |Cn,
where C is a positive constant independent of n and ". Consequently,

meas{|T ε i (u ε i )| n}  C n ! 0 as n ! 1 ,i =1, 2.
(5.46)

Since we have for ! > 0 and n 2 N, meas{|T

ε 0 i (u ε 0 i ) T ε 00 i (u ε 00 i )| !}  meas{|T ε 0 i (u ε 0 i )| n} +meas{|T ε 00 i (u ε 00 i )| n} +meas{|T k (T ε 0 i (u ε 0 i )) T k (T ε 00
i (u ε 00 i ))| !}, using similar arguments to the ones in [START_REF] Fulgencio | Uniqueness results for quasilinear elliptic problems in a two-component domain with L 1 data[END_REF], (5.37) and (5.46) allow one to show that {T ε i (u ε i )}, i =1 , 2, is Cauchy in measure. We can then find a subsequence of {T ε i (u ε i )}, i =1, 2, which converges almost everywhere. for all n m,w h e r eu 1 is the function given by Theorem 5.17. Moreover, we have b u i (x, •) 2 H 1 (Y i ),i =1, 2, with M Γ (b u 1 )=0, for a.e. x 2 Ω. (5.51)

Proof. The uniqueness of b u i ,f o ri =1, 2, is in the almost everywhere sense. We first prove (5.50). Let R 2 C 0 (R) with compact support verifying (5.49) and n m.

By (P1) of Proposition 5.10, Remark 5.16 Corollary 5.14, and using Theorem 5.17 together with the Lebesgue dominated convergence theorem, we obtain for i =1, 2, Define the set Ω n = {x 2 Ω || u 1 | <n }. The collection {Ω n } n2N is an increasing sequence of subsets of Ω.L e t{↵ n } n2N be a sequence of functions in C 0 (R) with compact support such that supp ↵ n ⇢ [ n, n] and ↵ n > 0 for all x 2 ( n, n).

T ε i (R(u ε i )rT n (u ε i )) = R(T ε i (u ε i ))T
Then, for any n 2 n 1 ,w eh a v e This proves (5.50). Finally, to show (5.51), note that from (5.57), we can deduce that for any R 2 W 1,1 (R) with compact support, the product R(u 1 )b u i belongs to L 2 (Ω,H 1 (Y i )),f o ri =1 , 2, with R(u 1 )M Γ (b u 1 )=0 . Then, since u 1 is finite a.e., we have (5.51).

↵ n 1 (u 1 )b u n 1 i = ↵ n 1 (u 1 )b u n 2 i ,i =1, 2.

Homogenization Results

In this section, we will discuss our homogenization results. We begin by proving the following proposition.

Proposition 5.20. Let u ε =( u ε 1 ,u ε 2 ) be the subsequence of renormalized solutions of (5.3) given by Corollary 5.14 and n 2 N.T h e n lim n!1 lim sup

ε!0 1 n Z {|u ε i |<n} A ε (x, u ε i )ru ε i ru ε i dx =0,i =1, 2 (5.58
) where S m is defined by (5.17), we have

lim n!1 lim sup ε!0 " 1 n Z Γ ε (u ε 1 u ε 2 )(T n (u ε 1 ) T n (u ε 2 )) d =0. ( 5
I ε 11 + I ε 21 + I ε 12 + I ε 22 + I ε 3 = I ε 4 , (5.60)
where

I ε i1 = 1 n Z Ω ε i S m (u ε i )A ε (x, u ε i )ru ε i rT n (u ε i ) dx, i =1, 2
I ε i2 = 1 n Z Ω ε i S 0 m (u ε i )A ε (x, u ε i )ru ε i ru ε i T n (u ε i ) dx, i =1, 2
I ε 3 = " 1 n Z Γ ε h ε (x)(u ε 1 u ε 2 )(S m (u ε 1 )T n (u ε 1 ) S m (u ε 2 )T n (u ε 2 )) d I ε 4 = Z Ω fS m (u ε ) T n (u ε ) n dx.
We study the b ehavior of each integral first as m approaches 1.A sf a ra s I ε i1 , I ε i2 and I ε 3 are concerned, the proof of Proposition 5.12 (see in particular (5.22), (5.23), (5.27) and (5.30)) gives that lim m!1

I ε i1 = 1 n Z {|u i ε |<n} A ε (x, u ε i )ru ε i ru ε i dx, i =1, 2 (
I ε 3 = " 1 n Z Γ ε h ε (x)(u ε 1 u ε 2 )(T n (u ε 1 ) T n (u ε 2 )) d . ( 5
I ε 4 = Z Ω f T n (u ε ) n dx.
(5.64)

We now lo ok at the limit of this as " tends to 0.B yc o n v e r g e n c e( 5 . 3 7 )o f Corollary 5.14 and (5.42) of Theorem 5.17, we have

lim sup ε!0 Z Ω f T n (u ε ) n dx =limsup ε!0 1 n Z Ω f Tn (u ε 1 ) dx +limsup ε!0 1 n Z Ω f Tn (u ε 2 ) dx = 1 n Z Ω f ✓ 1 u n 1 dx + 1 n Z Ω f ✓ 2 u n 1 dx = Z Ω f T n (u 1 ) n dx.
Now, we evaluate the limit of this last integral as n goes to 1.W ea g a i nu s e the Lebesgue Dominated Convergence Theorem. Note that by (5.21),

f T n (u 1 ) n  |f | 2 L 1 (Ω)
and since u 1 is finite a.e.,

T n (u 1 ) n ! 0 as n ! 1 , a.e. in Ω.

It follows that lim

Z {|u ε i |<n} A ε (x, u ε i )ru ε i ru ε i dx + " 1 n Z Γ ε (u ε 1 u ε 2 )(T n (u ε 1 ) T n (u ε 2 )) d ! =0.
Since all the integrals are nonnegative, this gives (5.58) and (5.59).

The following lemma states the convergence results that we need to identify the limit problem. Lemma 5.21. Let u ε =(u ε 1 ,u ε 2 ) be a renormalized solution of (5.3).W ec a n find a subsequence (still denoted by ")s u c ht h a tf o ra n yn 2 N the following convergences hold as " tends to 0: where

J ε i1 = Z Ω ε i S n (u ε i )A ε (x, u ε i )ru ε i rvdx, i=1, 2
J ε i2 = Z Ω ε i S 0 n (u ε i )A ε (x, u ε i )ru ε i ru ε i vdx, i=1, 2
J ε 3 = " 1 Z Γ ε h ε (x)(u ε 1 u ε 2 )(S n (u ε 1 ) S n (u ε 2 ))vd J ε 4 = Z Ω
fS n (u ε )vdx.

We will evaluate the limit of each term first as " goes to zero, and then as n approaches infinity. To do this, we use convergences (5.66) of Lemma 5.21, Proposition 5.20, and some properties of the unfolding operator in Proposition 5.10. By (P1) and (P2) of Proposition 5.10, and the fact that v has compact support, we have for i =1, 2, lim ε!0

J ε i1 =lim ε!0 Z Ω ε i S n (u ε i )A ε (x, T 2n (u ε i ))rT 2n (u ε i )rvdx =lim ε!0 1 |Y | Z Ω⇥Y i T ε i S n (u ε i )A ε (x
|J ε i2 |  lim n!1 lim sup ε!0 Z Ω ε i |S 0 n (u ε i )||A ε (x, u ε i )ru ε i ru ε i ||v| dx  2 • lim n!1 lim sup ε!0 kvk L 1 (Ω) 2n Z {|u ε i |<2n}
A ε (x, u ε i )ru ε i ru ε i dx.

Denoting

! 1 (n)=2• lim sup ε!0 1 2n Z {|u ε i |<2n} A ε (x, u ε i )ru ε i ru ε i dx,
we have by (5.58) of Proposition 5.20,

lim n!1 lim sup ε!0 J ε i2 kvk L 1 (Ω) lim n!1 ! 1 (n)=0,i =1, 2.
(5.76)

For J ε 3 ,b y( 5 . 3 1 ) ,w eo b t a i n

lim n!1 lim sup ε!0 |J ε 3 |  lim n!1 lim sup ε!0 " 1 Z Γ ε |h ε (x)||u ε 1 u ε 2 ||S n (u ε 1 ) S n (u ε 2 )||v| d  2 • lim n!1 lim sup ε!0 " 1 kvk L 1 (Ω) khk L 1 (Γ) 2n Z Γ ε (u ε 1 u ε 2 )(T 2n (u ε 1 ) T 2n (u ε 2 )) d .
If we set

! 2 (n)=2khk L 1 (Γ) lim sup ε!0 " 1 2n Z Γ ε (u ε 1 u ε 2 )(T 2n (u ε 1 ) T 2n (u ε 2 )) d ,
then by (5.59) of Proposition 5.20,

lim n!1 lim sup ε!0 J ε 3 kvk L 1 (Ω) lim n!1 ! 2 (n)=0.
(5.77)

For the integral on the right-hand side of (5.73), by (P1) and (P2) of Proposition 5.10, and Remark 5.16, and since v has a compact support, we have

lim ε!0 J ε 4 =lim ε!0 1 |Y | 2 X i=1 Z Ω⇥Y i T ε i fS n (u ε i )v dx dy =lim ε!0 1 |Y | 2 X i=1 Z Ω⇥Y i T ε i (f )S n T ε i (u ε i ) T ε i (v) dx dy.
which can be rewritten as (5.81) Now, let v ε i (x)="!(x)⇠ i x ε , i =1, 2, where

! 2 D(Ω) and ⇠ i 2 H 1 per (Y i ) \ L 1 (Y i ),i =1, 2.
Note that we have

T ε i (v ε i )="T ε i (!)⇠ i ! 0 strongly in L 2 (Ω ⇥ Y i ),i =1, 2.
(5.82) Moreover, since, for i =1, 2, rv ε i (x)="r!(x)⇠ i x ε + !(x)r y ⇠ i x ε ,w eg e t

T ε i (rv ε i )="⇠ i T ε i (r!)+r y ⇠ i T ε i (!) ! !r y ⇠ i ,i =1, 2, (5.83)
strongly in L 2 (Ω ⇥ Y i ). Taking = 2 and v ε =(v ε 1 ,v ε 2 ) as test function in (5.14) we have

L ε 11 + L ε 21 + L ε 12 + L ε 22 + L ε 3 = L ε 14 + L ε 24 , (5.84)
where

L ε i1 = Z Ω ε i 2 (u ε i )A ε (x, u ε i )ru ε 1 rv ε i dx, i =1, 2 L ε i2 = Z Ω ε i 0 2 (u ε i )A ε (x, u ε i )ru ε i ru ε i v ε i dx, i =1, 2 L ε 3 = " 1 Z Γ ε h ε (x)(u ε 1 u ε 2 )(v ε 1 2 (u ε 1 ) v ε 2 2 (u ε 2 )) d L ε i4 = Z Ω ε i f 2 (u ε i )v ε i dx, i =1, 2.
We study each term as " tends to 0.F o rL ε i1 ,u s i n gs i m i l a ra r g u m e n t st o show (5.75), and by (5.67) of Lemma 5.21 (written for S = 2 ), we obtain for i =1, 2, lim ε!0

L ε i1 =lim ε!0 1 |Y | Z Ω⇥Y i T ε i A ε (x, T n (u ε i )) T ε i 2 (u ε i )rT n (u ε i ) T ε i (rv ε i ) dx dy = 1 |Y | Z Ω⇥Y i
2 (u 1 )A(y, u 1 )(ru 1 + r y b u i )!r y ⇠ i dx dy.

(5.85)

For L ε i2 ,w eh a v ef o ri =1, 2,

|L ε i2 | = Z Ω ε i 0 2 (u ε i )A ε (x, u ε i )ru ε i ru ε i ⇣ "!(x)⇠ i ⇣ x " ⌘⌘ dx  "k!k L 1 (Ω) k⇠ i k L 1 (Y i )
Z

Ω ε i 0 2 (u ε i )A ε (x, u ε i )ru ε i ru ε i dx.
By (5.16) of Proposition 5.12 and the fact that A 2 L 1 (Ω ⇥ ( m, m)) N ⇥N , this last integral is uniformly bounded. As a consequence,

lim ε!0 L ε i2 =0.
(5.86)

For L ε i4 , by (P1) of Proposition 5.10 and Remark 5.16, we have

lim ε!0 L ε i4 =lim ε!0 1 |Y | Z Ω⇥Y i T ε i f 2 (u ε i )v ε i dx dy =lim ε!0 1 |Y | Z Ω⇥Y i T ε i (f ) 2 T ε i (u ε i ) T ε i (v ε i ) dx dy.
where b u n i is the function defined in (5.68) for i =1, 2. Taking into account (5.57), by Lemma 4.9 of [START_REF] Cioranescu | The periodic unfolding method in domains with holes[END_REF] (with D ε = A ε (x, u ε 1 ) and ⇣ ε = rT n (u ε 1 )), and (P2) of Proposition 5.10, we have

E i,n  1 |Y | Z Ω⇥Y i
A(y, u 1 )(rT n (u 1 )+r y b u n i )(rT n (u 1 )+r y b u n i ) dx dy

 lim inf ε!0 1 |Y | Z Ω⇥Y i T ε i (A ε (x, u ε i ))T ε i (rT n (u ε i ))T ε i (rT n (u ε i )) dx dy  lim sup ε!0 1 |Y | Z Ω⇥Y i T ε i (A ε (x, u ε i ))T ε i (rT n (u ε i ))T ε i (rT n (u ε i )) dx dy  lim sup ε!0 Z Ω ε i A ε (x, u ε i )rT n (u ε i )rT n (u ε i ) dx.
By (5.58) of Proposition 5.20,

lim n!1 1 n E i,n  lim n!1 lim sup ε!0 1 n Z Ω ε i A ε (x, u ε i )rT n (u ε i )rT n (u ε i ) dx =0.
Observing that for any 0 <n k<n+1 the ellipticity condition (A3.1) (see Remark 5.7 for a similar argument) gives (5.96)

1 |Y | Z Ω⇥Γ ✓ T ε 1 (T n (u ε 1 )) T ε 2 (T n (u ε 2 )) " ◆ 2 dx d y  lim sup ε!0 1 |Y | Z Ω⇥Γ ✓ T ε 1 (T n (u ε 1 )) T ε 2 (T n (u ε 2 )) " ◆ 2 dx d y  lim sup ε!0 1 " Z Γ ε (T n (u ε 1 ) T n (u ε 2 )) 2 d  lim sup ε!0 1 " Z Γ ε (u ε 1 u ε 2 )(T n (u ε 1 ) T n (u ε 2 )) d .
Since a Poincaré-Wirtinger inequality holds in W per (Y 1 ),( 5 . 9 6 )d e fi n e sa norm.

The following results give a characterization of b u i , i =1, 2, which allows us to obtain the homogenized problem in Ω. This kind of result is a classical and a quite simple step in the literature. For completeness, we prove it here in detail since the corresponding theorem in [START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF] was proved without giving details. With respect to the case of perforated domains studied in [START_REF] Donato | Homogenization of quasilinear elliptic problems with nonlinear Robin conditions and L 1 data[END_REF], the situation here has the additional difficulties due to the fact that the second component b u 2 does not necessarily have a zero average on Γ. @u 1 @x j (x),i =1, 2, (5.98)

(b) la fonction x 7 !

 7 B(x, r) est mesurable p.p. r 2 R, et vérifie les hypothèses suivantes :(A3.1) B(x, r)⇠ • ⇠ ↵|⇠| 2 ,a v e c↵ > 0, pour p.p. x 2 Ω, 8r 2 R, 8⇠ 2 R N ; (A3.2) pour tout k>0, B(x, r) 2 L 1 (Ω ⇥ ( k, k)) N ⇥N .

  Let B = {b 1 ,b 2 ,...,b N } be a basis for R N .I nt h i sm e t h o d ,o n ec a nd e fi n ea more general reference cell Y as

h 2 L 1 ∂ΩFigure 2 . 1 :

 2121 Figure 2.1: The two-component domain Ω

2 ,

 2 so that by Lebesgue Dominated Convergence Theorem,

Remark 2 . 10 (

 210 Stability). By adapting the proof of Theorem 2.8, it is possible to derive a stability result. More precisely, let us consider u ε ,ar e n o r -

Remark 3 . 3 .

 33 In the previous definition, conditions(3.9a) and (3.10a) are standard in the definition of renormalized solution. However, due to the presence of the boundary integral on Γ, (3.9b) and (3.10b) have to be added. In particular, since i (u i ) 2 L 1 (Γ) is not an assumption in Definition 3.2, we need (2.12b) to give sense to the integral on Γ in (3.11) (see[START_REF] Fulgencio | Uniqueness results for quasilinear elliptic problems in a two-component domain with L 1 data[END_REF] Remark 2]).

(3. 14 )

 14 We can clearly see from the definition of ✓ n that it is a continuous Lipschitz function verifying |✓ n (r)|  1 and |✓ 0 n (r)|  1 n , a.e. in R.( 3 . 1 5 ) Substituting S 1 = S 2 = ✓ n and = '(T 2n (u)) in (3.11) of Definition 3.2, we have Z Ω 1

. 21 )

 21 Combining (3.16)-(3.21), we obtain

  .1). The two-component domain is an open bounded set Ω of R N (with N 2), which is the union of two open subsets Ω ε 1 and Ω ε 2 ,a n do ft h e i rc o m m o n boundary (the interface) denoted Γ ε . The component Ω ε 2 is the union of the "periodic translated sets of "Y 2 , Y 2 being contained in the reference periodicity cell Y . The other component Ω ε

Theorem 4 . 4 (

 44 [START_REF] Gallouët | On the regularity of solutions to elliptic equations[END_REF]). Let N 2 and Ω be a bounded connected open subset of R N with a Lipschitz continuous boundary @Ω.S u p p o s eu is the unique solution of

Proposition 4 . 7 . 1 λ 2 )

 4712 w λ (y)= λ (y)+ y for a.e. y 2 Y. (4.41) For a.e. x 2 Ω,d e fi n e The solution λ of (4.15) satisfies Z R N \Γ # Ar λ r' dy + (' 1 ' 2 ) d = Z R N \Γ # A r' dy, (4.43) for any ' 2 L 2 (R N ) with supp ' compact and the restriction of ' in Y # i , denoted by ' i ,b e l o n g st oH 1 (Y # i ).

Theorem 4 . 10 .

 410 Let B and D be in M (↵, ,Y),w i t ht h ec o r r e s p o n d i n gh omogenized matrices B 0 and D 0 ,g i v e nb y(4.57).T h e nt h e r ee x i s t sc 0 > 0 dependent on ↵, , Y 1 and Y 2 such that |B 0 D 0 |  c 0 kB Dk L 1 (Y ) . (4.59) Proof. Let t D denote the transposed matrix of D.L e t λ,B i,ε , w λ,B i,ε , µ, t D i,ε and w µ, t D i,ε , i =1 , 2,b et h ef u n c t i o n sd e fi n e di n( 4 . 4 1 )a n d( 4 . 4 2 )f o rA = B and A = t D,r e s pe c t i v e l y .

Corollary 4 . 11 .

 411 Let A :(y, t) 2 Y ⇥ R 7 ! A(y, t) 2 R N ⇥N be a real matrix field with the following properties:(P1) A(•,t) belongs to M (↵, ,Y) for all t 2 R; (P2) A(•,t)={a ij } i,j=1,...,N is Y periodic for every t; (P3) there exists a function ! : R ! R such that 8 < : (a) ! is continuous, non-decreasing and !(t) > 0, 8t>0; (b) |A(y, s) A(y, t)|  !(|s t|) for a.e. y 2 Y, 8s, t 2 R.

  ,b yt h ec l a s s i c a lS t a m p a c c h i a ' sl e m m a( s e eL e m m a4 . 1i n[ 8 0 ] ) , there exists d>0 such that'(k 0 + d)=|A k 0 +d | =0,where d depends on C and k 0 . This gives us the desired result, with

  For a.e. z 2 R N ,w ed e n o t eb y[z] Y the integer part of z,i . e . ,[z] Y = (k 1 l 1 ,k 2 l 2 ,...,k N l N ),f o rs o m ek i 2 Z, i =1,...,N such that z [z] Y 2 Y .

Proposition 5 .

 5 10 ([45, 46]). For p 2 [1, 1),t h eo p e r a t o r sT ε i , i =1 , 2 are linear and continuous from L p (Ω ε i ) to L p (Ω ⇥ Y ).M o r e o v e r ,f o ri =1, 2, f o re v e r y', Lebesgue measurable on

  r s tt w oc o n v e r g e n c e si n (5.37) and convergence (5.38) follow from Proposition 5.11 and a standard diagonal process. To show the third one, we use the first one, estimate (2.21) of Proposition 5.13 and (P7) of Proposition 5.10. We obtain

Theorem 5 . 18 . Let b u n 1 2

 5182 L 2 (Ω,H 1 per (Y 1 )) and b u n 2 2 L 2 (Ω,H 1 (Y 2 )), n 2 N, be the functions given by Corol lary 5.14 with M Γ (b u n 1 )=0.T h e nt h e r ee x i s t s au n i q u em e a s u r a b l ef u n c t i o nb u i : Ω ⇥ Y i ! R,i =1, 2, such that for every R 2 C 0 (R) with compact support, verifying supp R ⇢ [ m, m],for some m 2 N, (5.49)we have R(u 1 )b u n i = R(u 1 )b u i a.e. in Ω ⇥ Y i ,(5.50)

εi 2 ⌘ 0 .

 20 (rT n (u ε i )) * R(u 1 )(rT n (u 1 )+r y b u n i ),T ε i (R(u ε i )rT m (u ε i )) = R(T ε i (u ε i ))T ε i (rT m (u ε i )) * R(u 1 )(rT m (u 1 )+r y b u m i ), both weakly in L 2 (Ω ⇥ Y i ).S i n c esupp R ⇢ [ m, m],t h e nf o ra n yn m, R(u ε i )rT n (u ε i )=R(u ε i )rT m (u ε i ),i =1, 2, R(u 1 )rT n (u 1 )=R(u 1 )rT m (u 1 ), and thus, R(u 1 )r y b u n i = R(u 1 )r y b u m i , 8n m, i =1, 2. Since R(u 1 ) is independent of y,f o rs o m eΦ m,n i (x) 2 L 2 (Ω) (independent of y), i =1, 2,w eh a v e R(u 1 )b u n i R(u 1 )b u m i = Φ m,n i (x),i =1, 2, 8n m.(5.52)Similarly, from (5.56), taking the average on Γ in both sides of (5.55), we have R(u 1 )M Γ (b u n 2 )=R(u 1 )M Γ (b u m 2 ), 8n m, which, using again (5.52) (now for i =2), gives Φ m,n As a consequence, (5.52) becomes R(u 1 )b u n i = R(u 1 )b u m i a.e. in Ω ⇥ Y i , 8n m, i =1, 2.

It follows that b u n 1 i = b u n 2 i

 2 a.e. in Ω n 1 ⇥ Y i ,i =1, 2, 8n 2 n 1 .Then, we can define the function bu i : Ω ⇥ Y i ! R for i =1, 2,a s b u i (x, y)=b u n i (x, y), 8(x, y) 2 Ω n ⇥ Y i ,n 2 N.(5.57)Since u 1 is finite almost everywhere,Ω = [n2N Ω n (up to a zero measure set).

Remark 5 . 19 . 1 ⌘ 2 ⌘

 51912 Let us point out that in the proof of the previous theorem, the fact that Φ m,n 0 in (5.52) follows from (5.56).H o w e v e r ,i no r d e rt o prove that Φ m,n 0 in (5.52) (which is essential in the proof of Theorem 5.18), we need additional and more delicate arguments since the average on Γ of b u n 2 ,f o rn 2 N,i sn o tn e c e s s a r i l yz e r o .

. 59 )

 59 Proof. Let m, n 2 N.W r i t i n g( 5 . 1 4 )o fD e fi n i t i o n5 . 6f o r= S m and v = 1 n T n (u ε ),

. 63 )

 63 For I ε 4 ,s i n c eS m (u ε ) ! 1 a.e. in Ω,a sm ! +1, while |S m (u ε |  1 a.e. in Ω and since fS m (u ε ) T n (u ε ) n  |f | 2 L 1 (Ω),we can apply the Lebesgue Dominated Convergence Theorem, which gives lim m!1

2 weakly in L 2 (

 22 ε i ) *✓ i T n (u 1 )weakly in L 2 (Ω),i =1, 2,T ε i (T n (u ε i )) ! T n (u 1 ) strongly in L 2 (Ω,H 1 (Y i )),i=1, 2,(5.66)and for any S 2 C 0 (R) with compact support and suppS ⇢ [ n, n],f o r i =1u ε i )rT n (u ε i ) * S(u 1 )(rT n (u 1 )+r y b u i ), weakly in L 2 (Ω ⇥ Y i ), T ε i S(u ε i ) T ε 1 T n (u ε 1 ) T ε 2 T n (u ε 2 ) " * S(u 1 )(b u 1 b u 2 ), weakly in L 2 (Ω ⇥ Γ),(5.67) where u 1 : Ω ! R is measurable and finite a.e., and bu i : Ω ⇥ Y i ! R,f o r i =1, 2 with b u i (x, •) 2 H 1 (Y i ) for a.e. x 2 Ω,a n dM Γ (b u 1 )=0.Proof. From (5.37) of Corollary 5.14, there is a subsequence of {"} (still denoted by ")s u c ht h a tf o ra n yn 2 N,w ec a nfi n du n1 2 H 1 0 (Ω), b u n 1 2 L 2 (Ω,H 1 per (Y 1 )) with M Γ (b u n 1 )=0a.e. in x 2 Ω,a n db u n 2 2 L 2 (Ω,H 1 (Y 2 )) such that the following convergences hold as " tends to 0:8 > > > > > > > > > < > > > > > > > > > : s Tn (u ε i ) *✓ i u n 1 weakly in L 2 (Ω),i =1, 2, T ε i T n (u ε i ) ! u n 1 strongly in L 2 (Ω,H 1 (Y i )),i=1, 2, T ε i rT n (u ε i ) * ru n 1 + r y b u n i weakly in L 2 (Ω ⇥ Y i ),i =1, 2, Ω ⇥ Γ).

(5. 68 )

 68 Then, convergences (5.66) follow from the first two convergences in (5.68) and identity (5.42) of Theorem 5.17. Now, we show (5.67). Let S 2 C 0 (R) with supp S ⇢ [ n, n].Then S is bounded, and by (5.41) of Theorem 5.17, we have for i =1, 2,S T ε i (u ε i ) ! S(u 1 ) in L 1 (Ω ⇥ Y i )weak-* and a.e. in Ω ⇥ Y i . (5.69) This convergence also holds when we replace Ω ⇥ Y i by Ω ⇥ Γ. Then, using Remark 5.16, by the third convergence in (5.68), we deduce for i =1, 2T ε i S(u ε i )rT n (u ε i ) = S T ε i (u ε i ) T ε i rT n (u ε i ) * S(u 1 )(rT n (u 1 )+r y b u n i ),

i

  , T 2n (u ε i ))rT 2n (u ε i )rv dx dy =lim A ε (x, T 2n (u ε i )) T ε i S n (u ε i )rT 2n (u ε i ) T ε i (rv) dx dy. Note that T ε i A ε (x, T 2n (u ε i )) is bounded and by (5.43) of Theorem 5.42,A ε y, T 2n (T ε i (u ε i )) * A y, T 2n (u 1 ) in L 1 (Ω ⇥ Y i ) weak-⇤ .(5.74)Hence, it can be shown by Lebesgue Dominated Convergence Theorem thatT ε i t A ε (x, T 2n (u ε i )) T ε i (rv) ! t A y, T 2n (u 1 ) rv, strongly in (L 2 (Ω ⇥ Y i )) N .Consequently, by (5.67) of Lemma 5.21 (written for S = S n ), for i =1, T 2n (u 1 ) rvS n (u 1 )(rT 2n (u 1 )+r y b u i ) dx dy = 1 |Y | Z Ω⇥Y i S n (u 1 )A y, T 2n (u 1 ) (rT 2n (u 1 )+r y b u i )rvdxdy. (5.75) On the other hand, for J ε i2 , i =1, 2,b y( 5 . 2 0 ) ,w eh a v e lim

  , u 1 )(ru 1 + r y b u i )r( 1 (u 1 )') dx dy = Z Ω f 1 (u 1 )' dx.

1 k Z {|u 1 u n 1 b u n 2 k 0 T ε 1 (T n (u ε 1 )) T ε 2 (

 11120112 |<k}⇥Y i A(y, u 1 )(rT k (u 1 )+r y b u i )(rT k (u 1 )+r y b u i ) dx dt  1 n E i,n+1 ,we obtain(5.71).It remains to show (5.95). By (5.38) of Corollary 5.14, we havekb L 2 (Ω⇥Γ)  lim inf ε!T n (u ε 2 )) " L 2 (Ω⇥Γ). Consequently, by (5.57), (P9) of Proposition 5.10, and (5.34) (written for f = u ε 1 and g = u ε 2 ), we obtain Z {|u 1 |<n}⇥Γ (b u 1 b u 2 ) 2 dx d y  1

By ( 5 2 )(T n (u ε 1 )

 521 T n (u ε 2 )) d =0.This proves (5.95) and then (5.72) by a monotony argument. This completes the proof.In order to state the next theorem, we introduce the space W per (Y 1 ),b yW per (Y 1 )={u 2 H 1 per (Y 1 ) | M Γ (u)=0} equipped with the norm kuk Wper(Y 1 ) = kruk L 2 (Y 1 ) .

Theorem 5 . 23 ., λ 2 )

 5232 Let λ =( λ 1 be the unique solution of the fol lowing variational problem:8 > > > > > > > > > < > > > > > > > > > :

Find

  λ (•,t)=( λ 1 (•,t), λ 2 (•,t)) 2 W per (Y 1 ) ⇥ H 1 (Y 2 ) such that , t)r y λ i (y, t)r y v i dy + Z Γ h(y)( λ 1 (y, t) λ 2 (y, t))(v 1 v 2 ) d t) r y v i dy for any v =(v 1 ,v 2 ) 2 H 1 per (Y 1 ) ⇥ H 1 (Y 2 ),(5.97) for all t 2 R and 2 R N . Using the assumptions and notations of Theorem 5.22, the function b u i , i =1, 2,c a nb ee x p r e s s e da s b u i (x, y)= N X j=1 j i (y, u 1 (x))

  j=1,...,N est Y périodique pour tout t; (P3) A(y, t) est localement lipschitzienne par rapport à la deuxième variable, i.e., pour tout r>0,i le x i s t eu n ec o n s t a n t ep o s i t i v eM r telle que |A(y, s) A(y, t)|  M r |s t| 8s, t 2 ( r, r), p.p. y 2 Y.

Alors, la matrice homogénéisée A 0 (voir (12))e s ta u s s il o c a l e m e n tl i p s c h i t z ienne, i.e., pour tout r>0,i le x i s t eu n ec o n s t a n t ep o s i t i v eC r telle que

  = F 1 and then to (4.32) with F = F 2 . Then for i =1 , 2,w e can find 2 <p i 0 < 2 ⇤ such that

					)
	and (4.30) as	(	div(Ar λ 2 )=G λ 2 + µ λ 2 Ar λ 2 • n 2 + h λ 2 =0	in Y 2 , on Γ,	(4.32)
	where µ λ 1 and µ λ 2 are as defined in (4.22) and (4.23), respectively. For i =1, 2,s e t
			F i = G λ i + µ λ i .		(4.33)
	Let us apply Theorem 4.4 (taking also into account Remark 4.5) first to
	(4.31) with F

  Definition 5.8. For i =1 , 2,a n df o ra n yf u n c t i o n' i Lebesgue measurable on Ω ε

i ,t h ep e r i o d i cu n f o l d i n go p e r a t o rT ε i is defined by

  . in Ω ⇥ Y i and on Ω ⇥ Γ,i =1, 2, (5.41)Proof. Let n 2 N.N o t et h a t( 5 . 3 9 )h o l d st r u e ,s i n c eT n (0) = 0. Then, from Corollary 5.14, we have as " tends to 0,

	T n (T i (u ε ε i )) ! u n 1	strongly in L 2 (Ω,H 1 (Y i )),i =1, 2.
	with
		T n (u 1 )=u n 1 , 8n 2 N,	(5.42)
	where u n 1 is given in Corollary 5.14, and
	T ε

i (A ε (x, T n (u ε i ))) ! A(y, T n (u 1 )) a.e. in Ω ⇥ Y i ,i =1, 2.

(5.43)

We now present the assumptions and some definitions for our problem. The domain Ω is a connected bounded open set in R N with its boundary @Ω.W e can write Ω as the disjoint unionΩ = Ω 1 [ Ω 2 [ Γ,where Ω 2 is an open set such that Ω 2 ⇢ Ω with a Lipschitz continuous boundary Γ and Ω 1 = Ω \ Ω 2 (see figure below).

(u 1 )A(y, u 1 )(ru 1 + r y b u i ) 2 (L 2 (Ω ⇥ Y i )) N ,i =1, 2,

Acknowledgments

Iw o u l dl i k et oe x p r e s sm ys i n c e r ea p p r e c i a t i o nt ot h ep e o p l ea n do r g a n i z ations who made this incredible journey possible.

We denote by u i = u| Ω i the restriction of u in Ω i , where u is any measurable function defined on Ω \ Γ.

We prescribe the following assumptions on f , h and B:

(A1) The function f is in L 1 (Ω).

(A2) The function h belongs to L 1 (Γ) and for some h 0 > 0, h(y) h 0 for a.e. y 2 Γ. (A4) B(x, r) is Locally Lipschitz with respect to r,t h a ti s ,f o ra n yc o m p a c t subset K of R,t h e r ee x i s t sM K > 0 such that |B(x, r) B(x, s)|  M K |r s|, 8r, s 2 K.

(3.

3)

The main goal of this research is to study the homogenization of the following quasilinear problem in the two-component domain described above:

on @Ω,

with a L 1 data and a not globally bounded matrix field.

More precisely, we prescribe the following assumptions:

(A2) h is a Y periodic function in L 1 (Γ) and there exists h 0 2 R such that 0 <h 0 <h(y) a.e. on Γ,a n ds e t h ε (x)=h ⇣ x " ⌘ a.e. on Γ ε ;

(A3) A :( y, t) 2 Y ⇥ R 7 ! A(y, t) 2 R N ⇥N is a real matrix field such that A(•,t)={a ij } i,j=1,...,N is Y periodic for every t, A is a Carathéodory function with the following properties:

(A3.1) A(y, t)⇠ • ⇠ ↵|⇠| 2 ,a . e . y 2 Y, 8t 2 R, 8⇠ 2 R N ;

(A3.2) A(y, t) 2 L 1 (Ω ⇥ ( k, k)) N ⇥N , 8k>0,

for every (x, t) 2 Ω ⇥ R.

We also introduce the following assumption, which is important in showing uniqueness results.

(A4) The matrix field A(y, t) is locally Lipschitz continuous with respect to the second variable, that is, for every r>0,t h e r ee x i s t sap o s i t i v e constant M r such that |A(y, s) A(y, t)| <M r |s t|, 8s, t 2 [ r, r], 8y 2 Y.

In the sequel, we use the following notations:

• e u i is the zero extension to the whole Ω of a function u i defined on Ω ε i , for i =1, 2;

,o faf u n c t i o nu defined in Ω;

• ⌫ ε i is the unit outward normal to Ω ε i for i =1, 2;

• ω is the characteristic function of a set !;

fdx,f o ra n yo pe ns e t! and for any f 2 L 1 (!).

We now define the Sobolev space that we will be working with, adapted to the fact that we are dealing with a jump on the interface. Definition 5.1. The functional space H ε is defined by

equipped with the norm

where

Remark 5.2. The norm (5.6) is equivalent to that of H 1 (Ω ε 1 ),withc onstants independent of " (see [START_REF] Monsurrò | Homogenization of a two-component composite with interfacial thermal barrier[END_REF] for details).

We have the following proposition on the equivalence of the norms of

2 ) and H ε , which is proved in [START_REF] Donato | Corrector results for a parabolic problem with a memory effect[END_REF].

We also have the following lemma from [START_REF] Monsurrò | Homogenization of a two-component composite with interfacial thermal barrier[END_REF]:

With this proposition, we are now in the position to define a renormalized solution of (5.3).

Definition 5.6. The function

and for any 2 C 1 (R) (or equivalently for any

)

Remark 5.7.

1. In the renormalized formulation presented in [START_REF] Fulgencio | Uniqueness results for quasilinear elliptic problems in a two-component domain with L 1 data[END_REF], convergences (5.13a)-(5.13b) are stated for k 2 N,w h i c hr e a d s

(5.15b)

Actual ly, convergences (5.13a)-(5.13b) can be deduced from convergences (5.15a)-(5.15b).I n d e e di ti ss u fficient to observe that if n  k<n+1 then the ellipticity condition (A3.1) gives

2 ) be a bounded sequence in H ε . Then there exist a subsequence (still denoted by "), and three functions v 1 2

Furthermore,

A priori estimates

We now prove some results that we will use when proving the homogenization results.

Proposition 5.12. Let u ε =( u ε 1 ,u ε 2 ) be a renormalized solution of (5.3). Then there exists a positive constant C such that for every k>0 and " > 0,

where C is independent of " and k.

Proof. Let u ε =(u ε 1 ,u ε 2 ) be a renormalized solution of (5.3). Fix k>0.F o r n>0,d e fi n et h ef u n c t i o nS n : R ! R by

( Z

(5.26)

We wish to pass to the limit as n approaches 1 on both sides of (5.26). The right-hand side goes to kkf k L 1 (Ω) .F o rt h efi r s ti n t e g r a lo f( 5 . 2 6 ) ,s i n c e for i =1 , 2 and n>kwe have

(5.27)

To pass to the limit of the integral on the boundary of (5.26), we write

(5.28)

From (5.24) and (5.12b) of Definition 5.6, we have

(5.29)

Then, by Lebesgue Dominated Convergence Theorem, we obtain

(5.30)

In addition, since S n is Lipschitz continuous with (5.20), we have

a.e. on Γ ε . Consequently, by (5.21),

where the integral on the right-hand side goes to zero as n approaches 1 by (5.13b) of Definition 5.6. As a result,

(5.32)

Combining (5.28), (5.30), and (5.32), we conclude that

Consequently, using (5.23), (5.27), and (5.33), we can pass to the limit in (5.26). We obtain

where C 1 is independent of " and k. Now, observe that for any functions f, g defined on Γ ε ,w eh a v e

since T k is an increasing Lipschitz continuous function with Lipschitz constant 1.

By the ellipticity of A,a s s u m p t i o no nh, and (5.34) (written for f = u ε 1 and g = u ε 2 ), we have

Hence, there exists a subsequence (still denoted by ")andu i :

(5.47)

Note that by (5.46), u i is finite a.e. for i =1, 2.

To show a pointwise convergence on Ω ⇥ Γ,l e tn>0.D e fi n e

Then by (5.39), (P9) of Proposition 5.10, and (5.36) of Proposition 5.13, we have for " small enough, and i =1, 2,

where C>0,i n d e pe n d e n to fn and ".

Then, using similar arguments as above, we have by a trace argument (up to a subsequence)

Let us prove now (5.42). The continuity of T n ,( 5 . 4 4 ) ,a n d( 5 . 4 7 )i m p l y

Since u n 1 is independent of y for any n>0, it follows that for i =1 , 2, u i is independent of y. Moreover, (5.48)

To end the proof, observe that convergence (5.43) can be proved as that given in Proposition 5.1 of [START_REF] Cabarrubias | Homogenization of a quasilinear elliptic problem with nonlinear Robin boundary conditions[END_REF] for perforated domains.

The following theorem, essential for the homogenization procedure done in the next section, makes use of some ideas of Proposition 5.3 of [START_REF] Donato | Homogenization of quasilinear elliptic problems with nonlinear Robin conditions and L 1 data[END_REF]. Here, the situation is more delicate due to the presence of the second component and the jump on the boundary (see Remark 5.19).

From Remark 5.16, convergence (5.38) of Corollary 5.14, and Theorem 5.17, using the boundedness of R,w eh a v e

weakly in L 2 (Ω ⇥ Γ).S i m i l a r l y ,w eh a v e

weakly in L 2 (Ω ⇥ Γ).

On the other hand, using (5.49), we obtain

which implies that

Hence, from convergences (5.53) and (5.54), we deduce that

Recall now that from Corollary 5.14,

(5.56)

Then, noting that R(u 1 ) and Φ m,n 1 are independent of y,t a k i n gt h ea v e r a g e on Γ in both sides of (5.52) (for i =1), we have

weakly in L 2 (Ω ⇥ Y i ), while by the fourth convergence in (5.68), it follows that for i =1, 2,

weakly in L 2 (Ω ⇥ Γ). Then (5.67) follows from Theorem 5.18 written for R = S.

We now identify the unfolded problem satisfied by the triple

. This is the most difficult proof, which is long and quite labourous. Then the triple

(5.70) In addition, for k>0,t h ef o ll o w i n gl i m i t sh o l d :

Proof. We do the proof in two steps. The first step is to obtain the unfolded equation (5.70). The next step is proving the convergences (5.94) and (5.95).

Step 1. Obtaining the unfolded equation (5.70). Let v 2 D(Ω) and n 2 N. Choosing v i = v and = S n (see (5.17)) as test functions in (5.14) of Definition 5.6, we have

Using (P4) of Proposition 5.10, (5.69) (with S = S n ), and the fact that f , S n (u 1 ),a n dv are independent of y,w eo b t a i n

(5.78)

Combining (5.75)-(5.78), we have

where

for some m>0.U s i n gv = 1 (u 1 )', where ' 2 H 1 0 (Ω) \ L 1 (Ω),a st h et e s t function in (5.79), we get

(5.80)

. Consequently, if we pass to the limit on both sides of (5.80) as n ! 1 ,w e obtain

Then by (P4) of Proposition 5.10, (5.69) (written for S = 2 ), and (5.82), we obtain lim

Using the definition of v ε , we can write L ε 3 as

To pass to the limit of this expression, we want to be able to express the term

This will enable us to use the second convergence in (5.67) of Lemma 5.21. But this is not a straightforward task since

To achieve our aim, we make use again of the functions S n .W ec h o o s en large enough, such that

(5.88)

Then, we can write

(5.89) By (5.31), we obtain

and (5.59) of Proposition 5.20 implies

(5.90) Note that (5.88) implies supp 2 ⇢ supp S n =[ 2n, 2n]. Hence, we can write

By Lemma 2.16 of [START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF], (P1) of Proposition 5.10, and Remark 5.16, we obtain

Note that, by (5.24),

and hence, (5.41) of Theorem 5.17 gives

Thus, combining this with (5.89) and (5.90), gives

Using similar arguments to obtain the preceding identity, we deduce that

It follows that

(5.92) Hence, passing to the limit of (5.84) as " goes to 0,u s i n gt h er e s u l t s( 5 . 8 5 ) -(5.87) and (5.92), we conclude that

for any ! 2 D(Ω) and for any

Note that 2 has a compact support, then from (5.51) of Theorem 5.18, we deduce that and

Hence, we obtain by density,

for every

Adding (5.81) and (5.93) gives the limit problem (5.70).

Step 2. Proof of the convergences (5.71) and (5.72).

Let us first prove that for n 2 N,

(5.95)

To prove convergence (5.94), for any n 2 N,a n di =1, 2 set

Note that we have the following convergences as " tends to 0:

where j =( j 1 , j 2 ) is the unique solution of (5.97) written for = e j with {e j } N j=1 being the canonical basis.

Proof. Let ' ⌘ 0 and Φ i (x, y)=!(x)v i (y) in (5.70) of Theorem 5.22, where ! 2 D(Ω) and v i 2 H 1 per (Y i ).S u b s t i t u t i n gt h e s ev a l u e s ,w eo b t a i n

Equivalently,

Since this holds for all ! 2 D(Ω) and u 1 is independent of y,w eh a v e

Let k>0 and choose 2 such that 2 (t)=1whenever |t|  k. Consequently,

),a n da . e . i n{x 2 Ω || u 1 (x)|  k}, for any k>0.S i n c eu 1 is finite a.e. in Ω,( 5 . 9 9 )h o l d sf o ra . e . x 2 Ω.

Note that by linearity with respect to ,w eh a v e

where j =( j 1 , j 2 ) is the unique solution of (5.97) with = e j . More particularly, if we take = ru 1 and t = u 1 (x), for some x 2 Ω, (5.100)

and denote the corresponding solution by b

(5.101) Let x 2 Ω.R e p l a c i n g( 5 . 1 0 0 )i n( 5 . 9 7 ) ,w eo b t a i n

A(y, u 1 )ru 1 r y v i dy. A(y, u 1 )ru 1 r y ' 1 dy.

Subtracting these last two equations, we have Z

By density, this holds for every ' 1 2 H 1 (Y 1 ). Then, (5.103) written for

The ellipticity of A implies

It follows that for some

Similary, taking v =( 0 , ' 2 ), where ' 2 2 D(Y 2 ),a st e s tf u n c t i o nf o rb o t h (5.99) and (5.102), we conclude that

for some ⇠ 2 (x) 2 L 2 (Ω) (independent of y). Consequently, from (5.101),

To show (5.98), we need to prove that

Case i =1.

Taking the average on Γ on both sides of (5.104) and using (5.51) of Theorem 5.18, we obtain

Since M Γ (b 1 )=0for a.e. x 2 Ω, it follows that ⇠ 1 ⌘ 0.

(5.107)

To show that ⇠ 2 ⌘ 0,n o t et h a tf r o m( 5 . 1 0 4 ) ,( 5 . 1 0 5 ) ,a n d( 5 .

Replacing these identities in (5.102) gives

Rearranging the terms, we obtain

This, together with (5.99), leads to

) such that v 1 >v 2 a.e. on Γ,w e conclude that ⇠ 2 ⌘ 0, since h>0 a.e. on Γ. This, together with (5.106) and (5.107), gives (5.98).

We are now able to describ e the homogenized problem in Ω.L e tu sfi r s t define, the homogenized matrix A 0 (t),i n t r o d u c e di n[ 6 8 ]f o rt h el i n e a rc a s e and given by A 0 (t)=A 0 1 (t)+A 0 2 (t), for every t 2 R, where

) the solution of (5.97). Standard arguments (see for instance [START_REF] Cioranescu | An introduction to homogenization[END_REF] or [START_REF] Cioranescu | The periodic unfolding method[END_REF]), provide the following uniform ellipticity:

(5.108)

We recall the following recent result from [START_REF] Donato | Some properties of an elliptic periodic problem with an interfacial resistance[END_REF], which is important for uniqueness results.

Theorem 5.24 ([41]). Suppose that the matrix field A(y, t) satisfies the assumptions (A3) and (A4). Then the homogenized matrix A 0 is locally Lipschitz, that is, for every r>0,t h e r ee x i s t sap o s i t i v ec o n s t a n tC r such that

Theorem 5.25 (The homogenized problem in Ω). Let u 1 be a cluster point of the sequence

on @Ω, (5.109)

that is,

, for any k>0, (5.110)

and for every

(5.112)

If in addition, (A4) holds, then u 1 is the unique renormalized solution of (5.109) and all of the sequences in Lemma 5.21 converge (not just a subsequence).

Proof. From Corollary 5.14 and (5.42) of Theorem 5.17, we have (5.110). To obtain (5.112), we take Φ i ⌘ 0 for i =1, 2 in (5.70) of Theorem 5.22. Hence, we have

Substituting the expressions for b u i , i =1, 2, from (5.98) of Theorem 5.23 and denoting 1 as to simplify the notation, we get

which is equivalent to (5.112).

To show (5.111), let us first take k = n 2 N. Choose ' = T 2n (u 1 ), Φ i = b u i {|u 1 |<2n} ,a n d i = S n , i =1 , 2, for (5.70) of Theorem 5.22. Substituting these functions, we have

where

We study the b ehavior of each integral as n approaches 1.

Note that, for i =1, 2,

By (5.94) of Theorem 5.22,

(5.114)

If we substitute (5.98) of Theorem 5.23 to K i , i =1, 2,t h e nw eg e t

Consequently, for i =1, 2,

A 0 i (u 1 )ru 1 ru 1 dx.

(5.115)

For L, using Theorem 5.18 (written for R = S n )a n d( 5 . 2 4 ) ,w eo b t a i n

We can conclude from (5.95) of Theorem 5.22 that

which is equivalent by a Cesaro argument (see [START_REF] Guibé | Remarks on the uniqueness of comparable renormalized solutions of elliptic equations with measure data[END_REF]Remark 6]) to

At last, in view of the ellipticity of the homogenized matrix A 0 (t) stated in (5.108), we obtain (5.111) (see Remark 5.7). Moreover, with the additional assumption (A4) on the matrix field A,the uniqueness of u 1 follows from Theorem 5.24 and [START_REF] Blanchard | Quasi-linear degenerate elliptic problems with L 1 data[END_REF]. Furthermore, since u 1 is unique, it follows that both b u 1 and b u 2 are uniquely determined by (5.98). As a consequence, the limit problem (5.70) has a unique solution and the convergences in (5.66) and (5.67) of Lemma 5.21 in fact hold for the whole sequence {"}.

Perspectives

In this thesis, we showed some results concerning a quasilinear problem in a two-component domain with L 1 data. In the first part, we obtained existence and uniqueness results in the framework of renormalized solutions. In the second part, we proved homogenization results and also studied some properties of the corresponding cell problem.

There are very interesting open problems related to this problem that we wish to study in the future.

As mentioned in Chapter 5, the homogenization of the following quasilinear problem with

for the cases 2 ( 1, 1] and 2 ( 1, 1) is not yet studied. These cases appear to be relatively easier compared to the case = 1 except probably for the case =1. This case is still an open problem even for the variational case, i.e., f 2 L 2 (Ω), which we also plan to look at in the future. In addition, we want to obtain some corrector results for all the cases of . We will first consider the corrector results for the case = 1 as it was what we considered in this thesis. It is worth mentioning that corrector results in the framework of renormalized solutions are yet to be studied. We can also study the existence, uniqueness and homogenization results for problem (2.1) with a nonlinear boundary condition. That is, we consider the following condition on Γ (B(x, u 1 )ru 1 )⌫ 1 = h(x, u 1 u 2 ), where the function h(y, t):R N ⇥ R ! R satisfies the following properties:

1. h is a Carathéodory function, 2. h is an increasing function in C 1 (R) such that h(y, 0) = 0 for all y 2 R N , 3. there exists a constant c 1 > 0 and an exponent q, with 1  q<2 if N =2, 3 and 1  q< N N 2

if N>3, such that @h @t (y, t)  c 1 (1 + |t| q 1 ) for a.e. y 2 Y and for all t 2 R, 4. there exists a constant c 2 > 0 such that th(y, t) c 2 |t| 2 for a.e. y 2 Y and for all t 2 R.

This kind of nonlinear boundary condition was considered by [START_REF] Donato | Homogenization of diffusion problems with a nonlinear interfacial resistance[END_REF], where the authors studied the existence, uniqueness and homogenization results for al i n e a rp r o b l e mi nt h ev a r i a t i o n a ls e t t i n g( t h a ti s ,t h ed a t af belongs to L 2 (Ω)).

Another aim is to consider the parabolic version of (2.1), i.e.,

@u 1 @t div(B(x, u 1 )ru 1 )=f in Ω 1 ⇥ (0,T), @u 2 @t div(B(x, u 2 )ru 2 )=f in Ω 2 ⇥ (0,T) (B(x, u 1 )ru 1 )⌫ 1 =(B(x, u 2 )ru 2 )⌫ 1 on Γ ⇥ (0,T), (B(x, u 1 )ru 1 )⌫ 1 = h(x)(u 1 u 2 ) on Γ ⇥ (0,T),

on @Ω ⇥ (0,T).

where f 2 L 1 (Ω) and T is a given time. Moreover, we would like to give a definition of a renormalized solution of the parabolic problem (5.118), and obtain existence and uniqueness results.

Then we can also study the homogenization of the corresponding periodic problem, that is 8 > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > : @u ε 1 @t div(B(x, u ε 1 )ru ε 1 )=f in Ω ε 1 ⇥ (0,T), @u ε 2 @t div(B(x, u ε 2 )ru ε 2 )=f in Ω 2 ⇥ (0,T) (B(x, u ε 1 )ru ε 1 )⌫ 1 =(B(x, u ε 2 )ru ε 2 )⌫ 1 on Γ ε ⇥ (0,T), (B(x, u 1 )ru 1 )⌫ 1 = h(x)(u 1 u 2 ) on Γ ⇥ (0,T),

on @Ω ⇥ (0,T).

where f 2 L 1 (Ω ⇥ (0,T)) and for i =1 , 2, g U 0,ε i converges weakly to ✓ i U 0 i in L 1 (Ω),f o rs o m eU 0 i 2 L 1 (Ω). Finally, another possible extension for problem (2.1) is considering a nonlinear problem with a Leray-Lions operator with p-growth in the same kind of domain and similar boundary conditions. That is, an operator of the form div(a(x, u, ru)), This operator was introduced by J. Leray and J.L. Lions in [START_REF] Leray | Quelques résultats de Visik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder[END_REF] and by J.L. Lions in [START_REF] Lions | Quelques méthodes de résolution des problèmes aux limites non linéaires[END_REF]. In the framework of renormalized solutions with this kind of operator, we refer to [START_REF] Lions | Sur les solutions renormalisées d'équations elliptiques[END_REF][START_REF] Maso | Renormalized solutions of elliptic equations with general measure data[END_REF] for the case of Dirichlet boundary condition and to [START_REF] Betta | Neumann problems for nonlinear elliptic equations with L 1 data[END_REF][START_REF] Betta | Uniqueness for Neumann problems for nonlinear elliptic equations[END_REF] for the case of Neumann boundary condition.