Homogenization of Elliptic Problems with Quadratic Growth and Nonhomogenous Robin Conditions in Perforated Domains

Transcription

1 Chin. Ann. Math. 37B6), 2016, DOI: /s y Chinese Annals of Mathematics, Series B c The Editorial Office of CAM and Springer-Verlag Berlin Heidelberg 2016 Homogenization of Elliptic Problems with Quadratic Growth and Nonhomogenous Robin Conditions in Perforated Domains Imen CHOURABI 1 Patrizia DONATO 2 Abstract This paper deals with the homogenization of a class of nonlinear elliptic problems with quadratic growth in a periodically perforated domain. The authors prescribe a Dirichlet condition on the exterior boundary and a nonhomogeneous nonlinear Robin condition on the boundary of the holes. The main difficulty, when passing to the it, is that the solution of the problems converges neither strongly in L 2 ) nor almost everywhere in. A new convergence result involving nonlinear functions provides suitable weak convergence results which permit passing to the it without using any extension operator. Consequently, using a corrector result proved in [Chourabi, I. and Donato, P., Homogenization and correctors of a class of elliptic problems in perforated domains, Asymptotic Analysis, 921), 2015, 1 43, DOI: /ASY ], the authors describe the it problem, presenting a it nonlinearity which is different for the two cases, that of a Neumann datum with a nonzero average and with a zero average. Keywords Homogenization, Elliptic problems, Quadratic growth, Nonhomogeneous Robin boundary conditions, Perforated domains 2000 MR Subject Classification 17B40, 17B50 1 Introduction In this paper, we study the homogenization of a class of a nonlinear elliptic problems containing a nonlinear term depending on the solution u and on its gradient u with quadratic growth. The problem is posed in the perforated domain =\T obtained by removing from an open bounded set of R N N 2) a closed set T representing a set of -periodic holes of size. We prescribe a Dirichlet condition on the exterior boundary Γ 0 and a nonhomogeneous nonlinear Robin condition on the boundary Γ 1 of the holes. More precisely, we study the asymptotic behavior, as tends to zero, of the bounded solution u of the following problem: diva x, u ) u )+λu = b x, u, u )+f in, A x, u ) u ) ν + γ ρ x)hu )=g on Γ 1, 1.1) u =0 onγ 0, where λ 0andν is the unit external normal vector with respect to. Manuscript received July 25, Université de Monastir, UR Analysis and Control of PDE, UR13ES64, Department of Mathematics Faculty of Sciences of Monastir 5019 Monastir, Tunisia. ichourabi@yahoo.fr 2 Normandie Université, Université de Rouen, Laboratoire de Mathématiques Raphaël Salem, UMR CNRS 6085, Avenue de l Université, BP 12, Saint Etienne de Rouvray, France. Patrizia.Donato@univ-rouen.fr

2 834 I. Chourabi and P. Donato The function f belongs to L m ) with m> N 2 and ρx) =ρ ) x,whereρ is a Y -periodic function that belongs to L T), Y and T being the reference cell and the reference hole, respectively. We assume that A x, t) =A x,t) is a bounded, uniformly elliptic, Y -periodic matrix field and that the function b x, t, ξ) isacarathéodory function on R R N with quadratic growth with respect to the third variable. We also suppose that g is defined by x ) g, if M T g) 0, g x) = x ) g, if M T g) =0, where g is a Y -periodic function in L r T)withr>N 1andM T g) denotes its mean over T. Let us mention that this type of equations appears in calculus of variations and stochastic control and the nonlinear term by, t, ξ) appears in the Euler equation of certain functionals. The homogenization of this kind of equation with a linear matrix field A x) involvinga term b x, t, ξ) continuous in x variables and with quadratic growth with respect to ξ has been studied in [4 5] for a fixed domain and in [19] for perforated domains with Neumann boundary condition and f = 0. In [20] the case where b x, t, ξ) is singular in t in a fixed domain has been studied. The homogenization result of problem 1.1) is stated in Theorem 2.1. The main feature of this result is that the expression of the it nonlinearity b 0 depends on the average of the nonhomogeneous boundary function g. This is due to the fact that, as proved in [10], the corrector results for the associated linear problem are different in the two cases M T g) 0 and M T g) =0. More precisely, according to these two cases, we derive two different it problems for the L 2 -it u 0 of the zero extension of u : 1) If M T g) 0org 0, the function u 0 is a solution of the problem diva 0 u 0 ) u 0 )+θλu 0 + c γ hu 0 )=b 0 u 0, u 0 )+ T Y M Tg)+θf in, u 0 =0 on, Y \T where θ = Y and A 0 t) is the homogenized matrix introduced in [1] see also [2 3, 8]) for quasilinear problems with Neumann conditions in perforated domains and the constant c γ is defined by T c γ = Y M Tρ), if γ =1, 0 if γ>1. The function b 0 is given by b 0 s, ξ) = 1 by, s, Cy, s)ξ)dy, s R, ξ R N, Y Y \T where {C,s)} is the usual corrector. 2) If M T g) =0withg 0) and A is independent of t, i.e., Ay, t) =Ay) iny,the function u 0 is a solution of the problem { diva 0 u 0 )+θλu 0 + c γ hu 0 )=b 0 u 0, u 0 )+θf in, u 0 =0 on,

3 Homogenization of Elliptic Problems in Perforated Domains 835 where A 0 is the now classical constant homogenized matrix introduced in [18]. The function b 0 is defined by b 0 t, ξ) = 1 by, s, Cy)ξ + χ g y))dy for every ξ R N, Y Y \T where C is the usual corrector independent of t) and the function χ g is the solution of the related problem posed in the reference cell with a nonhomogeneous Neumann data g. As in [4 5, 19 20], the main tool in the homogenization process is a corrector result. To do that, we construct a suitable associated linear problem. The homogenization and the corrector results for this linear problem have been already proved in [10]. We use here the result about correctors proved therein and we show that the corrector for the linear problem is also a corrector for our nonlinear problem in both cases M T g) 0andM T g) =0. Here, the main difficulty in particular, when passing to the it in the nonlinear problem, is due to the presence of the holes, and the solutions converge neither strongly in L 2 ) nor almost everywhere in. To overcome this difficulty, we do not use the extension operators as done in [19] for the case of homogeneous Neumann condition. We apply here the periodic unfolding method introduced in [12] see [13] for more details) and extended to perforated domains in [16 17] see also [11] for more general situations). Using the fact that the unfolding operator T for perforated domains transforms any function defined on into a function defined on a fixed domain, we prove a new technical convergence result involving nonlinear functions, stated in Theorem 4.1, which provides suitable weak convergence results. This technical tool allows to pass to the it and to prove the corrector result simplifying the proofs and the presentation, even in the case studied in [19]. This paper is organized as follows: In Section 2 we present the problem and state the main results. In Section 3 we recall some preinary results. Section 4 is devoted to the proof of the homogenization result. 2 Position of the Problem and Statement of the Main Results In order to state the main results of this paper, we recall some general notations introduced in [11] see also [12] for the unfolding periodic method in perforated domains). We denote by an open bounded set of R N N 2) with being Lipschitz continuous; Y =]0,l 1 [ ]0,l N [ the reference cell, where l i > 0 for all 1 i N; T, the reference hole, a compact set contained in Y and Y = Y \T the perforated reference cell, with T Lipschitz-continuous with a finite number of connected components; T = k + T ) the closed set of R N representing the holes, where k Ξ G = { ξ R N : ξ = N k i b i, k 1,,k N ) Z } N, Ξ = {ξ G, ξ + Y ) }, where i=1 b =b 1,,b N )isabasisinr N ; =\T the perforated domain; =interior k + Y ) ) and = \T the corresponding perforated set; k Ξ Λ =\ and Λ = \ the corresponding perforated set. As in [8, 17], we decompose see Figure 1) the boundary of the perforated domain as follows: =Γ 0 Γ 1, where Γ 1 = T and Γ 0 = \Γ 1. In the sequel, we also denote by

4 836 I. Chourabi and P. Donato Figure 1 The perforated domain and the reference cell Y χ E the characteristic function of a subset E of R N ; E the Lebesgue measure of a Lebesgue-measurable subset E of R N and E the N 1)- Hausdorff measure in R N of its boundary E; M Y v) = 1 Y vy) dy the average of any function v L 1 Y ); Y M T v) = 1 vy) dσ y the average of any function v L 1 T); T T ṽ or v) the extension by zero on of any function defined on ; ν the unit external normal vector with respect to Y \T or ; Mα, β, ) the set of matrix fields A =a ij ) 1 i,j N L )) such that { Ax)λ, λ) α λ ) 2, Ax)λ β λ, for x a.e in, any λ R N and α, β R, with0<α<β; θ = Y the proportion of material; Y c different strictly positive constants independent of. Let us recall that χ θ weakly* in L ), 2.1) as tends to zero. Our purpose is to study the asymptotic behavior, as tends to zero, of the following problem: diva x, u ) u )+λu = b x, u, u )+f in, A x, u ) u ) ν + γ ρ x)hu )=g on Γ 1, 2.2) u =0 onγ 0, wherewesupposethat H 1 ) λ 0. H 2 )0 f L m ) with m> N 2, f 0.

5 Homogenization of Elliptic Problems in Perforated Domains 837 H 3 ) The real matrix field A :y, t) Y R Ay, t) ={a ij y, t)} i,j=1,,n R N 2 satisfies the following conditions: 1) Ay, t) isy -periodic for every t and a Carathéodory function, i.e., Ay, ) is continuous for almost every y Y and A,t) is measurable for every t R; 2) A,t) Mα, β, Y ) for every t R; 3) A is Lipschitz continuous with respect to the second variable, i.e., K R : Ay, t) Ay, t 1 ) K t t 1 a.e. y Y, for t t 1 R. H 4 ) The function h is an increasing and continuously differentiable function such that for some positive constant C andanexponentq, one has h0) = 0, t R, h t) C1 + t q 1 ), with 1 q if N =2 and 1 q N N 2 if N>2. H 5 ) The function ρ is positive and Y -periodic, and it belongs to L T). H 6 ) The function g is Y -periodic and belongs to L s T), where s>n 1. H 7 ) The function b :y, t, ξ) Y R R N by, t, ξ) R satisfies the following conditions: 1) by, t, ξ) isy -periodic for every t, ξ) R R N, and is a Carathéodory function, i.e., by,, ) is continuous for a.e. y Y and b,t,ξ) is measurable for everyt, ξ) R R N ; 2) for some positive constant c 0, one has by, t, ξ) c ξ 2 ) for a.e. y Y, t R, ξ R N ; 3) d 1 and d 2 are continuous increasing functions with d 1 0) 0andd 2 0) = 0, such that by, t, ξ) by, t, ξ ) d 1 t )1 + ξ + ξ ) ξ ξ for a.e. y Y, t R, ξ,ξ R N and by, t, ξ) by, t,ξ) d 2 t t )1 + ξ 2 ) for a.e. y Y, t, t R, ξ R N. For almost every x in, every t in R and every ξ in R N,weset x ) x ) x ) A x, t) =A,t, b x, t, ξ) =b,t,ξ, ρ x) =ρ 2.3) and We introduce now the space x ) g, if M T g) 0, g x) = x ) g, if M T g) =0. V = {v H 1 ): v =0onΓ 0}, 2.4)

6 838 I. Chourabi and P. Donato equipped with the norm v V = v L 2 ) for every v V, and the variational formulation of problem 2.2) A x, u ) u ϕ dx + γ ρ hu )ϕ dσ + λ u ϕ dx Γ 1 = b x, u, u )ϕ dx + fϕ dx + g ϕ dσ, ϕ V L ). Γ 1 2.5) The existence of a solution to problem 2.5), under the assumptions H 1 ) H 7 ), has been proved in [9, Theorem 6.1] together with the boundedness of the solution and some uniform estimates for the sequence {u }. More precisely, it was proved that there exists a constant c such that u V c, u L ) c, 2.6) where c is independent of and depends only on α, m, s and the Sobolev embedding constant, with the numbers α, m and s being defined by H 2 ) H 3 )andh 6 ), respectively. We recall that see [1 2, 8, 10]) for every fixed t R, the homogenized matrix A 0 t) is defined by A 0 t)λ = 1 Ay, t) y ŵ λ y, t)dy, λ R N, 2.7) Y Y \T where ŵ λ y, t) =λy χ λ y, t) 2.8) and for every t R and λ R N, the function χ λ,t) is the solution to the problem diva,t) y χ λ,t)) = diva,t)λ) in Y \T, A,t)λ y χ λ,t)) ν =0 χ λ,t)isy-periodic, 1 Y \T Y \T χ λ,t)dy =0. on T, 2.9) We also define for any and every fixed t R, the corrector matrix C,t)=Cij,t)) 1 i,j n, given by see [10]) x ) C x, t) =C,t, a.e. in, C ij y, t) = ŵ 2.10) j y, t), i,j =1,,N, a.e. on Y, y i where {e j } N j=1 is the canonical basis of RN. We recall below its main properties. Proposition 2.1 see [10]) Under the assumption H 3 ),letu be the solution of problem 2.2). Then, there exists a constant c 1, independent of, such that for some r>2, C,t) L r ) c 1 for every and for any t R. 2.11)

7 Homogenization of Elliptic Problems in Perforated Domains 839 Moreover, the functions C,u ) are equi-integrable and i) C,u ) I weakly in L 2 )) N 2, ii) A,u )C,u ) A 0 u 0 ) weakly in L 2 )) N ) We can now state the main result of this paper. Theorem 2.1 Under the assumptions H 1 ) H 7 ),letu be the solution of problem 2.2). Then, there exists a subsequence {u } still denoted by ), afunctionu 0 H 1 0 ) L ) and a Carathéodory function b 0 : R R N R such that { i) ũ θu 0 weakly in L 2 ) and weakly in L ), ii) b,u, u )) b 0 u 0, u 0 ) on D ), where θ is defined by 2.1). The homogenized problems, depending on the mean of g, are the following ones: 1) If M T g) 0or g 0, the function u 0 is a solution of the problem diva 0 u 0 ) u 0 )+θλu 0 + c γ hu 0 )=b 0 u 0, u 0 )+ T Y M Tg)+θf in, u 0 =0 on, 2.13) 2.14) where the homogenized matrix A 0 t) is given by 2.7) for every fixed t R, and the constant c γ is defined by T c γ = Y M Tρ), if γ =1, 0, if γ> ) The function b 0 is given by b 0 s, ξ) = 1 by, s, Cy, s)ξ)dy, Y Y \T s R, ξ R N, 2.16) where the corrector matrix fields {C,s)} are defined by 2.10). Moreover, diva, ũ ) u ) diva 0 u 0 ) u 0 ) strongly on H 1 ). 2.17) 2) If M T g) =0with g 0) and A is independent of t, i.e., Ay, t) =Ay) in Y,the function u 0 is a solution of the problem { diva 0 u 0 )+θλu 0 + c γ hu 0 )=b 0 u 0, u 0 )+θf in, u 0 =0 on, 2.18) where the constant homogenized matrix A 0 is given by 2.7) independent of t). The function b 0 is defined by b 0 t, ξ) = 1 by, s, Cy)ξ + χ g y))dy for every ξ R N, 2.19) Y Y \T

8 840 I. Chourabi and P. Donato where C ) is defined by 2.10) independent of t) and the function χ g is the solution of the following problem: diva χ g )=0 in Y \T, A χ g ν = g χ g is Y-periodic, M Y χ g )=0. on T, 2.20) Moreover, diva u ) diva 0 u 0 ) strongly on H 1 ). 2.21) This result will be proved in Section 4. Remark 2.1 Let us point out the main novelty in this result. It concerns the fact that the presence of g in the nonhomogeneus boundary condition of problem 2.2) gives rise to two different it nonlinearities b 0 in the problem, according to the case M T g) 0 or M T g) =0. This is due to the fact that the corrector results for the associated linear problem are different in the two cases, as recalled in Theorem 3.2. We end this section by stating the following result, which shows that C is a corrector for the nonlinear problem 2.5). Corollary 2.1 Under the assumptions of Theorem 2.1 and the notation therein, we have the following assertions: 1) If M T g) 0or g =0,then u C,u ) u 0 L 1 )N =0. 2) If M T g) =0and A is independent of t, i.e., Ay, t) =At) in Y,then u C L ) u 0 y χ g =0. ) 1 )N Proof This corollary is a straightforward consequence of Theorem 3.4 and Theorem 4.1 proved in Sections 3 and 4, respectively. 3 Some Preinary Results In this section, we recall some homogenization and corrector results proved in [10]. To do that, we introduce as in [10] a linear operator L from H 1 ) to V ) verifying the following assumption: H 8 )If{ψ } is a sequence such that then ψ V c and ψ θψ 0 weakly in L 2 ), 3.1) L Z),ψ V,V = Z, ψ 0 H 1 ),H 1 0 ). 3.2)

9 Homogenization of Elliptic Problems in Perforated Domains 841 Remark 3.1 Let us point out that there exist many operators L verifying the assumption H 8 ), which can be constructed in different ways. For instance, the assumption H 8 ) is satisfied for L = P see Remark 4.3 of [10]), where P is the adjoint of the linear extension operators P introduced by Cioranescu D. and Saint Jean Paulin J. in [18]. Let us recall that for any sequence {v } in V,wehavethat { v V c and ṽ θv 0 weakly in L 2 )} P v v 0 weakly in H0 1 ). 3.3) Another different) operator can be defined by using the periodic unfolding method as done in [21] when studying the correctors for the wave equation in perforated domains via the above method. We refer to [10, Remark 2.3] for more details and comments. We recall first the following homogenization result. Theorem 3.1 see [10]) Under the assumptions H 1 ) H 6 ) and H 8 ), let Z be given in H 1 ) and let v be the unique solution of problem diva x, z ) v )+λv = L Z) in, A x, z ) v ) ν + γ ρ x)hz )=g on Γ 1, 3.4) v =0 on Γ 0, where the sequence {z } belongs to V and A, ρ and g are given by 2.3) and 2.4). Suppose that the sequence {z } satisfies 3.1), that is z V c and z θz 0 weakly in L 2 ), with z 0 H 1 0 ) and θ given by 2.1). Then, as tends to 0, we have the following convergences: { i) ṽ θv 0 weakly in L 2 ), ii) A,z ) v A 0 z 0 ) v 0 weakly in L 2 )) N. The function v 0 is the unique solution of the problem diva 0 z 0 ) v 0 )+θλv 0 = c γ hz 0 )+Z + T Y M Tg) in, v 0 =0 on, 3.5) 3.6) where the homogenized matrix A 0 t) and the constant c γ are defined by 2.7) and 2.15) respectively. Let us recall now the results concerning the convergence of the energies and the corrector, proved in [10] where we distinguish the two cases given in 2.4). Theorem 3.2 see [10]) Under the assumptions of Theorem 3.1, leta 0 be defined by 2.7) and χ g defined by 2.20). Letv and v 0 be the solutions of problems 3.6) and 3.4), respectively. 1) If M T g) 0or g 0, then A,z ) v v A 0 z 0 ) v 0 v 0 weakly in L 1 ). Moreover, if {C,z )} is defined by 2.10), we have v C,z ) v 0 L1 )N =0.

10 842 I. Chourabi and P. Donato 2) If M T g) =0with g 0) and A is independent of t, i.e., Ay, t) =Ay) in Y,then )) A ) v y χ g v A 0 v 0 + M Y A χ g )) v 0 weakly in L 1 ). Moreover, if C ) is defined by 2.10) independent of t), we have v C L ) v 0 y χ g =0. ) 1 )N The proof of the corrector result given in [10] is based on the proposition below, which will be needed in the sequel. Proposition 3.1 see [10]) Under the assumptions of Theorem 3.1 and with the notations therein, we have the following assertions: 1) If M T g) 0or g 0, then sup v C,z )Φ L2 c v )N 0 Φ L2 ) N, Φ C 0 )) N. 2) If M T g) =0with g 0) and A is independent of t, i.e., Ay, t) =At) in Y,then sup v C L )Φ y χ g ) c v 0 Φ L2 ), Φ 2 N C 0 )) N. )N In both cases, c = Cα, β) is a constant independent of Φ. We also recall the following property. Lemma 3.1 see [5]) Let {g } be a sequence of functions which converges weakly in L 1 ) to a function g 0 and let {t } be a sequence of equibounded and measurable functions, which converges almost pointwise in to a function t 0.Then g t dx = g 0 t 0 dx. We end this section by stating the following result. Proposition 3.2 Under assumption H 7 ),let{b } be the sequence of the Carathéodory functions given by 2.3). Then,thefunctionb 0 given by 2.16) or 2.19) satisfies H 7 ) and for any φ in C0 ))N and ϕ 0 in L ), one has the following assertions: 1) If M T g) =0or g =0,then [b x, ϕ 0,C φ)] b 0 ϕ 0,φ) weakly in L 1 ), 3.7) where {C,z )} is defined by 2.10). 2) If M T g) =0and A is independent of t, i.e., Ay, t) =Ay) in Y, [b x, ϕ 0,C φ + χ g )] b 0 ϕ 0,φ) weakly in L 1 ), 3.8) where C ) is defined by 2.10) independent of t) and the function χ g is the solution of problem 2.20). Proof The convergence 3.7) is a simple consequence of Theorem 2.6 of [19] see also [5] for the case of a fixed domain) and the convergence 3.8) can be deduced by the same arguments as those used to prove 3.7).

11 Homogenization of Elliptic Problems in Perforated Domains Proof of the Main Result 4.1 A technical result In this section, we give a preinary tool which will play an essential role in proving the corrector result stated in Theorem 4.2, and seems interesting by itself. To prove it, we use the periodic unfolding method, introduced in [12] see [13] for a general presentation and detailed proofs) and extended to perforated domains in [16 17] see [11] for more general situations and a comprehensive presentation). Theorem 4.1 Let {ψ } be a sequence satisfying 3.1). 1) The following convergences hold: ψ ψ 0 2 dx =0, ψ 2 dx = θ 2) Let p [1, + ) and {h } be a sequence in L p ) such that ψ 0 2 dx. 4.1) h h 0 weakly in L p ), 4.2) for some h 0 in L p ). Suppose further that F : R R is a continuous function such that F ψ ) L q ), with q p, + ) with 1 p + 1 =1, if p>1, p 4.3) q =+, if p =1. If for some positive constant c independent of, then h F ψ )dx = Moreover, if F ψ ) L q ) c 4.4) h 0 F ψ 0 )dx. 4.5) h h 0 strongly in L p ), 4.6) then In particular, h F ψ )dx = θ h 0 F ψ 0 )dx. 4.7) F ψ ) θfψ 0 ), 4.8) weakly in L p ) if p>1 and weakly in L ) if p =1. Proof Convergences 4.1) follow from Corollary 1.13, Corollary 1.19 and Theorem 2.13 of [11]. In order to show 4.5), let us first recall see [11]) that for any Lebesgue-measurable function φ on, the unfolding operator T is defined as [ x ) φ T φ)x, y) = ]Y + y a.e. for x, y) Y, 0 a.e. for x, y) Λ Y.

12 844 I. Chourabi and P. Donato In view of Proposition 1.14 of [11], the assumption 4.2) implies that there exists some function ĥ0 in L p Y ) such that T h ) ĥ0 weakly in L p Y ), 4.9) with 1 ĥ 0,y)dy = h ) Y Y Using again Theorem 2.13 of [11], we derive that T ψ ) ψ 0 strongly in L 2,H 1 Y )). Then, there exists a subsequence still denoted by {}) such that T ψ ) ψ 0 a.e. in Y, so that in view of the continuity of F, we get for a subsequence) T F ψ )) = F T ψ )) F ψ 0 ) a.e. in Y. 4.11) Moreover, using the properties of T see [11, Proposition 1.12 and Corollary 1.13]), we deduce that 1 h F ψ )dx = T Y h )x, y)t F ψ ))x, y) dx dy. 4.12) Y On the other hand, if p>1, thanks to 4.11), the Hölder inequality provides the equintegrability of T F ψ )) F ψ 0 ) p. Then, using 4.4) we can apply the Vitali s theorem to obtain T F ψ )) F ψ 0 ) strongly in L p Y ), and this convergence holds for the whole sequence, since the it is uniquely determined. Consequently, from 4.9), we have 1 T Y h )x, y)t F ψ ))x, y) dx dy = 1 ĥ 0 x, y)f ψ 0 )dxdy. 4.13) Y Y Y Since F ψ 0 ) is independent of y, in view of 4.10) this gives the result for p>1. Let p = 1 now. Then, thanks to 4.3) 4.4), 4.9) and 4.11), we can still pass to the it in the right-hand side of 4.12) by applying Lemma 3.1 in Y,withg = T h )and t = T ψ ). We obtain again 4.13) and conclude as in the previous case. Finally, observe that if 4.6) holds, from Corollary 1.19 of [11], one has ĥ0 = h 0 which implies that 1 ĥ 0 x, y)f ψ 0 )dxdy = θ h 0 F ψ 0 )dx, Y Y since θ = Y Y. This gives 4.7) and in particular 4.8) taking h = h 0 ), which concludes the proof.

13 Homogenization of Elliptic Problems in Perforated Domains A corrector result for the nonlinear problem In this section we prove Theorem 2.1. To do that, we adapt some arguments introduced in [4 5] for the case of oscillating coefficients in a fixed domain, and extended for the linear case in the periodically perforated domains in [19]. We make here an essential use of Theorem 4.1, proved in the previous section. The main idea is to define a suitable linear problem associated to a weak cluster point of the sequence of the solutions of problem 2.2) and then prove that the corrector for this linear problem is also a corrector for the original nonlinear problem. Observe first that from 2.6) there exists a subsequence {u } still denoted by ), which will be fixed from now on, and a function u 0 H 1 0 ) L ) such that ũ θu 0 weakly in L 2 ) and weakly in L ), 4.14) as tends to zero, where θ is defined by 2.1). Then, in the present situation, the suitable linear problem associated to problem 2.2) is the following one: diva x, u ) v )+λv = L diva 0 u 0 ) u 0 )+θλu 0 + c γ hu 0 ) T ) Y M Tg) in, A x, u ) v ) ν + γ ρ x)hu )=g on Γ 1, v =0 on Γ 0, where L is a linear operator satisfying H 8 ). Its variational formulation is A x, u ) v φ dx + λ v φ dx + γ = g φ dσ + L diva 0 u 0 ) u 0 )+θλu 0 Γ 1 +c γ hu 0 ) T ) Y M Tg),φ, φ V. V,V In view of Theorem 3.1, written for z = u and Γ 1 ρ hu )φ dσ Z = diva 0 u 0 ) u 0 )+θλu 0 + c γ hu 0 ) T Y M Tg), using 4.14) we deduce that 4.15) 4.16) ṽ θv 0 weakly in L 2 ), as tends to zero, where θ is defined by 2.1) and v 0 H0 1 ) is the unique solution of the equation diva 0 x, u 0 ) v 0 )+θλv 0 = diva 0 x, u 0 ) u 0 )+θλu 0. Hence, v 0 = u 0,sothat ṽ θu 0 weakly in L 2 ). 4.17) We approximate now the function u 0 H0 1) L ) by a sequence {u n } D) such that { i) un u 0 strongly in H0 1 ), as n +, 4.18) ii) u n L ) c,

14 846 I. Chourabi and P. Donato for any n, wherec is independent of n. Let us introduce, for any n N, the sequence {v n, } where the function v n, is the solution of the problem: diva x, u ) v n, )+λv n, = L diva 0 u 0 ) u n ) +θλu n + c γ hu 0 ) T ) Y M Tg) in, A x, u ) v n, )ν + γ ρ x)hu )=g on Γ 1, v n, =0 onγ 0, whose variational formulation is A x, u ) v n, φ dx + λ v n, φ dx + γ = g φ dσ + L diva 0 u 0 ) u n )+θλu n Γ 1 +c γ hu 0 ) T ) Y M Tg),φ, φ V. V,V Γ 1 ρ hu )φ dσ 4.19) Then, for any n, we apply again Theorem 3.1, written here for z = u and for Z = diva 0 u 0 ) u n )+θλu n + c γ hu 0 ) T Y M Tg). The same argument used to prove 4.17) gives as tends to zero, where θ is defined by 2.1). Moreover, ṽ n, θu n weakly in L 2 ), 4.20) v n, V c, 4.21) where c is independent of n and, and from the classical results of Stampacchia [24], for any fixed n, wehave v n, L ) c n, 4.22) where c n is a constant independent of. The following theorem is the essential tool to prove the corrector result for the nonlinear problem. Theorem 4.2 Under the assumptions H 1 ) H 7 ),letu be a sequence of solution of problem 2.5) and v be a solution of problem 4.19). Then, up to a subsequence, we have u v L2 )) N =0. Proof Let Φ be the function defined by Φ = ξ μ u v n, ) V L ), 4.23) where μ is a suitable positive constant to be chosen later on, and ξ μ s) =se μs2, s R N.

15 Homogenization of Elliptic Problems in Perforated Domains 847 Taking Φ as a test function in 2.5) and 4.19), after subtraction of two identities, we obtain A x, u ) u v n, ) u v n, )ξ μ u v n, )dx + λ u v n, ) 2 e μu vn,)2 dx = b x, u, u )ξ μ u v n, )dx + fξ μ u v n, )dx L diva 0 u n ) u n )+θλu n + c γ hu 0 ) T ) Y M Tg),ξ μ u v n, ). V,V Using H 1 ), H 3 ) and the property 2) of H 7 ), since ξ μ 0, we get α u v n, ) 2 ξ μu v n, )dx c u 2 ) ξ μ u v n, ) dx + fξ μ u v n, )dx L diva 0 u n ) u n )+θλu n ),ξ μ u v n, ) V,V c u v n, ) 2 +2 v n, 2 ) ξ μ u v n, ) dx + fξ μ u v n, )dx L diva 0 u n ) u n )+θλu n + c γ hu 0 ) T ) Y M Tg),ξ μ u v n, ). V,V Let us take μ = c2 0 α. For this choice, one gets αξ μs) 2c 2 0 ξ μ s) α 2 for any s, so that using again H 3 ), we get α u v n, ) 2 dx u v n, ) 2 αξ 2 μu v n, ) 2c 0 ξ μ u v n, ) ) dx c ) α A x, u ) v n, v n, ξ μ u v n, ) dx + fξ μ u v n, )dx L diva 0 u n ) u n )+θλu n + c γ hu 0 ) T ) Y M Tg),ξ μ u v n, ). 4.24) V,V From 2.6), 4.21) 4.22), the function u v n, satisfies 3.1) and the sequence {ξ μ u v n, )} is bounded in L ) for fixed n). Hence, applying Theorem 4.1 to ψ = u v n, and p = m, firstwithh = f and F = ξ μ, and then with h = c 0 and F = ξ μ,weobtain fξ μ u v n, )+c 0 ξ μ u v n, ) dx = θ fξ μ u 0 v n )dx + θc 0 ξ μ u 0 v n ) dx. 4.25) Observe now that in view of Theorem 3.2 we can apply Theorem 4.1 for p = 1 to the functions h = A x, u ) v n, v n,, ψ = u v n,, F = ξ μ. We get A x, u ) v n, v n, ξ μ u v n, ) dx = θa 0 u 0 ) v n v n ξ μ u 0 v n ) dx. 4.26) On the other hand, from 2.6) and 4.22) 4.23), using again Theorem 4.1, we deduce that z = ξ μ u v n, ) satisfies 3.1) with z 0 = ξ μ u 0 v n ).

16 848 I. Chourabi and P. Donato Then, the assumption H 8 ) given in Section 3 implies that = L diva 0 u n ) u n )+θλu n + c γ hu 0 ) T ) Y M Tg),ξ μ u v n, ) diva 0 u n ) u n )+θλu n + c γ hu 0 ) T Y M Tg),ξ μ u 0 v n ). = I n. H 1 ),H 1 0 ) V,V 4.27) Collecting 4.24) 4.27), we get α sup u v n, ) 2 dx 2 2θ c 0 A 0 u 0 ) v n v n ξ μ u 0 v n ) dx α + θ f + c 0 ) ξ μ u 0 v n ) dx + c u n u 0 H 1 0 ) + I n. 4.28) Hence, using H 2 ) H 3 ), 4.18) and 4.23) we deduce that α sup u v n, ) 2 dx = ) n 2 On the other hand, taking v n, v as test function in 4.16) and 4.19), subtracting the two identities and passing to the it, one can easily deduce that v n, v ) 2 dx α A x, u ) v n, v ) v n, v )) dx = A 0 u n ) u n u n A 0 u 0 ) u n u 0 )dx c u n u 0 H 1 0 ), where the right-hand side goes to zero as n, by virtue of 4.18). This, together with 4.29), gives the result, since u v ) 2 dx 2 u v n, ) 2 dx +2 v n, v ) 2 dx. 4.3 Proof of Theorem 2.1 We distinguish here the two cases given by 2.4). Let us treat first the case where M T g) 0org 0. Let {φ n } n N be a sequence in C0 ))N such that φ n u 0 strongly in L 2 )) N. 4.30)

17 Homogenization of Elliptic Problems in Perforated Domains 849 Then, for any ϕ H0 1) L ), using 3) of H 7 ), 2.6) and 2.16), we get [ ] b x, u, u )) ϕ b 0 u 0, u 0 )ϕ dx b x, u, u ) b x, u,c φ n ) ϕdx + [b x, u,c φ n )) b 0 u 0,φ n )]ϕ dx + b 0 u 0,φ n ) b 0 u 0, u 0 ) ϕ dx d 1 c) 1 + u + C φ n ) u C φ n ϕ dx + [b x, u,c φ n )) b 0 u 0,φ n )]ϕ dx + cd 1 c) 1 + φ n + u 0 ) φ n u 0 ϕ dx. = In + Jn + J n. 4.31) Let us pass to the it as tends to zero in the right-hand side of this inequality. Concerning the first term, by a similar argument as in [5, 19], using 2.6), 2.12), 4.30), and Proposition 3.1, we have sup I n d 1c) sup u + C φ n u ) u C φ n ϕ dx sup C φ n u L2 ) + C φ n u 2 L 2 ) ) c φ n u 0 L 2 ) + φ n u 0 2 L 2 ) ). This, together with 4.30), implies n For the third term J n independent of ), we have n sup In = ) sup J n c φ n u 0 L2 ) = ) n It remains to pass to the it in the second term. To this aim, we write Jn [b x, u 0,C φ n )) b 0 u 0,φ n )]ϕ dx + [b x, u 0,C φ n )) b x, u,c φ n )) ]dx [b x, u 0,C φ n )) b 0 u 0,φ n )]ϕ dx + c u u 0 dx, where we used the assumption H 7 ) and again 2.12) and 4.30). Consequently, by Proposition 3.2 and Theorem 4.1, n sup Jn = )

18 850 I. Chourabi and P. Donato Hence, by 4.31) 4.34) for any ϕ H0 1) L ), we have [ ] b x, u, u )) ϕ b 0 u 0, u 0 )ϕ dx = ) Moreover, from 2.1) and 4.14), we derive λ u ϕ dx = λ and ũ ϕ dx = λθ u 0 ϕ dx 4.36) fϕ dx = fχ ϕ dx = θ fϕ dx. 4.37) On the other hand, using the unfolding periodic method and arguing as in [10] see also [8]), we get ρ x)hu )ϕ dσ x = c γ hu)ϕ dx, 4.38) Γ 1 where c γ is defined by 2.15) and Γ 1 g x)ϕ dσ x = T Y M Tg) ϕ dx. 4.39) Let us prove now 2.17). For any ψ H0 1), from Theorem 4.2, the assumption H 3)andthe Hölder inequality, we obtain div A x, ũ ) u,ψ H 1 ),H0 1) = A x, u ) u ψ dx = A x, u ) v ψ dx + A x, u ) u v ) ψ dx = A x, u ) v ψ dx, 4.40) where v is the solution of problem 4.15). On the other hand, thanks to 3.5)ii) and 4.17), we can apply Theorem 3.1 written for z = u and z 0 = v 0 = u 0 ) to problem 4.15) and we have A x, u ) v ψ dx = A 0 u 0 ) u 0 ψ dx. This together with 4.40) gives 2.17). Hence, using 2.17) and 4.35) 4.39), we can pass to the it in 2.5) for any ϕ in H 1 0 ) L ) and obtain A 0 u 0 ) u 0 ϕ dx + θλ u 0 ϕ dx + c γ u 0 ϕ dx = b 0 u 0, u 0 )ϕ dx + T Y M Tg) ϕ dx + θ fϕ dx, ϕ H0 1 ) L ), 4.41)

19 Homogenization of Elliptic Problems in Perforated Domains 851 where b 0 is given by 2.16), i.e., the variational formulation of the homogenized problem 2.18). Let us consider now the case M T g) =0andthatA is independent of t. Using3)ofH 3 ), for ϕ H0 1) L ), we write here b x, u, u )) ϕ dx b 0 u 0, u 0 )ϕ dx b x, u, u ) b x, u,c φ n + χ g ) ϕ dx + [b x, u,c φ n + χ )) b 0 u 0,φ n )]ϕ dx + b 0 u 0,φ n ) b 0 u 0, u 0 ) ϕ dx b 1 c) 1 + u + C φ n + χ u C φ n χ g ϕ dx + [b x, u,c φ n + χ g )) b 0 u 0,φ n )]ϕ dx + cb 1 c) 1 + φ n + u 0 φ n u 0 ϕ )dx. Then, arguing as before and using here the results corresponding to this case, we have b x, u, u )) b 0 u 0, u 0 ) weakly in L 1 ), 4.42) where b 0 is now given by 2.19). On the other hand, 4.36) 4.38) still hold true and from Proposition 3.8 of [10], g x)ϕ dσ x =0. Γ 1 Finally, the convergence 2.21) follows as before by using 4.40). Hence, again we can pass to the it in 2.5) for this case, to obtain A 0 u 0 ) u 0 ϕ dx + θλ u 0 ϕ dx + c γ u 0 ϕ dx 4.43) = b 0 u 0, u 0 )ϕ dx + θ fϕ dx, ϕ H0 1 ) L ) with b 0 given by 2.19), which is the variational formulation of the homogenized problem 2.18). This ends the proof of Theorem 2.1. References [1] Artola, M. and Duvaut, G., Homogénéisation d une classe de problèmes non linéaires, C. R. Acad. Paris Série A, 288, 1979, [2] Artola, M. and Duvaut, G., Un résultat d homogénéisation pour une classe de problèmes de diffusion non linéaires stationnaires, Ann. Faculte Sci. Toulouse, IV, 1982, [3] Bendib, S., Homogénéisation d une classe de problèmes non linéaires avec des conditions de Fourier dans des ouverts perforés, Institut National Polytechnique de Lorraine, [4] Bensoussan, A., Boccardo, L., Dall Aglio, A. and Murat, F., H-convergence for quasilinear elliptic equations under natural hypotheses on the correctors, Proceedings of the Conference Composite Media and Homogenization Theory, Trieste, [5] Bensoussan, A., Boccardo, L. and Murat, F., H-convergence for quasi-linear elliptic equations with quadratic growth, Appl. Math. Optim., 26, 1992,

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