Homogenization and correctors for monotone problems in cylinders of small diameter

Available online at www.sciencedirect.com Ann. I. H. Poincaré – AN 30 (2013) 519–545 www.elsevier.com/locate/anihpc Homogenization and correctors for monotone problems in cylinders of small diameter Juan Casado-Díaz a,∗, François Murat b, Ali Sili c aDpto. de Ecuaciones Diferenciales y Análisis Numérico, Fac. de Matemáticas C. Tarfia s/n, 41012 Sevilla, Spain bLaboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Boîte courrier 187, 75252 Paris Cedex 05, France cDépartement de Mathématiques, Université du Sud Toulon-Var, BP 20132, 83957 La Garde Cedex, France Received 20 July 2010; received in revised form 19 October 2012; accepted 19 October 2012 Available online 9 November 2012 Abstract In this paper we study the homogenization of monotone diffusion equations posed in an N-dimensional cylinder which converges to a (one-dimensional) segment line. In other terms, we pass to the limit in diffusion monotone equations posed in a cylinder whose diameter tends to zero, when simultaneously the coefficients of the equations (which are not necessarily periodic) are also varying. We obtain a limit system in both the macroscopic (one-dimensional) variable and the microscopic variable. This system is nonlocal. From this system we obtain by elimination an equation in the macroscopic variable which is local, but in contrast with usual results, the operator depends on the right-hand side of the equations. We also obtain a corrector result, i.e. an approximation of the gradients of the solutions in the strong topology of the space Lpin which the monotone operators are defined. Résumé Dans cet article nous étudions l’homogénéisation d’équations de diffusion monotones posées dans un cylindre de dimension N qui converge vers un segment (qui est donc unidimensionnel). En d’autres termes, nous passons à la limite dans des équations de diffusion monotones posées dans un cylindre dont le diamètre tend vers zéro, quand en même temps les coefficients des équations (qui ne sont pas nécessairement périodiques) varient eux aussi. Nous obtenons un système limite en la variable macroscopique (unidimensionnelle) et en la variable microscopique. Ce système est non local. A partir de ce système nous obtenons par élimination une équation en la variable macroscopique qui est locale, mais dans laquelle, à la difference des résultats usuels, l’opérateur dépend du second membre des équations. Nous obtenons aussi un résultat de correcteur, c’est à dire une approximation des gradients des solutions dans la topologie forte de l’espace Lpdans lequel sont définis les opérateurs monotones. Keywords: Homogenization; Thin domains; Monotone problems *Corresponding author. E-mail addresses: [email protected] (J. Casado-Díaz), [email protected] (F. Murat), [email protected] (A. Sili). 0294-1449/$ – see front matter http://dx.doi.org/10.1016/j.anihpc.2012.10.004 L'Association Publications de l'Institut Henri Poincaré. Published by Elsevier B.V. All rights reserved. ©2012 L'Association Publications de l'Institut Henri Poincaré. Published by Elsevier B.V. All rights reserved. ©2012 L'Association Publications de l'Institut Henri Poincaré. Published by Elsevier B.V. All rights reserved. ©2012 520 J. Casado-Díaz et al. / Ann. I. H. Poincaré – AN 30 (2013) 519–545 1. Introduction We consider in this paper the homogenization, when the coefficients vary, of monotone problems posed in a cylinder of RNwith fixed length and small diameter. Specifically, we consider a bounded open interval I⊂Rand a bounded domain ω⊂RN−1. Defining the cylinder Ωεby Ωε=I×(εω), we are interested in the solutions of the monotone problem ⎧ ⎪ ⎨ ⎪ ⎩ −div ˜ Aε(˜x,∇˜uε)=˜ fε−div ˜ fεin Ωε, ˜ Aε(˜x,∇˜uε)−˜ fε˜νε=0inI×(ε∂ω), ˜uε=0in∂I ×(εω), (1.1) where ˜ Aε:Ω×RN→RNare Carathéodory functions which are monotone, uniformly p-coercive and with uniform (p −1)-growth, and where, for Ω=I×ω, there exist f∈Lp(Ω) and F∈Lp(Ω)N, such that ˜ fε˜x1,˜x=f˜x1,˜x ε,˜ fε˜x1,˜x=F˜x1,˜x ε,a.e. ˜x1,˜x∈Ωε.(1.2) In this problem, the Neumann boundary condition in the lateral boundary I×(ε∂ω) is crucial, while changing the Dirichlet boundary condition on the bases ∂I ×(εω) (as far as an H1a priori estimate is conserved) does not affect the limit equation. A problem similar to (1.1), but where the operators are linear, Fε≡0, N=3, and Ωεis a cylinder of fixed basis ω⊂R2and small height (which therefore converges to the two-dimensional set ω) has been considered in [4] and [10] (see also [9] for the elasticity problem). In this case, the limit problem, which is posed in the two-dimensional limit domain ω, has a structure which is similar to the structure of the problem posed in Ωε. This will also be the case for problem (1.1), whose limit is posed on the one-dimensional domain I, but, in contrast with usual results, the corresponding operator will depend on F. In order to study the homogenization of (1.1), we perform the change of variables (x1,x)=(˜x1,˜x/ε), which transforms Ωεin Ωas it is usual in the study of the behavior of solutions of partial differential problems posed in thin domains (see e.g. [2,4,5,9–14,17–19,22,23,25]). Defining uεby uε(x1,x)=˜uε(x1,εx), problem (1.1) is then transformed into a new problem which can be written in the variational form: ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ uε∈W1,p(Ω), uε=0on∂I ×ω,  Ω Aε(x, Dεuε)Dεvdx= Ω fvdx+ Ω FDεvdx, ∀v∈W1,p(Ω), v =0on∂I ×ω, (1.3) where Dεis the differential operator Dε=(∂ ∂x1,1 ε∇x)and where Aε:Ω×RN→RNare Carathéodory functions which, similarly to ˜ Aε, are uniformly p-coercive and with uniform (p −1)-growth. These conditions on Aεimply that Dεuεis bounded in Lp(Ω)N. Thus (see e.g. [17]) there exist u0∈W1,p 0(I) and u1∈Lp(I, W1,p(ω)/R)such that uεconverges weakly to u0in W1,p(Ω) and Dεuεconverges weakly to D0(u0,u 1)=(du0 dx1,∇xu1)in Lp(Ω)N. When Aε=Ais fixed, it has been proved in [17] (see also [18,19] for the elasticity problem) that (u0,u 1)is the solution of the following problem ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (u0,u 1)∈W1,p 0(I) ×LpI,W1,p(ω)/R,  Ω Ax,D0(u0,u 1)D0(v0,v 1)dx = Ω fv 0dx + Ω FD0(v0,v 1)dx, ∀(v0,v 1)∈W1,p 0(I) ×LpI,W1,p(ω)/R, (1.4) where D0(v0,v 1)=(dv0 dx1,∇xv1); in this problem both the macroscopic and microscopic variables x1and xappear. One can then wonder whether, when Aεdepends on ε, there exist a subsequence of ε, still denoted by ε, and a Carathéodory function A:Ω×RN→RNsatisfying the same conditions as Aε, such that for every f∈Lp(Ω) and J. Casado-Díaz et al. / Ann. I. H. Poincaré – AN 30 (2013) 519–545 521 every F∈Lp(Ω)N, the limit (u0,u 1)of the solutions uεof (1.1) is the solution of (1.4). We show in the present paper that this is not the case. In contrast we prove (Theorem 3.1 below) that the limit of (1.1) is the nonlocal problem ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (u0,u 1)∈W1,p 0(I) ×LpI,W1,p(ω)/R,  Ω Ax1,D 0(u0,u 1)(x1,.) D0(v0,v 1)dx = Ω fv 0dx + Ω FD0(v0,v 1)dx, ∀(v0,v 1)∈W1,p 0(I) ×LpI,W1,p(ω)/R, (1.5) where Ais no more a Carathéodory function A:Ω×RN→RN, but a nonlocal Carathéodory operator A:I× R×∇ W1,p(ω) →Lp(ω)N, which is measurable in the first variable and continuous in the two other ones (where ∇W1,p(ω) denotes the space of the derivatives ∇xvof functions v∈W1,p(ω)), but such that for a.e. x1∈I,the function Ax1,D 0(u0,u 1)(x1,.) =Ax1,du0 dx1 (x1), ∇xu1(x1,.) ∈Lp(ω)N at the point x∈ωdepends not only on ∇xu1(x1,x)but on all the points of ∇xu1(x1,z)for z∈ω. A similar effect has been obtained in [6] for the homogenization of elliptic periodic equations of the type −divA(x, x ε)∇uε. Eliminating u1in function of u0in the system (1.5), we obtain a local equation for u0, namely ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ u0∈W1,p 0(I),  I aFx1,du0 dx1dv0 dx1 dx1= I ω fdx v0dx1+ I ω F1dxdv0 dx1 dx, ∀v0∈W1,p 0(I), (1.6) where now aF:I×R→Ris a Carathéodory function, but which depends on the (N −1)-last entries Fof the right-side Fof (1.3). Thus, problem (1.6) is not sufficient for studying the effect of the right-hand side Fon the solutions of (1.3), and we must remain with (1.5) for this study. In addition to searching for the limit problem of (1.1), we are also interested in the present paper in obtaining a corrector result for (1.1). Our main result in this direction essentially establishes (see Theorem 3.8 below for the precise formulation) the existence of a (sub-)sequence of nonlocal operators Pε:I×R×∇ W1,p(ω) →Lp(ω)N such that Dεuε−Pε(x1,D 0(u0,u 1)) converges strongly to zero in Lp(Ω)N. Let us emphasize that here again the corrector Pεis nonlocal. In Section 4below we prove that when Aεdoes not depend on x1, the operator Aand the corrector Pεare actually local (other assumptions which also provide a local operator Acan be found in [8,13,14]). In contrast, we show in Section 5by means of two (periodic in x1) examples that even if Aεdoes not depend on x, the operator Ais nonlocal. Let us conclude this introduction by pointing now that we consider in the present paper the case of cylinders with fixed length and small diameter with Neumann boundary condition on the lateral boundary. Analogous results can be obtained by the same proofs in the case of cylinders with fixed bases and small height with Neumann boundary conditions on the two bases. 2. Notation and preliminaries We consider an integer number N⩾2. The vectors xof RNwill be decomposed as x=(x1,x), with x1∈R,x∈RN−1. The vectors of RN−1will be considered as elements of RNby identifying x∈RN−1with (0,x)∈RN. We define MNas the space of matrices of order N. We denote by e1∈RNthe vector (1,0). The N-dimensional measure of a set B⊂RNwill be denoted by |B|, while the (N −1)-dimensional measure of a set D⊂RN−1will be denoted by |D|N−1. If Xis a normed space and Xits dual, we denote by x,xthe duality pairing between x∈Xand x∈X. 522 J. Casado-Díaz et al. / Ann. I. H. Poincaré – AN 30 (2013) 519–545 For an open set Θ⊂Rmand a number q∈[1,+∞], we denote by W1,q (Θ) the usual Sobolev space. If Υis a subset of the boundary ∂Θ of Θ, we denote by W1,p Υ(Θ) the space of those functions of W1,p(Θ) which vanish on Υ. When m=N−1, the space of the gradients of the functions of W1,p(Θ) will be denoted by ∇W1,p(Θ). When p=2, we write H1(Θ) =W1,2(Θ),H1 Υ(θ) =W1,2 Υ(Θ),∇H1(Θ) =∇W1,2(Θ). We use the index to denote periodicity, for example C∞ ([0,1])is the space of the functions of C∞(R)which are periodic of period 1. For a bounded smooth connected open set ω⊂RN−1and a bounded interval I=]b,d[⊂R,wesetΩ=I×ω, Γ=({b}∪{d})×ω,Ωε=I×(εω),Γε=({b}∪{d})×(εω). In what follows, we consider a sequence of Carathéodory functions Aε:Ω×RN→RN. We define ˆ Eε:Ω× RN×RN→Rand ˇ Eε:Ω×RN→R(where Erefers to energy) by ˆ Eε(x,ξ,ζ)=Aε(x, ξ) −Aε(x, ζ )(ξ −ζ), ˇ Eε(x, ξ ) =Aε(x, ξ )ξ, for every ξ,ζ ∈RN,a.e.x∈Ω. We will assume that there exist p∈(1,+∞),α>0,β >0, σ∈(0,min{1,p−1}) and h1,h 2∈L1(Ω),h1,h 2⩾0, such that for every ξ,ζ ∈RNand a.e. x∈Ω,wehave Aε(x, 0)=0,(2.1) α|ξ−ζ|p⩽ˆ Eε(x,ξ,ζ), if p∈[2,+∞), (2.2) α|ξ−ζ|p⩽ˆ Eε(x,ξ,ζ)p 2h1+ˇ Eε(x, ξ ) +ˇ Eε(x, ζ )2−p 2,if p∈(1,2],(2.3) Aε(x, ξ) −Aε(x, ζ ) p⩽βh2+ˇ Eε(x, ξ ) +ˇ Eε(x, ζ )p−1−σˆ Eε(x,ξ,ζ) σ.(2.4) Remark 2.1. Using the fact that (2.1), and (2.2)or(2.3) imply that α|ξ|p⩽ˇ Eε(x, ξ ), if p∈[2,+∞), and α|ξ|p⩽h1+ˇ Eε(x, ξ ), if p∈(1,2], and the fact that (2.1) and (2.4) imply that there exist β∗>0 and h∗∈L1(Ω),h∗⩾0, such that ˇ Eε(x, ξ ) ⩽β∗|ξ|p+h∗, one can prove the following equivalences: In the case p∈[2,+∞),ifAεsatisfies (2.1), (2.2), and (2.4), then there exist ¯ β>0 and ¯ h2∈L1(Ω),¯ h2⩾0, such that for every ξ,ζ ∈RNand a.e. x∈Ω Aε(x, ξ) −Aε(x, ζ )⩽¯ β¯ h2+|ξ|p+|ζ|pp−1−σ p−σ|ξ−ζ|σ p−σ.(2.5) Reciprocally, if Aεsatisfies (2.1), (2.2), and if there exist ¯ β>0, σ∈(0,1)and ¯ h2∈L1(Ω),¯ h2⩾0, such that for every ξ,ζ ∈RNand a.e. x∈Ω Aε(x, ξ) −Aε(x, ζ )⩽¯ β¯ h2+|ξ|p+|ζ|pp−1−σ p|ξ−ζ|σ,(2.6) then there exist β>0 and h2∈L1(Ω),h2⩾0, such that Aεsatisfies (2.4). In the case p∈(1,2],ifAεsatisfies (2.1), (2.3), and (2.4), then there exist ¯α>0, ¯ β>0 and ¯ h1,¯ h2∈L1(Ω), ¯ h1,¯ h2⩾0, such that Aεsatisfies (2.5) and is such that for every ξ,ζ ∈RNand a.e. x∈Ω ¯α|ξ−ζ|2⩽ˆ Eε(x,ξ,ζ) ¯ h1+|ξ|p+|ζ|p2−p p.(2.7) Reciprocally, if Aεsatisfies (2.1), (2.6), and (2.7)forsome ¯α>0, ¯ β>0, σ∈(0,p −1), and ¯ h1,¯ h2∈L1(Ω), ¯ h1,¯ h2⩾0, then there exist α>0, β>0 and h 1,h  2∈L1(Ω),h 1,h  2⩾0, such that for every ξ,ζ ∈RNand a.e. x∈Ω α|ξ−ζ|p⩽ˆ Eε(x,ξ,ζ)p 2h 1+ˇ Eε(x, ξ ) +ˇ Eε(x, ζ )2−p 2,if p∈(1,2],(2.8) Aε(x, ξ) −Aε(x, ζ ) p⩽βh 2+ˇ Eε(x, ξ ) +ˇ Eε(x, ζ )2(p−1)−pσ 2ˆ Eε(x,ξ,ζ)pσ 2.(2.9) J. Casado-Díaz et al. / Ann. I. H. Poincaré – AN 30 (2013) 519–545 523 Remark 2.2. As a consequence of Remark 2.1 we get that the class of Carathéodory functions satisfying (2.1), (2.2) or (2.3), and (2.4) is not empty. Indeed the function defined by Aε(x, ξ) =aε(x)|ξ|p−2ξ, with p∈(1,+∞)and aε∈L∞(Ω) such that 0 <ˇα⩽aε(x) ⩽ˇ β<+∞, satisfies (2.1), (2.2)or(2.3), and (2.4)forsomeα>0, β>0, h=0 and σ=1forp⩾2, σ=p(p −1)/2for1<p<2. Classes of Carathéodory functions satisfying assumptions slightly more general than (2.1), (2.2)or(2.3), and (2.4) have been introduced in Section 7 of [7], where observations similar to the ones made in the above Remarks 2.2 and 2.1 can also be found. As it was done in [7], we prefer here to impose (2.1), (2.2)or(2.3), and (2.4) in place of the more classical assumptions (2.1), (2.2)or(2.7), and (2.6), because the assumptions written in the first form are stable by homogenization (see Theorem 3.1 below). We denote by Dε:W1,p(Ω) →Lp(Ω)Nand D0:W1,p(I) ×Lp(I, W 1,p(ω)) →Lp(Ω)N, the differential operators defined by Dεu=∂1ue1+1 ε∇xu, ∀u∈W1,p(Ω), (2.10) D0(u0,u 1)=du0 dx1 e1+∇ xu1,∀(u0,u 1)∈W1,p(I) ×LpI,W1,p(ω).(2.11) We denote by Ca generic positive constant, which only depends on p,N,α,β,σ,h1,h2,|ω|and |I|and can change from a line to another one. Our aim is to study the asymptotic behavior of the solutions uεof ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ uε∈W1,p Γ(Ω),  Ω Aε(x, Dεuε)Dεvdx= Ω fvdx+ Ω FDεvdx, ∀v∈W1,p Γ(Ω), (2.12) where f∈Lp(Ω),F∈Lp(Ω)N(we will see later that the boundary condition uε∈W1,p Γ(Ω) is not very important). As we already said in the Introduction, problem (2.12) is equivalent to (1.1) with ˜ fεand ˜ fεgiven by (1.2). Taking uεas test function in (2.12) and using Poincaré’s inequality, we deduce that the solutions uεof (2.12) satisfy  Ω |Dεuε|pdx ⩽Cfp Lp(Ω) +Fp Lp(Ω)N. In what follows, we will use the following lemma (see [17]). Theorem 2.3. If uεis a sequence in W1,p(Ω) such that  Ω |Dεuε|pdx ⩽C, then there exist u0∈W1,p(I),u1∈Lp(I, W 1,p(ω)) and a subsequence of uε(still denoted by uε)such that uεu 0in W1,p(Ω), (2.13) DεuεD 0(u0,u 1)in Lp(Ω)N.(2.14) 3. Homogenization In this section we perform the homogenization of (2.12). The main result of the present paper is contained in the following theorem which describes the asymptotic behavior of the solutions of (2.12). 524 J. Casado-Díaz et al. / Ann. I. H. Poincaré – AN 30 (2013) 519–545 Theorem 3.1. There exist an operator A:I×R×∇W1,p(ω) →Lp(ω)Nand a subsequence of ε, still denoted by ε, such that for every uε∈W1,p(Ω),u0∈W1,p(I),u1∈Lp(I, W1,p(ω)),fε∈Lp(Ω),Fε∈Lp(Ω)N,f∈Lp(Ω) and F∈Lp(Ω) which satisfy uεu 0in W1,p(Ω), (3.1) 1 ε∇xuε∇xu1in Lp(Ω)N−1,(3.2) fεf in Lp(Ω), (3.3) Fε→Fin Lp(Ω)N,(3.4)  Ω Aε(x, Dεuε)Dεvdx= Ω fεvdx+ Ω FεDεvdx, ∀v∈W1,p Γ(Ω), (3.5) we have Aε(x, Dεuε)Ax1,D 0(u0,u 1)in Lp(Ω)N.(3.6) Moreover, the functions u0,u1,fand Fare related by  Ω Ax1,D 0(u0,u 1)D0(v0,v 1)dx = Ω fv 0dx + Ω FD0(v0,v 1)dx, ∀(v0,v 1)∈W1,p 0(I) ×LpI,W1,p(ω).(3.7) The operator Aalso satisfies the following properties: The application x1→A(x1,s,∇xψ)∈Lp(ω)Nis measurable ∀(s, ψ) ∈R×W1,p(ω). (3.8) A(., 0,0)=0a.e. in I. (3.9) Denoting by ˆ E:I×R×∇ W1,p(ω) ×R×∇ W1,p(ω) →L1(ω)N,ˇ E:I×R×∇ W1,p(ω) →L1(ω)N,the operators ˆ E(x1,s 1,∇xψ1,s 2,∇xψ2)=A(x1,s 1,∇xψ1)−A(x1,s 2,∇xψ2)(s1−s2)e1+∇ x(ψ1−ψ2), ∀(s1,ψ 1), (s2,ψ 2)∈R×W1,p(ω), a.e. x1∈I, (3.10) ˇ E(x1,s,∇xψ)=A(x1,s,∇xψ)(se1+∇ xψ), ∀(s, ψ) ∈R×W1,p(ω), a.e. x1∈I, (3.11) we have for every s1,s 2∈R,ψ1,ψ 2∈W1,p(ω) and a.e. x1∈I α {x1}×ω|s1−s2|p+∇x(ψ1−ψ2) pdx ⩽ {x1}×ω ˆ E(x1,s 1,∇xψ1,s 2,∇xψ2)dx,if p∈[2,+∞), (3.12) α {x1}×ω|s1−s2|p+∇x(ψ1−ψ2) pdx ⩽ {x1}×ω ˆ E(x1,s 1,∇xψ1,s 2,∇xψ2)dxp 2 · {x1}×ωh1+ˇ E(x1,s 1,∇xψ1)+ˇ E(x1,s 2,∇xψ2)dx2−p 2 ,if p∈(1,2],(3.13) J. Casado-Díaz et al. / Ann. I. H. Poincaré – AN 30 (2013) 519–545 525  {x1}×ωA(x1,s 1,∇xψ1)−A(x1,s 2,∇xψ2) p dx ⩽β {x1}×ωh2+ˇ E(x1,s 1,∇xψ1)+ˇ E(x1,s 2,∇xψ2)dxp−1−σ p−1 · {x1}×ω ˆ E(x1,s 1,∇xψ1,s 2,∇xψ2)dxσ p−1 .(3.14) Remark 3.2. The properties (3.9), (3.12), (3.13) and (3.14) imply the existence of h0∈L1(I) and C>0, such that for every s1,s 2∈R,ψ1,ψ 2∈W1,p(ω) and a.e. x1∈I, the operator Asatisfies  {x1}×ωA(x1,s 1,∇xψ1) p dx⩽h0+C|s1|p+ {x1}×ω |∇xψ1|pdx,(3.15)  {x1}×ωA(x1,s 1,∇xψ1)−A(x1,s 2,∇xψ2) p dx ⩽h0+C|s1|+|s2|p+ {x1}×ω|∇xψ1|+|∇ xψ2|pdxp(p−1−σ) (p−1)(p−σ) ·|s1−s2|p+ {x1}×ω∇x(ψ1−ψ2) pdxσ (p−1)(p−σ) .(3.16) Remark 3.3. Thanks to (3.8), (3.9) and (3.14), we deduce that for every φ1∈Lp(I) and every φ∈Lp(I;W1,p(ω)), the function x1∈I→ A(x1,φ 1(x1), ∇xφ(x1, .))(x)is in Lp(I;Lp(ω))N. Thus, the term A(x1,D 0(u0,u 1)) which appears in (3.6), (3.7) has a meaning as a function of Lp(Ω)N. Remark 3.4. Observe that in Theorem 3.1 the sequence uεis not supposed to vanish on Γ. Thus, the boundary condition in Γis not important in the homogenization result, for example, it can be substituted by a Neumann condition. Eliminating u1from (3.7), Theorem 3.1 gives in particular the problem satisfied by the limit uof the sequence uε of solutions of (2.12). This is given by Corollary 3.5. We consider the subsequence of εand the operator A=(A1,A)given by Theorem 3.1. We define the operator R:Lp(ω)N−1→(W1,p(ω)/R)by RG,v 1= ω G∇xv1dx,∀G∈Lp(ω)N−1,∀v1∈W1,p(ω)/R. We also introduce U:I×R×(W 1,p(ω)/R)→W1,p(ω)/Rand a:I×R×(W1,p(ω)/R)→Rby  ω Ax1,s,∇xU(x1,s,η) ∇xv1dx=η,v1,∀v1∈W1,p(ω), a.e. x1∈I, a(x1,s,η)=A1x1,s,∇xU(x1,s,η) ,∀(s, η) ∈R×W1,p(ω)/R,a.e. x1∈I. Then, if uε,u0,u1,fε,Fε,fand Fare as in the statement of Theorem 3.1, the function u0satisfies the equation −d dx1 ax1,du0 dx1 ,RF(x1,.) = ωfx1,x−∂1F1x1,xdxin I. 526 J. Casado-Díaz et al. / Ann. I. H. Poincaré – AN 30 (2013) 519–545 Remark 3.6. Corollary 3.5 shows that for η∈Lp(I, (W 1,p(ω)/R))fixed, and F∈Lp(Ω)N−1such that RF=η in W1,p(ω)/R, for a.e. x1∈I, the limit problem of (2.12) is local in x1. In particular, defining a0:I×R→Rby a0(x1,s)=a(x1,s,0), ∀s∈R,a.e. x1∈R, Corollary 3.5 shows that for every f,F1∈Lp(Ω), the solution uεof ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ uε∈W1,p Γ(Ω),  Ω Aε(Dεuε)Dεvdx= Ω fvdx+ Ω F1∂1vdx, ∀v∈W1,p Γ(Ω), converges weakly in W1,p(Ω) to the unique solution u0of −d dx1 a0x1,du0 dx1= ωfx1,x−∂1F1x1,xdxin I, u0∈W1,p 0(I). This is similar to the homogenization result given in [10] for the case of a plate. In addition to Theorem 3.1, we also have a corrector result for the sequence of solutions uεof (2.12). This is given by Theorem 3.8 below, first we need to give the following definition. Definition 3.7. We consider the subsequence of εand the operator Agiven by Theorem 3.1. For every (s, ψ) ∈ R×W1,p(ω) and a.e. x1∈I, we define Wε(x1,s,∇xψ) as the solution of ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ Wε(x1,s,∇xψ)∈W1,p(Ω)/R,  Ω Aεx1,D εWε(x1,s,∇xψ)Dεvdx= Ω A(x1,s,∇xψ)Dεvdx, ∀v∈W1,p(Ω)/R. (3.17) We then define Pε:I×R×∇W1,p(ω) →Lp(ω)Nby Pε(x1,s,∇xψ)=DεWε(x1,s,∇xψ). Theorem 3.8. We consider the subsequence of εand the operator Agiven by Theorem 3.1. Then, there exist a constant C>0and a function h0∈L1(I), such that for uε,u0,u1,fε,Fε,fand Fas in the statement of Theorem 3.1 and for every step function Ψ=m j=1(sje1+∇ xψj)χ(ij−1,ij)×ω, with sj∈R,ψj∈W1,p(ω),1⩽j⩽m,b<i 0<···< im<d, we have limsup ε→0 (i0,im)×ωDεuε−Pε(x1,Ψ)  pdx ⩽im  i0h0+C du0 dx1 +|Ψ1|p +C ω|∇xu1|+Ψpdxdx1q · (i0,im)×ωD0(u0,u 1)−Ψ pdx1−q ,(3.18) with q=(p −1−σ)/(p−σ)if p∈[2,+∞),q=(p −2σ)/(2(p −σ)) if p∈(1,2]. Moreover, if uε=0on Γor if (3.5)holds for every v∈W1,p Γ(Ω), then we can take i0=b,im=d. J. Casado-Díaz et al. / Ann. I. H. Poincaré – AN 30 (2013) 519–545 527 Remark 3.9. The meaning of Theorem 3.8 is that taking Ψclose enough to D0(u0,u 1),Pε(x1,Ψ)is a good approximation of Dεuεin the strong topology of Lp(Ω)N(corrector result). In fact, if we formally take Ψ=D0(u0,u 1) in (3.18), we will deduce Dεuε−Pεx1,D 0(u0,u 1)→0inLp(Ω)N. However, we do not know if Pεis a Carathéodory function and thus Pε(x1,D 0(u0,u 1)) is not well defined. Proof of the results of Section 3 The proof of our results is an adaptation of L. Tartar’s method (see [20,24]). We start with the following result. Lemma 3.10. We consider uε,w ε∈W1,p(Ω),fε,g ε∈Lp(Ω),Fε,G ε∈Lp(Ω)N, which satisfy  Ω Aε(x, Dεuε)Dεvdx= Ω fεvdx+ Ω FεDεvdx, ∀v∈W1,p Γ(Ω), (3.19)  Ω Aε(x, Dεwε)Dεvdx= Ω gεvdx+ Ω GεDεvdx, ∀v∈W1,p Γ(Ω). (3.20) We assume that there exist u0,w 0∈W1,p(I),u1,w 1∈Lp(I, W1,p(ω)),T,S ∈Lp(Ω)N,f,g ∈Lp(Ω),F,G ∈ Lp(Ω)N, such that uεu 0,w εw 0in W1,p(Ω), (3.21) 1 ε∇xuε∇xu1,1 ε∇xwε∇xw1in Lp(Ω)N−1,(3.22) Aε(x, Dεuε)T, A ε(x, Dεwε)S in Lp(Ω)N,(3.23) fεf, g εg in Lp(Ω), (3.24) Fε→F, Gε→Gin Lp(Ω)N.(3.25) Then, T,Ssatisfy the following properties ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩  Ω TD 0(v0,v 1)dx = Ω fv 0dx + Ω FD0(v0,v 1)dx,  Ω SD0(v0,v 1)dx = Ω gv0dx + Ω GD0(v0,v 1)dx, ∀(v0,v 1)∈W1,p 0(I) ×LpI,W1,p(ω). (3.26) For a.e. x1∈I, we have α {x1}×ωD0(u0−w0,u 1−w1) pdx⩽ {x1}×ω (T −S)D0(u0−w0,u 1−w1)dx,if p∈[2,+∞), (3.27) α {x1}×ωD0(u0−w0,u 1−w1) pdx ⩽ {x1}×ω (T −S)D0(u0−w0,u 1−w1)dxp 2 · {x1}×ωh1+TD 0(u0,u 1)+SD0(w0,w 1)dx2−p 2 ,if p∈(1,2],(3.28) 534 J. Casado-Díaz et al. / Ann. I. H. Poincaré – AN 30 (2013) 519–545 Definition 4.2. We assume that the functions Aεdo not depend on x1and we consider the subsequence of εand the function Agiven by Lemma 4.5. We define P ε:ω×RN→RN−1by ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ P ε(., ξ) ∈∇W1,p(ω),  ω A εx,ξ 1+P εx,ξ∇xψdx = ω Ax,ξ∇xψdx , ∀ψ∈W1,p(ω), (4.8) for every ξ∈RN. Theorem 4.3. Under the assumptions of Definition 4.2, there exist a constant C>0and a function h0∈L1(I) such that for uε,u0,u1,fε,Fε,fand Fas in the statement of Theorem 3.1 and for every step function Φ=m j=1(sje1+ n l=1ηjlχKl)χ(ij−1,ij), with sj∈R,ηjl ∈RN−1,b<i 0<···<i m<d,Kl⊂¯ωcompact, |Kl1∩Kl2|N−1=0if l1= l2,¯ω=n l=1Kl, we have lim ε→0 (i0,im)×ω ∂1uε−du0 dx1 p dx =0,(4.9) limsup ε→0 (i0,im)×ω 1 ε∇xuε−P εx,Φ p dx ⩽ (i0,im)×ωhc+C du0 dx1 +|∇ xu1|+|Φ|pdxq · (i0,im)×ω du0 dx1 −Φ1 p +∇xu1−Φ pdx1−q ,(4.10) with q=(p −1−σ)/(p−σ)if p∈[2,+∞),q=(p −2σ)/(2(p −σ)) if p∈(1,2]. Proof of the results of Section 4 We start with the following lemma which can be proved reasoning similarly to Lemma 3.10. Lemma 4.4. We assume that the functions Aε(and then h)do not depend on x1. We consider ψε,η ε∈W1,p(ω), s1,s 2∈R,F,G ∈Lp(ω)N−1, which satisfy  ω A εx,s 1e1+∇ xψε∇xvdx= ω F∇xvdx,∀v∈W1,p(ω), (4.11)  ω A εx,s 2e1+∇ xηε∇xvdx= ω G∇xvdx,∀v∈W1,p(ω). (4.12) We assume there exist ψ,η ∈W1,p(ω),T=(T1,T), S =(S1,S)∈Lp(ω)N, such that ∇xψε∇xψ, ∇xηε∇xηin Lp(ω)N−1,(4.13) Aεx,s 1e1+∇ xψεT, A εx,s 2e1+∇ xηεS in Lp(ω)N.(4.14) Then, T,Ssatisfy  ω T∇xvdx= ω F∇xvdx, ω S∇xvdx= ω G∇xvdx,∀v∈W1,p(ω). (4.15) The functions Tand Ssatisfy the following inequalities a.e. in ω J. Casado-Díaz et al. / Ann. I. H. Poincaré – AN 30 (2013) 519–545 535 |T−S|p⩽βh2+T(s 1e1+∇ xψ)+S(s2e1+∇ xη)p−1−σ p−1 ·(T −S)(s1−s2)e1+∇ x(ψ −η) σ p−1,(4.16) α|s1−s2|p+∇x(ψ −η) p⩽(T −S)(s1−s2)e1+∇ x(ψ −η),(4.17) if p∈[2,∞), and α|s1−s2|p+∇x(ψ −η) p⩽(T −S)(s1−s2)e1+∇ x(ψ −η)p 2 ·h1+T(s 1e1+∇ xψ)+S(s2e1+∇ xη)2−p 2,(4.18) if p∈(1,2]. Moreover, for every ϑ∈W1,∞(¯ω), we have lim ε→0 ω ˆ Eεx,s 1e1+∇ xψε,s 2e1+∇ xηεϑdx = ω (T −S)(s1−s2)e1+∇ x(ψ −η)ϑdx. (4.19) Using this lemma we can also prove the following result reasoning similarly to the proof of Theorem 3.1. Lemma 4.5. We assume that the functions Aεdo not depend on x1. Then, there exist a subsequence of ε, still denoted by ε, and Carathéodory function A:ω×RN→RNsatisfying (4.4), (4.7)and (4.5)or (4.6)depending if p∈[2,+∞) or p∈(1,2], such that for every (s, F )∈R×Lp(ω)N−1, the sequence ψε∈W1,p(ω)/Rof solutions of the Neumann problems  ω A εx,se 1+∇ xψε∇xv1dx= ω F∇xv1dx,∀v1∈W1,p(ω)/R,(4.20) converges weakly in W1,p(ω)/Rto the solution ψof  ω Ax,se 1+∇ xψ∇xv1dx= ω F∇xv1dx,∀v1∈W1,p(ω)/R,(4.21) and satisfies Aεx,se 1+∇ xψεA x,se 1+∇ xψin Lp(ω)N.(4.22) Remark 4.6. For every s∈R, the function A s:ω×RN−1→RN−1defined by A sx,η=Ax,se 1+η,∀η∈RN−1,a.e. x∈ω, is the H-limit (see [20]) of the sequence (A ε)s:ω×RN−1→RN−1defined as A εsx,η=A εx,se 1+η,∀η∈RN−1,a.e. x∈ω. Proof of Theorem 4.1.We consider the subsequence of εgiven by Lemma 4.5, extracting a subsequence if necessary, we can assume that Theorem 3.1 holds. For s∈R,wetakeφs(x1)=sx1. Then, for ψ∈W1,p(ω), we define uε∈ W1,p(Ω) as uε(x) =φs(x1)+εψε(x),a.e.inω, with ψε∈W1,p(ω)/Rthe solution of  ω A εx,se 1+∇ xψε∇xv1dx= ω Ax,se 1+∇ xψ∇xv1dx,∀v1∈W1,p(ω)/R.(4.23) By Lemma 4.5,Dεuε=se1+∇ xψεconverges weakly in Lp(ω)Nto D0(φs(x1), ψ) =se1+∇ xψand Aεx,se 1+∇ xψεA x,D 0(φs,ψ) in Lp(ω). (4.24) Moreover, we have  Ω Aεx,D εuεDεvdx= Ω FDεvdx,∀v∈W1,p Γ(Ω), 536 J. Casado-Díaz et al. / Ann. I. H. Poincaré – AN 30 (2013) 519–545 with F1=0 and F=A(se1+∇ xψ). Then, from Theorem 3.1, we also deduce Aεx,D εuεAx1,D 0(φs,ψ) . By (4.24), we get Ax,se 1+∇ xψ=A(x1,s,∇xψ), for every (s, ψ) ∈R×W1,p(Ω),a.e.inΩ. This proves (4.1). Since this equality defines the operator A, we deduce that the sequence given in Theorem 3.1 can be taken as the subsequence given in Lemma 4.5, without extracting any subsequence. 2 Proof of Theorem 4.3.Let us only prove the case p∈[2,+∞), the case p∈(1,2]is analogous. For s∈Rand ψ∈W1,p(ω)/R, we define ψε∈W1,p(ω)/Ras the solution of (4.23). By Definition 3.7 of Pε,itis then easy to check that Pε(x1,s,ψ ε)−se1−∇ xψε→0inLp(Ω). (4.25) On the other hand, for ξ∈RN,K⊂¯ωcompact and ϑ∈W1,p(ω), with ϑ⩾χK, assertion (4.19) with s1=s,s2=ξ, ∇xηε=P ε(., ξ),T=A(x,se 1+∇ xψ),S=A(x,ξ) and the properties of Aεand Agive the existence of C>0 and h0∈L1(ω) such that limsup ε→0 ω∇xψε−P εx,ξ pϑdx  ⩽ ωh0+C|s|+|ξ|+|∇ xψ|pp−1−σ p−σ|s−ξ|p+∇xψ−ξ p1 p−σϑdx . If ϑdecreases to χKwe get limsup ε→0 K∇xψε−P εx,ξ pϑdx  ⩽C Kh0+C|s|+|ξ|+|∇ xψ|pp−1−σ p−σ|s−ξ|p+∇xψ−ξ p1 p−σdx.(4.26) We now consider Φ=m j=1(sje1+n l=1ηjlχKl)χ(ij−1,ij)as in the statement of Theorem 4.3 and Ψ=m j=1(sje1+ ∇xψj)χ(pj−1,pj), with ψ1,...,ψ m∈W1,p(ω).From(4.26) and Holder’s inequality, we easily get limsup ε→0 Ic×ωPε(x1,Ψ)−Φ1e1−P εx,Φ pdx ⩽ m  j=1 n  l=1 limsup ε→0 (ij−1,ij)×KlPε(x1,s j,ψ j)−sje1−P εx,η jl pdx ⩽ Ic×ωh0+C|E|+|Ψ|pdxp−1−σ p−σ Ic×ω |E−Ψ|pdx1 p−σ . From (3.18) we then deduce limsup ε→0 Dεuε−Φ1e1−P εx,Φ Lp(Ic×ω)N ⩽limsup ε→0 Dεuε−Pε(x1,Ψ)  Lp(Ic×ω)N+limsup ε→0 Pε(x1,Ψ)−Φ1e1−P εx,Φ Lp(Ic×ω)N ⩽ Ich0+C du0 dx1 +|Ψ1|p +C ω|∇xu1|+Ψpdxdx1p−1−σ p(p−σ) J. Casado-Díaz et al. / Ann. I. H. Poincaré – AN 30 (2013) 519–545 537 · Ic×ωD0(u0,u 1)−Ψ pdx1 p(p−σ) + Ic×ωh0+C|Φ|+|Ψ|pdxp−1−σ p(p−σ)  Ic×ω |Φ−Ψ|pdx1 p(p−σ) . Taking in this inequality Ψconverging to D0(u0,u 1)in Lp(Ic×ω) we deduce limsup ε→0 Dεuε−Φ1e1−P εx,Φ Lp(Ic×ω)N ⩽ Ic×ωh0+CD0(u0,u 1)+|Φ|pdxp−1−σ p(p−σ)  Ic×ωD0(u0,u 1)−E pdx1 p(p−σ) . This proves (4.10). To obtain (4.9), it is enough to use limsup ε→0    ∂1uε−du0 dx1   Lp(Ic×ω) ⩽limsup ε→0 Dεuε−Φ1e1−P εx,Φ Lp(Ic×ω)N+limsup ε→0    du0 dx1 −Φ1   Lp(Ic×ω) , and then to use the previous inequality with Econverging to D0(u0,u 1).2 5. Some examples with nonlocal limit In the previous section, we have shown that if the functions Aεdo not depend on x1, the limit problem of (2.12)is local. We show here that this assertion is not true when Aεdepends on x1even, if they do not depend on x. For this purpose, we consider a function A∈L∞ (0,1;MN), such that there exists α>0, which satisfies A(y1)ξξ ⩾α|ξ|2,∀ξ∈RN,a.e. y1∈R.(5.1) Then, we consider the homogenization problem ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ uε∈H1 Γ(Ω),  Ω Ax1 δεDεuεDεvdx= Ω fvdx+ Ω FDεvdx, ∀v∈H1 Γ(Ω), (5.2) where fbelongs to L2(Ω),Fbelongs to L2(Ω)Nand δε>0 satisfies lim ε→0δε=0.(5.3) Remark 5.1. The homogenization of the nonlinear problem ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ uε∈W1,p Γ(Ω),  Ω Ax1 δε ,D εuεDεvdx= Ω fvdx+ Ω FDεvdx, ∀v∈W1,p Γ(Ω), (5.4) can be performed using the same arguments which we will use here, but this complicates the exposition and it is not necessary for our purpose. To perform the homogenization of (5.2), we will use the two-scale convergence method of G. Nguetseng and G. Allaire (see [1,21]). The following is the definition of the two-scale convergence adapted to our problem. 538 J. Casado-Díaz et al. / Ann. I. H. Poincaré – AN 30 (2013) 519–545 Definition 5.2. Let uεbe a bounded sequence in L2(Ω), we say that uεtwo-scale converges to ˆu∈L2(I ×(0,1)×ω), and we write uε 2e ˆu, if for every φ∈L∞ (0,1), and every ϕ∈L2(Ω),wehave lim ε→0 Ω uε(x)φx1 δεϕ(x)dx = Ω 1  0 ˆux1,y 1,xφ(y1)ϕ(x) dy1dx. Remark 5.3. If a bounded sequence uεin L2(Ω) two-scale converges to ˆu∈L2(I ×(0,1)×ω), then uεconverges weakly in L2(Ω) to the function ugiven by u(x) = 1  0 ˆux1,y 1,xdy1,a.e. x∈Ω. Analogously to the well known two-scale compactness theorem for a sequence which is bounded in H1(Ω) (see [1,21]), we can prove in our case the following lemma. Lemma 5.4. We consider a sequence uε∈H1 Γ(Ω) such that for some u0∈H1 0(I) uεu 0in H1 Γ(Ω), (5.5)  Ω |Dεuε|2dx ⩽C(5.6) and we assume ∃lim ε→0 ε δε =λ∈[0,+∞].(5.7) Then, for a subsequence (still denoted by uε), we have: i) If λ=0, there exist ˆu0∈L2(I, H1 (0,1)/R)and ˆu1∈L2(I ×(0,1), H 1(ω)/R)such that Dεuε 2e du0 dx1 +∂y1ˆu0e1+∇ xˆu1.(5.8) ii) If λ∈(0,+∞), there exists ˆu1∈L2(I, H1 ((0,1)×ω)/R)such that Dεuε 2e du0 dx1 +λ∂y1ˆu1e1+∇ xˆu1.(5.9) iii) If λ=+∞, there exist ˆu0∈L2(I, H 1 ((0,1), L2(ω)/R)) and u1∈L2(I, H1(ω)) such that Dεuε 2e du0 dx1 +∂y1ˆu0e1+∇ xu1.(5.10) Remark 5.5. In view of (5.6) and of Theorem 2.3, for a subsequence, there exists a function u1∈L2(I, H1(ω)) such that DεuεD 0(u0,u 1)in L2(Ω)N. This function u1appears in (5.10) in the case iii), while in the cases i) and ii), the functions u1are given in terms of the functions ˆu1by u1(x) = 1  0 ˆu1x1,y 1,xdy1,a.e. x∈Ω(5.11) (see Remark 5.3). J. Casado-Díaz et al. / Ann. I. H. Poincaré – AN 30 (2013) 519–545 539 From Lemma 5.4 we can now deduce the following result. Theorem 5.6. We assume (5.7)and we consider the solution uεof (5.2). Then, we have: i) If λ=0,weget(5.8), with u0,ˆu0,ˆu1the solutions of ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (u0,ˆu0,ˆu1)∈H1 0(I) ×L2I,H1 (0,1)/R×L2I×(0,1), H 1(ω)/R,  Ω 1  0 A(y1)du0 dx1 +∂y1ˆu0e1+∇ xˆu1dv0 dx1 +∂y1ˆv0e1+∇ xˆv1dy1dx = Ω fv 0dx + Ω 1  0 Fdv0 dx1 e1+∇ xˆv1dy1dx, ∀(v0,ˆv0,ˆv1)∈H1 0(I) ×L2I,H1 (0,1)/R×L2I×(0,1), H 1(ω)/R. (5.12) ii) If λ∈(0,+∞),weget(5.9), with u0,ˆu1the solutions of ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (u0,ˆu1)∈H1 0(I) ×L2I,H1 (0,1)×ω/R,  Ω 1  0 A(y1)du0 dx1 +λ∂y1ˆu1e1+∇ xˆu1dv0 dx1 +λ∂y1ˆv1e1+∇ xˆv1dy1dx = Ω fv 0dx + Ω 1  0 Fdv0 dx1 e1+∇ xˆv1dy1dx, ∀(v0,ˆv1)∈H1 0(I) ×L2I,H1 (0,1)×ω/R. (5.13) iii) If λ=+∞,weget(5.10), with u0,ˆu0,u1the solutions of ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (u0,ˆu0,u 1)∈H1 0(I) ×L2I,H1 0,1,L 2(ω)/R×L2I,H1(ω)/R,  Ω 1  0 A(y1)du0 dx1 +∂y1ˆu0e1+∇ xu1dv0 dx1 +∂y1ˆv0e1+∇ xv1dy1dx = Ω fv 0dx + Ω Fdv0 dx1 e1+∇ xv1dx, ∀(v0,ˆv0,v 1)∈H1 0(I) ×L2I,H1 (0,1), L2(ω)/R×L2I,H1(ω)/R. (5.14) Proof. We only prove the case λ=0, the cases λ∈(0,+∞)and λ=+∞are similar. We consider a subsequence of ε,u0∈H1 0(I),ˆu0∈L2(I, H 1 (0,1)/R),ˆu1∈L2(I ×(0,1), H 1(ω)/R)such that (5.8) holds. For ϕ0,ˆϕ0∈C∞ 0(I),ˆ U0∈C∞ ([0,1]),ˆϕ1∈C∞ c(I, C∞(¯ω)),ˆ U1∈C∞ ([0,1]), we take as test function in (5.2) the sequence vε∈H1 Γ(Ω) defined by vε(x) =ϕ0(x1)+δεˆϕ0(x1)ˆ U0x1 δε+εˆϕ1(x) ˆ U1x1 δε,a.e. x∈Ω. Using vε(x) =ϕ0(x1)+rε, Dεvε(x) =dϕ0 dx1 (x1)+ˆϕ0(x1)dˆ U0 dy1x1 δεe1+∇ xˆϕ1(x) ˆ U1x1 δε+Rε(x), where rεand Rεconverge strongly to zero in L2(Ω) and L2(Ω)Nrespectively, we get 540 J. Casado-Díaz et al. / Ann. I. H. Poincaré – AN 30 (2013) 519–545  Ω Ax1 δεDεuεdϕ0 dx1 (x1)+ˆϕ0(x1)dˆ U0 dy1x1 δεe1+∇ xˆϕ1(x) ˆ U1x1 δεdx = Ω fϕ 0dx + Ω Fdϕ0 dx1 (x1)+ˆϕ0(x1)dˆ U0 dy1x1 δεe1+∇ xˆϕ1(x) ˆ U1x1 δεdx +Oε, where Oεtends to zero. Using (5.8) to pass to the limit in this equality we deduce  Ω 1  0 A(y1)du0 dx1 +∂y1ˆu0e1+∇ xˆu1dϕ0 dx1 +ˆϕ0 dˆ U0 dy1e1+∇ xˆϕ1ˆ U1dxdy1 = Ω fϕ 0dx + Ω 1  0 Fdϕ0 dx1 +ˆϕ0 dˆ U0 dy1e1+∇ xˆϕ1ˆ U1dxdy1, for every ϕ0,ˆϕ0,ˆ U0,ˆϕ1and ˆ U1, as above. By linearity and density, this implies that u0,ˆu0and ˆu1are the solutions of (5.12), and then, by uniqueness, that it is not necessary to extract any subsequence. 2 Remark 5.7. When λ∈(0,+∞), Theorem 5.6 can be deduced from the results obtained in [22] (in [22] F=0, but to assume F= 0 does not make the problem more difficult). Other homogenization results for thin structures with periodic coefficients can be found in [2,3,5,11]. Remark 5.8. For λ=0, the above theorem means that the asymptotic behavior of uεis as if we consider δε=δ fixed, and we take the limit first in εand then in δ, i.e. as we make first the reduction of dimension and then the homogenization. It is possible to obtain a general result in this direction assuming that the frequency of the oscillations in x1is smaller than 1 ε. Specifically, the following result holds: Assume Aεsatisfying the assumptions in Section 2 and such that lim ε→0sup 0⩽h⩽1 Aε(. +εhe1,ξ)−Aε(., ξ ) Lp(Ic×ω) =0,∀IcI. For f∈Lp(Ω) and F∈Lp(Ω)N, we define u0,ε,u1,ε as the solutions of ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (u0,ε,u 1,ε)∈W1,p(I) ×LpI,W1,p(ω)/R,  Ω Aεdu0,ε dx1 e1+∇ xu1,εdv0 dx1 e1+∇ xv1dx = Ωfv 0+Fdv0 dx1 e1+∇ xv1dx, ∀(v0,v 1)∈W1,p(I) ×LpI,W1,p(ω)/R. (5.15) Then, we have Dεuε−du0,ε dx1 e1+∇ xu1,ε→0inLp(Ω), where uεis the solution of (2.12). This reduces the homogenization of (2.12) to the homogenization of (5.15). We will not prove this result because we will not use it. When λ∈(0,+∞), Theorem 5.6 means that the reduction of dimension and the homogenization hold simultaneously. When λ=∞, Theorem 5.6 means that we can perform first the homogenization and then the reduction of dimension. Clearly in this case the problem for u0and u1is local. We will see that the other two cases give nonlocal problems in general. Namely we give two examples, with λ=0 and λ=1 in which the limit problem of (2.12)is nonlocal. In the two examples we assume N=2, ω=(0,1), and we set x=x2. J. Casado-Díaz et al. / Ann. I. H. Poincaré – AN 30 (2013) 519–545 541 Example 1 We define A∈L∞ ((0,1), M2)by A(y1)=11 1γ(y 1),a.e. y1∈(0,1), (5.16) with γ∈L∞ (0,1), such that there exists ν>0, with γ>1+νa.e. in (0,1). We consider δεsuch that lim ε→0 ε δε =0. Theorem 5.9. For the above choice of Aand δε, the limit problem of (5.2)is (3.7)where A:R×∇H1(ω) →L2(ω)2 is the nonlocal operator given by As, dψ dx2=s+dψ dx2e1+s+γ∗dψ dx2 +ˆγ−γ∗ 1  0 dψ dx2 (t) dt,a.e. in ω, (5.17) with γ∗=1  0 dy1 γ(y 1)−1 ,ˆγ=1  0 dy1 γ(y 1)−1−1 +1.(5.18) Proof. For f∈L2(Ω) and F∈L2(Ω)2, we define (u0,ˆu0,ˆu1)as the solution of (5.12) and u1by (5.11). We know that if uεis the solution of (2.12) then (2.13), (2.14) hold. Taking in (5.12), v0=0, ˆv0=0, we deduce du0 dx1 +∂y1ˆu0+γ(y 1)∂x2ˆu1=F2a.e. in I×(0,1)×ω. (5.19) Using now v0=0 and ˆv1=0in(5.12) we deduce that there exists a function r∈L2(I) such that du0 dx1 +∂y1ˆu0+ 1  0 ∂x2ˆu1dx2=r(x1)a.e. in I×(0,1). Taking in this expression the value of ∂x2ˆu1given by (5.19), we get du0 dx1 +∂y1ˆu0+1 γ¯ F2−du0 dx1 −∂y1ˆu0=r(x1)a.e. in I×(0,1), with ¯ F2(x1)= 1  0 F2(x1,x 2)dx 2,a.e. x1∈I. Thus, we have ∂y1ˆu0=γ γ−1r−du0 dx1 −1 γ−1¯ F2a.e. in I×(0,1). (5.20) Integrating this equality with respect to y1and using that ˆu0is periodic with respect to y1, we easily deduce r(x1)=ˆγ−1 ˆγ du0 dx1 +1 ˆγ ¯ F2, which substituted in (5.20) proves ∂y1ˆu0=γ ˆγ−11 γ−1¯ F2−du0 dx1.(5.21) 542 J. Casado-Díaz et al. / Ann. I. H. Poincaré – AN 30 (2013) 519–545 Using the expression (5.21)of∂y1ˆu0in (5.19)wehave ∂x2ˆu1=1 γF2−du0 dx1−1 ˆγ−1 γ1 γ−1¯ F2−du0 dx1, a.e. in I×(0,1)×ω, which integrated in (0,1)with respect to y1gives ∂x2u1=1 γ∗F2−du0 dx1+1 ˆγ−1 γ∗¯ F2−du0 dx1,a.e. in Ω. (5.22) Integrating now in (0,1)with respect to x2, we get ¯ F2−du0 dx1 =ˆγ 1  0 ∂x2u1(x1,t)dt, a.e. in I, which substituted in (5.22) implies du0 dx1 +γ∗∂x2u1+ˆγ−γ∗ 1  0 ∂x2u1(x1,t)dt =F2,a.e. in Ω. (5.23) On the other hand, taking in (5.12)ˆv0=0, ˆv1=0wehave  Ωdu0 dx1 +∂x2u1dv0 dx1 dx1= Ωfv 0+F1 dv0 dx1dx1,(5.24) for every v0∈H1 0(I).From(5.23) and (5.24) we conclude that u0,u1satisfy (3.7) with Agiven by (5.17). 2 Example 2 We take δε=ε,ω=(0,1), and we define A∈L∞ ((0,1), M2)by A(y1)=10 0γ(y 1),a.e. in (0,1), (5.25) with γ∈L∞ (0,1), such that for some α>0, we have γ>αa.e. in R. Theorem 5.10. For the above choice of Aε, the limit problem of (2.12)is (3.7)where A:R×∇H1(ω) ×L2(0,1)2 is a nonlocal operator given by As, dψ dx2(x) =se1+A2dψ dx2(x2), a.e. x∈Ω, (5.26) with A2:L2(ω) →L2(ω) defined by A2(H)(x2)=2 π2 ∞  k=11 0H(t)sin(kπt) dt k21 0ψk(y1)dy 1 sin(kπx2), a.e. x2∈(0,1). (5.27) Here ψk∈H1 (0,1)is the solution of 1  0 dψk dy1 dv dy1 dy1+k2π2 1  0 γψ kvdy 1= 1  0 vdy 1,∀v∈H1 (0,1). (5.28) Proof. For f∈L2(Ω) and F∈L2(Ω)2, we define u0,ˆu1as the solutions of (5.13) with λ=1 and u1by (5.11). We know that if uεis the solution of (2.12) then (2.13), (2.14) hold. J. Casado-Díaz et al. / Ann. I. H. Poincaré – AN 30 (2013) 519–545 543 Taking in (5.13)v0=0, we deduce that for a.e. x1∈I, the function ˆu1(x1,.,.)∈H1((0,1)2), periodic with respect to y1, satisfies 1  0 1  0 (∂y1ˆu1∂y1ˆv1+γ∂ x2ˆu1∂x2ˆv1)dy 1dx2= 1  0 1  0 F2∂x2ˆv1dy1dx2, ∀ˆv1∈H1(0,1)2,periodic with respect to y1. Using that the functions cos(kπx2), with k∈N, are a basis of H1(0,1)(they are the eigenfunctions corresponding to the operator (d2 dx2 2 )−1with Neumann boundary condition), we look for a Fourier expansion for ˆu1, ˆu1= ∞  k=1 ηk(y1)cos(kπx2). We get ∂x2ˆu1(x1,y 1,x 2)=2π2 ∞  k=1 k2 1  0 F2(x1,t)sin(kπt) dt ψk(y1)sin(kπx2), a.e. in I×(0,1)2. Integrating with respect to y1, we conclude that u1satisfies A2∂x2u1(x1,.) (x2)=F2(x), a.e. x∈Ω, (5.29) with A2defined by (5.27). On the other hand, taking ˆv0=0in(5.13) we deduce  Ω du0 dx1 dv0 dx1 dx = Ωfv 0+F2 dv0 dx1dx. (5.30) From (5.29) and (5.30) we conclude that u0,u1satisfy (3.7) with Agiven by (5.27). 2 Remark 5.11. We observe that if the operator A2is local, i.e. if there exists c:Ω→Rsuch that A2(H ) =c(x1,x 2)H , for every H∈L2(0,1), then, by (5.27)cis a positive constant and 1  0 ψk(y1)dy 1=1 ck2π2,∀k⩾1.(5.31) The following result proves that this only holds if γis constant. Proposition 5.12. The solution ψkof (5.28)satisfies (5.31)if and only if γ=ca.e. in (0,1). Proof. It is clear that if γis constant then (5.31) holds. For the reciprocate, we assume that (5.31) hold. Taking ψkas test function in (5.28) and using (5.31) we deduce k2 1  0 dψk dy1 2 dy1+k4π2 1  0 γ|ψk|2dy1=1 cπ2.(5.32) Therefore, up to a subsequence, there exists ψ∈L2(0,1)such that k2ψkconverges weakly to ψin L2(0,1). Then, taking ϕ∈H1 (I) as test function in (5.28) we deduce 1  0 dψk dy1 dϕ dy1 dy1+π2 1  0 γk2ψkϕdy 1= 1  0 ϕdy 1.(5.33)