Renata Bunoiu

Karim Ramdani
email: karim.ramdani@inria.fr

Claudia Timofte
email: claudia.timofte@g.unibuc.ro

T-COERCIVITY FOR THE ASYMPTOTIC ANALYSIS OF SCALAR PROBLEMS WITH SIGN-CHANGING COEFFICIENTS IN THIN PERIODIC DOMAINS

Keywords: 2010 Mathematics Subject Classification. 35B27, 78M40, 35Q60, 35J20 Sign-changing coefficients, T-coercivity, thin domains, asymptotic analysis

We study a scalar problem in thin periodic composite media formed by two materials, a classical one and a metamaterial (also known as negative material). By applying T-coercivity methods and homogenization techniques specific to the thin periodic domains under consideration, for two geometric settings, we derive the homogenized limit problems, which both exhibit dimension-reduction effects.

Introduction

Metamaterials (also known as negative or left-handed materials) are artificial composite materials exhibiting negative dielectric permittivity and magnetic permeability for some frequencies, and hence behaving as negative refractive index materials (see the review papers by Shamonina and Solymar [START_REF] Shamonina | Metamaterials: How the subject started[END_REF] and Smith, Pendry and Wiltshire [START_REF] Smith | Metamaterials and negative refractive index[END_REF]). This property leads to new super lens effects and explains the growing interest for them in the last decade. From a mathematical point a view, Bouchitté et al. proposed a rigorous derivation and a mathematical justification of negative materials in electromagnetics using homogenization techniques [START_REF] Bouchitté | Multiscale nanorod metamaterials and realizable permittivity tensors[END_REF][START_REF] Bouchitté | Homogenization of finite metallic fibers and 3d-effective permittivity tensor[END_REF][START_REF] Bouchitté | Homogenization of a wire photonic crystal: the case of small volume fraction[END_REF] and similar results have been established by Ávila et al. [START_REF] Ávila | Multiscale modeling of elastic waves: theoretical justification and numerical simulation of band gaps[END_REF] in phononics. The well-posedness of scalar problems involving both classical dielectric materials and metamaterials (and, hence, leading to differential operators with sign-changing coefficients) has been studied by Bonnet-Ben Dhia et al. [START_REF] Bonnet-Ben Dhia | T -coercivity for scalar interface problems between dielectrics and metamaterials[END_REF][START_REF] Bonnet-Ben Dhia | Time harmonic wave diffraction problems in materials with sign-shifting coefficients[END_REF], Chesnel and Ciarlet Jr. [START_REF] Chesnel | T-coercivity and continuous Galerkin methods: application to transmission problems with sign changing coefficients[END_REF], Chung and Ciarlet Jr. [START_REF] Chung | A staggered discontinuous Galerkin method for wave propagation in media with dielectrics and meta-materials[END_REF], Nicaise and Venel [START_REF] Nicaise | A posteriori error estimates for a finite element approximation of transmission problems with sign changing coefficients[END_REF], Nguyen [START_REF] Nguyen | Asymptotic behavior of solutions to the Helmholtz equations with sign changing coefficients[END_REF][START_REF] Nguyen | Limiting absorption principle and well-posedness for the Helmholtz equation with sign changing coefficients[END_REF]. The case of Maxwell's system with sign changing coefficients has also been investigated by Bonnet-Ben Dhia, Chesnel and Ciarlet Jr. [7], Fernandes and Raffetto [START_REF] Fernandes | Existence, uniqueness and finite element approximation of the solution of time-harmonic electromagnetic boundary value problems involving metamaterials[END_REF], Oliveri and Raffetto [START_REF] Oliveri | A warning about metamaterials for users of frequency-domain numerical simulators[END_REF] and Nguyen and Sil [START_REF] Nguyen | Limiting Absorption Principle and Well-Posedness for the Time-Harmonic Maxwell Equations with Anisotropic Sign-Changing Coefficients[END_REF]. More recently, homogenization for composite materials involving both positive and negative materials has been considered in [START_REF] Bonnetier | Homogenization of the eigenvalues of the Neumann-Poincaré operator[END_REF][START_REF] Bunoiu | Homogenization of Maxwell's equations and related scalar problems with sign-changing coefficients[END_REF][START_REF] Bunoiu | Homogenization of materials with sign changing coefficients[END_REF].

Our goal in this article is to study a scalar problem in specific periodic thin domains, namely composite materials formed by two constituents, a classical one and a metamaterial, occupying a thin three-dimensional region denoted Ω ε . The results are valid in R d (d 2), but the three-dimensional case is physically more relevant, if we think, for instance, to the practical applications mentioned below.

The domain Ω ε is divided in two open subdomains, denoted by Ω ε 1 and Ω ε 2 . The subdomain Ω ε 2 , assumed to be disconnected, is formed by ε-periodically distributed inclusions which do not touch the boundary of Ω ε . We shall consider two distinct geometrical settings, which both correspond to important physical applications, like demultiplexer devices and antennas [START_REF] Ourir | Homogenization of Thin and Thick Metamaterials and Applications[END_REF]. In the first geometrical setting (see Figure 1), the inclusions are ε-periodically distributed in an horizontal layer of height ε, while in the second one (see Figure 2), the inclusions are ε-periodically distributed at the interior of a thin vertical rod with thickness of order ε.

For both of these geometries, we analyze the well-posedness of the corresponding microscopic problems (see (2.4) and (5.2)) and the asymptotic behavior of their solutions as the small parameter ε tends to zero. The main difference with respect to the classical scalar case is the presence of two materials with conductivities of different sign. Consequently, classical methods based on ellipticity arguments to prove well-posedness and obtain energy estimates do not apply for the problems under study. To cope with this difficulty, we apply T-coercivity technics (see Definition 3.1; we refer the reader to the papers of Bonnet-Ben Dhia et al. [START_REF] Bonnet-Ben Dhia | T -coercivity for scalar interface problems between dielectrics and metamaterials[END_REF][START_REF] Bonnet-Ben Dhia | Two-field and three-field formulations for wave transmission between media with opposite sign dielectric constants[END_REF] for more details), which have been already successfully used in the context of homogenization by two of the authors in [START_REF] Bunoiu | Homogenization of materials with sign changing coefficients[END_REF][START_REF] Bunoiu | Homogenization of Maxwell's equations and related scalar problems with sign-changing coefficients[END_REF]. The first difference from these last two references is the degenerate geometry considered in this work. The second difference concerns the construction of the T-coercive operators. More precisely, like in [START_REF] Bunoiu | Homogenization of materials with sign changing coefficients[END_REF], the T-coercive operators involved here are constructed using suitably chosen extension operators from one subdomain (the one occupied by the positive or negative material) into the other. Defining the contrast κ between the two materials as in (2.3), we need to distinguish, throughout the whole paper, between the cases of large contrasts and small contrasts, each one involving a particular extension operator (and correspondingly a particular T-coercive operator). We propose in this paper a similar and quite direct proof for both cases, whereas [START_REF] Bunoiu | Homogenization of materials with sign changing coefficients[END_REF] was limited to the case of large contrasts, while the proof proposed in [START_REF] Bunoiu | Homogenization of Maxwell's equations and related scalar problems with sign-changing coefficients[END_REF] was more involved and indirect. The use of alternative approaches to study homogenization of these problems with sign-changing coefficients, like the analysis of the spectrum of the Neumann-Poincaré operator developed in Bonnetier et al. [START_REF] Bonnetier | Homogenization of the eigenvalues of the Neumann-Poincaré operator[END_REF], is also probably possible.

Once the well-posedness proved, by using the two-scale convergence method (see Nguetseng [START_REF] Nguetseng | A general convergence result for a functional related to the theory of homogenization[END_REF] and Allaire [START_REF] Allaire | Homogenization and two-scale convergence[END_REF]) adapted to thin periodic domains (see, for instance, Gahn and Neuss-Radu [START_REF] Gahn | Singular limit for reactive diffusive transport through an array of thin channels in case of critical diffusivity[END_REF] and Jerez-Hanckes et al. [START_REF] Jerez-Hanckes | Derivation of cable equation by multiscale analysis for a model of myelinated axons[END_REF]), we derive for both geometries depicted in Figures 1 and2 the associated homogenized problems (see Theorem 4.3 and Theorem 5.3). Let us emphasize that they both exhibit dimension reduction effects, 3D → 2D and 3D → 1D, respectively. Nevertheless, the values of the homogenized coefficients keep track of the lost variables, at a local scale, through the cell problems (see Remarks 4.4 and 5.4).

For mathematical studies of diffusion problems in thin periodic media, we refer, for instance, to [START_REF] Bhattacharya | Effective transmission conditions for reactiondiffusion processes in domains separated by thin channels[END_REF][START_REF] Bunoiu | Upscaling of a double porosity problem with jumps in thin porous media[END_REF][START_REF] Caillerie | Homogénéisation des équations de la diffusion stationnaire dans les domaines cylindriques aplatis[END_REF][START_REF] Damlamian | Homogenization limits of diffusion equations in thin domains[END_REF][START_REF] Fatima | Homogenization and dimension reduction of filtration combustion in heterogeneous thin layers[END_REF][START_REF] Gahn | Correctors and error estimates for reaction-diffusion processes through thin heterogeneous layers in case of homogenized equations with interface diffusion[END_REF][START_REF] Gahn | Singular limit for reactive diffusive transport through an array of thin channels in case of critical diffusivity[END_REF][START_REF] Gahn | Effective interface conditions for processes through thin heterogeneous layers with nonlinear transmission at the microscopic bulk-layer interface[END_REF][START_REF] Gaudiello | A two-dimensional electrostatic model of interdigitated comb drive in longitudinal mode[END_REF][START_REF] Hoang | Diffusion in a highly heterogeneous thin domain[END_REF][START_REF] Jerez-Hanckes | Derivation of cable equation by multiscale analysis for a model of myelinated axons[END_REF][START_REF] Mel'nik | Asymptotic analysis of boundary-value problems in thin perforated domains with rapidly varying thickness[END_REF][START_REF] Neuss-Radu | Effective transmission conditions for reaction-diffusion processes in domains separated by an interface[END_REF] and the references therein. For elasticity problems in related thin periodic domains, we refer to [START_REF] Caillerie | Thin elastic and periodic plates[END_REF][START_REF] Canon | Modelling of thin elastic plates with small piezoelectric inclusions and distributed electronic circuits. Models for inclusions that are small with respect to the thickness of the plate[END_REF][START_REF] Canon | Modeling of thin isotropic elastic plates with small piezoelectric inclusions and distributed electric circuits[END_REF][START_REF] Damlamian | Homogenization limits of the equations of elasticity in thin domains[END_REF][START_REF] Griso | Homogenization via unfolding in periodic layer with contact[END_REF][START_REF] Griso | Asymptotic analysis for domains separated by a thin layer made of periodic vertical beams[END_REF]. For flow problems in thin porous media, we refer, for example, to [START_REF] Amaziane | Homogenization of a single phase flow through a porous medium in a thin layer[END_REF][START_REF] Anguiano | Darcy's laws for non-stationary viscous fluid flow in a thin porous medium[END_REF][START_REF] Anguiano | Homogenization of Bingham flow in thin porous media[END_REF][START_REF] Brillard | Boundary homogenization in perforated domains for adsorption problems with an advection term[END_REF][START_REF] Gómez | Asymptotics for models of non-stationary diffusion in domains with a surface distribution of obstacles[END_REF].

This article is organized as follows: in Section 2, we introduce the microscopic problem (2.4) stated in the thin periodic layer and we fix the notation. In Section 3, we prove the well-posedness of problem (2.4), using appropriate extension operators and the T-coercivity technics. The homogenization result for the case of a thin periodic layer is stated and proved in Section 4. Finally, in Section 5, we address the case of a three-dimensional thin periodic rod. We start by describing more precisely the geometry of the domain Ω ε , which represents a two-phase thin periodic composite medium (see Figure 1). Let ω be a smooth and bounded domain in R 2 . We denote the independent variable x ∈ R 3 by x = (x 1 , x 2 , x 3 ) = (x, x 3 ) and we define

Ω ε = ω × (0, ε) = {x = (x, x 3 ) ∈ R 3 : x ∈ ω, 0 < x 3 < ε}.
(2.1)

Here, ε ∈ (0, 1) is a sequence of strictly positive numbers such that 1 ε ∈ N * . This small parameter is related to the characteristic dimension of our domain. Thus, Ω ε is a thin heterogeneous layer, its thickness, the periodicity of its heterogeneities and their size being of order ε. More precisely, the microscopic structure of Ω ε consists of an exact number of replicated unit cells Y = (0, 1) 3 , rescaled with ε. The reference cell is given by Y = Y 1 ∪ Y 2 , where Y 1 and Y 2 are two non-empty disjoint connected open subsets of Y such that Y 2 ⊂ Y . We assume that the boundary ∂Y 2 is Lipschitz continuous. For each k ∈ Z 3 , we define the shifted cells

Y k 1 = k + Y 1 and Y k 2 = k + Y 2 .
We also define, for each ε, the set of indexes

Z ε = k ∈ Z 3 : εY k 2 ⊂ Ω ε . Finally, we set Ω ε 2 = ∪ k∈Zε εY k 2 and Ω ε 1 = Ω \ Ω ε 2 .
We denote by n ε the unit outward normal to Ω ε 2 . The boundary of the domain Ω ε is split into three parts: the lateral boundary Σ

ε,D = {x ∈ R 3 : x ∈ ∂ω, 0 < x 3 < ε}, the top boundary Σ ε,N + = {x ∈ R 3 : x ∈ ω, x 3 = ε} and the bottom boundary Σ ε,N -= {x ∈ R 3 : x ∈ ω, x 3 = 0}.
Given two real constants a 1 and a 2 such that a 1 a 2 < 0, let a ∈ L ∞ (R 3 ) denote the 1-periodic function in the variables y 1 and y 2 defined by a(y) = a 1 1 Y1 (y) + a 2 1 Y2 (y).

For simplicity and without loss of generality, we assume that

a 1 > 0, a 2 < 0, (2.2)
and we define the contrast κ as the positive number

κ = a 1 a 2 = a 1 |a 2 | . (2.3) Setting a ε (x) = a x ε , ∀x ∈ Ω ε ,
our goal is to analyze the asymptotic behavior, as ε → 0, of the solution u ε of the problem

-div(a ε ∇u ε ) = f in Ω ε , u ε = 0 on Σ ε,D , ∂u ε ∂ν ε ± = 0 on Σ ε,N ± , (2.4)
where the function f ∈ L 2 (ω) is given and ν ε ± is the unit outward normal to Σ ε,N ± . Let us emphasize that this particular choice of the function f is classical in the framework of asymptotic analysis in thin domains. Nevertheless, more general right-hand sides can be considered, see, for instance, [41, Section 2].

It is worth noticing that the above scalar problem is nothing but the problem obtained from the homogenization problem

-div(a ε ∇u ε ) = f in Ω ε , u ε = 0 on Σ ε,D , a ε ∇u ε • ν ε ± = 0 on Σ ε,N ± ,
in the special case where the 3x3 matrix a = (a ij ) 1 i,j 3 is given by

a ε (x) = a x ε , a(y) = a(y)I 3 . (2.5)
To write the variational formulation of problem (2.4), for every positive ε < 1, we introduce the Hilbert space

V ε = v ∈ H 1 (Ω ε ) : v = 0 on Σ ε,D , (2.6) endowed with the norm v V ε = ∇v L 2 (Ω ε ) , for any v ∈ V ε .
The variational formulation of problem (2.4) is as follows: find

u ε ∈ V ε such that A ε (u ε , v) = ε (v), ∀v ∈ V ε , (2.7)
where the bilinear form A ε : V ε × V ε → R and the linear form ε : V ε → R are given by

A ε (u, v) = Ω ε a ε ∇u • ∇v dx = a 1 Ω ε 1 ∇u • ∇v dx + a 2 Ω ε 2D ∇u • ∇v dx, ε (v) = Ω ε f v dx.

Well-posedness of the microscopic problem in a thin periodic layer via T-coercivity technics

Since the bilinear form involved in the above weak formulation is not coercive, the well-posedness of (2.7) is far from being obvious. To overcome this difficulty, an issue is to make use of T-coercivity technics (see, for instance, [START_REF] Bonnet-Ben Dhia | T -coercivity for scalar interface problems between dielectrics and metamaterials[END_REF][START_REF] Bonnet-Ben Dhia | Two-field and three-field formulations for wave transmission between media with opposite sign dielectric constants[END_REF]). In the framework of homogenization, this method has already been successfully used for scalar problems set in a fixed domain with periodic geometry; see [START_REF] Bunoiu | Homogenization of materials with sign changing coefficients[END_REF] for the case of a Dirichlet scalar problem (case of large contrasts only) and [START_REF] Bunoiu | Homogenization of Maxwell's equations and related scalar problems with sign-changing coefficients[END_REF] for the one of Dirichlet and Neumann scalar problems and vector Maxwell's equations for extreme contrasts, large or small. It is worth mentioning reference [START_REF] Bonnetier | Homogenization of the eigenvalues of the Neumann-Poincaré operator[END_REF], in which homogenization of scalar problems with sign-changing coefficients is achieved using another approach, namely the analysis of the spectrum of the Neumann-Poincaré operator.

3.1. Background on T-coercivity. We start by recalling the definition of Tcoercivity.

Definition 3.1. Let V be a Hilbert space endowed with the norm • and let T ∈ L(V ) be a bounded linear operators on V . A bilinear form a(•, •) defined on

V × V is called T-coercive if there exists γ > 0 such that |a(u, Tu)| γ u 2 , ∀u ∈ V.
The T-coercivity result used throughout the paper is detailed below (Theorem 3.2). Note that this result is stated in the special case of symmetric bilinear forms (only such bilinear forms are involved in this work) depending on a small parameter. In particular, it is slightly different from the result given in [17, Theorem 2.1]) which holds for arbitrary bilinear forms. The T-coercivity operators needed to prove well-posedness for the three problems with sign -changing coefficients involved in our study (namely the microscopic problem, the cell problems and the twoscale limit problem) are constructed using suitable extension operators (from the background medium into the inclusions or vice-versa). It is worth noticing that such extension operators are generally used in homogenization theory to obtain compactness results, while they are used here to obtain well-posedness and energy estimates for the solutions of our problems. We also emphasize that the construction of suitable extension operators from the inclusions into the exterior domain is more technical than the construction of extension operators from the outer domain into the inclusions. Indeed, this requires to adapt an idea used in [START_REF] Cazeaux | Homogenization of a model for the propagation of sound in the lungs[END_REF] to obtain uniform energy estimates of gradient type for these operators (see (3.13) and (3.21)). Theorem 3.2. Let V be a Hilbert space equipped with the norm • and let A ε (•, •) be a bilinear form on V satisfying the following conditions:

(1) A ε (•, •) is symmetric: A ε (u, v) = A ε (v, u). (2) A ε (•, •) is uniformly continuous: there exists M > 0 such that |A ε (u, v)| M u v , ∀u, v ∈ V. (3.1)
(3) There exists a family (T ε ) ε>0 of uniformly bounded linear operators on V and γ > 0 such that

|A ε (u, T ε u)| γ u 2 , ∀u ∈ V. (3.2)
Then, given a family ( ε ) ε>0 in V , the space of linear forms on V , the variational problem:

Find u ε ∈ V such that A ε (u ε , v) = ε (v) for all v ∈ V (3.3)
admits a unique solution u ε ∈ V for all ε > 0 and there exists C > 0 independent of ε such that

u ε C ε V . (3.4)
Proof. First of all, we prove that (T ε ) ε>0 is a family of uniformly boundedly invertible operators, i.e. that T ε ∈ L(V ) is an isomorphism and there exists a constant

C > 0 independent of ε such that (T ε ) -1 C. Let us introduce the bounded operator A ε ∈ L(V ) associated to the bilinear form A ε (•, •), (A ε u, v) = A ε (u, v), ∀u, v ∈ V.

Equation (3.1) implies that

A ε u = sup v =0 (A ε u, v) v = sup v =0 A ε (u, v) v M u .
Consequently, (A ε ) ε>0 is a family of uniformly bounded operators (with

A ε M ). Since (T ε ) * A ε (T ε ) * A ε = T ε A ε ,
the family ((T ε ) * A ε ) ε>0 is also uniformly bounded (recall that (T ε ) ε>0 is uniformly bounded by assumption). Moreover, thanks to (3.2), the Lax-Milgram theorem implies that (T ε ) * A ε is an isomorphism, and the estimates

γ u 2 |A ε (u, T ε u)| = | (A ε u, T ε u) | = | ((T ε ) * A ε u, u) | (T ε ) * A ε u u , show that ((T ε ) * A ε ) -1
1/γ. The family ((T ε ) * A ε )) ε>0 is thus uniformly boundedly invertible. As a consequence, A ε is injective and has closed range. On the other hand, since A ε (•, •) is symmetric, (3.2) shows that A ε T ε is uniformly boundedly invertible, as well. Hence, A ε is onto. Consequently, A ε and T ε define two isomorphisms on V and are uniformly boundedly invertible, as can it be seen from the identities (

A ε ) -1 = T ε (A ε T ε ) -1 and (T ε ) -1 = (A ε T ε ) -1 A ε .
Let us prove now that the variational problem

(3.3) is well-posed. Since T ε is invertible, (3.3) is equivalent to the variational problem Find u ε ∈ V such that A ε (u ε , T ε v) = ε (T ε v) for all v ∈ V . (3.5)
By assumption, the bilinear form

A ε (•, •) := A ε (•, T ε •) satisfies A ε (u, u)| γ u 2 , ∀u ∈ V.
Applying the Lax-Milgram theorem to the coercive bilinear form A ε (•, •) and the linear form ε (•) := ε (T ε •), we obtain the existence of a unique solution u ε ∈ V for (3.5). Moreover, since (T ε ) ε>0 is uniformly bounded, the estimate (3.4) holds.

The result above is used in the next two sections to establish the well-posedness of a scalar problem set in the reference cell (namely the cell problem (4.10) involved in the definition of the homogenized matrix) and of problem (2.4) (or equivalently (2.7)) for extreme contrasts.

3.2.

Well-posedness in the reference cell. Let H 1 per (Y ) denote the subset of H 1 (Y ) constituted of functions being 1-periodic with respect to the variables y 1 and y 2 :

H 1 per (Y ) = {v ∈ H 1 (Y ) | v is 1-periodic in y 1 and y 2 }. Setting M(u) = 1 |Y | Y u dy = Y u dy, let W per (Y ) be the Hilbert space W per (Y ) = u ∈ H 1 per (Y ) : M(u) = 0 , (3.6)
endowed with the norm u Wper(Y

) := ∇u L 2 (Y ) . Given f ∈ L 2 (Y )
, we investigate in this section the well-posedness of the following cell-problem.

Find u ∈ W per (Y ) such that -div(a∇u) = f , in Y , (3.7)
whose variational formulation reads

Find u ∈ W per (Y ) such that A(u, v) = Y f v dy for all v ∈ W per (Y ), (3.8)
where A(•, •) denotes the symmetric bilinear form defined on H 1 (Y ) by

A(u, v) = Y a(y)∇u • ∇v dy = a 1 Y1 ∇u • ∇v dy + a 2 Y2 ∇u • ∇v dy, (3.9)
for all u, v ∈ H 1 (Y ).

In the sequel, we denote the restrictions to Y 1 and Y 2 of a given function u ∈ H 1 (Y ) by u 1 := u |Y1 and u 2 := u |Y2 .

To prove the well-posedness of (3.7) (or, equivalently, (3.8)), we need to distinguish between the case of large contrasts (the positive number κ defined by (2.3) is large enough) and the one of small contrasts (κ is small enough), as each one is treated using a specific T-coercivity operator.

Large contrasts. Let P denote the harmonic extension operator from

Y 1 to Y 2 . In other words, given u ∈ H 1 (Y 1 ), Pu ∈ H 1 (Y ) is defined by setting Pu = u in Y 1 , while in Y 2 , Pu ∈ H 1 (Y 2 ) is the unique solution of the Dirichlet boundary value problem -∆(Pu) = 0 in Y 2 , Pu = u on ∂Y 2 .
Clearly, there exists κ Y > 0 such that

∇(Pu) 2 L 2 (Y ) κ Y ∇u 2 L 2 (Y1) , ∀u ∈ H 1 (Y 1 ). (3.10) Proposition 3.3. For u ∈ H 1 (Y ), let T Y u := T Y u -M T Y u , (3.11)
with

T Y u = u 1 in Y 1 , -u 2 + 2Pu 1 in Y 2 .
Then, the following assertions hold:

(1) T Y ∈ L(W per (Y )).

(2) T Y y = y -M(y).

(3) For κ > κ Y , where κ is defined in (2.3), there exists γ > 0 such that for all u ∈ W per (Y ): 3. For all u ∈ W per (Y ) and for all η > 0, we have by Young's inequality

A(u, T Y u) = Y a(y)∇u(y) • ∇(T Y u)(y) dy γ ∇u 2 L 2 (Y ) . ( 3
A(u, T Y u) = a 1 Y1 |∇u| 2 dy + |a 2 | Y2 |∇u| 2 dy + 2a 2 Y2 ∇u • ∇(Pu) dy a 1 ∇u 2 L 2 (Y1) + |a 2 | ∇u 2 L 2 (Y2) -|a 2 |η ∇u 2 L 2 (Y2) - |a 2 | η ∇(Pu) 2 L 2 (Y2) .
Consequently, using (3.10), we obtain that

A(u, T Y u) |a 2 | κ - κ Y η ∇u 2 L 2 (Y1) + (1 -η) ∇u 2 L 2 (Y2) .
Thus, if the contrast satisfies κ > κ Y , we can choose η ∈]κ Y /κ, 1[ and get the existence of a constant γ > 0 such that for all u ∈ W per (Y ),

A(u, T Y u) γ ∇u 2 L 2 (Y ) = γ u 2 Wper(Y ) .
Remark 3.4. Let us emphasize that the results of the above proposition have been already obtained in [START_REF] Bunoiu | Homogenization of materials with sign changing coefficients[END_REF]Theorem 2.5]. However, the authors omitted to subtract the average term (the validity of the result was not affected).

Small contrasts. We denote by H 1 0,Σ (Y ) the space of functions in H 1 (Y ) which are zero on the lateral boundary Σ D of the domain Y (the top and bottom boundaries are denoted Σ N ± ). Let Q denote the harmonic extension operator from

H 1 (Y 2 ) onto H 1 0,Σ (Y ). In other words, given u ∈ H 1 (Y 2 ), Qu ∈ H 1 0,Σ (Y ) is defined by setting Qu = u in Y 2 , while in Y 1 , Qu is the unique solution of the boundary value problem -∆(Qu) = 0 in Y 1 , Qu = u on ∂Y 2 , Qu = 0 on Σ D , ∂(Qu) ∂ν = 0 on Σ N ± .
Setting

M 2 (u) = 1 |Y 2 | Y2 u dy, let H 1 mean (Y 2 ) = {u ∈ H 1 (Y 2 ) | M 2 (u) = 0}.
It can be easily seen that there exists κ Y > 0 such that

∇(Qu) 2 L 2 (Y ) κ Y ∇u 2 L 2 (Y2) , ∀u ∈ H 1 mean (Y 2 ). (3.13) Proposition 3.5. For u ∈ H 1 (Y ), set T Y u := T Y u -M T Y u , (3.14)
where

T Y u = u 1 -2Q(u 2 -M 2 (u 2 )) in Y 1 , -u 2 + 2M 2 (u 2 ) in Y 2 . (3.15)
Then, the following assertions hold:

(1) T Y ∈ L(W per (Y )).

(2) For κ < 1/κ Y , where κ Y is defined in (3.13) there exists γ > 0 such that for all u ∈ W per (Y ): The continuity of the map T Y is straightforward.

A(u, T Y u) = Y a(y)∇u(y) • ∇(T Y u)(y) dy γ ∇u 2 L 2 (Y ) . ( 3
2. For all u ∈ W per (Y ) and for all η > 0, we have by Young's inequality

A(u, T Y u) = A(u, T Y u) = a 1 Y1 |∇u| 2 dy + |a 2 | Y2 |∇u| 2 dy -2a 1 Y2 ∇u • ∇(Q(u 2 -M 2 (u 2 ))) dy a 1 ∇u 2 L 2 (Y1) + |a 2 | ∇u 2 L 2 (Y2) -a 1 η ∇u 2 L 2 (Y1) - a 1 η ∇(Q(u 2 -M 2 (u 2 ))) 2 L 2 (Y1) .
Consequently, using (3.13), we obtain that

A(u, T Y u) a 1 ∇u 2 L 2 (Y1) + |a 2 | ∇u 2 L 2 (Y2) -a 1 η ∇u 2 L 2 (Y1) - a 1 κ Y η ∇u 2 L 2 (Y2) ,
that is,

A(u, T Y u) a 1 (1 -η) ∇u 2 L 2 (Y1) + 1 κ - κ Y η ∇u 2 L 2 (Y1) .
Thus, if the contrast satisfies κ < 1/κ Y , we can choose η ∈]κκ Y , 1[ and get the existence of a constant γ > 0 such that for all u ∈ W per (Y ):

A(u, T Y u) γ ∇u 2 L 2 (Y ) = γ u 2 Wper(Y ) .
We can now state a well-posedness result for the cell problem (3.7) for extreme contrasts.

u Wper(Y ) = ∇u L 2 (Y ) C f L 2 (Y ) . (3.18)
Proof. The result follows immediately from Theorem 3.2 using the T-coercivity operators of Proposition 3.3 and Proposition 3.5, respectively, for large or small contrasts.

3.3.

Well-posedness of the problem in Ω ε . For every function u ∈ H 1 (Ω ε ), we set

u 1 := u| Ω ε 1 , u 2 := u| Ω ε 2 . Let H 1 0,Σ (Ω ε 1
) be the space of functions u in H 1 (Ω ε 1 ) and such that u = 0 on the lateral boundary Σ ε,D of Ω ε .

Like for the cell problem considered in the previous section, we need to distinguish the cases of large and small contrasts to study the well-posedness of (2.4) (or equivalently of its variational formulation (2.7)). The case of large contrasts can be easily adapted from [START_REF] Bunoiu | Homogenization of materials with sign changing coefficients[END_REF] (even though the domain considered here is thin) and therefore we only state the results for the sake of completeness. For the case of small contrasts, we propose a new proof which is more elementary than the ones given for the case of a fixed domain in [START_REF] Bunoiu | Homogenization of Maxwell's equations and related scalar problems with sign-changing coefficients[END_REF] (using the T-coercivity) and in [START_REF] Bonnetier | Homogenization of the eigenvalues of the Neumann-Poincaré operator[END_REF] (via the analysis of the spectrum of the Neumann-Poincaré operator).

Large contrasts.

As in [17, Proposition 2.4], one can prove the following result. Proposition 3.7. There is a family of linear bounded extension operators

(P ε ) ε>0 from Ω ε 1 to Ω ε such that (1) P ε ∈ L(H 1 0,Σ (Ω ε 1 ), V ε ). (2) For all u ∈ H 1 0,Σ (Ω ε 1 ), (P ε u)(x) = u(x) for all x ∈ Ω ε 1 . (3) If κ Y is the constant defined in (3.10), one has for all u ∈ H 1 0,Σ (Ω ε 1 ), ∇(P ε u) 2 L 2 (Ω ε ) κ Y ∇u 2 L 2 (Ω ε 1 ) . (3.19)
According to [17, Theorem 2.6], we also have the following result.

Proposition 3.8. For u ∈ H 1 (Ω ε ), set

T ε u = u 1 in Ω ε 1 , -u 2 + 2P ε u 1 in Ω ε 2 .
(3.20)

Then, the following assertions hold:

(1) (T ε ) ε>0 is a family of uniformly bounded linear operators of L(V ε ).

(2) For κ > κ Y , where κ Y is defined in (3.10), there exists γ > 0 such that for all ε > 0 and all u ∈ V ε ,

A ε (u, T ε u) = Ω ε a ε (x)∇u(x) • ∇(T ε u)(x) dx γ ∇u 2 L 2 (Ω ε ) . Small contrasts. Given u ∈ H 1 (Ω ε 2 ), let M ε 2 (u) ∈ H 1 (Ω ε 2 ) denote the piecewise constant function defined on each inclusion εY k 2 ⊂ Ω ε , k ∈ Z ε , by M ε 2 (u)(x) = 1 |εY k 2 | εY k 2 u dx, ∀x ∈ εY k 2 .
We set [START_REF] Cazeaux | Homogenization of a model for the propagation of sound in the lungs[END_REF]Lemma 2.3], one has the following result. Proposition 3.9. There is a family of linear bounded extension operators

H 1 mean (Ω ε 2 ) = u ∈ H 1 (Ω ε 2 ) | M ε 2 (u) = 0 . As in
(Q ε ) ε>0 from Ω ε 2 to Ω ε such that (1) Q ε ∈ L(H 1 mean (Ω ε 2 ), V ε ). (2) For all u ∈ H 1 mean (Ω ε 2 ), (Q ε u)(x) = u(x) for all x ∈ Ω ε 2 . (3) If κ Y is the constant defined in (3.13), one has for all u ∈ H 1 mean (Ω ε ): ∇(Q ε u) 2 L 2 (Ω ε ) κ Y ∇u 2 L 2 (Ω ε 2 ) . (3.21) Proposition 3.10. For u ∈ H 1 (Ω ε ), let T ε u = u 1 -2Q ε (u 2 -M ε 2 (u 2 )) in Ω ε 1 , -u 2 + 2M ε 2 (u 2 ) in Ω ε 2 . (3.22)
Then, the following assertions hold:

(1) (T ε ) ε>0 is a family of uniformly bounded linear operators of L(V ε ).

(2) For 0 < κ < 1/κ Y , where κ Y is defined in (3.13), there exists γ > 0 such that for all ε > 0 and all u ∈ V ε ,

A ε (u, T ε u) = Ω ε a ε (x)∇u(x) • ∇(T ε u)(x) dx γ ∇u 2 L 2 (Ω ε ) . Proof. 1. Let u be given in V ε . Since u 2 -M ε 2 (u 2 ) ∈ H 1 mean (Ω ε 2 ), Q ε (u 2 -M ε 2 (u 2 )
) is well-defined and so is T ε u. Moreover, T ε u is continuous across the interface ∂Ω ε 1 ∩ ∂Ω ε 2 and satisfies T ε u |Σ ε,D = 0. Hence, T ε u ∈ V ε and one clearly has T ε ∈ L(V ε ) and is uniformly bounded.

2. For all u ∈ H 1 (Ω ε ) and for all η > 0, we have

A ε (u, T ε u) = Ω ε a ε (x)∇u(x) • ∇(T ε u)(x) dx = a 1 Ω ε 1 |∇u 1 | 2 dx + |a 2 | Ω ε 2 |∇u 2 | 2 dx -2a 1 Ω ε 1 ∇u 1 • ∇ (Q ε (u 2 -M ε 2 (u 2 )) dx a 1 ∇u 1 2 L 2 (Ω ε 1 ) + |a 2 | ∇u 2 2 L 2 (Ω ε 2 ) -a 1 η ∇u 1 2 L 2 (Ω ε 1 ) - a 1 η ∇ (Q ε (u -M ε 2 (u)) 2 L 2 (Ω ε 1 )
, where we have used Young's inequality for the last estimate. Consequently, using (3.21), we obtain that

|A ε (u, T ε u)| a 1 (1 -η) ∇u 1 2 L 2 (Ω ε 1 ) + 1 κ - κ Y η ∇u 2 2 L 2 (Ω ε 2 ) .
Thus, if the contrast satisfies 0 < κ < 1/κ Y , we can choose η ∈]κκ Y , 1[ and get the existence of a constant γ > 0 such that

|A ε (u, T ε u)| γ ∇u 2 L 2 (Ω ε ) , ∀u ∈ H 1 (Ω ε ).
We can now state a well-posedness result for problem (2.4) for extreme contrasts. then, for every given f ∈ L 2 (ω), problem (2.4) (or, equivalently, its variational formulation (2.7)) admits a unique solution u ε ∈ V ε and there exists a constant C > 0 independent of ε such that for all ε > 0,

∇u ε L 2 (Ω ε ) C √ ε f L 2 (ω) . (3.24)
Proof. The claimed result follows immediately from Theorem 3.2, Proposition 3.8 and Proposition 3.10 and from the hypothesis on f (f does not depend on x 3 ), which yields

f L 2 (Ω ε ) = √ ε f L 2 (ω) .

Homogenization results in the case of a thin periodic layer

Our goal in this section is to pass to the limit, with ε → 0, in the variational formulation (2.7) of problem (2.4). To this end, we use the two-scale convergence method [START_REF] Nguetseng | A general convergence result for a functional related to the theory of homogenization[END_REF][START_REF] Allaire | Homogenization and two-scale convergence[END_REF], adapted to thin periodic domains (see, for instance, [START_REF] Neuss-Radu | Effective transmission conditions for reaction-diffusion processes in domains separated by an interface[END_REF]).

Let C ∞ per (Y ) be the space of infinitely differentiable functions in R 3 that are 1periodic in the first two variables y 1 and y 2 . We recall the definition of the weak two-scale convergence from [START_REF] Neuss-Radu | Effective transmission conditions for reaction-diffusion processes in domains separated by an interface[END_REF].

Definition 4.1. A sequence (v ε ) in L 2 (Ω ε ) weakly two-scale converges to v ∈ L 2 (ω × Y ) if one has 1 ε Ω ε v ε (x)ψ x, x ε dx → ω×Y v(x, y)ψ(x, y) dx dy, for all ψ ∈ C ∞ 0 (ω, C ∞ per (Y ))
. Then, we denote

v ε 2-s --v.
Using the a priori estimate (3.24), Poincaré inequality in Ω ε (see, for instance, [START_REF] Mel'nik | Asymptotic analysis of boundary-value problems in thin perforated domains with rapidly varying thickness[END_REF], Section 2.1) and [START_REF] Neuss-Radu | Effective transmission conditions for reaction-diffusion processes in domains separated by an interface[END_REF]Prop. 4.4(i)], it follows that there exist u ∈ H 1 0 (ω) and u ∈ L 2 (ω, W per (Y )) (recall that W per (Y ) is defined in (3.6)), such that, up to a subsequence, for ε → 0, we obtain

u ε 2-s --u, ∇ xu ε 2-s --∇ x u + ∇ ȳ u, ∂ x3 u ε 2-s --∂ y3 u. (4.1)
The special form of the limits in convergences (4.1) 2 and (4.1) 3 suggests to introduce the following notation: to every w ∈ H 1 (ω), whose gradient ∇ xw(x) has two components, we associate the tridimensional vector ∇w(x) defined by ∇w(x) = (∇ xw(x), 0).

We introduce the space

H = H 1 0 (ω) × L 2 (ω, W per (Y )
) and for all V = (v, v) ∈ H we define the norm The unique solution u ε of the variational problem (2.7) converges, in the sense of (4.1), to the unique solution (u, u) ∈ H of the well-posed two-scale limit problem ω×Y a(y)(∇u

V 2 H = ∇v + ∇ y v 2 L 2 (ω×Y ) , which is equivalent to the norm ∇v 2 L 2 (ω) + ∇ y v 2 L 2 (
+ ∇ y u) • (∇ϕ + ∇ y Φ) dx dy = ω f (x)ϕ(x) dx, (4.2)
for all ϕ ∈ H 1 0 (ω) and Φ ∈ L 2 (ω, H 1 per (Y )). Proof. According to Theorem 3.11, the variational problem (2.7) admits a unique solution u ε which satisfies the energy estimate (3.24). Dividing (2.7) by ε and then choosing the test function (recalling that x = (x,

x 3 )) v(x, x 3 ) = ϕ(x) + εψ x, x ε , (4.3)
with ϕ ∈ D(ω), ψ ε ∈ C ∞ 0 (ω, C ∞ per (Y ))
, we can pass to the limit as ε → 0. Using the convergences (4.1), we obtain by standard density arguments the two-scale limit problem (4.2).

We prove now the well-posedness of the limit problem (4.2). Define on H the bilinear form

B(U, V) := ω×Y a(y) ∇u(x) + ∇ y û(x, y) • ∇v(x) + ∇ y v(x, y) dx dy,
for all U = (u, û) ∈ H and all V = (v, v) ∈ H. Given F ∈ H (H denotes here the dual space of H), consider the variational problem

Find U = (u, û) ∈ H such that B(U, V) = F, V H ,H , ∀V = (v, v) ∈ H. (4.4)
We remark that the two-scale limit problem (4.2) fits in the frame of problem (4.4), for which we shall prove the well-posedness successively for large and small contrasts.

(i) Large contrasts. Let T be the operator defined on H by

T U = (u, T Y û) , ∀U = (u, û) ∈ H, (4.5)
where T Y ∈ L(W per (Y )) is defined in (3.11). Since T Y ∈ L(W per (Y )), we clearly have T ∈ L(H). For every U = (u, û) ∈ H, we note that

∇u(x) + ∇ y û(x, y) = ∇ y ∇u(x) • y c + û(x, y) ,
where we have set y c := y -M(y). Therefore,

B(U, T U) = ω×Y a(y)∇ y ∇u(x) • y c + û(x, y) • ∇ y ∇u(x) • y c + T Y û(x, y) dx dy.
According to Proposition 3.3, we have T Y y c = y c and, thus,

∇ y ∇u(x) • y c + T Y û(x, y) = ∇ y ∇u(x) • T Y y c + T Y û(x, y) = ∇ y T Y ∇u(x) • y c + û(x, y) .
Consequently, setting U x(y) := ∇u(x) • y c + û(x, y), we have

B(U, T U) = ω×Y a(y)∇ y U x(y) • ∇ y T Y U x(y) dx dy = ω A(U x, T Y U x) dx,
where A(•, •) is defined in (3.9). Using the T-coercivity inequality (3.12) in the above relation, we obtain that, for κ > κ Y , there exists γ > 0 such that

B(U, T U) γ ω ∇ y U x 2 L 2 (Y ) dx = γ ∇u(x) + ∇ y û(x, y) 2 L 2 (ω×Y ) = γ U 2 H .
(ii) Small contrasts. As above, the idea is to use the T-coercivity of the reference cell problem given in (3.14) to derive a suitable two-scale T-coercivity operator T . However, since affine functions are not anymore stable by the extension operator Q used to define T Y , the proof needs to be adapted. Let T be the operator defined on H by

T U = (u, T Y û + T Y ψ -ψ) , ∀U = (u, û) ∈ H, (4.6)
where T Y ∈ L(W per (Y )) is defined in (3.14) and

ψ := ∇u • y c .
Let us first check that T ∈ L(H). Obviously, we only need to prove that T Y û + T Y ψ -ψ ∈ W per (Y ) for all U = (u, û) ∈ H. First of all, we have

M (T Y û + T Y ψ -ψ) = -M(ψ) = -∇u • M(y c ) = 0.
Next, by the definition of T Y (see (3.14)), T Y û is 1-periodic in the variables y 1 and y 2 . Furthermore, it follows from the definition of T Y (see (3.15)) that T Y ψ = ψ on the lateral boundary Σ D of ∂Y . Hence,

T Y ψ -ψ = T Y ψ -M( T Y ψ) -ψ = -M( T Y ψ), on Σ D .
From the above relation we deduce that T Y ψ -ψ is constant on Σ D , and thus,

T Y û + (T Y ψ -ψ) is 1-periodic in y 1 and y 2 . Consequently, T Y û + T Y ψ -ψ ∈ W per (Y ), which shows that T ∈ L(H).
Moreover, for every U = (u, û) ∈ H, one can easily check that

B(U, T U) = ω×Y a(y)∇ y ψ(x, y) + û(x, y) • ∇ y T Y ψ(x, y) + T Y û(x, y) dx dy.
Hence, setting U x(y) := ψ(x, y) + û(x, y), we have

B(U, T U) = ω×Y a(y)∇ y U x(y) • ∇ y T Y U x(y) dx dy = ω A(U x, T Y U x) dx.
Using the T-coercivity inequality (3.16) in the above relation, we obtain that, for κ < 1/κ Y , there exists γ > 0 such that

B(U, T U) γ ω ∇ y U x 2 L 2 (Y ) dx = γ ∇u(x) + ∇ y û(x, y) 2 L 2 (ω×Y ) = γ U 2 H .
From the uniqueness of (u, u) ∈ H, all the above convergences hold for the whole sequence. Here, a hom is the constant homogenized 2 × 2 matrix whose entries are defined, for i, j ∈ {1, 2}, by

a hom ij = Y a ij - 3 k=1 a ik ∂χ j ∂y k dy, (4.9)
where the coefficients (a ij ) 1 i,j 3 are those of the (diagonal) matrix a defined in (2.5).

The function χ = (χ 1 , χ 2 ) ∈ (H 1 per (Y )) 2 is the weak solution of the cell problem (j = 1, 2), -div y a(y)(∇ y χ j -e j ) = 0, in Y,

(∇ y χ j -e j ) • ν ± = 0, on Σ N ± , M Y (χ j ) = 0, (4.10)
where Σ N + = (0, 1) 2 × {1}, Σ N -= (0, 1) 2 × {0}, and ν ± = (0, 0, ±1). Proof. By choosing ϕ = 0 in the two-scale limit problem (4.2), we obtain ω×Y a(y)(∇u + ∇ y u) • ∇ y Φ dx dy = 0, (4.11) for all Φ ∈ L 2 (ω, H 1 per (Y )). Choosing Φ ∈ L 2 (ω, H 1 per (Y )) such that Φ = 0 on ω × ∂Y and integrating by parts with respect to y, we are formally led to -div y (a(y)∇ y u) = div y (a(y)∇u) in ω × Y.

Taking in (4.11) a test function Φ ∈ L 2 (ω, H 1 per (Y )) which is zero on Σ N -(and, respectively, which is zero on Σ N + ), we formally obtain

∇ y u • ν + = -∇u • ν + on ω × Σ N + , ∇ y u • ν -= -∇u • ν -on ω × Σ N
-. The linearity of the problem suggests us to search u(x, y) = -∇u(x) • χ(y), where the vector χ(y) = (χ 1 (y), χ 2 (y), χ 3 (y)) belonging to (H 1 per (Y )) 3 has to be determined. Recalling that ∇u(x) = (∇ xu(x), 0), we notice that only the first two components of χ will play a role in our analysis. Inserting this factorization into the equation, we therefore obtain the two local problems (4.10).

By choosing now Φ = 0 in (4.2), we obtain We integrate this last equality by parts with respect to x and, by using (4.8) and (4.10), we are led to the homogenized problem (4.7) with the homogenized coefficients given by (4.9).

Remark 4.4. The expected dimension reduction effect can be seen in the fact that the 2 × 2 homogenized matrix a hom is lower dimensional with respect to the initial matrix. However, information coming from the vertical direction of the initial problem is preserved. Indeed, the value of the coefficients (4.9) is influenced by the vertical local variable y 3 , through the solution of the cell problem (4.10). Consequently, despite the fact that the derivative in the third direction does not explicitly appear in the homogenized problem (4.7), its solution u is implicitly influenced by the third local variable y 3 . Due to the diagonal form of the matrix a = (a ij ) 1 i,j 3 (see (2.5)), the homogenized coefficients (4.9) can be written as In this section, we study the case of a thin rod. We start by describing the geometry of the domain Ω ε , which represents now a two-phase thin rod (see Figure 2). We denote the independent variable x ∈ R 3 by x = (x 1 , x 2 , x 3 ) and we define

Ω ε = εY × (0, L), (5.1)
where Y = (0, 1) 2 , L > 0 and ε ∈ (0, 1) is a sequence of strictly positive numbers such that L ε ∈ N * . This small parameter is related to the characteristic dimension of our domain. Thus, Ω ε is a thin heterogeneous rod, its thickness, the periodicity of its heterogeneities and their size being of order ε. More precisely, the microscopic structure of Ω ε consists of an exact number of replicated unit cells Y = Y ×(0, 1) = (0, 1)

= k + Y α , for α ∈ {1, 2}. We also define, for each ε, Z ε = k ∈ Z 3 : εY k 2 ⊂ Ω ε ; we set Ω ε 2 = ∪ k∈Zε εY k 2 and Ω ε 1 = Ω \ Ω ε 2 .
The boundary of the domain Ω ε is split into three parts: Σ ε N , the lateral boundary of the domain Ω ε , Σ ε,N = {x ∈ R 3 : (x 1 , x 2 ) ∈ ∂εY , 0 < x 3 < L}, and Σ ε,D + , Σ ε,D -, the top and the bottom boundaries, respectively, Σ ε,D

+ = {x ∈ R 3 : (x 1 , x 2 ) ∈ εY , x 3 = L}, Σ ε,D -= {x ∈ R 3 : (x 1 , x 2 ) ∈ εY , x 3 = 0}.
The goal in this section is to analyze the asymptotic behavior, as ε → 0, of the solution u ε of the problem

-div(a ε ∇u ε ) = f in Ω ε , u ε = 0 on Σ ε,D ± , ∂u ε ∂ν ε = 0 on Σ ε,N , (5.2)
where the function f ∈ L 2 ((0, L)) is given and ν ε is the unit outward normal to the lateral boundary Σ ε,N . Here, we have set as before

a ε (x) = a x ε , ∀x ∈ Ω ε ,
where a(y) denotes the 1-periodic in the variable y 3 function defined on Y by

a(y) = a 1 1 Y1 (y) + a 2 1 Y2 (y),
with a 1 , a 2 ∈ R and a 1 a 2 < 0.

To write the variational formulation of problem (5.2), we introduce, for every positive ε < 1, the Hilbert space

V ε = v ∈ H 1 (Ω ε ) : v = 0 on Σ ε,D ± , endowed with the norm v V ε = ∇v L 2 (Ω ε ) , for any v ∈ V ε .
The variational formulation of problem (5.2) is the following: find

u ε ∈ V ε such that A ε (u ε , v) = ε (v), ∀v ∈ V ε , (5.3

) where the bilinear form A

ε : V ε × V ε → R and the linear form ε : V ε → R are given by A ε (u, v) = Ω ε a ε ∇u • ∇v dx, ε (v) = Ω ε f v dx .
As in the case of the thin periodic layer, the aim is to prove the well-posedness of the variational problem (5.3) and then to pass to the limit with ε → 0 in this problem. The proofs follow the same steps as in the previous case, with the use of an appropriate weakly two-scale convergence (see Definition 5.2 below) and with a slight modification in the definition of the extension operator Q, as follows. In this case, we denote by H 1 0,Σ (Y ) the space of functions in H 1 (Y ) which are zero on the upper and lower boundaries Σ D ± of the domain Y (the lateral boundary is denoted Σ Then, for any ε ∈ (0, 1), the variational problem (5.3) has a unique solution u ε ∈ V ε . Moreover, there exists a constant C > 0, independent of ε, such that ∇u ε L 2 (Ω ε ) ≤ Cε f L 2 (ω) .

N ). Let Q denote the harmonic extension operator from H 1 (Y 2 ) onto H 1 0,Σ (Y ). In other words, given u ∈ H 1 (Y 2 ), Qu ∈ H 1 0,Σ (
(5.5)

To pass to the limit, with ε → 0, in the variational formulation (5.3) of problem (5.2), we use again the two-scale convergence method [START_REF] Nguetseng | A general convergence result for a functional related to the theory of homogenization[END_REF][START_REF] Allaire | Homogenization and two-scale convergence[END_REF], adapted to the thin periodic rod case. Let C ∞ per (Y ) be the space of infinitely differentiable functions in R 3 that are 1-periodic in the variable y 3 . We recall the definition of the weakly two-scale convergence (see, for instance, [START_REF] Jerez-Hanckes | Derivation of cable equation by multiscale analysis for a model of myelinated axons[END_REF]). Using the a priori estimate (5.5) and well-adapted compactness results (see, for instance, [START_REF] Jerez-Hanckes | Derivation of cable equation by multiscale analysis for a model of myelinated axons[END_REF]), it follows that there exist u ∈ H 1 0 (0, L) and u ∈ L 2 ((0, L), H 1 per (Y )), with M Y ( u) = 0, such that, up to a subsequence, for ε → 0, we obtain

u ε 2-s --u, ∇ x u ε 2-s --∇ y u, ∂ x3 u ε 2-s -- du dx 3 + ∂ y3 u, (5.6)
where x = (x 1 , x 2 ) and y = (y 1 , y 2 ). Here, the space H 1 per (Y ) is defined by H 1 per (Y ) = {v ∈ H 1 (Y ) : v is 1-periodic in y 3 }. The special form of the limits in convergences (5.6) 2 and (5.6) 3 suggests the introduction of the following notation: to every w ∈ H 1 (0, L), we associate the three-dimensional vector ∇w(x 3 ) defined by ∇w(x 3 ) = 0, 0, dw dx 3 .

Let W per (Y ) = {v ∈ H 1 per (Y ) : M Y (v) = 0}. We introduce the space H = H 1 0 (0, L) × L 2 ((0, L), W per (Y )) and for all V = (v, v) ∈ H we define the norm where Σ N denotes the lateral boundary of the reference cell Y and ν is the unit outward normal to Σ N .

Remark 5.4. Because of the dimension-reduction effect in the first two directions, the homogenized matrix reduces in this case to a scalar. However, the value of this scalar coefficient a hom is influenced by the local variables y 1 and y 2 through the solution χ 3 of the cell problem (5.10).

2 .Figure 1 .

 21 Figure 1. Example of a thin periodic layer and the corresponding reference cell Y .

. 16 ). 1 .

 161 ProofFor every u ∈ W per (Y ), T Y u clearly defines a function of H 1 (Y ) (since it is continuous across the interface ∂Y 2 ) satisfying by construction M(T Y u) = 0. Moreover, we have T Y u = u -M T Y u on the boundary ∂Y , and hence T Y u is 1-periodic in y 1 and y 2 . Consequently, T Y u ∈ W per (Y ) for every u ∈ W per (Y ).

Theorem 3 . 6 .

 36 Let κ Y and κ Y denote the constants in (3.10) and (3.13), respectively. If the contrast κ:= a1 a2 satisfies κ > κ Y or κ < 1/κ Y ,(3.17)then, for every given f ∈ L 2 (Y ), problem (3.7) admits a unique solution and there exists a constant C > 0 such that

Theorem 3 . 11 .

 311 Let κ Y and κ Y denote the constants in (3.19) and (3.21), respectively. If the contrast κ := | a1 a2 | satisfies κ > κ Y or κ < 1/κ Y , (3.23)

Theorem 4 . 3 .

 43 Under the hypotheses of Theorem 4.2, the unique solution u ε of the variational problem (2.7) converges, in the sense of (4.1), to (u, u) ∈ H, where u is the unique solution of the homogenized problem-div x a hom ∇ xu(x) = f (x) in ω, χ j (y) ∀(x, y) ∈ ω × Y.(4.8)

  ω×Y a(y) ∇u + ∇ y u • ∇ϕ dx dy = ω f (x)ϕ(x) dx.(4.12)

5 .

 5 can be further simplified in the particular, but relevant, situation in which the reference cell Y is invariant under rotation of angle π/2 (see[START_REF] Bunoiu | Homogenization of materials with sign changing coefficients[END_REF] Proposition 4.2]). The case of a thin periodic rod

Figure 2 .

 2 Figure 2. Example of a thin periodic rod and the corresponding reference cell Y .

2 L 2 Theorem 5 . 1 .

 2251 Y ) is defined by setting Qu = u in Y 2 , while, in Y 1 , Qu is the unique solution of the boundary value problem-∆(Qu) = 0 in Y 1 , Qu = u on ∂Y 2 , Qu = 0 on Σ D ± , ∂(Qu) ∂ν = 0 on Σ N .With the notationM 2 (u) = 1 |Y 2 | Y2 u dyand settingH 1 mean (Y 2 ) = u ∈ H 1 (Y 2 ) | M 2 (u) = 0 , it can be easily seen that there exists κ Y > 0 such that ∇(Qu) (Y ) κ Y ∇u 2 L 2 (Y2) , ∀u ∈ H 1 mean (Y 2 ).(5.4)Next we have a well-posedness result. Assume that the contrast κ := a1 a2 satisfiesκ > κ Y or κ < 1/κ Y ,where κ Y and κ Y denote the constants in (3.19) and (5.4) respectively.

Definition 5 . 2 .

 52 A sequence (v ε ) in L 2 (Ω ε ) weakly two-scale converges to v ∈ L 2 ((0, L) × Y ) if one 3 , y)ψ(x 3 , y) dx 3 dy, for all ψ ∈ C ∞ 0 ((0, L), C ∞ per (Y )). Then, we denotev ε 2-s --v.

V 2 H 2 L 2 (Theorem 5 . 3 . 3 = 9 )

 2225339 = ∇v + ∇ y v (0,L)×Y ) . Assume that the contrast κ := a1 a2 satisfies κ > κ Y or κ < 1/κ Y , where κ Y and κ Y denote the constants in (3.[START_REF] Caillerie | Homogénéisation des équations de la diffusion stationnaire dans les domaines cylindriques aplatis[END_REF]) and (5.4), respectively. Then, the unique solution u ε of the variational problem (5.3) converges, in the sense of (5.6), to (u, u) ∈ H, where u is the unique solution of the homogenized problem-a hom d 2 u dx 2 f on (0, L), u(0) = u(L) = 0 (5.7)andu(x 3 , y) = -du dx 3 (x 3 )χ 3 (y), ∀(x 3 , y) ∈ (0, L) × Y.(5.8)Here, a hom is the scalar defined byThe function χ 3 ∈ H 1 per (Y ) is the weak solution of the cell problem -div y (a(y)(∇ y χ 3 -e 3 )) = 0 in Y, (∇ y χ 3 -e 3 ) • ν = 0 on Σ N , M(χ 3 ) = 0, (5.10)

  .12) Proof. 1. For every u ∈ W per (Y ), T Y u clearly defines a function of H 1 (Y ) (since it is continuous across the interface ∂Y 2 ) satisfying M T Y u = 0. Moreover, we have T Y u = u -M T Y u on the boundary ∂Y , and hence T Y u is 1-periodic in y 1 and y 2 . Consequently, T Y u ∈ W per (Y ) for every u ∈ W per (Y ). The continuity of the map T Y is straightforward. 2. Since affine functions are harmonic, they are invariant by P and hence T Y y = y.

  ω×Y ) (see, for instance, [26, Lemma 5.4]). Assume that the contrast κ := a1 a2 satisfies κ > κ Y or κ < 1/κ Y , where κ Y and κ Y denote the constants in (3.19) and (3.21), respectively.

	Theorem 4.2.

  3 , rescaled with ε.One has Y = Y 1 ∪ Y 2 ,where Y 1 and Y 2 are two non-empty disjoint connected open subsets of Y such that Y 2 ⊂ Y . We assume that ∂Y 2 is Lipschitz continuous. For each k ∈ Z 3 , we denote Y k α

Acknowledgments. The authors would like to thank Lucas Chesnel and Mahran Rihani for fruitful discussions on T-coercivity technics and for suggesting the idea used in (4.6) to cope with the fact that affine functions are not preserved by the extension operator. This paper was completed during the visit of the third author at Institut Elie Cartan de Lorraine, whose warm hospitality is gratefully acknowledged.