Renata Bunoiu
email: renata.bunoiu@univ-lorraine.fr

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

HOMOGENIZATION OF A THERMAL PROBLEM WITH FLUX JUMP

The goal of this paper is to analyze, through homogenization techniques, the effective thermal transfer in a periodic composite material formed by two constituents, separated by an imperfect interface where both the temperature and the flux exhibit jumps. Following the hypotheses on the flux jump, two different homogenized problems are obtained. These problems capture in various ways the influence of the jumps: in the homogenized coefficients, in the right-hand side of the homogenized problem, and in the correctors.

Introduction

In the last two decades, the study of the macroscopic properties of heterogeneous composite materials with imperfect contact between their constituents has been a subject of major interest for engineers, mathematicians, physicists. In particular, the problem of thermal transfer in such heterogeneous media has attracted the attention of a broad category of researchers, due to the fact that the macroscopic properties of a composite can be strongly influenced by the imperfect bonding between its components. This imperfect contact can be generated by various causes: the presence of impurities at the boundaries, the presence of a thin interphase, the interface damage, chemical processes.

The homogenization of a thermal problem in a two-component composite with interfacial barrier, with jump of the temperature and continuity of the flux, was studied for the first time in the pioneering work [START_REF] Auriault | Macroscopic modelling of heat transfer in composites with interfacial thermal barrier[END_REF], where the asymptotic expansion method was used. Many mathematical studies were performed since then, in order to rigorously justify the convergence results. Various mathematical methods were used: the energy method in [START_REF] Donato | Homogenization of two heat conductors with an interfacial contact resistance[END_REF] and [START_REF] Monsurrò | Homogenization of a two-component composite with interfacial thermal barrier[END_REF], the two-scale convergence method in [START_REF] Ene | Model of diffusion in partially fissured media[END_REF], and more recently the unfolding method for periodic homogenization in [START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF] and [START_REF] Timofte | Multiscale analysis of diffusion processes in composite media[END_REF], to cite just a few of them. The main common point of all these studies is the fact that at the interface between the two components the flux of the temperature is continuous, the temperature field has a jump and the flux is proportional to this jump. Several cases are studied, following the order of magnitude with respect to the small parameter ε of the resistance generated by the imperfect contact between the constituents, leading to completely different macroscopic problems. Here, ε is a small real parameter related to the characteristic size of the two constituents. In some cases, an effect of the imperfect conditions is observed in the coefficients of the homogenized matrix, via the local problems; in other cases, there is no effect at all in the homogenized problem.

For similar homogenization problems of parabolic or hyperbolic type, we refer the reader to [START_REF] Donato | Homogenization of the wave equation in composites with imperfect interface: a memory effect[END_REF] and [START_REF] Jose | Homogenization of a parabolic problem with an imperfect interface[END_REF]. For the case when both components of the composite material are connected, we refer to [START_REF] Le Nguyen | Homogenization of heat transfer process in composite materials[END_REF], [START_REF] Polisevski | Homogenization cases of heat transfer in structures with interfacial barriers[END_REF], [START_REF] Polisevski | Heat conduction through a first-order jump interface[END_REF], [START_REF] Polisevski | Diffusion in an intermediate model of fractured porous media[END_REF], [START_REF] Timofte | Multiscale analysis of diffusion processes in composite media[END_REF] and [START_REF] Timofte | Multiscale modeling of heat transfer in composite materials[END_REF]. Also, for problems involving jumps in the solution in other contexts, such as heat transfer in polycrystals with interfacial resistance, linear elasticity problems or problems modeling the electrical conduction in biological tissues, see [START_REF] Amar | Evolution and memory effects in the homogenization limit for electrical conduction in biological tissues[END_REF], [START_REF] Amar | A hierarchy of models for the electrical conduction in biological tissues via two-scale convergence: The nonlinear case[END_REF], [START_REF] Donato | Homogenization of an elastic double-porosity medium with imperfect interface via the periodic unfolding method[END_REF], [START_REF] Ene | Microstructure models for composites with imperfect interface via the periodic unfolding method[END_REF], [START_REF] Ene | Homogenization of a thermoelasticity model for a composite with imperfect interface[END_REF], [START_REF] Ene | Homogenization results for a dynamic coupled thermoelasticity problem[END_REF], [START_REF] Hummel | Homogenization for heat transfer in polycrystals with interfacial resistances[END_REF] and [START_REF] Timofte | Multiscale analysis in nonlinear thermal diffusion problems in composite structures[END_REF].

Our goal in this paper is to analyze the effective thermal transfer in a periodic composite material formed by two constituents, one connected and the other one disconnected, separated by an imperfect interface where both the temperature and the flux exhibit jumps. This mathematical model is not restricted to the thermal transfer, but can be used in other contexts, too. Transmission problems involving jumps in the solutions or in the fluxes are encountered in various domains, such as linear elasticity, theory of semiconductors, the study of photovoltaic systems or problems in media with cracks (see, for instance, [START_REF] Auriault | Homogenization of Coupled Phenomena in Heterogenous Media[END_REF], [START_REF] Benveniste | Imperfect soft and stiff interfaces in two-dimensional elasticity[END_REF], [START_REF] Brinkman | A Drift-Diffusion-Reaction Model for Excitonic Photovoltaic Bilayers: Asymptotic Analysis and a 2-D HDG Finite Element Scheme[END_REF], [START_REF] Hashin | Thin interphase-imperfect interface in elasticity with application to coated fiber composites[END_REF] and [START_REF] Khludnev | Analysis of Cracks in Solids[END_REF]). Formal methods of averaging were widely used in the literature to deal with such imperfect transmission problems. Still, obtaining rigorous results based on the homogenization theory is a difficult task in many cases. Some results were nevertheless obtained for problems with flux jump, by using homogenization techniques. We mention here the results obtained in [START_REF] Gahn | Homogenization of reaction-diffusion processes in a two-component porous medium with a nonlinear flux condition at the interface, and application to metabolic processes in cells[END_REF] for reaction-diffusion problems in porous media, in [START_REF] Ijioma | Pattern formation in reverse smouldering combustion: a homogenization approach[END_REF] for problems arising in the combustion theory and in [START_REF] Fellner | A discontinuous Poisson-Boltzmann equation with interfacial transfer: homogenisation and residual error estimate[END_REF] for a problem corresponding to the Gouy-Chapman-Stern model for an electric double layer.

Here, we consider a composite material occupying an open bounded set Ω in R N (N ≥ 2), with a Lipschitz-continuous boundary ∂Ω. We assume that Ω is formed by two parts denoted Ω ε 1 and Ω ε 2 , occupied by two materials with different thermal characteristics, separated by an imperfect interface Γ ε . We also assume that the phase Ω ε 1 is connected and reaches the external fixed boundary ∂Ω and that Ω ε 2 is disconnected: it is the union of domains of size ε, periodically distributed in Ω with periodicity ε. In such a domain, we study the asymptotic behavior, as ε tends to zero, of the solution u ε = (u ε 1 , u ε 2 ) of the following problem:

                     -div (A ε ∇u ε 1 ) = f in Ω ε 1 , -div (A ε ∇u ε 2 ) = f in Ω ε 2 , A ε ∇u ε 1 • n ε = h ε ε (u ε 1 -u ε 2 ) -G ε on Γ ε , A ε ∇u ε 2 • n ε = h ε ε (u ε 1 -u ε 2 ) on Γ ε , u ε 1 = 0 on ∂Ω.
The main novelty brought by our paper consists in allowing the presence, apart from the discontinuity in the temperature field, of a jump in the thermal flux across the imperfect interface Γ ε , given by the function G ε . Two different representative cases are studied here, following the conditions imposed on G ε (stated explicitly in Section 2, Case 1 and Case 2). Let us mention that such functions were already encountered in a different context, more precisely in [START_REF] Cioranescu | The periodic unfolding method in domains with holes[END_REF] and [START_REF] Chourabi | Homogenization and correctors of a class of elliptic problems in perforated domains[END_REF] for the case of the perforated domains with non homogeneous Neumann boundary conditions on the perforations. For a similar problem of homogenization in a perforated domain with Fourier boundary conditions we refer the reader to [START_REF] Belyaev | Averaging in a perforated domain with an oscillating third boundary condition[END_REF]. After passage to the limit with the unfolding method, we obtain here two different unfolded problems (stated in Theorem 4.1 and Theorem 4.7), corresponding to the above mentioned cases for the flux jump function G ε . In Corollary 4.2 and Theorem 4.9, we then give the corresponding homogenized problems. In both situations, the homogenized matrix A hom is constant and it depends on the function describing the jump of the solution. This phenomenon was already observed in some cases without flux jump. Moreover, for the first case studied here, we notice in the right-hand side the presence of a new source term distributed all over the domain Ω and depending on the flux jump function. For the second case, we notice that the influence of the jump in the flux is captured by the correctors only and so this jump plays no role in the homogenized problem; nevertheless, in Remark 4.10 we mention a case when the homogenized problem depends on this jump, too. This type of result is to be compared with the Neumann problem in perforated domains (see [START_REF] Cioranescu | The periodic unfolding method in domains with holes[END_REF], [START_REF] Chourabi | Homogenization and correctors of a class of elliptic problems in perforated domains[END_REF]), where similar phenomena occur.

Various other scalings of the temperature and flux jumps, leading to different macroscopic problems, will be studied in a forthcoming paper. Moreover, for the study of the homogenization of a two permeability problem with flux jump, we refer to [START_REF] Bunoiu | On the homogenization of a two permeability problem with flux jump[END_REF].

The paper is organized as follows: in Section 2, we introduce the microscopic problem and we fix the notation. In Section 3, we review the definition and the basic properties of the unfolding operators we shall use in our proofs. We state and prove the main homogenization results of this paper in Section 4. Corrector results are given in Section 5. We end our paper by a few concluding remarks and some references. is Lipschitz continuous and that Y 2 is connected. In fact, our results can be extended to the case in which the set Y 2 has a finite number of connected components, as in [START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF].

Throughout the paper, the small parameter ε will take its values in a positive real sequence tending to zero and C will be a positive constant independent of ε, whose value can change from line to line.

For each k ∈ Z N , we denote

Y k = k + Y and Y k α = k + Y α , for α = 1, 2.
We also define for each ε,

Z ε = k ∈ Z N : ε Ȳ k 2 ⊂ Ω and we set Ω ε 2 = k∈Zε εY k 2 and Ω ε 1 = Ω \ Ωε 2 . The boundary of Ω ε
2 is denoted by Γ ε and n ε is the unit outward normal to Ω ε 2 . Our goal is to describe the asymptotic behavior, as ε → 0, of the solution u ε = (u ε 1 , u ε 2 ) of the following problem:

(2.1)

                     -div (A ε ∇u ε 1 ) = f in Ω ε 1 , -div (A ε ∇u ε 2 ) = f in Ω ε 2 , A ε ∇u ε 1 • n ε = h ε ε (u ε 1 -u ε 2 ) -G ε on Γ ε , A ε ∇u ε 2 • n ε = h ε ε (u ε 1 -u ε 2 ) on Γ ε , u ε 1 = 0 on ∂Ω. Remark 2.1 We notice that (2.2) A ε ∇u ε 1 • n ε -A ε ∇u ε 2 • n ε = -G ε ,
which clearly shows that the flux of the solution exhibits a jump across Γ ε .

The function f ∈ L 2 (Ω) is given. Let h be a Y -periodic function in L ∞ (Γ) such that there exists h 0 ∈ R with 0 < h 0 < h(y)
a.e. on Γ. We set

h ε (x) = h x ε a.e. on Γ ε . For α, β ∈ R, with 0 < α ≤ β, let M(α, β, Y ) be the set of all the matrices A ∈ (L ∞ (Y )) N ×N
with the property that, for any

ξ ∈ R N , α|ξ| 2 ≤ (A(y)ξ, ξ) ≤ β|ξ| 2 , almost everywhere in Y .
For a Y -periodic smooth symmetric matrix A ∈ M(α, β, Y ), we set

A ε (x) = A x ε a.e. in Ω.
Let g be a Y -periodic function that belongs to L 2 (Γ). We define

g ε (x) = g x ε a.e. on Γ ε .
For the given function G ε in (2.1), we consider the following two relevant forms (see [START_REF] Chourabi | Homogenization and correctors of a class of elliptic problems in perforated domains[END_REF]):

Case 1: G ε = εg x ε , if M Γ (g) = 0. Case 2: G ε = g x ε , if M Γ (g) = 0.
Here, M Γ (g) = 1 |Γ| Γ g(y) dy denotes the mean value of the function g on Γ.

In order to write the variational formulation of problem (2.1), we introduce, for every positive ε < 1, the Hilbert space

(2.3) H ε = V ε × H 1 (Ω ε 2 ).
The space

V ε = v ∈ H 1 (Ω ε 1 ), v = 0 on ∂Ω is endowed with the norm v V ε = ∇v L 2 (Ω ε 1 ) , for any v ∈ V ε and the space H 1 (Ω ε
2 ) is equipped with the usual norm. On the space H ε , we consider the scalar product

(2.4) (u, v) H ε = Ω ε 1 ∇u 1 ∇v 1 dx + Ω ε 2 ∇u 2 ∇v 2 dx + 1 ε Γ ε (u 1 -u 2 )(v 1 -v 2 ) dσ x where u = (u 1 , u 2 ) and v = (v 1 , v 2 )
belong to H ε . The norm generated by the scalar product (2.4) is given by

(2.5) v 2 H ε = ∇v 1 2 L 2 (Ω ε 1 ) + ∇v 2 2 L 2 (Ω ε 2 ) + 1 ε v 1 -v 2 2 L 2 (Γ ε ) .
The variational formulation of problem (2.1) is the following one: find

u ε ∈ H ε such that (2.6) a(u ε , v) = l(v), ∀v ∈ H ε ,
where the bilinear form a : H ε × H ε → R and the linear form l :

H ε → R are given by a(u, v) = Ω ε 1 A ε ∇u 1 ∇v 1 dx + Ω ε 2 A ε ∇u 2 ∇v 2 dx + Γ ε h ε ε (u 1 -u 2 )(v 1 -v 2 ) dσ x and l(v) = Ω ε 1 f v 1 dx + Ω ε 2 f v 2 dx + Γ ε G ε v 1 dσ x ,
respectively.

We state now an existence and uniqueness result and some necessary a priori estimates for the solution of the variational problem (2.6).

Theorem 2.2 For any ε ∈ (0, 1), the variational problem (2.6) has a unique solution u ε ∈ H ε .

Moreover, there exists a constant C > 0, independent of ε, such that

(2.7) ∇u ε 1 L 2 (Ω ε 1 ) ≤ C, ∇u ε 2 L 2 (Ω ε 2 ) ≤ C and (2.8) u ε 1 -u ε 2 L 2 (Γ ε ) ≤ Cε 1/2 .
Proof. The proof of this theorem follows exactly the same steps as in [START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF] and [START_REF] Jose | Homogenization of a parabolic problem with an imperfect interface[END_REF].

3 The periodic unfolding method for a two-component domain

In this section, we shall briefly recall the definitions and the main properties of the unfolding operators T ε 1 and T ε 2 , introduced, for a two-component domain, by P. Donato et al. in [START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF] (see, also, [START_REF] Cioranescu | The periodic unfolding method in domains with holes[END_REF], [START_REF] Cioranescu | The periodic unfolding method in homogenization[END_REF], [START_REF] Cioranescu | The periodic unfolding method in perforated domains[END_REF] and [START_REF] Donato | Homogenization of diffusion problems with a nonlinear interfacial resistance[END_REF]) and of the boundary unfolding operator T ε b , introduced in [START_REF] Cioranescu | The periodic unfolding method in perforated domains[END_REF] and [START_REF] Cioranescu | Asymptotic behavior of elliptic problems in perforated domains with nonlinear boundary conditions[END_REF]. The main feature of these operators is that they map functions defined on the oscillating domains Ω ε 1 , Ω ε 2 and, respectively, Γ ε , into functions defined on the fixed domains

Ω × Y 1 , Ω × Y 2 and Ω × Γ, respectively. For x ∈ R N , we denote by [x] Y its integer part k ∈ Z N , such that x -[x] Y ∈ Y and we set {x} Y = x -[x] Y for a.e. x ∈ R N . So, for almost every x ∈ R N , we have x = ε x ε Y + x ε Y
. For defining the above mentioned periodic unfolding operators, we consider the following sets (see [START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF]):

Z ε = k ∈ Z N | εY k ⊂ Ω , Ω ε = int k∈ Zε ε Ȳ k , Λ ε = Ω \ Ω ε , Ω ε α = k∈ Zε εY k α , Λ ε α = Ω ε α \ Ω ε α , Γ ε = ∂ Ω ε 2 .
Definition 3.1 For any Lebesgue measurable function ϕ on Ω ε α , α ∈ {1, 2}, we define the periodic unfolding operators by the formula

T ε α (ϕ)(x, y) =      ϕ ε x ε Y + εy for a.e. (x, y) ∈ Ω ε × Y α 0 for a.e. (x, y) ∈ Λ ε × Y α If ϕ is a function defined in Ω, for simplicity, we write T ε α (ϕ) instead of T ε α (ϕ| Ω ε α ).
For any function φ which is Lebesgue-measurable on Γ ε , the periodic boundary unfolding operator T ε b is defined by

T ε b (φ)(x, y) =      φ ε x ε Y + εy for a.e. (x, y) ∈ Ω ε × Γ 0 for a.e. (x, y) ∈ Λ ε × Γ Remark 3.2 We notice that if ϕ ∈ H 1 (Ω ε α ), then T ε b (ϕ) = T ε α (ϕ)| Ω ε ×Γ .
We recall here some useful properties of these operators (see, for instance, [START_REF] Cioranescu | The periodic unfolding method in domains with holes[END_REF], [START_REF] Donato | Homogenization of diffusion problems with a nonlinear interfacial resistance[END_REF] and [START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF]).

Proposition 3.3 For p ∈ [1, ∞) and α = 1, 2, the operators T ε α are linear and continuous from

L p (Ω ε α ) to L p (Ω × Y α ) and (i) if ϕ and ψ are two Lebesgue measurable functions on Ω ε α , one has T ε α (ϕψ) = T ε α (ϕ)T ε α (ψ); (ii) for every ϕ ∈ L 1 (Ω ε α ), one has 1 |Y | Ω×Yα T ε α (ϕ)(x, y) dx dy = Ω ε α ϕ(x) dx = Ω ε α ϕ(x) dx - Λε ϕ(x) dx; (iii) if {ϕ ε } ε ⊂ L p (Ω) is a sequence such that ϕ ε -→ ϕ strongly in L p (Ω), then T ε α (ϕ ε ) -→ ϕ strongly in L p (Ω × Y α ); (iv) if ϕ ∈ L p (Y α ) is Y -periodic and ϕ ε (x) = ϕ(x/ε), then T ε α (ϕ ε ) -→ ϕ strongly in L p (Ω × Y α ); (v) if ϕ ∈ W 1,p (Ω ε α ), then ∇ y (T ε α (ϕ)) = εT ε α (∇ϕ) and T ε α (ϕ) belongs to L 2 Ω; W 1,p (Y α ) . Moreover, for every ϕ ∈ L 1 (Γ ε ), one has Γ ε ϕ(x) dσ x = 1 ε|Y | Ω×Γ T ε b (ϕ)(x, y) dx dσ y .
The following result was proven, for our geometry, in [START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF].

Lemma 3.4 If u ε = (u ε 1 , u ε 2 ) is a sequence in H ε , then (3.1) 1 ε|Y | Ω×Γ |T ε 1 (u ε 1 ) -T ε 2 (u ε 2 )| 2 dx dσ y ≤ Γ ε |u ε 1 -u ε 2 | 2 dσ x .
Moreover, if ϕ ∈ D(Ω), then, for ε small enough, we have

ε Γ ε h ε (u ε 1 -u ε 2 )ϕ dσ x = Ω×Γ h(y) (T ε 1 (u ε 1 ) -T ε 2 (u ε 2 )) T ε α (ϕ) dx dσ y , with α = 1 or α = 2.
We also recall here some general compactness results obtained in [START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF] for bounded sequences in

H ε . Lemma 3.5 Let u ε = (u ε 1 , u ε
2 ) be a bounded sequence in H ε . Then, there exists a constant C > 0, independent of ε, such that

(3.2) T ε 1 (∇u ε 1 ) L 2 (Ω×Y 1 ) ≤ C, T ε 2 (∇u ε 2 ) L 2 (Ω×Y 2 ) ≤ C, T ε 2 (u ε 1 ) -T ε 1 (u ε 2 ) L 2 (Ω×Γ) ≤ Cε. Theorem 3.6 Let u ε = (u ε 1 , u ε
2 ) be a bounded sequence in H ε . Then, up to a subsequence, still denoted by ε, there exist

u 1 ∈ H 1 0 (Ω), u 2 ∈ L 2 (Ω), u 1 ∈ L 2 Ω, H 1 per (Y 1 ) and u 2 ∈ L 2 Ω, H 1 (Y 2 ) such that (3.3) T ε 1 (u ε 1 ) -→ u 1 strongly in L 2 Ω, H 1 (Y 1 ) , T ε 1 (∇u ε 1 ) ∇u 1 + ∇ y u 1 weakly in L 2 (Ω × Y 1 ), T ε 2 (u ε 2 ) u 2 weakly in L 2 (Ω, H 1 (Y 2 )), T ε 2 (∇u ε 2 ) ∇ y u 2 weakly in L 2 (Ω × Y 2 ) , u ε α |Y α | |Y | u α weakly in L 2 (Ω), α = 1, 2,
where M Γ ( u 1 ) = 0 for almost every x ∈ Ω and • denotes the extension by zero of a function to the whole of the domain Ω. Moreover, we have u 1 = u 2 and

(3.4) 1 ε [T ε 1 (u ε 1 ) -M Γ (T ε 1 (u ε 1 ))] y Γ ∇u 1 + u 1 weakly in L 2 Ω, H 1 (Y 1 ) ,
with y Γ = y -M Γ (y) and

(3.5) 1 ε [T ε 2 (u ε 2 ) -M Γ (T ε 2 (u ε 2 ))] u 2 weakly in L 2 Ω, H 1 (Y 2 ) .

Homogenization results

In this section, we pass to the limit, with ε → 0, in the variational formulation (2.6) of the problem (2.1). To this end, we use the periodic unfolding method and the general compactness results given in Section 3.

We start by emphasizing again that by applying the general results stated in Theorem 3.6 to the solution u ε = (u ε 1 , u ε 2 ) of the variational problem (2.6), which is bounded in H ε , we obtain, at the macroscale, u 1 = u 2 . In what follows, we shall denote their common value by u. We notice that u belongs to H 1 0 (Ω). Moreover, using the priori estimates (2.7)-(2.8) and the general compactness results (3.2)-

(3.3), we know that there exist u ∈ H 1 0 (Ω), u 1 ∈ L 2 (Ω, H 1 per (Y 1 )), u 2 ∈ L 2 (Ω, H 1 (Y 2 )
) such that M Γ ( u 1 ) = 0 and up to a subsequence, for ε → 0, we have:

(4.1) T ε 1 (u ε 1 ) → u strongly in L 2 (Ω, H 1 (Y 1 )), T ε 1 (∇ u ε 1 ) ∇ u + ∇ y u 1 weakly in L 2 (Ω × Y 1 ), T ε 2 (u ε 2 ) u weakly in L 2 (Ω, H 1 (Y 2 )), T ε 2 (∇ u ε 2 ) ∇ y u 2 weakly in L 2 (Ω × Y 2 ), u ε α |Y α | |Y | u weakly in L 2 (Ω), α = 1, 2.
Moreover, one has

(4.2) T ε 1 (u ε 1 ) -T ε 2 (u ε 2 ) ε u 1 -ū2 weakly in L 2 (Ω × Γ), where ū2 ∈ L 2 (Ω, H 1 (Y 2 )) is defined by ū2 = u 2 -y Γ ∇u -ξ Γ , for some ξ Γ ∈ L 2 (Ω). Let W per (Y 1 ) = {v ∈ H 1 per (Y 1 ) | M Γ (v) = 0}.
We consider the space

V = H 1 0 (Ω) × L 2 (Ω; W per (Y 1 )) × L 2 Ω, H 1 (Y 2 ) ,
endowed with the norm

V 2 V = ∇v + ∇ y v 1 2 L 2 (Ω×Y 1 ) + ∇v + ∇ y v2 2 L 2 (Ω×Y 2 ) + v 1 -v2 2 L 2 (Ω×Γ) , for all V = (v, v 1 , v 2 ) ∈ V.
For the passage to the limit, we have to distinguish between two cases, following the form of the function G ε .

Case 1:

G ε = εg x ε , if M Γ (g) = 0. Theorem 4.1 The unique solution u ε = (u ε 1 , u ε 2 )
of the variational problem (2.6) converges, in the sense of (4.1), to the unique solution (u, u 1 , ū2 ) ∈ V of the following unfolded limit problem:

1 |Y | Ω×Y 1 A(y)(∇u+∇ y u 1 )(∇ϕ+∇ y Φ 1 ) dx dy + 1 |Y | Ω×Y 2 A(y)(∇u+∇ y ū2 )(∇ϕ+∇ y Φ 2 ) dx dy+ (4.3) 1 |Y | Ω×Γ h(y)( u 1 -ū2 )(Φ 1 -Φ 2 ) dx dσ y = Ω f (x)ϕ(x) dx + |Γ| |Y | M Γ (g) Ω ϕ(x) dx, for all ϕ ∈ H 1 0 (Ω), Φ 1 ∈ L 2 (Ω, H 1 per (Y 1 )) and Φ 2 ∈ L 2 (Ω, H 1 (Y 2 )).
Proof. In order to obtain the limit problem (4.3), we first unfold the variational formulation (2.6) and by using Lemma 3.4 we get

1 |Y | Ω×Y 1 T ε 1 (A ε )T ε 1 (∇u ε 1 )T ε 1 (∇v 1 ) dx + 1 |Y | Ω×Y 2 T ε 2 (A ε )T ε 2 (∇u ε 2 )T ε 2 (∇v 2 ) dx+ 1 |Y | Ω×Γ h(y) T ε 1 (u ε 1 ) -T ε 2 (u ε 2 ) ε T ε 1 (v 1 ) -T ε 2 (v 2 ) ε dσ x = 1 |Y | Ω×Y 1 T ε 1 (f )T ε 1 (v 1 ) dx+ 1 |Y | Ω×Y 2 T ε 2 (f )T ε 2 (v 2 ) dx + 1 ε 1 |Y | Ω×Γ T ε b (G ε )T ε b (v 1 ) dσ x .
Then, for α = 1, 2, we choose in this unfolded problem the admissible test functions

(4.4) v α = ϕ(x) + εω α (x)ψ α x ε , with ϕ, ω α ∈ D(Ω), ψ 1 ∈ H 1 per (Y 1 ), ψ 2 ∈ H 1 (Y 2
) and for which we obviously have

(4.5) T ε α (v α ) → ϕ(x) strongly in L 2 (Ω × Y α ) and (4.6) T ε α (∇v α ) → ∇ϕ(x) + ∇ y Φ α strongly in L 2 (Ω × Y α ),
where Φ α (x, y) = ω α (x)ψ α (y).

Then, the passage to the limit with ε → 0 is classical, by using convergences (4.1)-(4.2), (4.5)-(4.6) and the ideas in [START_REF] Donato | Homogenization of diffusion problems with a nonlinear interfacial resistance[END_REF]. The only term which requires more attention is in the right-hand side, the integral term involving the function G ε . For this term, we have:

1 ε 1 |Y | Ω×Γ T ε b (G ε )T ε b (v 1 ) dσ x = 1 |Y | Ω×Γ T ε b g x ε T ε b ϕ(x) + εω 1 (x)ψ 1 x ε dσ x = 1 |Y | Ω×Γ g(y)T ε b (ϕ)(x, y) dx dσ y + ε 1 |Y | Ω×Γ g(y)T ε b (ω 1 )(x, y)T ε b (ψ 1 )(x, y) dx dσ y → |Γ| |Y | M Γ (g) Ω ϕ(x) dx.
By the density of

D(Ω) ⊗ H 1 per (Y 1 ) in L 2 (Ω, H 1 per (Y 1 )) and of D(Ω) ⊗ H 1 (Y 2 ) in L 2 (Ω, H 1 (Y 2
)), we get (4.3).

We notice that our limit problem (4.3) is similar with the one obtained in [START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF] (see relation (3.43)), the only difference being the right-hand side, in which an extra term involving the function g arises. More precisely, our right-hand side writes

Ω F (x)ϕ(x) dx, with F (x) = f (x) + |Γ| |Y | M Γ (g).
The extra term is, in fact, just a real constant and this allows us to prove the uniqueness of the solution of problem (4.3) exactly as in [START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF], since the presence of this constant term does not change the linearity nor the continuity of its right-hand side. Thus, due to the uniqueness of (u, u 1 , ū2 ) ∈ V, all the above convergences hold true for the whole sequence, which ends the proof of the theorem.

Corollary 4.2

The function u ∈ H 1 0 (Ω) defined by (4.1) is the unique solution of the following homogenized equation:

(4.7) -div (A hom ∇u) = f + |Γ| |Y | M Γ (g) in Ω,
where A hom is the homogenized matrix whose entries are given, for i, j = 1, . . . , N , by

(4.8) A hom ij = 1 |Y | Y 1 a ij - N k=1 a ik ∂χ j 1 ∂y k dy + 1 |Y | Y 2 a ij - N k=1 a ik ∂χ j 2 ∂y k dy, in terms of χ j 1 ∈ H 1 per (Y 1
) and χ j 2 ∈ H 1 (Y 2 ), j = 1, . . . , N , the weak solutions of the following cell problems:

(4.9)                  -div y (A(y)(∇ y χ j 1 -e j )) = 0, y ∈ Y 1 , -div y (A(y)(∇ y χ j 2 -e j )) = 0, y ∈ Y 2 , (A(y)∇ y χ j 1 ) • n = (A(y)∇ y χ j 2 ) • n, y ∈ Γ, (A(y)(∇ y χ j 1 -e j )) • n = h(y)(χ j 1 -χ j 2 ), y ∈ Γ. M Γ (χ j 1 ) = 0.
where n denotes the unit outward normal to Y 2 .

Proof. The proof of this result is classical. Indeed, by choosing successively ϕ = 0 and

Φ 1 = Φ 2 = 0 in (4.
3), we obtain:

1 |Y | Ω×Y 1 A(y)(∇u + ∇ y u 1 )∇ y Φ 1 dx dy+ 1 |Y | Ω×Y 2 A(y)(∇u + ∇ y ū2 )∇ y Φ 2 dx dy+ (4.10) 1 |Y | Ω×Γ h(y)( u 1 -ū2 )(Φ 1 -Φ 2 ) dx dσ y = 0 and 1 |Y | Ω×Y 1 A(y)(∇u + ∇ y u 1 )∇ϕ dx dy + 1 |Y | Ω×Y 2 A(y)(∇u + ∇ y ū2 )∇ϕ dx dy = (4.11) Ω f (x)ϕ(x) dx + |Γ| |Y | M Γ (g) Ω ϕ(x) dx.
We search now u 1 and ū2 in the usual form (4.12)

u 1 (x, y) = - N j=1 ∂u ∂x j (x)χ j 1 (y), (4.13) ū2 (x, y) = - N j=1 ∂u ∂x j (x)χ j 2 (y).
Standard computations lead to the homogenized limit problem, in which the term containing g gives a contribution only to the right-hand side, without affecting the cell problems and the homogenized matrix.

Remark 4.3 The right scaling ε in front of the function g ε prescribed at the interface Γ ε leads in the limit to the presence of a new source term distributed all over the domain Ω.

Remark 4.4 It is possible to study our initial problem (2.1) also for a nonzero function g with mean-value M Γ (g) equal to zero. But, in this situation, there is no contribution of g in the right-hand side of the homogenized equation and, thus, the limit problem is the same as in the case with no g at all in the microscopic problem.

Remark 4.5 We remark that the homogenized matrix A hom depends on the function h. So, the effect of the two jumps involved in our microscopic problem is recovered in the homogenized problem, in the right-hand side and also in the left-hand side (through the homogenized coefficients).

Remark 4.6 All the previous results still hold true for the case in which the set Y 2 is not connected, but consists on a finite number of connected components.

Case 2: G ε (x) = g x ε , if M Γ (g) = 0.
Theorem 4.7 The unique solution u ε = (u ε 1 , u ε 2 ) of the variational problem (2.6) converges, in the sense of (4.1), to the unique solution (u, u 1 , ū2 ) ∈ V of the following unfolded limit problem:

1 |Y | Ω×Y 1 A(y)(∇u+∇ y u 1 )(∇ϕ+∇ y Φ 1 ) dx dy + 1 |Y | Ω×Y 2 A(y)(∇u+∇ y ū2 )(∇ϕ+∇ y Φ 2 ) dx dy+ (4.14) 1 |Y | Ω×Γ h(y)( u 1 -ū2 )(Φ 1 -Φ 2 ) dx dσ y = Ω f (x)ϕ(x) dx+ 1 |Y | Ω×Γ g(y)Φ 1 (x, y) dx dσ y , for all ϕ ∈ H 1 0 (Ω), Φ 1 ∈ L 2 (Ω, H 1 per (Y 1 )), Φ 2 ∈ L 2 (Ω, H 1 (Y 2 )).
Proof. In order to get the problem (4.14), we pass to the limit in the unfolded form of the variational formulation (2.6) with the same test functions (4.4) as in Theorem 4.1, which verify (4.5) and (4.6). The only difference is that now the limit of the term involving the function G ε is different. More precisely, we have:

1 ε 1 |Y | Ω×Γ T ε b (G ε )T ε b (v 1 ) dσ x = 1 ε 1 |Y | Ω×Γ T ε b g x ε T ε b ϕ(x) + εω 1 (x)ψ 1 x ε dσ x = 1 ε 1 |Y | Ω×Γ g(y)T ε b (ϕ)(x, y) dx dσ y + 1 |Y | Ω×Γ g(y)T ε b (ω 1 )(x, y)T ε b (ψ 1 )(x, y) dx dσ y → 1 |Y | Ω×Γ g(y)ω 1 (x)ψ 1 (y) dx dσ y ,
since M Γ (g) = 0. By using the density of D(Ω) ⊗ H 1 per (Y 1 ) in the space L 2 (Ω, H 1 per (Y 1 )) and of D(Ω) ⊗ H 1 (Y 2 ) in the space L 2 (Ω, H 1 (Y 2 )), we get the unfolded limit problem (4.14).

Due to the uniqueness of (u, u 1 , ū2 ) ∈ V, which is proven by the Lax-Milgram theorem, all the above convergences hold true for the whole sequence, which ends the proof of the theorem. This cannot be the case here, since this term involves now explicitly both variables x and y.

We have to understand the contribution in the homogenized problem of this nonstandard term generated by the discontinuity of the flux in the initial problem. Actually, it will be seen in the next theorem that, apart from the classical solutions χ j 1 and χ j 2 of the cell problems (4.9), we are led to introduce in where χ j 1 and χ j 2 are defined by (4.9) and the function (η 1 , η 2 ) is the unique solution of the cell problem

                 -div y (A(y)∇η 1 ) = 0 in Y 1 , -div y (A(y)∇η 2 ) = 0 in Y 2 , A(y)∇η 1 • n = h(y)(η 1 -η 2 ) -g(y) on Γ, A(y)∇η 2 • n = h(y)(η 1 -η 2 ) on Γ, M Γ (η 1 ) = 0.
The function u ∈ H 1 0 (Ω) is the unique solution of the following homogenized equation

(4.15) -div (A hom ∇u) = f in Ω,
where A hom is the homogenized matrix whose entries are given in (4.8).

Proof. By choosing ϕ = 0 in (4.14), we obtain:

1 |Y | Ω×Y 1 A(y)(∇u + ∇ y u 1 )∇ y Φ 1 dx dy+ 1 |Y | Ω×Y 2 A(y)(∇u + ∇ y ū2 )∇ y Φ 2 dx dy+ (4.16) 1 |Y | Ω×Γ h(y)( u 1 -ū2 )(Φ 1 -Φ 2 ) dx dσ y = 1 |Y | Ω×Γ g(y)Φ 1 (x, y) dx dσ y .
We point out that the presence of the term

+ ∇ y ū2 ) • n = h(y)( u 1 -ū2 ) a.e. on Ω × Γ, (4.20)
A(y)(∇u

+ ∇ y u 1 ) • n = h(y)( u 1 -ū2 ) -g(y)
a.e. on Ω × Γ.

We point out here that we also have a discontinuity type condition: In the classical case with jump in the solution and with continuity of the flux, the use of the standard correctors χ j 1 and χ j 2 defined in (4.9) is enough in order to express the functions u 1 and ū2 in terms of the function ∇u. The presence of the function g in relations (4.20) and (4.21) suggests us to search u 1 and ū2 in the following nonstandard form:

(4.22) u 1 (x, y) = - N j=1 ∂u ∂x j (x)χ j 1 (y) + η 1 (y), (4.23) ū2 (x, y) = - N j=1 ∂u ∂x j (x)χ j 2 (y) + η 2 (y),
where χ j 1 and χ j 2 are defined by (4.9) and the functions η 1 , η 2 have to be found. To this end, we introduce (4.22) and (4.23) in (4.17)-(4.21) and we obtain:

(4.24)                  -div y (A(y)∇η 1 ) = 0 in Y 1 , -div y (A(y)∇η 2 ) = 0 in Y 2 , A(y)∇η 1 • n = h(y)(η 1 -η 2 ) -g(y) on Γ, A(y)∇η 2 • n = h(y)(η 1 -η 2 ) on Γ, M Γ (η 1 ) = 0.
We obviously have (4.25)

A(y)∇η 1 • n -A(y)∇η 2 • n = -g(y)
and then we notice that the new local problem (4.24) is an imperfect transmission problem, involving both the discontinuities in the solution and in the flux, given in terms of h and g, respectively.

By the Lax-Milgram theorem, the problem (4.24) has a unique solution in the space

H = W per (Y 1 ) × H 1 (Y 2 ),
endowed with the scalar product

(η, ζ) H = (∇η 1 , ∇ζ 1 ) L 2 (Y 1 ) + (∇η 2 , ∇ζ 2 ) L 2 (Y 2 ) + (η 1 -η 2 , ζ 1 -ζ 2 ) L 2 (Γ)
.

By choosing now Φ 1 = Φ 2 = 0 in (4.14), we get:

1 |Y | Ω×Y 1 A(y)(∇u + ∇ y u 1 )∇ϕ dx dy + 1 |Y | Ω×Y 2 A(y)(∇u + ∇ y ū2 )∇ϕ dx dy = (4.26) Ω f (x)ϕ(x) dx.
Integrating by parts with respect to x, we obtain, for x ∈ Ω,

-div x 1 |Y | Y 1 A(y)(∇u + ∇ y u 1 ) dy + 1 |Y | Y 2 A(y)(∇u + ∇ y ū2 ) dy = f (x).
By using here the particular form (4.22) and (4.23) of the functions u 1 and ū2 and the definition (4.8) of the matrix A hom , we get:

(4.27) -div x (A hom ∇u) = f + div x 1 |Y | Y 1 A(y)∇η 1 (y)dy + 1 |Y | Y 2 A(y)∇η 2 (y)dy in Ω,
which leads immediately to the homogenized problem (4.15). We notice that this problem does not involve the function g, because the second term of the right-hand side in (4.27) actually vanishes.

Remark 4.10 All the above results can be extended to the case in which A ε is a sequence of matrices in M(α, β, Ω) such that

(4.28) T ε α (A ε ) → A strongly in L 1 (Ω × Y ),
for some matrix A = A(x, y) in M(α, β, Ω × Y ). The heterogeneity of the medium modeled by such a matrix induces different effects in our limit problems (4.3) and (4.14) respectively. In both cases, since the correctors χ j α depend also on x, the new homogenized matrix A hom

x is no longer constant, but it depends on x. A more interesting effect arises in the second case. As we have seen in Theorem 4.9, if the matrix A depends only on the variable y, the functions η α are independent of x and there is no contribution of the term containing g in the decoupled form of the limit problem. So, the limit equation is the same as that corresponding to the case with no jump on the flux in the microscopic problem. Now, the dependence of A on x prevails this phenomenon to occur, and, hence, the function g brings an explicit contribution in the homogenized problem, which becomes

-div x (A hom x ∇u) = f + div x 1 |Y | Y 1 A(x, y)∇η 1 (x, y)dy + 1 |Y | Y 2 A(x, y)∇η 2 (x, y)dy in Ω.
A similar effect was observed in the homogenization of the Neumann problem in perforated domains (see [START_REF] Cioranescu | The periodic unfolding method in domains with holes[END_REF]).

Corrector results

Our goal in this section is to state corrector results for the problem (2.1). To this end, let us start by recalling the definition of the adjoints of the unfolding operators T ε α , for α = 1, 2 (see [START_REF] Cioranescu | The periodic unfolding method in domains with holes[END_REF] and [START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF]). We recall now some useful properties of the averaging operators U ε α (see [START_REF] Cioranescu | The periodic unfolding method in domains with holes[END_REF] and [START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF]).

Proposition 5.2 For p ∈ [1, ∞) and α = 1, 2, the operators U ε α are linear and continuous from L p (Ω × Y α ) to L p (Ω ε α ) and (i) U ε α (φ) -φ L p (Ω ε α ) → 0 for every φ ∈ L p (Ω); (ii) if ϕ ε ∈ L p (Ω ε α ), then the following statements are equivalent:

• T ε α (ϕ ε ) → ϕ strongly in L p (Ω × Y α ) and

Λ ε α |ϕ ε | p dx → 0;
• ϕ ε -U ε α ( ϕ) L p (Ω ε α ) → 0. We are now in the position to state the convergence of the energy and corrector results for the solution u ε = (u ε 1 , u ε 2 ) of problem (2.1). We shall consider separately the above mentioned representative cases corresponding to the hypotheses imposed on the jump function G ε .

Case 1. In this case, since the term containing the flux jump gives a contribution only to the right-hand side of the limit equation, without affecting the solutions of the cell problems and the homogenized matrix, we are led to standard corrector results, whose proof follows exactly the same steps as in [START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF], Theorem 4.4. Moreover, the following corrector result holds true:

(5.5)

∇u ε 1 -∇u + N j=1 U ε 1 ∂u ∂x j U ε 1 ∇ y χ j 1 L 2 (Ω ε 1 )
-→ 0

Conclusions

Via the periodic unfolding method, the effective thermal transfer in a periodic composite material formed by two constituents, separated by an imperfect interface where both the temperature and the flux exhibit jumps, was analyzed. Depending on the hypotheses imposed on the jump of the flux, two different homogenized problems were obtained. The influence of the jumps of the flux and of the temperature field is captured at the limit in various ways: in the homogenized coefficients, in the right-hand side of the homogenized problem, and in the correctors.

Remark 4 . 8

 48 Let us point out that the term 1 |Y | Ω×Γ g(y)Φ 1 (x, y) dx dσ y in (4.14) represents the main difference with respect to the unfolded equation (4.3), where the term involving g is a nonzero constant, recovered explicitly in the right-hand side of the homogenized equation (4.7).

  (4.22)-(4.23) two additional scalar terms η 1 and η 2 , verifying a new imperfect transmission cell problem (see (4.24)).

Theorem 4 . 9

 49 The solution (u, u 1 , ū2 ) ∈ V of (4.14) is such that:u 1 (x, y) =χ j 2 (y) + η 2 (y),

(4. 21 )

 21 A(y)(∇u + ∇ y u 1 ) • n = A(y)(∇u + ∇ y ū2 ) • n -g(y) a.e. on Ω × Γ.

Definition 5 . 1

 51 For p ∈ [1, ∞], the averaging operators U ε α : L p (Ω × Y α ) → L p (Ω ε α ), with α = 1, 2, are defined by U ε α (φ)(x) = for a.e. x ∈ Ω ε α , 0 for a.e. x ∈ Λ ε α .It is not difficult to see that these averaging operators are almost left-inverses of the corresponding unfolding operators T ε α , i.e., for any ϕ ∈ L p (Ω ε α ), one has for a.e. x ∈ Ω ε α , 0 for a.e. x ∈ Λ ε α .

Theorem 5 . 3 Ω ε 1 A ε ∇u ε 1 ∇u ε 1 dx + Ω ε 2 A ε ∇u ε 2 ∇u ε 2 dx = 1 |Y | Ω×Y 1 AT ε 1 (∇ u ε 1 )T ε 2 (∇ u ε 2 )

 5312111122 Under the assumptions of Theorem 4.1, if uε = (u ε 1 , u ε 2 ) is the unique solution of problem (2.1), then lim ε→0 (y)(∇u + ∇ y u 1 )(∇u + ∇ y u 1 ) dx dy+ (5.1) 1 |Y | Ω×Y 2 A(y)(∇u + ∇ y u 2 )(∇u + ∇ y u 2 ) dx dy ≤ Ω A hom ∇u∇u dx, → ∇ u + ∇ y u 1 strongly in L 2 (Ω × Y 1 ) → ∇ y u 2 strongly in L 2 (Ω × Y 2 ).

  N the reference cell in R N . We assume that Y 1 and Y 2 are two non-empty disjoint connected open subsets of Y such that Y 2 ⊂ Y and Y = Y 1 ∪ Y 2 . We also suppose that Γ = ∂Y 2

	2 Setting of the problem
	Let Ω be a bounded open set in R N (N	2), with a Lipschitz continuous boundary ∂Ω and
	Y = (0, 1)

Acknowledgments. Both authors acknowledge the support from the Laboratoire Éuropéen Associé CNRS Franco-Roumain "Math-Mode".

and

(5.6)

-→ 0.

Case 2. In this case, the result is similar to the one stated in Theorem 5.3, but now the functions η α , solution of the local problem (4.24), appear in the correctors of the solution

Theorem 5.4 Under the assumptions of Theorem 4.7, if

is the unique solution of problem (2.1), then (5.1)-(5.4) hold true. Moreover, we have the following corrector result:

-→ 0 and

(5.8)

Proof. The proof of relations (5.1)-(5.4) follows the classical steps. The main difference with respect to the previous case is that, in order to get relation (5.1), we need the following convergence result:

This result is a direct consequence of Proposition 3.5 in [START_REF] Cioranescu | The periodic unfolding method in domains with holes[END_REF], the fact that M Γ (g) is zero and that the function u belongs to the space H 1 0 (Ω). We now notice that

and

. Then, we derive (5.7) and (5.8) by using convergences (5.3), (5.4), Proposition 5.2 and relations (4.22), (4.23).

Remark 5.5 Let us notice that similar corrector results can be stated in the case in which the matrix A depends both on x and y, as in Remark 4.10.


Renata Bunoiu
email: renata.bunoiu@univ-lorraine.fr

Claudia Timofte
email: clau-dia.timofte@g.unibuc.ro

On the homogenization of a two-conductivity problem with flux jump

Keywords: homogenization, imperfect interface, the periodic unfolding method AMS subject classification: 35B27, 80M35, 80M40

In this paper, we study the homogenization of a thermal diffusion problem in a highly heterogeneous medium formed by two constituents. The main characteristics of the medium are the discontinuity of the thermal conductivity over the domain as we go from one constituent to another and the presence of an imperfect interface between the two constituents, where both the temperature and the flux exhibit jumps. The limit problem, obtained via the periodic unfolding method, captures the influence of the jumps in the limit temperature field, in an additional source term, and in the correctors, as well.

Introduction

Our goal in this paper is to analyze the effective thermal transfer in a periodic composite material formed by two constituents occupying a domain Ω in R N (N ≥ 2), divided in two open subdomains, denoted by Ω ε 1 and Ω ε 2 , and separated by an imperfect interface Γ ε . We assume that the phase Ω ε 1 is connected and reaches the external fixed boundary ∂Ω and that Ω ε 2 is disconnected, being the union of domains of size ε, periodically distributed in Ω with period of order ε, where ε is a positive real number less than one. Nevertheless, if N ≥ 3, our results still hold true if the domain Ω ε 2 is connected, too. The order of magnitude of the thermal conductivity of the material occupying the domain Ω ε 2 is of order ε 2 , while the conductivity of the material occupying the domain Ω ε 1 is supposed to be of order one. Our problem presents various sources of singularities: the geometric one related to the interspersed periodic distribution of the components, the material one related to the conductivities and the ones generated by the presence of an imperfect interface between the two materials. All these singularities are described in terms of ε.

More precisely, we study the asymptotic behavior, as the small parameter ε tends to zero, of the solution u ε = (u ε 1 , u ε 2 ) of the following problem:

                   -div (A ε ∇u ε 1 ) = f in Ω ε 1 , -div (ε 2 A ε ∇u ε 2 ) = f in Ω ε 2 , A ε ∇u ε 1 • n ε = εh ε (u ε 1 -u ε 2 ) -G ε on Γ ε , ε 2 A ε ∇u ε 2 • n ε = εh ε (u ε 1 -u ε 2 ) on Γ ε , u ε 1 = 0 on ∂Ω.
The case G ε = 0, which corresponds to a continuous flux, proportional to the jump of the temperature field across the imperfect interface, has attracted, in the last two decades, the interest of a broad category of researchers. In the pioneering work [START_REF] Auriault | Macroscopic modelling of heat transfer in composites with interfacial thermal barrier[END_REF], using the asymptotic expansion method, the authors study the homogenization of a thermal problem in a two-component composite with interfacial barrier in the particular case in which the conductivities of the two components are both of order one. For this problem, the convergence results were rigorously justified later by using various mathematical methods: the energy method in [START_REF] Donato | Homogenization of two heat conductors with an interfacial contact resistance[END_REF] and [START_REF] Monsurrò | Homogenization of a two-component composite with interfacial thermal barrier[END_REF], the two-scale convergence method in [START_REF] Ene | Model of diffusion in partially fissured media[END_REF] and the unfolding method for periodic homogenization in [START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF], [START_REF] Timofte | Multiscale analysis of diffusion processes in composite media[END_REF], [START_REF] Timofte | Multiscale modeling of heat transfer in composite materials[END_REF] and [START_REF] Le Nguyen | Homogenization of heat transfer process in composite materials[END_REF], to quote just a few of them. Also, for problems involving jumps in the solution in other contexts, such as heat transfer in polycrystals with interfacial resistance, linear elasticity problems or problems modeling the electrical conduction in biological tissues, see [START_REF] Amar | Evolution and memory effects in the homogenization limit for electrical conduction in biological tissues[END_REF], [START_REF] Amar | A hierarchy of models for the electrical conduction in biological tissues via two-scale convergence: The nonlinear case[END_REF], [START_REF] Ene | Microstructure models for composites with imperfect interface via the periodic unfolding method[END_REF], [START_REF] Ene | Homogenization of a thermoelasticity model for a composite with imperfect interface[END_REF], [START_REF] Ene | Homogenization results for a dynamic coupled thermoelasticity problem[END_REF], [START_REF] Hummel | Homogenization for heat transfer in polycrystals with interfacial resistances[END_REF] and [START_REF] Timofte | Multiscale analysis in nonlinear thermal diffusion problems in composite structures[END_REF]. The case corresponding to the scaling of the conductivities considered in this paper is addressed, among others, in [START_REF] Arbogast | Derivation of the double porosity model of single phase flow via homogenization theory[END_REF], [START_REF] Polisevski | Diffusion in an intermediate model of fractured porous media[END_REF], [START_REF] Polisevski | Heat conduction through a first-order jump interface[END_REF], [START_REF] Polisevski | Homogenization cases of heat transfer in structures with interfacial barriers[END_REF], [START_REF] Donato | Homogenization of an elastic double-porosity medium with imperfect interface via the periodic unfolding method[END_REF], [START_REF] Gruais | Heat transfer models for two-component media with interfacial jump[END_REF], [START_REF] Ainouz | Homogenized double porosity models for poro-elastic media with interfacial flow barrier[END_REF], [START_REF] Ainouz | Homogenization of a dual-permeability problem in two-component media with imperfect contact[END_REF].

After passing to the limit with respect to the small parameter ε, a regularised model of diffusion is obtained, which in fact is a special case of the double-porosity model, introduced in [START_REF] Rubinstein | On the problem of the process of propagation of heat in heterogeneous media[END_REF] in the frame of the heat transfer and in [START_REF] Barenblatt | Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks (strata)[END_REF] in the context of the flow in porous media. For a review of such models in various types of fissured porous media, see, for instance, [START_REF] Showalter | Microstructure Models of Porous Media, Homogenization and Porous Media[END_REF] and the references therein.

In this paper, we consider the case G ε = 0, which corresponds to a discontinous flux as well.

We study here two representative cases for the jump function G ε , stated explicitly in Section 2, relations (2.2) and (2.3), which both lead to different modified reguralized models of diffusion.

More precisely, in the first case, a new global source term, macroscopically distributed over the entire equivalent domain, appears in the right-hand side of the homogenized equation (3.8). In the second case, the novelty brought by the presence of the flux jump is the emergence of the new non homogeneous Neumann cell problem (3.25) and the presence of its solution in the corrector (3.24). We notice that this jump plays no role in the homogenized problem (3.23). Nevertheless, in Remark 3.11 we mention a case in which the homogenized problem depends on this jump, too. This last result is to be compared with the Neumann problem in perforated domains (see [START_REF] Cioranescu | The periodic unfolding method in domains with holes[END_REF] and [START_REF] Chourabi | Homogenization and correctors of a class of elliptic problems in perforated domains[END_REF]), where the same cases of G ε are considered on the boundary of the holes and a similar phenomenon occurs. More recently, this type of functions G ε is encountered in [START_REF] Bunoiu | Homogenization of a Thermal Problem with Flux Jump, Networks and Heterogeneous Media[END_REF] for the study of a thermal problem with flux jump, involving conductivities of order one and a scaling of the jump in the temperature field of order ε -1 . We also point out here the effect of the jump of the solution, which is recovered in the corrector (3.10) and in the weak limit (3.15), via the solution of the local Robin problem (3.13). This phenomenon was already noticed in the case in which G ε = 0. For transmission problems involving jump in the flux in other contexts, such as linear elasticity, theory of semiconductors, the study of photovoltaic systems, combustion theory or heat transfer problems, see [START_REF] Auriault | Homogenization of Coupled Phenomena in Heterogenous Media[END_REF], [START_REF] Benveniste | Imperfect soft and stiff interfaces in two-dimensional elasticity[END_REF], [START_REF] Brinkman | A Drift-Diffusion-Reaction Model for Excitonic Photovoltaic Bilayers: Asymptotic Analysis and a 2-D HDG Finite Element Scheme[END_REF], [START_REF] Fellner | A discontinuous Poisson-Boltzmann equation with interfacial transfer: homogenisation and residual error estimate[END_REF], [START_REF] Hashin | Thin interphase-imperfect interface in elasticity with application to coated fiber composites[END_REF], [START_REF] Ijioma | Pattern formation in reverse smouldering combustion: a homogenization approach[END_REF].

Let us notice that if in the microscopic problem the temperature and the flux are continuous across Γ ε and if moreover the thermal conductivities of the two materials are both of order one, we have a standard transmission problem, and, then, the limit process leads to a single-diffusion equation. However, if we assume that the thermal conductivities of the two materials are both of order one, but we keep our jump conditions at the interface, then the limit problem is a system of two coupled equations. The case G ε = 0 is studied in [START_REF] Ene | Model of diffusion in partially fissured media[END_REF] and the limit process leads to the celebrated Barenblatt system, introduced in [START_REF] Barenblatt | Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks (strata)[END_REF]. The case G ε = 0, which leads to a modified Barenblatt model, is addressed in [START_REF] Bunoiu | On the Homogenization of a Diffusion Problem with Flux Jump[END_REF].

The rest of the paper is organized as follows: in Section 2, we introduce the microscopic problem and we fix the notation. In Section 3, we state and prove the main homogenization results of this paper. Corrector results are given, too. We end our paper with a few concluding remarks, an appendix in which we review the definition and the basic properties of the unfolding operators and their adjoints, and some references.

Setting of the problem

Let Ω be a bounded open set in R N (N 2), with a Lipschitz continuous boundary ∂Ω and let Y = (0, 1) N be the reference cell in R N . We suppose that Y 1 and Y 2 are two non-empty disjoint connected open subsets of Y such that Y 2 ⊂ Y and Y = Y 1 ∪ Y 2 . We also assume that Γ = ∂Y 2 is Lipschitz continuous and that Y 2 is connected.

Throughout the paper, the small parameter ε takes values in a positive real sequence tending to zero and C is a positive constant independent of ε, whose value can change from line to line.

For each k ∈ Z N , we denote Y k = k + Y and Y k α = k + Y α , for α = 1, 2. We also define for each ε, Z ε = k ∈ Z N : εY k 2 ⊂ Ω and we set Ω ε 2 = k∈Zε εY k 2 and Ω ε 1 = Ω \ Ω ε 2 . The boundary of Ω ε
2 is denoted by Γ ε and n ε is the unit outward normal to Ω ε 2 . Our goal in this paper is to analyze the asymptotic behavior, as ε → 0, of the solution

u ε = (u ε 1 , u ε 2 ) of the following problem:                    -div (A ε ∇u ε 1 ) = f in Ω ε 1 , -div (ε 2 A ε ∇u ε 2 ) = f in Ω ε 2 , A ε ∇u ε 1 • n ε = εh ε (u ε 1 -u ε 2 ) -G ε on Γ ε , ε 2 A ε ∇u ε 2 • n ε = εh ε (u ε 1 -u ε 2 ) on Γ ε , u ε 1 = 0 on ∂Ω.
(2.1)

Remark 2.1 We remark that the flux of the solution is discontinuous across Γ ε . Indeed, we have

A ε ∇u ε 1 • n ε -ε 2 A ε ∇u ε 2 • n ε = -G ε .
The function f ∈ L 2 (Ω) is given. Let g be a Y -periodic function that belongs to L 2 (Γ). We define

g ε (x) = g x ε a.e. on Γ ε .
For the given function G ε in (2.1), we consider the following two relevant situations (see, also, [START_REF] Bunoiu | Homogenization of a Thermal Problem with Flux Jump, Networks and Heterogeneous Media[END_REF], [START_REF] Cioranescu | The periodic unfolding method in domains with holes[END_REF] and [START_REF] Chourabi | Homogenization and correctors of a class of elliptic problems in perforated domains[END_REF]):

Case 1 : G ε (x) = εg x ε , if M Γ (g) = 0, (2.2)
Case 2 : G ε (x) = g x ε , if M Γ (g) = 0. (2.3)
Here, M Γ (g) = 1 |Γ| Γ g(y) dy denotes the mean value of the function g on Γ.

Moreover, we make the following assumptions on the data:

(H1) h is a Y -periodic function such that h ∈ L ∞ (Γ) and there exists h 0 ∈ R with 0 < h 0 < h(y) a.e. on Γ. We set

h ε (x) = h x ε a.e. on Γ ε . (H2) For λ, µ ∈ R, with 0 < λ ≤ µ, let M(λ, µ, Y ) be the set of all the matrices A ∈ (L ∞ (Y )) N ×N such that for any ξ ∈ R N , λ|ξ| 2 ≤ (A(y)ξ, ξ) ≤ µ|ξ| 2 , almost everywhere in Y .
For a Y -periodic symmetric matrix A ∈ M(λ, µ, Y ), we set

A ε (x) = A x ε a.e. in Ω.
In order to write the variational formulation of problem (2.1), we introduce, for every positive ε < 1, the Hilbert space

H ε = V ε × H 1 (Ω ε 2 ). The space V ε = v ∈ H 1 (Ω ε 1 ), v = 0 on ∂Ω is endowed with the norm v V ε = ∇v L 2 (Ω ε 1 )
, for any v ∈ V ε , and the space H 1 (Ω ε

2 ) is equipped with the standard norm. On the space H ε , we consider the scalar product

(u, v) H ε = Ω ε 1 ∇u 1 ∇v 1 dx + Ω ε 2 ε 2 ∇u 2 ∇v 2 dx + ε Γ ε (u 1 -u 2 )(v 1 -v 2 ) dσ x (2.4)
where u = (u 1 , u 2 ) and v = (v 1 , v 2 ) belong to H ε . The norm generated by the scalar product (2.4) is given by

v 2 H ε = ∇v 1 2 L 2 (Ω ε 1 ) + ε 2 ∇v 2 2 L 2 (Ω ε 2 ) + ε v 1 -v 2 2 L 2 (Γ ε ) . (2.5)
The variational formulation of problem (2.1) is the following one: find

u ε ∈ H ε such that a(u ε , v) = l(v), ∀v ∈ H ε , (2.6)
where the bilinear form a :

H ε × H ε → R and the linear form l : H ε → R are given by a(u, v) = Ω ε 1 A ε ∇u 1 ∇v 1 dx + ε 2 Ω ε 2 A ε ∇u 2 ∇v 2 dx + ε Γ ε h ε (u 1 -u 2 )(v 1 -v 2 ) dσ x and l(v) = Ω ε 1 f v 1 dx + Ω ε 2 f v 2 dx + Γ ε G ε v 1 dσ x ,
respectively.

We recall in the next lemma a result from [START_REF] Donato | Homogenization of an elastic double-porosity medium with imperfect interface via the periodic unfolding method[END_REF], which is a key argument allowing us to prove an existence and uniqueness result and a priori estimates for the solution of the variational problem (2.6). In the sequel, unless otherwise mentioned, by C we denote a positive constant which is independent of ε and whose value can change from line to line.

Lemma 2.2 For every v given in the space H ε , the following inequalities hold true:

v 1 L 2 (Ω ε 1 ) ≤ C v H ε and v 2 L 2 (Ω ε 2 ) ≤ C v H ε .
Proof. The first inequality is a direct consequence of the definition (2.5), together with the Poincaré inequality applied to functions from the space V ε , namely

v 1 L 2 (Ω ε 1 ) ≤ C ∇v 1 L 2 (Ω ε 1 ) . (2.7)
In order to prove the second inequality, we need the following inequalities (see [START_REF] Donato | Homogenization of an elastic double-porosity medium with imperfect interface via the periodic unfolding method[END_REF]):

v 2 L 2 (Ω ε 2 ) ≤ C(ε ∇v 2 L 2 (Ω ε 2 ) + ε 1 2 v 2 L 2 (Γ ε ) ) (2.8)
and

ε 1 2 v 1 L 2 (Γ ε ) ≤ C(ε ∇v 1 L 2 (Ω ε 1 ) + v 1 L 2 (Ω ε 1 )
).

(2.9)

The triangular inequality applied in (2.8), together with (2.9) and (2.7), imply

v 2 L 2 (Ω ε 2 ) ≤ C(ε ∇v 2 L 2 (Ω ε 2 ) + ε 1 2 v 2 -v 1 L 2 (Γ ε ) + ε 1 2 v 1 L 2 (Γ ε ) ) ≤ C(ε ∇v 1 L 2 (Ω ε 1 ) + ε ∇v 2 L 2 (Ω ε 2 ) + ε 1 2 v 1 -v 2 L 2 (Γ ε ) + v 1 L 2 (Ω ε 1 ) ) ≤ C( ∇v 1 L 2 (Ω ε 1 ) + ε ∇v 2 L 2 (Ω ε 2 ) + ε 1 2 v 1 -v 2 L 2 (Γ ε ) ),
and the second inequality then follows, by using the definition (2.5).

Theorem 2.3 For any ε ∈ (0, 1), the variational problem (2.6) has a unique solution u ε ∈ H ε .

Moreover, there exists a constant C > 0, independent of ε, such that

u ε 1 L 2 (Ω ε 1 ) ≤ C, u ε 2 L 2 (Ω ε 2 ) ≤ C (2.10)
and

∇u ε 1 L 2 (Ω ε 1 ) ≤ C, ε ∇u ε 2 L 2 (Ω ε 2 ) ≤ C, ε 1/2 u ε 1 -u ε 2 L 2 (Γ ε ) ≤ C. (2.11)
Proof. In order to prove the existence and the uniqueness of the solution for problem (2.6), we apply the Lax-Milgram theorem for the space H ε endowed with the norm (2.5). Due to the hypotheses (H1) and (H2), we easily get that the bilinear form a is coercive and continuous.

Indeed, we have

a(v, v) ≥ C v 2 H ε , ∀v ∈ H ε , and
a(u, v) ≤ C u H ε v H ε , ∀u, v ∈ H ε .
Let us prove now that the linear form l is continuous, i.e.

l(v) ≤ C v H ε , ∀v ∈ H ε .
One obviously has

|l(v)| ≤ f L 2 (Ω ε 1 ) v 1 L 2 (Ω ε 1 ) + f L 2 (Ω ε 1 ) v 2 L 2 (Ω ε 2 ) + Γ ε G ε (x)v 1 (x) dσ x .
(2.12)

According to Proposition 3.8 in [START_REF] Chourabi | Homogenization and correctors of a class of elliptic problems in perforated domains[END_REF], we obtain the estimate of the last term in (2.12) as follows:

(i) if G ε satisfies (2.2), then Γ ε G ε (x)v 1 (x) dσ x = Γ ε εg x ε v 1 (x) dσ x ≤ ε C ε (|M Γ (g)| + ε) ∇v 1 L 2 (Ω ε 1 ) ≤ C ∇v 1 L 2 (Ω ε 1 ) ; (ii) if G ε satisfies (2.3), then Γ ε G ε (x)v 1 (x) dσ x = Γ ε g x ε v 1 (x) dσ x ≤ C ε (|M Γ (g)| + ε) ∇v 1 L 2 (Ω ε 1 ) ≤ C ∇v 1 L 2 (Ω ε 1 ) , since M Γ (g) = 0.
Coming back to (2.12), we obtain:

|l(v)| ≤ C( v 1 L 2 (Ω ε 1 ) + v 2 L 2 (Ω ε 2 ) + ∇v 1 L 2 (Ω ε 1 ) ).
By using Lemma 2.2 and the definition (2.5), we get the continuity of l. Thus, the Lax-Milgram theorem applies.

In order to obtain the a priori estimates (2.10) and (2.11), we take v = u ε in the variational formulation (2.6). By using the coerciveness of a and the continuity of l, we obtain

u ε H ε ≤ C,
which obviously imply (2.11). Estimates (2.10) are then obtained by applying Lemma 2.2.

Homogenization results

Our goal in this section is to pass to the limit, with ε → 0, in the variational formulation (2.6) of the problem (2.1). To this end, we make use of the periodic unfolding method and the general compactness results given in the appendix of this paper.

More precisely, using the a priori estimates (2.10)-(2.11) and the general compactness results from Proposition 5.6, it follows that there exist

u 1 ∈ H 1 0 (Ω), u 1 ∈ L 2 (Ω, H 1 per (Y 1 )), u 2 ∈ L 2 (Ω, H 1 (Y 2 )
) such that M Γ ( u 1 ) = 0 and up to a subsequence, for ε → 0, we get:

T ε 1 (u ε 1 ) → u 1 strongly in L 2 (Ω, H 1 (Y 1 )), T ε 1 (∇u ε 1 ) ∇ u 1 + ∇ y u 1 weakly in L 2 (Ω × Y 1 ), T ε 2 (u ε 2 ) u 2 weakly in L 2 (Ω, H 1 (Y 2 )), εT ε 2 (∇u ε 2 ) ∇ y u 2 weakly in L 2 (Ω × Y 2 ), u ε 1 |Y 1 |u 1 weakly in L 2 (Ω), u ε 2 Y 2 u 2 (x, y) dy weakly in L 2 (Ω), (3.1)
where the space H 1 per (Y 1 ) is defined by

H 1 per (Y 1 ) = {v ∈ H 1 (Y 1 ) | v is Y-periodic}.
Let W per (Y 1 ) = {v ∈ H 1 per (Y 1 ) | M Γ (v) = 0}. We introduce the space V = H 1 0 (Ω) × L 2 (Ω; W per (Y 1 )) × L 2 Ω, H 1 (Y 2 ) ,
equipped with the norm

V 2 V = ∇v + ∇ y v 1 2 L 2 (Ω×Y 1 ) + ∇ y v 2 2 L 2 (Ω×Y 2 ) + v -v 2 2 L 2 (Ω×Γ) , for all V = (v, v 1 , v 2 ) ∈ V.
In order to pass to the limit in (2.6), we need to distinguish between two cases, depending on the form of the function G ε .

Case 1:

G ε = εg x ε , if M Γ (g) = 0. Theorem 3.2 The unique solution u ε = (u ε 1 , u ε 2 )
of the variational problem (2.6) converges, in the sense of (3.1), to the unique solution (u 1 , u 1 , u 2 ) ∈ V of the following unfolded limit problem:

Ω×Y 1 A(y)(∇u 1 + ∇ y u 1 )(∇ϕ + ∇ y Φ 1 ) dx dy + Ω×Y 2 A(y)∇ y u 2 ∇ y Φ 2 dx dy+ Ω×Γ h(y)(u 1 -u 2 )(ϕ -Φ 2 ) dx dσ y = Ω×Y 1 f (x)ϕ(x) dx dy + Ω×Y 2 f (x)Φ 2 (x, y) dx dy+ |Γ|M Γ (g) Ω ϕ(x) dx, (3.2)
for all ϕ ∈ H 1 0 (Ω), Φ 1 ∈ L 2 (Ω, H 1 per (Y 1 )) and Φ 2 ∈ L 2 (Ω, H 1 (Y 2 )).
Proof. In order to get the limit problem (3.2), we unfold the variational formulation (2.6).

Then, by using Proposition 5.3 and Lemma 5.4, we obtain

lim ε→0 Ω×Y 1 T ε 1 (A ε )T ε 1 (∇u ε 1 )T ε 1 (∇v 1 ) dx + Ω×Y 2 T ε 2 (A ε )T ε 2 (ε∇u ε 2 )T ε 2 (ε∇v 2 ) dx+ Ω×Γ h(y)(T ε 1 (u ε 1 ) -T ε 2 (u ε 2 ))(T ε 1 (v 1 ) -T ε 2 (v 2 )) dσ x = lim ε→0 Ω×Y 1 T ε 1 (f )T ε 1 (v 1 ) dx + Ω×Y 2 T ε 2 (f )T ε 2 (v 2 ) dx + 1 ε Ω×Γ T ε b (G ε )T ε b (v 1 ) dσ x .
In this unfolded problem, we choose the admissible test functions

v 1 = ϕ(x) + εω 1 (x)ψ 1 x ε , v 2 = ω 2 (x)ψ 2 x ε , (3.3) with ϕ, ω 1 , ω 2 ∈ D(Ω), ψ 1 ∈ H 1 per (Y 1 ), ψ 2 ∈ H 1 (Y 2 )
. It is not difficult to see that we have the following convergences:

T ε 1 (v 1 ) → ϕ(x) strongly in L 2 (Ω × Y 1 ), (3.4)
T ε 1 (∇v 1 ) → ∇ϕ(x) + ∇ y Φ 1 strongly in L 2 (Ω × Y 1 ), (3.5)
T ε 2 (v 2 ) → Φ 2 (x, y) strongly in L 2 (Ω × Y 2 ), (3.6)
and

T ε 2 (ε∇v 2 ) → ∇ y Φ 2 strongly in L 2 (Ω × Y 2 ), (3.7)
where Φ 1 (x, y) = ω 1 (x)ψ 1 (y) and Φ 2 (x, y) = ω 2 (x)ψ 2 (y).

The passage to the limit with ε → 0 is standard, by using convergences (3.1) and (3.4)-(3.7).

The only term which needs more attention is the one involving the function G ε . For this term, we get:

1 ε Ω×Γ T ε b (G ε )T ε b (v 1 ) dσ x = Ω×Γ T ε b g x ε T ε b ϕ(x) + εω 1 (x)ψ 1 x ε dσ x = Ω×Γ g(y)T ε b (ϕ)(x, y) dx dσ y + ε Ω×Γ g(y)T ε b (ω 1 )(x, y)T ε b (ψ 1 )(x, y) dx dσ y → |Γ|M Γ (g) Ω ϕ(x) dx.
By the density of

D(Ω) ⊗ H 1 per (Y 1 ) in L 2 (Ω, H 1 per (Y 1 )) and of D(Ω) ⊗ H 1 (Y 2 ) in L 2 (Ω, H 1 (Y 2
)), we obtain (3.2).

We notice that our limit problem (3.2) is similar with the one obtained in [START_REF] Polisevski | Homogenization cases of heat transfer in structures with interfacial barriers[END_REF], the only difference being the right-hand side, in which an extra constant term involving the function g arises. Indeed, our right-hand side actually writes

Ω F (x)ϕ(x) dx + Ω×Y 2 f (x)Φ 2 (x, y) dx dy, with F (x) = |Y 1 |f (x) + |Γ|M Γ (g).
The existence and the uniqueness for the solution of problem (3.2) is a consequence of the Lax-Milgram theorem. Due to the uniqueness of (u 1 , u 1 , u 2 ) ∈ V, all the above convergences hold true for the whole sequence.

Theorem 3.3 The unique solution u ε = (u ε 1 , u ε 2 )
of the variational problem (2.6) converges, in the sense of (3.1), to (u 1 , u 1 , u 2 ) ∈ V, where u 1 is the unique solution of the homogenized problem -div

(A hom ∇u 1 (x)) = f (x) + |Γ|M Γ (g) in Ω, u 1 = 0 on ∂Ω (3.8)
and

u 1 (x, y) = - N j=1 ∂u 1 ∂x j (x)χ j 1 (y) in Ω × Y 1 , (3.9)
u 2 (x, y) = u 1 (x) + f (x)χ 2 (y) in Ω × Y 2 .
(3.10)

Here, A hom is the constant homogenized matrix whose entries are defined, for i, j = 1, . . . , N by

A hom ij = Y 1 a ij - N k=1 a ik ∂χ j 1 ∂y k dy. (3.11)
The vectorial function χ j 1 ∈ H 1 per (Y 1 ) ( j = 1, . . . , N ) and the scalar function χ 2 ∈ H 1 (Y 2 ) are the weak solutions of the following cell problems:

       -div y (A(y)(∇ y χ j 1 -e j )) = 0 in Y 1 , (A(y)(∇ y χ j 1 -e j )) • n = 0 on Γ, M Γ (χ j 1 ) = 0 (3.12) and -div y (A(y)∇ y χ 2 ) = 1 in Y 2 , A(y)∇ y χ 2 • n + hχ 2 = 0 on Γ, (3.13)
where n denotes the unit outward normal to Y 2 .

Moreover, the weak solution (u ε 1 , u ε 2 ) of problem (2.6) verifies: Proof. By choosing ϕ = 0 in the unfolded limit problem (3.2), we obtain:

u ε 1 |Y 1 |u 1 weakly in L 2 (Ω) (3.
Ω×Y 1 A(y)(∇u 1 + ∇ y u 1 )∇ y Φ 1 dx dy + Ω×Y 2 A(y)(∇ y u 2 )∇ y Φ 2 dx dy- Ω×Γ h(y)(u 1 -u 2 )Φ 2 dx dσ y = Ω×Y 2 f Φ 2 dx dy. (3.16)
Then, taking Φ 2 = 0 in (3.16), we get:

Ω×Y 1 A(y)(∇u 1 + ∇ y u 1 )∇ y Φ 1 dx dy = 0, which implies -div y (A(y)∇ y u 1 ) = div y (A(y)∇u 1 ) in Ω × Y 1 and A(y)(∇u 1 + ∇ y u 1 ) • n = 0 on Ω × Γ.
Classical results from the theory of homogenization then imply (3.9) and (3.12).

By choosing now Φ 1 = 0 in (3.16), we get:

Ω×Y 2 A(y)(∇ y u 2 )∇ y Φ 2 dx dy - Ω×Γ h(y)(u 1 -u 2 )Φ 2 dx dσ y = Ω×Y 2 f Φ 2 dx dy, which implies -div y (A(y)∇ y u 2 ) = f in Ω × Y 2 and A(y)∇ y u 2 • n = h(y)(u 1 -u 2 ) on Ω × Γ.
This suggests us to search the function u 2 of the form

u 2 (x, y) = u 1 (x) + f (x)χ 2 (y) in Ω × Y 2 .
By replacing this form of u 2 in the two previous equations, we obtain

-div y (A(y)(f (x)∇ y χ 2 )) = f (x) in Ω × Y 2 ,
and

A(y) (f (x)∇ y χ 2 ) • n = -h(y)f (x)χ 2 (y) on Ω × Γ,
which imply that the scalar function χ 2 is the solution of the Robin cell problem (3.13).

By choosing now Φ 1 = Φ 2 = 0 in (3.2), we obtain:

Ω×Y 1 A(y)(∇u 1 + ∇ y u 1 )∇ϕ dx dy + Ω×Γ h(y)(u 1 -u 2 )ϕ dx dσ y = |Y 1 | Ω f (x)ϕ(x) dx + |Γ|M Γ (g) Ω ϕ(x) dx. (3.17)
We have the equality

Ω×Γ h(y)(u 1 -u 2 )ϕ dx dσ y = - Ω×Γ h(y)f (x)χ 2 (y)ϕ(x) dx dσ y = - Γ h(y)χ 2 (y) dσ y Ω f (x)ϕ(x) dx = Γ A(y)∇ y χ 2 (y) • n dσ y Ω f (x)ϕ(x) dx = Y 2 div y (A(y)∇ y χ 2 (y)) dy Ω f (x)ϕ(x) dx = Y 2 (-1) dy Ω f (x)ϕ(x) dx = -|Y 2 | Ω f (x)ϕ(x) dx,
and then relation (3.17) becomes:

Ω×Y 1 A(y)(∇u 1 + ∇ y u 1 )∇ϕ dx dy = Ω f (x)ϕ(x) dx + |Γ|M Γ (g) Ω ϕ(x) dx.
We integrate this last equality by parts with respect to x and, by using (3.9) and (3.12), we are led to the homogenized problem (3.8).

Remark 3.4 Due to the right scaling ε in front of the function g ε given at the interface Γ ε , we obtain at the limit a new source term distributed all over the domain Ω. Our initial problem (2.1) can be also studied for a nonzero function g with mean-value M Γ (g) equal to zero. In this situation, there is no contribution of g in the right-hand side of the homogenized equation and, thus, the limit problem is the same as in the case with no g at all in the microscopic problem.

Remark 3.5 The solution u 1 (x) of problem (3.8) represents the contribution coming from the first material distributed in Ω ε 1 and the solution u 2 (x, y), verifying relation (3.10), is an additional contribution coming from the second material distributed in Ω ε 2 . This shows that the diffusion within the domain Ω × Y 2 has more than a local character. The limit problems keep information from the two different materials, but on two different scales, and this is an important particularity of such models.(see e.g. [START_REF] Allaire | Homogenization and two-scale convergence[END_REF]) We remark that the homogenized matrix A hom and, so, the solution u 1 , are independent of the function h, while the limit u 2 depends on h, via the function χ 2 .

Remark 3.6 All the above results remain true for the case in which the set Y 2 is not connected, but consists on a finite number of connected components, as in [START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF].

We can also state the convergence of the energy and corrector results for the solution u ε = (u ε 1 , u ε 2 ) of problem (2.1). They are obtained in a classical way, by adapting to our case the proof of Proposition 4.7 in [START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF]. We have the following result: Theorem 3.7 Under the assumptions of Theorem 3.1, if

u ε = (u ε 1 , u ε 2 )
is the unique solution of problem (2.1), then

lim ε→0 Ω ε 1 A ε ∇u ε 1 ∇u ε 1 dx + Ω ε 2 ε 2 A ε ∇u ε 2 ∇u ε 2 dx = Ω×Y 1 A(y)(∇u 1 + ∇ y u 1 )(∇u 1 + ∇ y u 1 ) dx dy+ Ω×Y 2 A(y)∇ y u 2 ∇ y u 2 dx dy, (3.18) lim ε→0 Λ ε 1 |∇u ε 1 | 2 dx + Λ ε 2 |∇u ε 2 | 2 dx = 0, (3.19) T ε 1 (∇u ε 1 ) → ∇ u 1 + ∇ y u 1 strongly in L 2 (Ω × Y 1 ) (3.20)
and

T ε 2 (ε∇u ε 2 ) → ∇ y u 2 strongly in L 2 (Ω × Y 2 ). (3.21)
Moreover, the following corrector result holds true:

∇u ε 1 -∇u 1 + N j=1 U ε 1 ∂u 1 ∂x j U ε 1 ∇ y χ j 1 L 2 (Ω ε 1 )
-→ 0 and

∇u ε 2 -∇u 1 -f (x) U ε 2 (∇ y χ 2 ) L 2 (Ω ε 2 ) -→ 0.
Let us analyse now the second relevant situation for the jump function G ε .

Case 2:

G ε (x) = g x ε , if M Γ (g) = 0.
Theorem 3.8 The unique solution u ε = (u ε 1 , u ε 2 ) of the variational problem (2.6) converges, in the sense of (3.1), to the unique solution (u 1 , u 1 , u 2 ) ∈ V of the following unfolded limit problem:

Ω×Y 1 A(y)(∇u 1 + ∇ y u 1 )(∇ϕ + ∇ y Φ 1 ) dx dy + Ω×Y 2 A(y)∇ y u 2 ∇ y Φ 2 dx dy+ Ω×Γ h(y)(u 1 -u 2 )(ϕ -Φ 2 ) dx dσ y = Ω×Y 1 f (x)ϕ(x) dx dy+ Ω×Y 2 f (x)Φ 2 (x, y) dx dy + Ω×Γ g(y)Φ 1 (x, y) dx dσ y , (3.22)
for all ϕ ∈ H 1 0 (Ω), Φ 1 ∈ L 2 (Ω, H 1 per (Y 1 )), Φ 2 ∈ L 2 (Ω, H 1 (Y 2 )).
Proof.

To obtain the problem (3.22), we pass to the limit in the unfolded form of the variational formulation (2.6) with the test functions (3.3), which satisfy (3.4)-(3.7). The only difference with respect to the proof of Theorem 3.2 is the passage to the limit in the term involving the function G ε . More precisely, we have now:

1 ε Ω×Γ T ε b (G ε )T ε b (v 1 ) dσ x = 1 ε Ω×Γ T ε b g x ε T ε b ϕ(x) + εω 1 (x)ψ 1 x ε dσ x = 1 ε Ω×Γ g(y)T ε b (ϕ)(x, y) dx dσ y + Ω×Γ g(y)T ε b (ω 1 )(x, y)T ε b (ψ 1 )(x, y) dx dσ y → Ω×Γ g(y)ω 1 (x)ψ 1 (y) dx dσ y .
from the fact that M Γ (g) = 0. By using the density of

D(Ω) ⊗ H 1 per (Y 1 ) in L 2 (Ω, H 1 per (Y 1 )) and of D(Ω) ⊗ H 1 (Y 2 ) in L 2 (Ω, H 1 (Y 2
)), we are led to the unfolded limit problem (3.22).

Due to the uniqueness of (u 1 , u 1 , u 2 ) ∈ V, which can be proven by the Lax-Milgram theorem, all the above convergences hold true for the whole sequence and our theorem is proven. Remark 3.9 Let us notice that the term Ω×Γ g(y)Φ 1 (x, y) dx dσ y in (3.22) constitutes the main difference with respect to the unfolded equation (3.2), where the term involving g is a nonzero constant, appearing explicitly in the right-hand side of the homogenized equation (3.8). This is not the case here, since this term involves now explicitly both variables x and y. Our task now is to understand the contribution in the homogenized problem of this non standard term generated by the discontinuity of the flux in the microscopic problem. As we shall see in Theorem 3.10, apart from the cell problems (3.12) and (3.13), an additional non homogeneous Neumann cell problem needs to be introduced. Theorem 3.10 The unique solution u ε = (u ε 1 , u ε 2 ) of the variational problem (2.6) converges, in the sense of (3.1), to (u 1 , u 1 , u 2 ) ∈ V, where u 1 is the unique solution of the homogenized problem -div

(A hom ∇u 1 (x)) = f (x) in Ω, u 1 = 0 on ∂Ω (3.23)
and

u 1 (x, y) = - N j=1 ∂u 1 ∂x j (x)χ j 1 (y) + η(y), (3.24)
u 2 (x, y) = u 1 (x) + f (x)χ 2 (y).
Here, A hom is the homogenized matrix whose entries are given by (3.11) and the functions χ j 1 and χ 2 are defined by (3.12) and (3.13). The Y -periodic function η is the unique solution of the following non homogeneous Neumann cell problem: Proof. By taking ϕ = 0 in the unfolded limit problem (3.22), we obtain:

       -div y (A(y)∇ y η) = 0 in Y 1 , A(y)∇ y η • n = -g(y) on Γ, M Γ (η) = 0. ( 3
Ω×Y 1 A(y)(∇u 1 + ∇ y u 1 )∇ y Φ 1 dx dy + Ω×Y 2 A(y)∇ y u 2 ∇ y Φ 2 dx dy- Ω×Γ h(y)(u 1 -u 2 )Φ 2 dx dσ y = Ω×Y 2 f (x)Φ 2 (x, y) dx dy + Ω×Γ g(y)Φ 1 (x, y) dx dσ y . (3.26)
By choosing Φ 1 = 0 in (3.26), we have

Ω×Y 2 A(y)∇ y u 2 ∇ y Φ 2 dx dy - Ω×Γ h(y)(u 1 -u 2 )Φ 2 dx dσ y = Ω×Y 2 f (x)Φ 2 dx dy.
By taking now suitable test functions Φ 2 , we obtain

-div y (A(y)∇ y u 2 ) = f in Ω × Y 2 and A(y)∇ y u 2 • n = h(y)(u 1 -u 2 ) on Ω × Γ. (3.27)
We then find the functions u 2 and χ 2 exactly like in the proof of Theorem 3.3. Now, let us take Φ 2 = 0 in (3.26). We obtain

Ω×Y 1 A(y)(∇u 1 + ∇ y u 1 )∇ y Φ 1 dx dy = Ω×Γ g(y)Φ 1 dx dσ y .
By taking suitable test functions Φ 1 , we obtain

-div y (A(y)∇ y u 1 ) = div y (A(y)∇u 1 ) in Ω × Y 1 , (3.28)
A(y)(∇ x u 1 + ∇ y u 1 ) • n = -g(y) on Ω × Γ. (3.29)
We remark that (3.27) and (3.29) imply that we also have a discontinuity type condition:

A(y)(∇ x u 1 + ∇ y u 1 ) • n -A(y)∇ y u 2 • n = -h(y)(u 1 -u 2 ) -g(y) on Ω × Γ.
The presence of the function g in relation (3.29) suggests us to search u 1 in the following non standard form:

u 1 (x, y) = - N j=1 ∂u 1 ∂x j (x)χ j 1 (y) + η(y), (3.30)
where the functions χ j 1 are defined by (3.12) and the function η remains to be found. To this end, we replace u 1 given by (3.30) in (3.28)- (3.29). We obtain: is satisfied, thanks to the hypothesis (2.3) imposed on the function g.

       -div y (A(y)(-∇ x u 1 ∇ y χ 1 + ∇ y η(y)) = div y (A(y)∇ x u 1 ) in Ω × Y 1 , A(y) (∇ x u 1 -∇ x u 1 ∇ y χ 1 + ∇ y η) • n = -g(y) on Ω × Γ, M Γ (η) = 0.
By choosing now Φ 1 = Φ 2 = 0 in (3.22), we get:

Ω×Y 1 A(y)(∇u 1 + ∇ y u 1 )∇ϕ dx dy + Ω×Γ h(y)(u 1 -u 2 )ϕ dx dσ y = Ω×Y 1 f (x)ϕ(x) dx dy. (3.33)
Also, since

u 1 (x) -u 2 (x, y) = -f 2 (x)χ 2 (y) in Ω × Y 2 ,
we have, as in the proof of Theorem 3.3, the equality

Ω×Γ h(y)(u 1 -u 2 )ϕ dx dσ y = -|Y 2 | Ω f (x)ϕ(x) dx
and relation (3.33) then becomes:

Ω×Y 1 A(y)(∇u 1 + ∇ y u 1 )∇ϕ dx dy = Ω f (x)ϕ(x) dx.
We integrate this last equality by parts with respect to x and, using (3.30) and the definition (3.11) of the matrix A hom , we obtain

-div x (A hom ∇u 1 ) = f + div x Y 1
A(y)∇η(y) dy in Ω, which leads immediately to the homogenized problem (3.23). We notice that this problem does not involve the function g. Nevertheless, the influence of the flux jump g appears in the corrector function u 1 , via the cell problem (3.25).

Remark 3.11

The above results can be generalized to the case in which A ε is a sequence of matrices in M(λ, µ, Ω) such that

T ε α (A ε ) → A strongly in L 1 (Ω × Y ),
for some matrix A = A(x, y) in M(λ, µ, Ω × Y ). The heterogeneity of the medium described by such a matrix generates different effects in our limit problems (3.2) and (3.22), respectively.

In both situations, due to the fact that the correctors χ j 1 depend also on the variable x, the new homogenized matrix A hom x is no longer constant, but it depends on x. A more interesting effect occurs in the second case. As proven in Theorem 3.10, if the matrix A depends only on the variable y, the function η is independent of x and there is no contribution of the term containing g in the decoupled form of the limit problem. So, the limit equation is the same as that corresponding to the case with no jump of the flux in the microscopic problem. Now, the dependence of A on x prevents this phenomenon to occur, and, as a consequence, the function g gives an explicit contribution in the homogenized problem, which becomes

-div x (A hom x ∇u) = f + div x Y 1
A(x, y)∇η(x, y) dy in Ω.

A similar effect was observed in the homogenization of the Neumann problem in perforated domains (see [START_REF] Cioranescu | The periodic unfolding method in domains with holes[END_REF]).

In this second case too, a corrector result similar to the one stated in Theorem 3.7 holds true. The main difference now is that the function η, solution of the cell problem (3.25), appears in the correctors of the solution u ε = (u ε 1 , u ε 2 ) of problem (2.1), as well.

Theorem 3.12 Under the assumptions of Theorem 3.8, if

u ε = (u ε 1 , u ε 2 )
is the unique solution of problem (2.1), then (3.18)-(3.21) hold true. Moreover, we have the following corrector result:

∇u ε 1 -∇u 1 + N j=1 U ε 1 ∂u 1 ∂x j U ε 1 ∇ y χ j 1 -U ε 1 (∇ y η) L 2 (Ω ε 1 ) -→ 0 and ∇u ε 2 -∇u 1 -f (x) U ε 2 (∇ y χ 2 ) L 2 (Ω ε 2 ) -→ 0.
Remark 3.13 Let us point out that similar corrector results can be stated in the case in which the matrix A depends both on x and y, as in Remark 3.11.

Conclusions

Using the periodic unfolding method, the effective thermal transfer in a periodic composite material formed by two constituents, with different thermal properties, was analyzed. The main features of the considered composite material were the discontinuity of the thermal conductivity over the domain as we go from one constituent to another and the presence of an imperfect interface between the two constituents, where both the temperature and the flux exhibit jumps.

The limit problem captures the influence of the jumps in the limit temperature field, in an additional source term, and in the correctors, as well.

Appendix

We briefly recall here the definitions and the main properties of the unfolding operators T ε 1 and T ε 2 , introduced, for a two-component domain, by P. Donato et al. in [START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF] (see, also, [START_REF] Cioranescu | The periodic unfolding method in domains with holes[END_REF], [START_REF] Cioranescu | The periodic unfolding method in homogenization[END_REF], [START_REF] Cioranescu | The periodic unfolding method in perforated domains[END_REF] and [START_REF] Donato | Homogenization of diffusion problems with a nonlinear interfacial resistance[END_REF]) and of the boundary unfolding operator T ε b , introduced in [START_REF] Cioranescu | The periodic unfolding method in perforated domains[END_REF] and [START_REF] Cioranescu | Asymptotic behavior of elliptic problems in perforated domains with nonlinear boundary conditions[END_REF]. The main particularity of these operators is that they map functions defined on the oscillating domains Ω ε 1 , Ω ε 2 and, respectively, Γ ε , into functions defined on the fixed domains Ω × Y 1 , Ω × Y 2 and Ω × Γ, respectively. For x ∈ R N , we denote by [x] Y its integer part k ∈ Z N , such that x -[x] Y ∈ Y and we set {x} Y = x -[x] Y for x ∈ R N . So, for every x ∈ R N , we have x = ε x ε Y + x ε Y . For defining the above mentioned periodic unfolding operators, we consider the following sets (see [START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF]): We give now a few useful properties of these operators (see, e.g., [START_REF] Cioranescu | The periodic unfolding method in domains with holes[END_REF], [START_REF] Donato | Homogenization of diffusion problems with a nonlinear interfacial resistance[END_REF] and [START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF]). (iii) if {ϕ ε } ε ⊂ L p (Ω) is a sequence such that ϕ ε -→ ϕ strongly in L p (Ω), then T ε α (ϕ ε ) -→ ϕ strongly in L p (Ω × Y α );

Z ε = k ∈ Z N | εY k ⊂ Ω , Ω ε = int k∈ Zε εY k , Λ ε = Ω \ Ω ε , Ω ε α = k∈ Zε εY k α , Λ ε α = Ω ε α \ Ω ε α , Γ ε = ∂ Ω ε 2 .
(iv) if ϕ ∈ L p (Y α ) is Y -periodic and ϕ ε (x) = ϕ(x/ε), then T ε α (ϕ ε ) -→ ϕ strongly in L p (Ω × Y α ); Finally, we give for α ∈ {1, 2} the definition of the adjoints U α of the unfolding operators and we state some useful properties for them (see [START_REF] Cioranescu | The periodic unfolding method in domains with holes[END_REF] and [START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF]). • ϕ ε -U ε α ( ϕ) L p (Ω ε α ) → 0.

Remark 3 . 1

 31 We notice that in (3.1) we omitted to write |Y |. Indeed, since Y is the unit cube, we have |Y | = 1 and, in order to simplify the presentation, in the sequel we shall not write it.

14 ) and u ε 2 |Y 2 |u 1 + Y 2 χ 2

 14222 (y) dy f weakly in L 2 (Ω).(3.15)

. 25 )

 25 Convergences (3.14) and (3.15) still hold true, with u 1 solution of (3.23).

(3. 31 )-

 31 By using (3.12), we deduce that the scalar function η is the unique Y -periodic solution of the div y (A(y)∇ y η) = 0 in Y 1 , A(y)∇ y η • n = -g(y) on Γ, M Γ (η) = 0.(3.32) We observe that (3.32) is a non homogeneous Neumann problem. The compatibility condition Γ η(y) dy = 0.

Definition 5 . 1 Remark 5 . 2

 5152 For any Lebesgue measurable function ϕ on Ω ε α , α ∈ {1, 2}, we define the periodic unfolding operators by the formulaT ε α (ϕ)(x, y) = for a.e. (x, y) ∈ Ω ε × Y α 0 for a.e. (x, y) ∈ Λ ε × Y α If ϕ is a function defined in Ω, for simplicity, we write T ε α (ϕ) instead of T ε α (ϕ| Ω ε α ).For any function ϕ which is Lebesgue-measurable on Γ ε , the periodic boundary unfolding operator T ε b is defined by T ε b (ϕ)(x, y) for a.e. (x, y) ∈ Ω ε × Γ 0 for a.e. (x, y) ∈ Λ ε × Γ We notice that if ϕ ∈ H 1 (Ω ε α ), then T ε b (ϕ) = T ε α (ϕ)| Ω ε ×Γ .

Proposition 5 . 3

 53 For p ∈ [1, ∞) and α ∈ {1, 2}, the operators T ε α are linear and continuous from L p (Ω ε α ) to L p (Ω × Y α ) and (i) if ϕ and ψ are two Lebesgue measurable functions on Ω ε α , one hasT ε α (ϕψ) = T ε α (ϕ)T ε α (ψ); (ii) for every ϕ ∈ L 1 (Ω ε α ), one has 1 |Y | Ω×Yα T ε α (ϕ)(x, y) dx dy =

Definition 5 . 7 Proposition 5 . 8

 5758 For p ∈ [1, ∞], the averaging operators U ε α : L p (Ω × Y α ) → L p (Ω ε α ),are given by for a.e. x ∈ Ω ε α , 0 for a.e. x ∈ Λ ε α . It is not difficult to see that these averaging operators are almost left-inverses of the corresponding unfolding operators T ε α , i.e., for any ϕ ∈ L p (Ω ε α ), we have for a.e. x ∈ Ω ε α , 0 for a.e. x ∈ Λ ε α . For p ∈ [1, ∞), the operators U ε α are linear and continuous from L p(Ω × Y α ) to L p (Ω ε α ) and (i) U ε α (φ) -φ L p (Ω ε α ) → 0 for every φ ∈ L p (Ω); (ii) if ϕ ε ∈ L p (Ω ε α ), then the following statements are equivalent:• T ε α (ϕ ε ) → ϕ strongly in L p (Ω × Y α ) and Λ ε α |ϕ ε | p dx → 0;

Acknowledgements. Both authors acknowledge the support from the Laboratoire Européen Associé CNRS Franco-Roumain "Math-Mode".

(v) if ϕ ∈ W 1,p (Ω ε α ), then ∇ y (T ε α (ϕ)) = εT ε α (∇ϕ) and T ε α (ϕ) belongs to L 2 Ω; W 1,p (Y α ) . Moreover, for every ϕ ∈ L 1 (Γ ε ), one has

The following result was proven, for our geometry, in [START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF].

) is a sequence in H ε and ϕ ∈ D(Ω), then, for ε small enough and α ∈ {1, 2} we have

We remind some general compactness results obtained in [START_REF] Donato | Homogenization of an elastic double-porosity medium with imperfect interface via the periodic unfolding method[END_REF] for bounded sequences in H ε .

2 ) be a bounded sequence in H ε . Then, there exists a constant C > 0, independent of ε, such that

2 ) be a bounded sequence in H ε . Then, up to a subsequence, still denoted by ε, there exist

where M Γ ( u 1 ) = 0 for almost every x ∈ Ω and • denotes the extension by zero of a function to the whole of the domain Ω.