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 [4], 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 [17] and [31], the two-scale convergence method in [19], and more recently the unfolding method for periodic homogenization in [16] and [35], 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.
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, [3], [6], [7], [25] and [29]).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 [24] for reaction-diffusion problems in porous media, in [27] for problems arising in the combustion theory and in [23] 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: 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 [9] and [13] 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 [5].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 [9], [13]), 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 [8].
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 [16].
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 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 which clearly shows that the flux of the solution exhibits a jump across Γ ε .
a.e. on Γ.We set with the property that, for any For a Y -periodic smooth symmetric matrix A ∈ M(α, β, Y ), we set Let g be a Y -periodic function that belongs to L 2 (Γ).We define For the given function G ε in (2.1), we consider the following two relevant forms (see [13]): 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 The space 2 ) is equipped with the usual norm.On the space H ε , we consider the scalar product belong to H ε .The norm generated by the scalar product (2.4) is given by The variational formulation of problem (2.1) is the following one: find where the bilinear form a : H ε × H ε → R and the linear form l : respectively.
We state now an existence and uniqueness result and some necessary a priori estimates for the solution of the variational problem (2.6).
Moreover, there exists a constant C > 0, independent of ε, such that Proof.The proof of this theorem follows exactly the same steps as in [16] and [28].
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 [16] (see, also, [9], [10], [11] and [15]) and of the boundary unfolding operator T ε b , introduced in [11] and [12].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 . For defining the above mentioned periodic unfolding operators, we consider the following sets (see [16]): 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 For any function φ which is Lebesgue-measurable on Γ ε , the periodic boundary unfolding operator T ε b is defined by We recall here some useful properties of these operators (see, for instance, [9], [15] and [16]).
Moreover, if ϕ ∈ D(Ω), then, for ε small enough, we have We also recall here some general compactness results obtained in [16] for bounded sequences in 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.Moreover, we have u 1 = u 2 and with y Γ = y − M Γ (y) and

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)- ) such that M Γ ( u 1 ) = 0 and up to a subsequence, for ε → 0, we have: Moreover, one has We consider the space endowed with the norm For the passage to the limit, we have to distinguish between two cases, following the form of the function G ε .
Case 1: 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: 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 Then, for α = 1, 2, we choose in this unfolded problem the admissible test functions ) and for which we obviously have 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 [15].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: By the density of )), we get (4.3).
We notice that our limit problem (4.3) is similar with the one obtained in [16] (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 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 [16], 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: where A hom is the homogenized matrix whose entries are given, for i, j = 1, . . ., N , by ) and χ j 2 ∈ H 1 (Y 2 ), j = 1, . . ., N , the weak solutions of the following cell problems: 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 3), we obtain: We search now u 1 and ū2 in the usual form (4.12) 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.
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: 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: 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 The function u ∈ H 1 0 (Ω) is the unique solution of the following homogenized equation where A hom is the homogenized matrix whose entries are given in (4.8).
Proof.By choosing ϕ = 0 in (4.14), we obtain: We point out that the presence of the term 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: 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: 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 endowed with the scalar product .
By using here the particular form (4. 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 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 [9]).

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 [9] and [16]).We recall now some useful properties of the averaging operators U ε α (see [9] and [16]).

Remark 4 . 8
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).

Remark 4 . 10
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.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 ),
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