Two-temperature homogenized eigenfunctions of conduction through domains with jump interfaces

HAL Id: hal-01874134

https://hal.archives-ouvertes.fr/hal-01874134

Submitted on 14 Sep 2018

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires

Two-temperature homogenized eigenfunctions of conduction through domains with jump interfaces

Isabelle Gruais, Dan Poliševski, Alina Stefan

To cite this version:

Isabelle Gruais, Dan Poliševski, Alina Stefan. Two-temperature homogenized eigenfunctions of con- duction through domains with jump interfaces. Applicable Analysis, Taylor & Francis, 2020, 99 (13), pp.2361-2370. �10.1080/00036811.2018.1563292�. �hal-01874134�

Two-temperature homogenized eigenfunctions of conduction through

domains with jump interfaces

Isabelle Gruais, Dan Poliˇsevski and Alina S¸tefan

Abstract. In this paper we study the asymptotic behavior of the eigen- value problem solutions of the conduction process in an ε-periodic domain formed by two components separated by a first-order jump interface. We prove that when ε → 0 the limits of the eigenvalues and eigenfunctions of this problem verify a certain (effective) two-temperature eigenvalue prob- lem. Moreover, we show that the effective eigenvalue problem has only eigenvalues which come from the homogenization process.

Keywords: interfacial jump, conduction, eigenvalue, two-scale conver- gence, homogenization

MSC 2010: 35B27, 35P15, 49R05, 74A50, 80M40

1 Introduction

During the last decades there was a steady interest for the homoge- nization of problems with interfacial thermal barriers (see [2], [14], [7], [5]) or equivalent (see [13], [11], [8], [15]). Meanwhile, it was also studied the asymptotic behavior of the eigenvalue problems in ε-periodic domains (see [12]) , even for ε-periodically perforated domains (see [17]), where the key ingredient was the prolongation operator which was introduced by [6].

Here we continue our works [9], [10] and [16] by studying the asymptotic behavior of the eigenvalue problem solutions of the conduction process in an ε-periodic domain formed by two components separated by a first-order jump interface. We prove that when ε→0 the limits of the eigenvalues and eigenfunctions of this problem verify a certain (effective) two-temperature eigenvalue problem. Moreover, we show that the effective eigenvalue problem has only eigenvalues which come from the homogenization process. Our key ingredient is a pair of prolongation operators, corresponding to each component of the domain. Otherwise, we mainly follow the methods of the two-scale theory (see [1]). We have to remark that the procedure presented here can straightforwardly be generalized to any n-component ε-periodic domain with interfacial jumps of the first order.

The paper is organized as follows: in Section 2 we study the eigenvalue problem in theε-periodic domain; in Section 3 we present the key prolonga- tion operators, the a priori estimates and the specific compactness results;

Section 4 is devoted to the derivation of the effective eigenvalue problem and its connection with the homogenization process.

2 The eigenvalue problem

Let Ω be an open connected bounded set in R^N (N ≥ 3), locally lo- cated on one side of the boundary ∂Ω, a Lipschitz manifold composed of a finite number of connected components. For any ε ∈ (0,1),Ω has two ε-periodically ditributed components. For convenience, the periodicity is described by using the cubeY = (0,1)^N,as follows:

LetY_a⊂⊂Y be a Lipschitz open set such thatY_b=Y\Y_ais connected.

For anyε∈(0,1) we denote

Zε={k∈Z^N : εk+εY ⊆Ω}. (1) The twoε−periodic components of Ω are defined by:

Ωεa= int



 [

k∈Zε

(εk+εYa)



 (2)

Ω_εb = Ω\Ωεa. (3)

Denoting Γ :=∂Ya=∂Ya∩∂Yb, the interface between Ωεaand Ωεb have the property:

Γε:= [

k∈Zε

(εk+εΓ) =∂Ωεa=∂Ωεa∩∂Ω_εb. (4) Let us remark that Ω_εb is connected and all the boundaries are at least locally Lipschitz. Also, the inward normal on ∂Ya, denoted by ν, has the property

ν^ε(x) =ν

ε⁻¹x , ∀x∈Γε, (5) where

ε⁻¹x is formed by the fractional parts of the components of ε⁻¹x.

We have to introduce the Hilbert space Hε=

n

u∈L²(Ω) :u _Ω

εa∈H¹(Ωεa), u _Ω

εb∈H¹(Ω_εb), u= 0 on ∂Ω o

(6) endowed with the scalar product

(u, v)_Hε = Z

Ωεa

∇u∇v+ Z

Ωεb

∇u∇v+ε Z

Γε

[u][v], (7)

where [u] =γεau−γεbu and γεau, γεbu are the traces of u on Γε defined in H¹(Ωεa) and H¹(Ω_εb),respectively.

Our domain has the following well-known property [8]:

Lemma 2.1. For any v ∈ H_ε there exists C > 0, independent of ε, such that

|v|_L2(Ωεb)≤C|∇v|_L2(Ωεb), (8) ε^1/2|γ_εαv|_L2(Γε)≤C

|v|_L2(Ω_εα)+ε|∇v|_L2(Ω_εα)

, α ∈ {a, b}, (9)

|v|_L2(Ω_εa)≤C

ε^1/2|γ_εav|_L2(Γε)+ε|∇v|_L2(Ω_εa)

. (10)

Remark 2.1. Taking in account theL²−norm of the jump onΓε the results of the previous Lemma have important consequences:

ε^1/2 [v]

L²(Γε)≤C

v

L²(Ω)+ε

∇v

L²(Ωεa)+ε

∇v L²(Ωεb)

, (11)

|v|_L2(Ωεa)≤C|v|_H

ε,∀v∈H_ε. (12)

Next, we introduce the data of our problem: the transmission factor h^ε(x) = h(x/ε) and the symmetric conductivities a^ε_ij(x) = aij(x/ε) and b^ε_ij(x) =b_ij(x/ε),whereh, a_ij and b_ij belong toL^∞_per(Y) and have the prop- erty that there exists δ >0 such that

h≥δ, a.e. on Y, (13)

a_ijξ_jξ_i≥δξ_iξ_i and b_ijξ_jξ_i ≥δξ_iξ_i, ∀ξ∈R^N, a.e. on Y. (14) We consider the following eigenvalue problem:

Find λ^ε∈R^∗ such that ∃u^ε∈H_ε\ {0}verifying the equations

−div (a^ε∇u^ε) =λ^εu^ε, in Ωεa, (15)

−div (b^ε∇u^ε) =λ^εu^ε, in Ω_εb, (16) and the transmission conditions

a^ε_ij∂u^ε

∂x_jν_i^ε=b^ε_ij∂u^ε

∂x_jν_i^ε=εh^ε(γεau^ε−γεbu^ε) on Γε. (17) The variational formulation of the problem (15)-(17) is the following:

Find λ^ε∈R^∗ such that ∃u^ε∈Hε\ {0}verifying

G_ε(u^ε, v) :=

Z

Ωεa

a^ε_ij∂u^ε

∂xj

∂v

∂xi

+ Z

Ωεb

b^ε_ij∂u^ε

∂xj

∂v

∂xi

+ε Z

Γε

h^ε[u^ε][v] =λ^ε(u^ε, v),

∀v∈H_ε (18)

where (·,·) denotes the inner product in L²(Ω).

Using the procedure of [3], we introduce the operatorT^ε ∈ L L²(Ω), H_ε by denotig with T^ε(u), foru∈L²(Ω), the unique solution of the problem Z

Ωεa

a^ε_ij∂T^ε(u)

∂xj

∂v

∂xi

+ Z

Ω_εb

b^ε_ij∂T^ε(u)

∂xj

∂v

∂xi

+ε Z

Γε

h^ε[T^ε(u)][v] = Z

Ω

uv,∀v∈H_ε.

Defining ˜T^ε:L²(Ω)→L²(Ω) by ˜T^ε=J^ε◦T^ε,whereJ^εis the inclusion of Hε into L²(Ω), we see that the eigenvalue problem (18) is equivalent to the eigenvalue problem

T˜^εu^ε=µ^εu^ε, viaµ^ε= 1 λ^ε.

Lemma 2.2. The inclusion J^ε:H_ε→L²(Ω) is a compact operator.

Proof. As for anyv∈H_ε we have J^εv

_Lε(Ω)= v

_Ω

εa+ v

_Ω

εb ≤C v

_H

ε,

is sufficient to prove that the bounded sequences fromH_εcontain a conver- gent subsequence inL²(Ω).

Let {v_n}_n a bounded sequence in H_ε. We note v^a_n = v_n

Ωεa and v_n^b = vn

Ω_εb. Since {v_n^a}_n is a bounded sequence in H¹(Ωεa), from the Rellich’s theorem there existv^a∈H¹(Ωεa) and a subsequence, still denoted by{n}, such that

v_n^a→v^a strongly inL²(Ω_εa).

Further, {v_n^b}_n being bounded in H¹(Ω_εb), again the Rellich’s theorem implies the existence of somev^b ∈H¹(Ωεb) such that on a sub-subsequence it holds

v^b_n→v^b strongly inL²(Ω_εb).

It follows that v0 =

v^a in Ω_εa, v^b in Ωεb

⇒v0∈Hε⊂L²(Ω).

The proof is completed as

vn−v0

2

L²(Ω) =

v_n^a−v^a

2

L²(Ωεa)+

v^b_n−v^b

2

L²(Ω_εb)→0.

We see now that ˜T^ε is a self-adjoint, compact operator in L²(Ω) and recalling for instance [4] it follows that there exist{λ^ε_k}_k, eigenvalues of the problem (18), with the property

0< λ^ε₁ ≤λ^ε₂ ≤...→ ∞

and {u^ε_k}_k, the corresponding eigenfunctions, which are complete and or- thonormal inL²(Ω).

In the following sections we shall study the behaviour of (λ^ε_k, u^ε_k) when ε→0.

3 A priori estimates

We begin this section by proving the boundedness of {λ^ε_k}_ε, the eigen- values of (18). For every k∈N let us denote

Hε,k ={S subspace ofHε,dimS=k}.

Applying the Minimum-maximum principle (see [3]), λ^ε_k can be estimated via the Rayleigh quotient:

λ^ε_k = min

S∈H_ε,kmax

v∈S

Gε(v, v) v

2 L²(Ω)

≤

≤ C min

S∈H_ε,kmax

v∈S

∇v

2

L²(Ωεa)+

∇v

2

L²(Ω_εb)+ε [v]

2 L²(Γε)

v

2 L²(Ω)

. (19)

In order to further estimate λ^ε_k with respect to εwe introduce two prolon- gation operators.

First, for any v ∈ H¹(Ya) let w ∈ H¹(Yb) be the only solution of the Dirichlet problem:

−∇w= 0 in Y_b, (20)

w= 0 on ∂Y, w=v on Γ. (21)

We have to introduce here P_a(v)∈H¹(Y) by P_a(v) =

v inY_a, w inY_b. It has the property

Pa(v)

H¹(Y) ≤C v

H¹(Ya). (22) Denoting u^ε_k(y) := u(εk+εy), for any k∈Zε,y∈Y_a and u∈H¹(Ω_εa), we define our first prolongation operator, P_a^ε :H¹(Ωεa)→H₀¹(Ω), by

P_a^ε(u)(x) =







u(x) forx∈Ωεa

P_a(u^ε_k) {^x_ε}

forx∈(εk+εY_b), k∈Zε

0 forx∈Ω\S

k∈Zε(εk+εY) Rescaling (22) we easily obtain

P_a^ε(u)

H¹(Ω) ≤C

u

L²(Ωεa)+ε

∇u L²(Ωεa)

, ∀v∈H¹(Ωεa). (23) Second, for any v ∈H¹(Yb) let w ∈H¹(Yb) be the only solution of the Dirichlet problem:

−∇w= 0 inY_a, (24)

w=v on Γ. (25) For this component we introduceP_b(v)∈H¹(Y) by

Pb(v) =

w inY_a, v inY_b, which has a property similar to (22):

P_b(v)

H¹(Y)≤C v

H¹(Yb). (26) Denoting u^ε_k(y) := u(εk +εy), for any k ∈ Zε, y ∈ Yb, u ∈ H¹(Ωεb), u = 0 on∂Ω, we define our second prolongation operator, P_b^ε : {u ∈ H¹(Ω_εb), u=0 on∂Ω} →H₀¹(Ω), by

P_b^ε(u)(x) =

u(x) forx∈Ω_εb Pb(u^ε_k) {^x_ε}

forx∈(εk+εYa), k∈Zε. Rescaling (26) we get

P_b^ε(u)

H¹(Ω)≤C u

L²(Ωεb)+ε

∇u L²(Ωεb)

,

∀u∈H¹(Ωεb), u=0 on∂Ω. (27) We can now estimate the terms of (19).

The first term gives:

∇v

2 L²(Ωεa)

v

2 L²(Ω)

≤

∇P_a^ε(v)

2 L²(Ωεa)

P_a^ε(v)

2 L²(Ω)

· P_a^ε(v)

2 L²(Ω)

v

2 L²(Ω)

< C P_a^ε(v)

2 L²(Ω)

v

2 L²(Ω)

,

the constant being the first eigenvalue of the problem (20)-(21). Using (23) we obtain

∇v

2 L²(Ωεa)

v

2 L²(Ω)

< C1+ε²C2

Z

Ωεa

a^ε∇v∇v

v

2 L²(Ω)

. (28)

The second term gives:

∇v

2 L²(Ωεb)

v

2 L²(Ω)

≤

∇P_b^ε(v)

2 L²(Ωεb)

P_b^ε(v)

2 L²(Ω)

· P_b^ε(v)

2 L²(Ω)

v

2 L²(Ω)

< C P_b^ε(v)

2 L²(Ω)

v

2 L²(Ω)

,

the constant being the first eigenvalue of the problem (24)-(25). Using (27) we obtain

∇v

2 L²(Ωεb)

v

2 L²(Ω)

< C1+ε²C2

Z

Ω_εb

b^ε∇v∇v v

2 L²(Ω)

. (29)

The third term can be estimated from (11), that is

[v]

2 L²(Γε)

v

2 L²(Ω)

< C₁+ε²C₂ Z

Ωεa

a^ε∇v∇v

v

2 L²(Ω)

+ε²C₂ Z

Ωεb

b^ε∇v∇v

v

2 L²(Ω)

. (30)

Adding the estimates (28)-(30) we get λ^ε_k ≤C₁+ε²C₂λ^ε_k,

which obviously implies that λ^ε_k is bounded for sufficiently smallε.

Setting v = u^ε_k in (18) and using the coerciveness of G_ε(·,·) and the orthonormality of {u^ε_k}_k we find

Gε(u^ε_k, u^ε_k) = Z

Ωεa

a^ε_ij∂u^ε_k

∂xj

∂u^ε_k

∂xi

dx+ Z

Ωεb

b^ε_ij∂u^ε_k

∂xj

∂u^ε_k

∂xi

dx+

+ε Z

Γε

h^ε[u^ε_k]²ds=λ^ε_k, ∀u^ε ∈Hε. (31) As {λ^ε_k}_ε is bounded, we find that

{u^ε_k}_ε is bounded in H_ε. (32) Using the previously obtained result together with inequalities (8)-(10), it follows that there exists C >0 such that

|∇u^ε_k|_L2(Ωεa)≤C, |∇u^ε_k|_L2(Ωεb)≤C, |[u^ε_k]|_L2(Γε)≤C. (33) Applying to the properties of the two-scale convergence theory [1], a specific compactness result follows.

Theorem 3.1. There exist λ_k ∈ R^?+, u^a_k∈H¹(Ω), u^b_k ∈ H₀¹(Ω) and η_k^α∈ L²

Ω;He_per¹ (Yα)

, α ∈ {a, b}, such that the following convergences hold on some subsequence

χ^ε_αu^ε_k* χ^2s αu^α_k, (34) χ^ε_α∇u^ε_k* χ^2s α(∇_xu^α_k +∇_yη^α_k(·, y)), (35)

λ^ε_k→λ_k, (36)

where χ^ε_α :L²(Ωεα)→L²(Ω)andχα :L²(Ω×Yα)→L²(Ω×Y), α∈ {a, b}, denote the straight prolongations with zero; sometimes they can be identified with the corresponding characteristic functions.

4 The homogenization process

Passing (18) to the limit on the subsequence on which the convergences of Theorem 3.1 hold, we obtain like in [8]:

Lemma 4.1. For any ϕa ∈C^∞( ¯Ω), ϕ_b ∈ D(Ω) and ψα ∈ D(Ω;C_per^∞(Yα)), α∈ {a, b}, it holds

Z

Ω×Y_a

a_ij ∂u^a_k

∂x_j +∂η^a_k

∂y_j

∂ϕ_a

∂x_i + ψ_a

∂y_i

+ Z

Ω×Y_b

b_ij ∂u^b_k

∂x_j +∂η^b_k

∂y_j

∂ϕ_b

∂x_i + ψ_b

∂y_i

+ +eh

Z

Ω

(u^a_k−u^b_k)(ϕa−ϕb) =λk

Z

Ω×Y

χau^a_kϕa+χbu^b_kϕb (37) where eh is defined by

eh= Z

Γ

h(y)ds. (38)

Proof. Forϕ_α and ψ_α, α∈ {a, b}like in the hypotheses, we set v in (18) as follows:

v(x) =

ϕ_a(x) +εψ_a x,x

ε

, ϕ_b(x) +εψ_b x,x

ε

. (39)

We obtain

λ^ε(χ^ε_au^ε, ϕa) +λ^ε(χ^ε_au^ε, ϕ_b) +O(ε) = X

α∈{a,b}

Z

Ωεα

α^ε_ij∂u^ε

∂x_j ∂ϕα

∂x_i + ∂ψα

∂y_i

+

+ε Z

Γε

h^ε(γεau^ε−γεbu^ε) [ϕa−ϕb+ε(ψa−ψb)]. (40) The proof is completed by following the same steps as in the proof presented in [8].

In order to present the next results we have also to introduce the Hilbert space

H:=

H¹(Ω)×H₀¹(Ω)

×h

L²(Ω,He_per¹ (Y_a)×L²(Ω,He_per¹ (Y_b))i

, (41) endowed with the scalar product

(((ua, u_b),(ηa, η_b)),((ϕa, ϕ_b),(ψa, ψ_b)))_H = X

α∈{a,b}

Z

Ω

∇u_α∇ϕ_α+

+ Z

Ω

(u_a−u_b) (ϕ_a−ϕ_b) + X

α∈{a,b}

Z

Ω×Y_α

∇_yη_α∇_yψ_α. (42) Using density arguments, Lemma 4.1 yields:

Theorem 4.1. λ_k∈R^∗+ and (u^a_k, u^b_k),(η_a, η_b)

∈H\ {0} verify:

Z

Ω×Ya

a_ij ∂u^a_k

∂x_j +∂η^a_k

∂y_j

∂ϕa

∂x_i + ψa

∂y_i

+ Z

Ω×Y_b

b_ij ∂u^b_k

∂x_j +∂η_k^b

∂y_j

∂ϕ_b

∂x_i + ψ_b

∂y_i

+

+eh Z

Ω

(u^a_k−u^b_k)(ϕ_a−ϕ_b) =λ_k Z

Ω×Y

χ_au^a_kϕ_a+χ_bu^b_kϕ_b ,

∀((ϕ_a, ϕ_b),(ψ_a, ψ_b))∈H. (43) We can present now the main result of this paper.

Theorem 4.2. If (λ^ε_k, u^ε_k) is a solution of (18) then the limits of the con- vergences (34)-(36), that isλ_k∈R^∗+and (u^a_k, u^b_k)∈

H¹(Ω)×H₀¹(Ω)

\ {0}, put together a solution of the following effective eigenvalue problem:

Find (λ_k,(u^a_k, u^b_k))∈R^∗+×

H¹(Ω)×H₀¹(Ω)

\ {0} such that

G^hom((u^a_k, u^b_k),(ϕ_a, ϕ_b)) =λ_k Z

Ω

m u^a_kϕ_a+ (1−m)u^b_kϕ_b,

∀(ϕa, ϕb)∈H¹(Ω)×H₀¹(Ω), (44) where m=

Ya

,

G^hom((u^a_k, u^b_k),(ϕ_a, ϕ_b)) = Z

Ω

ˆ aij

∂u^a_k

∂x_j

∂ϕa

∂x_i + Z

Ω

ˆbij

∂u^b_k

∂x_j

∂ϕ_b

∂x_i +eh Z

Ω

(u^a_k−u^b_k)(ϕa−ϕ_b), (45) the effective coefficients αˆ_ij,α∈ {a, b}, are defined by

ˆ α_ij =

Z

Yα

α_ij +α_ik∂e^a_j

∂yk

dy, ∀i, j∈ {1,2, ..., N}, (46) and e^α_k ∈ He_per¹ (Y_α), k ∈ {1,2, ..., N}, is the unique solution of the local- periodic problem

− ∂

∂yi

αij

∂(e^α_k +y_k)

∂yj

= 0 in Yα, (47)

α_ij∂(e^α_k+y_k)

∂y_j ν_i = 0 on Γ. (48)

Moreover, there are no eigenvalues of the problem (44) except those ob- tained as limits of the homogenization process.

Proof. The first assertion follows from Theorem 4.1 using standard homog- enization procedures.

Next, suppose that there exists an eigenvalue of the problem (44), µ ∈ R^∗+, µ6=λ_k, ∀k≥1, where{λ_k}_krepresents the eigenvalues of the problem (44) obtained as limits of the homogenization process. It follows that there existsk such that λ_k < µ < λ_k+1. Let w= (w^a, w^b)∈

H¹(Ω)×H₀¹(Ω)

\ {0} be an eigenfunction associated toµ.

Let us introducew^ε∈Hε, the unique the solution to the problem G_ε(w^ε, v) =µ

Z

Ωεa

w^av+µ Z

Ωεb

w^bv, ∀v ∈H_ε. (49) Settingv=w^ε in the previous relation we find that

G_ε(w^ε, w^ε)≤C w^ε

L²(Ω).

Applying the coercivity property ofG_ε, we find that the sequence {w^ε}_ε is bounded in H_ε. Under these circumstances, there exists w^∗ = (w^∗_a, w_b^∗) ∈ H¹(Ω)×H₀¹(Ω) such that

P_a^εw^ε* w^∗_a inH¹(Ω) and P_b^εw^ε * w_b^∗ inH₀¹(Ω).

Homogenizing the problem (49) we obtain

ε→∞lim G_ε(w^ε, v) =G^hom(w^∗, v) =µ(w, v) =G^hom(w, v),∀v∈H₀¹(Ω), from which follows

w=w^∗ and Z

Ω

(w^ε)²→ Z

Ω

m(w^a)²+ (1−m)(w^b)² 6= 0. (50)

Next, we define ˆw^ε = w^ε−

k

X

i=1

Z

Ω

w^εu^ε_i

u^ε_i, where u^ε_i ∈ Hε is an eigen- function associated toλi. We note that

Z

Ω

ˆ

w^εu^ε_j = 0, ∀j ≤k,and hence G^ε( ˆw^ε,wˆ^ε)≥λ^ε_k+1

Z

Ω

( ˆw^ε)². (51)

As Z

Ω

w^εu^ε_i = Z

Ω

χ^ε_aP_a^εw^εu^ε_i + Z

Ω

(1−χ^ε_a)P_b^εw^εu^ε_i, ∀i∈ {1, ..., k}, the passage to the limit yields

Z

Ω

w^εu^ε_i → Z

Ω

mw^au^a_i + (1−m)w^bu^b_i = 0,

which obviously implies Z

Ω

( ˆw^ε)²→ Z

Ω

m(w^a)²+ (1−m)(w^b)². (52) Passing (49) to the limit in we finally get

Gε( ˆw^ε,wˆ^ε)→G^hom(w, w) =µ Z

Ω

m(w^a)²+ (1−m)(w^b)² <

< λk+1

Z

Ω

m(w^a)²+ (1−m)(w^b)², (53)

which is in contradiction with (51), via (52).

Remark 4.1. The formulation of the effective two-temperature eigenvalue problem follows from Theorem (4.2):

Find λ∈R^∗ such that ∃(u^a, u^b)∈

H¹(Ω)×H₀¹(Ω)

\ {0} verifying

−div (ˆa∇u^a) =m λ u^a in Ω, (54)

−div ˆb∇u^b

= (1−m)λ u^b in Ω, (55) and the boundary condition

ˆ a_ij∂u^a

∂xj

n_i= 0 on ∂Ω, (56)

where n is the outward normal on ∂Ω.

Acknowledgements. The authors gratefully acknowledge partial sup- port from the International Network GDRI ECO-Math.

References

[1] G. Allaire, Homogenization and two-scale convergence, S.I.A.M. J.

Math. Anal. 23 (1992), 1482–151.

[2] J.L. Auriault, H. Ene, Macroscopic modelling of heat transfer in com- posites with interfacial thermal barrier, Internat. J. Heat Mass Transfer 37 (1994), 2885–2892.

[3] I. Babuˇska, J. Osborn, Eigenvalue Problems, Handbook of numerical analysis, vol II, Elsevier Science Publishers, North-Holland, 1991.

[4] H. Brezis, Analyse fonctionnelle. Th´eorie et applications, 1983, Dunod, Paris.

[5] R. Bunoiu, C. Timofte, Homogenization of a thermal problem with flux jump, Networks and Heterogeneous Media 11(4):545-562 DOI:

10.3934/nhm.2016009 (2016), 1–22.

[6] D. Cioranescu and J. Saint-Jean-Paulin, Homogenization in open sets with holes, J. Math. Anal. Appl. 71 (2)(1979), 590–607

[7] P. Donato, S. Monsurr`o, Homogenization of two heat conductors with interfacial contact resistance, Anal. Appl. 2 (3) (2004), 247–273.

[8] H.I. Ene, D. Poliˇsevski, Model of diffusion in partially fissured media, Z.A.M.P. 53 (2002), 1052–1059.

[9] I. Gruais, D. Poliˇsevski, Heat transfer models for two-component media with interfacial jump, Applicable Analysis, 96(2) (2015), 247–260.

[10] I. Gruais, D. Poliˇsevski, Model of two-temperature convective transfer in porous media, Z.Angew. Math. Phys.(J. Appl. Math. Phys.), 68:143., https://doi.org/10.1007/s00033-017-0889-2 (2017), 1–10.

[11] H.K. Hummel, Homogenization for heat transfer in polycrystals with interfacial resistances, Appl. Anal., 75 (3-4) (2000), 403–424.

[12] S. Kesavan, Homogenization of elliptic eigenvalue problems, Appl.

Math. and Optim. 5 (1979), Part I, 153–167, Part II, 197–216.

[13] R. Lipton, Heat conduction in fine scale mixtures with interfacial con- tact resistance, SIAM J. Appl. Math. 58 (1) (1998), 55–72.

[14] S. Monsurr`o, Homogenization of a two-component composite with in- terfacial thermal barrier, Adv. Math. Sci. Appl., 13 (1) (2003), 43–63.

[15] D. Poliˇsevski, The Regularized Diffusion in Partially Fractured Porous Media, Current Topics in Continuum Mechanics, Vol. 2, Ed. Academiei, Bucharest, 2003.

[16] D. Poliˇsevski, R. Schiltz-Bunoiu, A. St˘anescu, Heat transfer with first- order interfacial jump in a biconnected structure , Bull. Math. Soc. Sci.

Math. Roumanie 58(4) (2015), 463–473.

[17] M. Vanninathan, Homogenization of eigenvalue problems in perforated domains, Proceedings of the Indian Academy of Sciences. Mathematical Sciences 90(3) (1981), 239–271.