Diffusion through a composite medium with a semi-periodic structure and high contrasting diffusivity

Abstract

We address the homogenization of the stationary diffusion equation in a composite medium with two components $M_{ε}$ and $B_{ε}$ having respectively $A^{ε} (x)$ and $α_{ε} A^{ε} (x)$ as diffusivity coefficients. We assume periodic distribution with size ε of the “holes” $B_{ε}$ but no periodicity is assumed on the matrices $A^{ε}$ . The high contrast between the two components is characterized by the assumption that the sequence $α_{ε}$ decreases towards zero. We study the three regimes corresponding to the limits $α : = {lim}_{ε \to 0} \frac{α_{ε}}{ε}$ . It is shown in particular that in the case $α = 0$ , the inclusions $B_{ε}$ behave as holes on the macroscopic diffusion process.

Keywords

homogenization high contrast conductivity degenerate equation semi-periodic structure

1. Introduction and statement of the results

In the present work we are interested in the homogenization of the following degenerate elliptic equation $\begin{matrix} (1.0) & \{\begin{matrix} u_{ε} - div ((α_{ε}^{2} χ_{B_{ε}} + χ_{M_{ε}}) A^{ε} (x) \nabla u_{ε}) = f (x) & in Ω, \\ u_{ε} = 0 & on \partial Ω, \end{matrix} \end{matrix}$ where $A^{ε}$ denotes a sequence of conductivity matrices and f the source term, the bounded regular open set Ω of $R^{3}$ is the reference configuration of the heterogeneous medium, $B^{ε}$ and $M^{ε}$ denote respectively the part of the medium with a poor conductivity $α_{ε}^{2} A^{ε} (x)$ , being $α_{ε}$ a sequence of positive numbers decreasing to zero, and the part with a high conductivity $A^{ε} (x)$ . The medium has a semi-periodic structure since the ε-periodicity is only assumed on the distribution of the inclusions $B^{ε}$ while the matrices are not necessarily periodic. Hence, the present work is a generalization of our previous one [18] which was concerned by the same problem in the periodic setting (the matrices were assumed to be periodic).

Throughout the paper, $χ_{E}$ denotes the indicator of the set E while $| E |$ denotes its Lebesgue measure. We will denote any positive constant by the same letter C.

From the technical point of view, the assumption on the periodic distribution of the holes allows us to use the two-scale convergence method see [1,13,15] and also [10] for a similar approach. The general non-periodic setting corresponding to an arbitrary distribution of the inclusions is still an open problem to our knowledge.

The main interest of our study is to establish a link between the homogenization in the presence of holes and the homogenization in the case of a medium with a “soft” component having a lower diffusivity compared to another component of the medium, the “stiff” one, with a high conductivity. Homogenization of degenerating equations both in the scalar and in the vectorial case were intensively studied in the literature during the recent years, see for instance [5–9,16,17]. The pioneer work in this framework was the paper [3] by Arbogast, Douglas and Hornung which deals with the double porosity model.

To homogenize equation (1.0) by giving the limit of $u_{ε}$ as ε goes to zero, we will use the same approach as in [8,17] and [16]: first, we seek for the equation satisfied by the two-scale limit of the sequence $u_{ε}$ . Such equation is in general a system of several equations involving both the macroscopic variable x and the microscopic one y. In a second step, we proceed to eliminate the variable y taking advantage from the linearity of the problem so that we obtain the homogenized equation satisfied by the average u of the two-scale limit of $u_{ε}$ .

We now make more precise the geometric setting of the problem together with the main notations and then we state and prove our main result.

The reference configuration of the composite medium is the bounded open set $Ω \subset R^{3}$ . In the sequel we denote by Y and by B respectively the cube and the ball defined by $\begin{matrix} (1.1) & Y = {(- \frac{1}{2}, \frac{1}{2})}^{3}, B = B (0, r), M = Y ∖ \overline{B}, \end{matrix}$ assuming that the radius r is such that $0 < r < \frac{1}{2}$ . We define the set of inclusions $B_{ε}$ and the matrix $M_{ε}$ by: $\begin{array}{l} (1.2) & B_{ε}^{i} = ε B (0, r) + ε i, B_{ε} = ⋃_{i \in I_{ε}} B_{ε}^{i}, \\ (1.3) & Y_{ε}^{i} = ε Y + ε i, M_{ε}^{i} = Y_{ε}^{i} ∖ {\overline{B}}_{ε}^{i}, M_{ε} = Ω ∖ {\overline{B}}_{ε} = ⋃_{i \in I_{ε}} M_{ε}^{i}, \end{array}$ where $\overline{B}$ denotes the closure of B and $\begin{matrix} (1.4) & I_{ε} = {i = (i_{1}, i_{2}, i_{3}) \in Z^{3}, Y_{ε}^{i} \subset Ω} . \end{matrix}$

Let $A^{ε}$ be a sequence of measurable $3 \times 3$ matrices satisfying the following assumptions: there exist two strictly positive constants α and β such that for almost all $x \in Ω$ , for all $ξ \in R^{3}$ and for all $ε > 0$ , $\begin{array}{l} (1.5) & | A^{ε} (x) ξ | ⩽ β | ξ |, \\ (1.6) & A^{ε} (x) ξ ξ ⩾ α | ξ |^{2} . \end{array}$ We also assume that the two-scale limit (see [1]) $A (x, y) \in L^{2} (Ω \times Y)$ of the sequence $A^{ε}$ which exists by virtue of (1.5) fulfills the following assumption $\begin{matrix} (1.7) & \forall i, j = 1, 2, 3, lim_{ε \to 0} \int_{Ω} {| A_{i j}^{ε} (x) |}^{2} d x = \int_{Ω} \int_{Y} {| A_{i j} (x, y) |}^{2} d x d y . \end{matrix}$

Assume also $\begin{matrix} (1.8) & f \in L^{2} (Ω) . \end{matrix}$ Our goal is the study of the asymptotic analysis of problem (1.0) as ε goes to zero in order to get the homogenized problem. As usual, we first write equation (1.0) in a variational form: $\begin{matrix} (1.9) & \{\begin{matrix} u_{ε} \in H_{0}^{1} (Ω), \\ \int_{Ω} u_{ε} ϕ d x + \int_{Ω} (α_{ε}^{2} χ_{B_{ε}} A^{ε} (x) + χ_{M_{ε}} A^{ε} (x)) \nabla u_{ε} \nabla ϕ (x) d x = \int_{Ω} f ϕ d x, \\ \forall ϕ \in H_{0}^{1} (Ω) . \end{matrix} \end{matrix}$ For a given $ε > 0$ , the Lax–Milgram Theorem implies existence and uniqueness of the solution $u_{ε}$ of (1.9).

As in [8,16,17], we use the two-scale convergence techniques to get a preliminary formulation of the homogenized equation mixing both the microscopic and the macroscopic variables. In a second step, we derive the final version of the homogenized equation involving the macroscopic variable only.

We denote by $D (K)$ the space of infinitely differentiable functions with compact support in the open set K and by $H_{#}^{1} (Y)$ the space of functions in $H_{Loc}^{1} (R^{3})$ which are Y-periodic. Similarly, $C_{#} (Y)$ (resp. $C_{#}^{\infty} (Y)$ ) denotes the space of continuous (resp. infinitely differentiable) functions in $R^{3}$ which are Y-periodic.

Statement of the results. Our main result is the following theorem.

Theorem 1.1.
Let $u_{ε}$ be the solution of ( 1.9 ) and let $A (x, y)$ be the two-scale limit of the sequence $A^{ε}$ . Then, the sequence $u_{ε}$ admits an extension ${\tilde{u}}_{ε}$ to the inclusions $B_{ε}$ such that $\begin{matrix} (1.10) & \{\begin{matrix} {\tilde{u}}_{ε} ⇀ u & in H_{0}^{1} (Ω), \\ u_{ε} ⇀ u (x) + f (x) \int_{B} \hat{v} (x, y) d y & in L^{2} (Ω), \end{matrix} \end{matrix}$ where the pair $(u, \hat{v})$ is the unique solution of the following system $\begin{matrix} (1.11) & \{\begin{matrix} u \in H_{0}^{1} (Ω), \\ m (x) u (x) - div A^{0} (x) \nabla u (x) = m (x) f (x) & in Ω, \\ \hat{v} \in L^{\infty} (Ω, H_{0}^{1} (B)) and a.e. x \in Ω, \\ \hat{v} (x, \cdot) - α^{2} {div}_{y} A (x, \cdot) \nabla_{y} \hat{v} (x, \cdot) = 1 & in B . \end{matrix} \end{matrix}$ The volume fraction of the material is given by $\begin{matrix} (1.12) & \{\begin{matrix} m (x) = 1 & if {lim}_{ε \to 0} \frac{α_{ε}}{ε} = + \infty, \\ m (x) = 1 - | B | & if {lim}_{ε \to 0} \frac{α_{ε}}{ε} = 0, \\ m (x) = 1 - θ (x) & if {lim}_{ε \to 0} \frac{α_{ε}}{ε} = α \in] 0, + \infty [, \end{matrix} \end{matrix}$ where $θ (x) : = \int_{B} \hat{v} (x, y) d y$ .

The entries of the homogenized matrix $A^{0}$ are given by $\begin{matrix} (1.13) & A_{i, j}^{0} (x) = \int_{M} A (x, y) (\nabla_{y} w_{j} (x, y) + e_{j}) (\nabla_{y} w_{i} (x, y) + e_{i}) d y, \forall i, j = 1, 2, 3, \end{matrix}$ where ${(e_{i})}_{i = 1, 2, 3}$ is the canonical basis of $R^{3}$ and $w_{i}$ denotes the unique solution of the cell problem $\begin{matrix} (1.14) & \{\begin{matrix} a.e. x \in Ω, \\ - {div}_{y} A (x, \cdot) (\nabla_{y} w_{i} (x, \cdot) + e_{i}) = 0 & in M = Y ∖ \overline{B}, \\ A (x, \cdot) (\nabla_{y} w_{i} (x, \cdot) + e_{i}) \cdot n = 0 & on \partial B, \\ w_{i} (x, \cdot) is Y -periodic . \end{matrix} \end{matrix}$
Remark 1.1.
Applying the maximum principle to the solution $\hat{v}$ of (1.13) (see the proof of the results in the next section), we see that Theorem 1.1 states that the volume fraction of the material is proportional to the intensity of the conduction in the inclusions $B_{ε}$ . When ${lim}_{ε \to 0} \frac{α_{ε}}{ε} = 0$ , the conductivity in $B_{ε}$ is weakest compared to the other two cases. In this case, the inclusions $B_{ε}$ behave exactly as holes and the volume fraction is then given by $m (x) = 1 - | B | = | Y | - | B |$ which is minimal and coincides with the well known volume fraction in the case of holes (see [1,2,4,11,12]). The volume fraction is maximum ( $m (x) = 1 = | Y |$ ) when the conductivity in $B_{ε}$ is the largest, that is when ${lim}_{ε \to 0} \frac{α_{ε}}{ε} = + \infty$ . In the critical case ${lim}_{ε \to 0} \frac{α_{ε}}{ε} = α$ , $α > 0$ , the volume fraction $m (x)$ is given by $m (x) = 1 - θ (x)$ .

Note also that system (1.11) is actually a coupled system only in the critical case $α \in] 0, + \infty [$ . Indeed, the first equation in (1.11) is then coupled with the second one through $m (x) = 1 - θ (x) = 1 - \int_{B} \hat{v} (x, y) d y$ .

2. Apriori estimates and the limit problems

2.1. Apriori estimates

Lemma 2.1.
The sequence of solutions of ( 1.9 ) satisfies the following apriori estimates for ε small enough: $\begin{array}{l} (2.1) & \int_{Ω} (α_{ε}^{2} | \nabla u_{ε} |^{2} χ_{B_{ε}} + | \nabla u_{ε} |^{2} χ_{M_{ε}}) d x ⩽ C, \\ (2.2) & ‖ u_{ε} ‖_{L^{2} (Ω)} ⩽ C, \\ (2.3) & α_{ε} ‖ u_{ε} ‖_{H^{1} (Ω)} ⩽ C . \end{array}$
Proof.
Taking $ϕ = u_{ε}$ in (1.9) and applying the Young inequality in the right hand side, we obtain by virtue of the uniform coerciveness of the sequence $A^{ε}$ (hypothesis (1.6)): $\begin{matrix} (2.4) & E_{ε} : = \int_{Ω} {| u_{ε} (x) |}^{2} d x + \int_{Ω} (α_{ε}^{2} | \nabla u_{ε} |^{2} χ_{B_{ε}} + | \nabla u_{ε} |^{2} χ_{M_{ε}}) d x ⩽ C, \end{matrix}$ which gives immediately (2.1), (2.2) and (2.3) for small ε ( $0 < α_{ε} ⩽ 1$ ). □

To get the homogenized equation, we need the following classical extension result one can find for instance in [12]:
An extension lemma.
There exists an extension ${\tilde{u}}_{ε}$ of the solution $u_{ε}$ in $B_{ε}$ such that for all ε: $\begin{array}{l} (2.5) & {\tilde{u}}_{ε} \in H_{0}^{1} (Ω), {\tilde{u}}_{ε} = u_{ε} in M_{ε}, \\ (2.6) & ‖ {\tilde{u}}_{ε} ‖_{H_{0}^{1} (Ω)}^{2} ⩽ C \int_{M_{ε}} (| u_{ε} |^{2} + | \nabla u_{ε} |^{2}) d x, \end{array}$ where the constant C does not depend on ε.

We now proceed to investigate the limit problems. We begin with the critical case.
2.2. The limit problem in the case ${lim}_{ε \to 0} \frac{α_{ε}}{ε} = α \in] 0, + \infty [$

Estimates (2.1), (2.2) and (2.6) imply that the extension ${\tilde{u}}_{ε}$ of $u_{ε}$ is bounded in $H_{0}^{1} (Ω)$ so that, up to extracting a subsequence, one can assume that there exists $u \in H_{0}^{1} (Ω)$ such that $\begin{matrix} (2.7) & {\tilde{u}}_{ε} ⇀ u weakly in H_{0}^{1} (Ω), \end{matrix}$ and $\begin{matrix} (2.8) & {\tilde{u}}_{ε} \to u strongly in L^{2} (Ω) . \end{matrix}$ By hypothesis, $α_{ε}$ is of the same order than ε so that the sequence $ε \nabla u_{ε}$ is bounded in ${(L^{2} (Ω))}^{3}$ by virtue of estimate (2.3). It is then a classical result from the two-scale convergence (see [1]) to conclude that, up to a subsequence, there exists $v_{0} (x, y) \in L^{2} (Ω, H_{#}^{1} (Y))$ such that: $\begin{matrix} (2.9) & u_{ε} ⇀ ⇀ v_{0}, and ε \nabla u_{ε} ⇀ ⇀ \nabla_{y} v_{0} . \end{matrix}$ Let us prove that $v_{0}$ does not depend on the variable y in $Ω \times M$ where $M = Y ∖ \overline{B}$ .

Since the sequence $\nabla u_{ε} χ_{M_{ε}}$ is bounded in ${(L^{2} (Ω))}^{3}$ , it admits (up to a subsequence) a two-scale limit $K (x, y)$ . Taking $ϕ (x, y) \in {(D (Ω; D (M)))}^{3}$ , we get: $\begin{array}{rcl} \int_{Ω} \nabla u_{ε} χ_{M_{ε}} ϕ (x, \frac{x}{ε}) d x & = & \sum_{i \in I_{ε}} \int_{M_{i}^{ε}} \nabla u_{ε} ϕ (x, \frac{x}{ε}) d x \\ = & - \sum_{i \in I_{ε}} \int_{M_{i}^{ε}} u_{ε} ({div}_{x} ϕ (x, \frac{x}{ε}) + \frac{1}{ε} {div}_{y} ϕ (x, \frac{x}{ε})) d x \\ (2.10) & = & - \int_{Ω} u_{ε} χ_{M_{ε}} ({div}_{x} ϕ (x, \frac{x}{ε}) + \frac{1}{ε} {div}_{y} ϕ (x, \frac{x}{ε})) d x, \end{array}$ so that multiplying equation (2.10) by ε and passing to the limit as ε goes to zero, we get with the help of (2.9): $\begin{matrix} (2.11) & \int_{Ω \times M} v_{0} (x, y) {div}_{y} ϕ (x, y) d y d x = 0 . \end{matrix}$ Since ϕ is an arbitrary test function, we infer $\begin{matrix} (2.12) & \nabla_{y} v_{0} (x, y) = 0 in Ω \times M . \end{matrix}$ Hence, $\begin{matrix} (2.13) & v_{0} (x, y) = v_{00} (x) in Ω \times M . \end{matrix}$

Since $v_{0} (x, y) \in L^{2} (Ω, H_{#}^{1} (Y))$ , we deduce that the function $v (x, y)$ defined by $\begin{matrix} (2.14) & v (x, y) : = v_{0} (x, y) - v_{00} (x) in Ω \times Y, \end{matrix}$ fulffils $\begin{matrix} (2.15) & v (x, y) \in L^{2} (Ω; H_{0}^{1} (B)) \end{matrix}$ and that (2.9) may be written as $\begin{matrix} (2.16) & u_{ε} ⇀ ⇀ v (x, y) + v_{00} (x), and ε \nabla u_{ε} ⇀ ⇀ \nabla_{y} v . \end{matrix}$ We now prove that the function $v_{00} (x)$ coincides with the function u defined in (2.8). Indeed, by virtue of (2.5), one can write: $\begin{matrix} (2.17) & {\tilde{u}}_{ε} χ_{M_{ε}} = u_{ε} χ_{M_{ε}} in Ω . \end{matrix}$ Passing to the two-scale limit in (2.17) and using (2.8) and (2.16), we infer $\begin{matrix} (2.18) & u (x) χ_{M} (y) = (v (x, y) + v_{00} (x)) χ_{M} (y) = v_{00} (x) χ_{M} (y) in Ω \times Y, \end{matrix}$ so that $\begin{matrix} (2.19) & u (x) = v_{00} (x) in Ω . \end{matrix}$ As a consequence, the first convergence in (2.16) takes the form $\begin{matrix} (2.20) & u_{ε} ⇀ ⇀ v (x, y) + u (x), u \in H_{0}^{1} (Ω), v \in L^{2} (Ω; H_{0}^{1} (B)) . \end{matrix}$ Let us note that the second convergence in (2.16) allows us to identify easily the two-scale limit of the sequence $α_{ε} \nabla u_{ε} χ_{B_{ε}}$ of the gradients related to the soft inclusions. Indeed, using (2.16), we get $\begin{matrix} (2.21) & α_{ε} \nabla u_{ε} χ_{B_{ε}} = \frac{α_{ε}}{ε} ε \nabla u_{ε} χ_{B_{ε}} ⇀ ⇀ α \nabla_{y} v χ_{B} (y) . \end{matrix}$ Finally, in order to exhibit the corrector allowing us to identify the homogenized problem, it remains to characterize the two-scale limit $K (x, y)$ of the sequence $\nabla u_{ε} χ_{M_{ε}}$ . This may be done in a classical way, see [1]: we choose a test function ϕ such that $\begin{matrix} (2.22) & ϕ \in {(D (Ω; C_{#}^{\infty} (Y)))}^{3}, supp ϕ (x, \cdot) |_{Y} \subset M, {div}_{y} ϕ = 0 . \end{matrix}$ We have: $\begin{array}{rcl} \int_{Ω} \nabla u_{ε} χ_{M_{ε}} ϕ (x, \frac{x}{ε}) d x & = & \sum_{i \in I_{ε}} \int_{M_{i}^{ε}} \nabla u_{ε} ϕ (x, \frac{x}{ε}) d x \\ = & - \sum_{i \in I_{ε}} \int_{M_{i}^{ε}} u_{ε} ({div}_{x} ϕ (x, \frac{x}{ε}) + \frac{1}{ε} {div}_{y} ϕ (x, \frac{x}{ε})) d x \\ (2.23) & = & - \int_{Ω} {\tilde{u}}_{ε} χ_{M_{ε}} {div}_{x} ϕ (x, \frac{x}{ε}) d x . \end{array}$ Using convergence (2.8) and an integration by parts, we get at the limit $\begin{matrix} (2.24) & \int_{Ω \times M} K (x, y) ϕ (x, y) d y d x = \int_{Ω \times M} \nabla_{x} u (x) ϕ (x, y) d y d x . \end{matrix}$ Hence, $(K (x, y) - \nabla_{x} u (x)) χ_{M} (y)$ is orthogonal to the divergence free vectors (with respect to y); thus it coincides with a gradient and there exists $u_{1} \in L^{2} (Ω; H_{#}^{1} (M)) : = {u \in L^{2} (Ω; H^{1} (M)), u (x, \cdot) is Y -periodic}$ such that $\begin{matrix} (2.25) & K (x, y) - \nabla_{x} u (x) = \nabla_{y} u_{1} (x, y) in Ω \times M . \end{matrix}$ Therefore the two-scale limit $K (x, y)$ is given by $\begin{matrix} (2.26) & K (x, y) = \nabla_{x} u (x) + \nabla_{y} u_{1} (x, y) in Ω \times M, u_{1} \in L^{2} (Ω; H_{#}^{1} (M)) . \end{matrix}$

We are now in a position to exhibit a corrector, i.e. an appropriate test function allowing us to pass to the limit $ε \to 0$ in (1.9).

We take a test function ϕ in (1.9) in the following form: $\begin{matrix} (2.27) & ϕ (x) = \bar{u} (x) + ε {\bar{u}}_{1} (x, \frac{x}{ε}) + \frac{ε}{α_{ε}} \bar{v} (x, \frac{x}{ε}), \end{matrix}$ with $\bar{u} \in H_{0}^{1} (Ω)$ , ${\bar{u}}_{1} \in D (Ω; C_{#}^{\infty} (M))$ , $\bar{v} \in D (Ω; H_{0}^{1} (B))$ . We extend ${\bar{u}}_{1} (x, \cdot)$ by periodicity to the whole of $R^{3}$ and $\bar{v} (x, \cdot)$ by zero outside B and then by periodicity to the whole of $R^{3}$ . Denoting by $^{t} X$ the transpose of the matrix X, we get: $\begin{array}{l} \int_{Ω} u_{ε} (\bar{u} (x) + ε {\bar{u}}_{1} (x, \frac{x}{ε}) + \frac{ε}{α_{ε}} \bar{v} (x, \frac{x}{ε})) + \nabla u_{ε} {χ_{M_{ε}}}^{t} A^{ε} (x) (\nabla \bar{u} (x) + ε \nabla_{x} {\bar{u}}_{1} (x, \frac{x}{ε}) + \nabla_{y} {\bar{u}}_{1}) \\ + α_{ε}^{2} \nabla u_{ε} {χ_{B_{ε}}}^{t} A^{ε} (x) (\nabla \bar{u} (x) + ε \nabla_{x} {\bar{u}}_{1} (x, \frac{x}{ε}) + \nabla_{y} {\bar{u}}_{1} (x, \frac{x}{ε}) \\ + \frac{ε}{α_{ε}} \nabla_{x} \bar{v} (x, \frac{x}{ε}) + \frac{1}{α_{ε}} \nabla_{y} \bar{v} (x, \frac{x}{ε})) d x \\ (2.28) & = \int_{Ω} f (x) (\bar{u} (x) + ε {\bar{u}}_{1} (x, \frac{x}{ε}) + \frac{ε}{α_{ε}} \bar{v} (x, \frac{x}{ε})) d x . \end{array}$

Thanks to the assumption (1.7) which makes the coefficients of the matrix A to be admissible for the strong two-scale convergence (see [1,15]), one can use the convergences proved above to pass to the limit in (2.28); we then obtain the following equation $\begin{array}{l} \int_{Ω \times Y} (u (x) + v (x, y)) (\bar{u} (x) + \frac{1}{α} \bar{v} (x, y)) d y d x \\ + \int_{Ω \times Y} χ_{M} (y) A (x, y) (\nabla u (x) + \nabla_{y} u_{1} (x, y)) (\nabla \bar{u} (x) + \nabla_{y} {\bar{u}}_{1} (x, y)) d y d x \\ + \int_{Ω \times Y} χ_{B} (y) A (x, y) α \nabla_{y} v (x, y) \nabla_{y} \bar{v} (x, y) d y d x \\ (2.29) & = \int_{Ω \times Y} f (x) (\bar{u} (x) + \frac{1}{α} \bar{v} (x, y)) d y d x . \end{array}$

By a classical density argument, we can extend system (2.29) to any test functions $(\bar{u}, {\bar{u}}_{1}, \bar{v}) \in H_{0}^{1} (Ω) \times L^{2} (Ω; H_{#}^{1} (M)) \times L^{2} (Ω; H_{0}^{1} (B))$ .

To prove that (2.29) is a well posed problem, we need to establish the following lemma.

Lemma 2.2.
The two-scale limit $A (x, y)$ of the sequence $A^{ε}$ is a bounded and coercive matrix: for a.e. $(x, y) \in Ω \times Y$ and for all $ξ \in R^{3}$ , $\begin{matrix} (2.30) & A (x, y) ξ ξ ⩾ α | ξ |^{2} and | A (x, y) ξ | ⩽ \frac{β^{2}}{α} | ξ |, \end{matrix}$ where α and β are the constants defined in ( 1.5 )–( 1.6 ).
Proof.
The proof of the lemma follows ideas from [14] (see also [19]).

By the use of inequality (1.5), we infer for all $ϕ \in C_{#}^{\infty} (Y)$ , for all non-negative function $ψ \in L^{2} (Ω; C_{#}^{\infty} (Y))$ , for almost all $(x, y) \in Ω \times Y$ and for all ε, $\begin{matrix} (2.31) & A^{ε} (x) \nabla_{y} ϕ (\frac{x}{ε}) \nabla_{y} ϕ (\frac{x}{ε}) ψ (x, \frac{x}{ε}) ⩾ α {| \nabla_{y} ϕ (\frac{x}{ε}) |}^{2} ψ (x, \frac{x}{ε}) . \end{matrix}$ Integrating (2.31) and then passing to the limit $ε \to 0$ , we get $\begin{matrix} (2.32) & \int_{Ω} \int_{Y} A (x, y) \nabla_{y} ϕ (y) \nabla_{y} ϕ (y) ψ (x, y) d x d y ⩾ α \int_{Ω} \int_{Y} {| \nabla_{y} ϕ (y) |}^{2} ψ (x, y) d x d y . \end{matrix}$

In particular, we can choose in (2.32) a test function ψ such that $ψ (x, \cdot)$ has a compact support and ϕ such that $ϕ (y) = ξ \cdot y$ on the support of ψ. Hence, we deduce that $\begin{matrix} A (x, y) ξ ξ ⩾ α | ξ |^{2} a.e. (x, y) \in Ω \times Y . \end{matrix}$ By a similar way, starting from the inequality $\begin{matrix} {(A^{ε})}^{- 1} ξ ξ ⩾ \frac{α}{β^{2}} | ξ |^{2}, \end{matrix}$ which is a consequence of the two inequalities (1.5) and (1.6) (see the proof in [14], Proposition 4), one can prove the second inequality in (2.30). This ends the proof of Lemma 2.2. □
Proof of the uniqueness of problem (2.29).
The matrix A being coercive and bounded, the existence and uniqueness of the solution of (2.29) is an immediate consequence of the Lax–Milgram Theorem since the left hand side of (2.29) defines a coercive continuous bilinear form on the space $H_{0}^{1} (Ω) \times L^{2} (Ω; H_{#}^{1} (M)) \times L^{2} (Ω; H_{0}^{1} (B))$ while the right hand side defines a continuous linear form. □

We now seek for a formulation of the homogenized problem involving only the macroscopic variable x. Taking $\bar{u} = {\bar{u}}_{1} = 0$ in (2.29) and using an obvious localization argument we obtain that for almost every $x \in Ω$ , the function $v (x, \cdot)$ is the unique solution of $\begin{matrix} (2.33) & \{\begin{matrix} v (x, \cdot) \in H_{0}^{1} (B), \\ \int_{B} ((u (x) + v (x, y)) \frac{1}{α} \bar{v} (y) + α A (x, y) \nabla_{y} v (x, y) \nabla_{y} \bar{v} (y)) d y = \int_{B} f (x) \frac{1}{α} \bar{v} (y) d y, \\ \forall \bar{v} \in H_{0}^{1} (B) . \end{matrix} \end{matrix}$ One can check easily that $v (x, y)$ is given in terms of u and f by $\begin{matrix} (2.34) & v (x, y) = (f (x) - u (x)) \hat{v} (x, y), \end{matrix}$ where $\hat{v} \in L^{\infty} (Ω; H_{0}^{1} (B))$ is the unique solution of $\begin{matrix} (2.35) & \{\begin{matrix} a.e. x \in Ω, \\ \hat{v} (x, \cdot) - α^{2} {div}_{y} A (x, \cdot) \nabla_{y} \hat{v} (x, \cdot) = 1, \\ \hat{v} (x, \cdot) \in H_{0}^{1} (B) . \end{matrix} \end{matrix}$

We now seek for $u_{1}$ in terms of u. We choose $\bar{v} = \bar{u} = 0$ in equation (2.29) so that $u_{1}$ is the unique solution of the equation $\begin{matrix} (2.36) & \{\begin{matrix} u_{1} \in L^{2} (Ω; H_{#}^{1} (M)), \\ \int_{Ω \times M} A (x, y) (\nabla u (x) + \nabla_{y} u_{1} (x, y)) \nabla_{y} {\bar{u}}_{1} (x, y) d y d x = 0, \\ \forall {\bar{u}}_{1} \in L^{2} (Ω; H_{#}^{1} (M)) . \end{matrix} \end{matrix}$

It is then quite easy to check (see also [1] and [4]) by the use of a similar localization argument used to establish (2.34) that $u_{1}$ is given by $\begin{matrix} (2.37) & u_{1} (x, y) = \sum_{i = 1}^{3} \frac{\partial u}{\partial x_{i}} (x) w_{i} (x, y), \end{matrix}$ where $w_{i} (x, y) \in L^{\infty} (Ω; H_{#}^{1} (M))$ is the unique solution of $\begin{matrix} (2.38) & \{\begin{matrix} a.e. x \in Ω, \\ - {div}_{y} A (x, \cdot) (\nabla_{y} w_{i} (x, \cdot) + e_{i}) = 0 & in M, \\ w_{i} (x, \cdot) \in H_{#}^{1} (M), \\ A (x, \cdot) (\nabla_{y} w_{i} (x, \cdot) + e_{i}) \cdot n = 0 & on \partial B, \end{matrix} \end{matrix}$ where ${(e_{i})}_{i = 1, 2, 3}$ denotes the canonical basis of $R^{3}$ .

The final form of the homogenized problem is now easy to get thanks to (2.34) and (2.37). Indeed, taking $\bar{v} = {\bar{u}}_{1} = 0$ in (2.29) and using (2.34) and (2.37), we get: $\begin{array}{l} \int_{Ω \times Y} ((u (x) + (f (x) - u (x)) \hat{v} (x, y)) \bar{u} (x) \\ + χ_{M} (y) A (x, y) (\nabla u (x) + \sum_{i = 1}^{3} \frac{\partial u}{\partial x_{i}} (x) \nabla_{y} w_{i} (x, y)) \nabla \bar{u} (x)) d y d x \\ (2.39) & = \int_{Ω \times Y} f (x) \bar{u} (x) d y d x \forall \bar{u} \in H_{0}^{1} (Ω) . \end{array}$

Defining the function $θ (x)$ and the entries of the matrix $A^{0} (x)$ by: $\begin{array}{l} (2.40) & \begin{aligned} θ (x) = \int_{B} \hat{v} (x, y) d y, \\ A_{i, j}^{0} (x) = \int_{M} A (x, y) (\nabla_{y} w_{j} (x, y) + e_{j}) (\nabla_{y} w_{i} (x, y) + e_{i}) d y, i, j = 1, 2, 3, \end{aligned} \end{array}$ we see that u is the unique solution of the problem $\begin{matrix} (2.41) & \{\begin{matrix} u \in H_{0}^{1} (Ω), \\ (1 - θ (x)) u (x) - div A^{0} (x) \nabla u (x) = (1 - θ (x)) f (x) & in Ω . \end{matrix} \end{matrix}$ Since the coerciveness of the matrix $A^{0}$ is a well known result, for instance see [4], to prove the uniqueness of the solution of (2.41), it is sufficient to prove that $(1 - θ (x)) ⩾ 0$ a.e. $x \in Ω$ . We obtain the last property by the use of the maximum principle for the solution $\hat{v}$ of (2.35). Indeed, according to the maximum principle, one has $\begin{matrix} (2.42) & a.e. x \in Ω, θ (x) : = \int_{B} \hat{v} (x, y) d y < \int_{B} 1 d y = | B | < 1, \end{matrix}$ so that $\begin{matrix} (2.43) & (1 - θ (x)) > 0 a.e. x \in Ω . \end{matrix}$ Note that the first strict inequality in (2.42) is justified by the fact that 1 is not the solution of (2.35) so that we have $\hat{v} (x, \cdot) < 1$ for almost every $x \in Ω$ . The second strict inequality in (2.42) is due to the assumption $| Y | = 1$ . For arbitrary values of $| Y |$ , instead of that inequality, one would obtain $\begin{matrix} (2.44) & | B | < | Y | . \end{matrix}$ Inequality (2.44) is therefore exactly the one we should have to prove since for general values of $| Y |$ , we have to replace in equation (2.41) the coefficient $1 - θ (x)$ by $| Y | - θ (x)$ .
2.3. The limit problem in the case ${lim}_{ε \to 0} \frac{α_{ε}}{ε} = 0$

The assumption $α_{ε} = o (ε)$ leads to a lack of compactness since the sequence $ε \nabla u_{ε}$ is no longer bounded in $L^{2} (Ω)$ . Using the same calculations as the ones used in the previous subsection, one check that only the first convergence in (2.9) may be kept by virtue of the boundedness of $u_{ε}$ in $L^{2} (Ω)$ while convergence (2.20) takes now the form $\begin{matrix} (2.45) & u_{ε} ⇀ ⇀ v (x, y) + u (x), u \in H_{0}^{1} (Ω), v \in L^{2} (Ω \times Y), v = 0 in Ω \times M . \end{matrix}$ Of course the convergences related to the “stiff” part $M_{ε}$ are not affected by the lack of compactness which arises in the soft part $B_{ε}$ so that we still have the following convergence obtained from (2.25) and (2.26) (recall that K is the two-scale limit of $\nabla u_{ε} χ_{M_{ε}}$ in (2.25)): $\begin{matrix} (2.46) & \nabla u_{ε} χ_{M_{ε}} ⇀ ⇀ (\nabla_{x} u (x) + \nabla_{y} u_{1} (x, y)) χ_{M} (y), u \in H_{0}^{1} (Ω), u_{1} \in L^{2} (Ω; H_{#}^{1} (M)) . \end{matrix}$ Hence, it remains only to find the two-scale limit of the sequence $α_{ε} \nabla u_{ε} χ_{B_{ε}}$ .

For that aim, we consider the sequence $(\frac{α_{ε}}{ε} u_{ε})$ which converges strongly to zero in $L^{2} (Ω)$ since $\frac{α_{ε}}{ε}$ goes to zero while $u_{ε}$ is bounded in $L^{2} (Ω)$ . Hence, it also two-scale converges to zero. On the other hand, $ε \nabla (\frac{α_{ε}}{ε} u_{ε}) = α_{ε} \nabla u_{ε}$ is bounded in $L^{2} (Ω)$ . Hence, there exists $h (x, y) \in L^{2} (Ω \times Y)$ such that $(\frac{α_{ε}}{ε} u_{ε}) ⇀ ⇀ h (x, y)$ and $ε \nabla (\frac{α_{ε}}{ε} u_{ε}) ⇀ ⇀ \nabla_{y} h (x, y)$ . Thus $h = 0$ and $α_{ε} \nabla u_{ε} ⇀ ⇀ 0$ . As a consequence one has $\begin{matrix} (2.47) & α_{ε} \nabla u_{ε} χ_{B_{ε}} ⇀ ⇀ 0 . \end{matrix}$ The previous calculations suggest us to take a test function in (1.9) in the form $\begin{matrix} (2.48) & ϕ (x) = \bar{u} (x) + ε {\bar{u}}_{1} (x, \frac{x}{ε}) + \bar{v} (x, \frac{x}{ε}), \end{matrix}$ with $\begin{matrix} (2.49) & \bar{u} \in H_{0}^{1} (Ω), {\bar{u}}_{1} \in D (Ω; H_{#}^{1} (M)), \bar{v} \in D (Ω \times B), \end{matrix}$ assuming $\bar{v} (x, \cdot)$ extended by zero and by periodicity to the whole of $R^{3}$ .

Equation (2.28) then becomes $\begin{array}{l} \int_{Ω} u_{ε} (\bar{u} (x) + ε {\bar{u}}_{1} (x, \frac{x}{ε}) + \bar{v} (x, \frac{x}{ε})) + \nabla u_{ε} {χ_{M_{ε}}}^{t} A^{ε} (x) (\nabla \bar{u} (x) + ε \nabla_{x} {\bar{u}}_{1} (x, \frac{x}{ε}) + \nabla_{y} {\bar{u}}_{1}) \\ + α_{ε}^{2} \nabla u_{ε} {χ_{B_{ε}}}^{t} A^{ε} (x) (\nabla \bar{u} (x) + ε \nabla_{x} {\bar{u}}_{1} (x, \frac{x}{ε}) + \nabla_{y} {\bar{u}}_{1} (x, \frac{x}{ε}) + \nabla_{x} \bar{v} + \frac{1}{ε} \nabla_{y} \bar{v} (x, \frac{x}{ε})) d x \\ (2.50) & = \int_{Ω} f (x) (\bar{u} (x) + ε {\bar{u}}_{1} (x, \frac{x}{ε}) + \bar{v} (x, \frac{x}{ε})) d x . \end{array}$ Thanks to the assumptions $α_{ε} \to 0$ and $\frac{α_{ε}}{ε} \to 0$ , one can pass to the limit in (2.50) with the help of (2.47) to get $\begin{array}{l} \int_{Ω \times Y} (u (x) + v (x, y)) (\bar{u} (x) + \bar{v} (x, y)) d y d x \\ + \int_{Ω \times Y} χ_{M} (y) A (x, y) (\nabla u (x) + \nabla_{y} u_{1} (x, y)) (\nabla \bar{u} (x) + \nabla_{y} {\bar{u}}_{1} (x, y)) d y d x \\ (2.51) & = \int_{Ω \times Y} f (x) (\bar{u} (x) + \bar{v} (x, y)) d y d x . \end{array}$ By a density argument, equation (2.51) can be extended to all test functions $\bar{u} \in H_{0}^{1} (Ω)$ , ${\bar{u}}_{1} \in L^{2} (Ω; H_{#}^{1} (Y))$ , ${\bar{u}}_{1} = 0$ in B, $\bar{v} \in L^{2} (Ω \times B)$ . Choosing $\bar{u} = {\bar{u}}_{1} = 0$ in (2.51), we obtain easily that $\begin{matrix} (2.52) & v (x, y) = f (x) - u (x) in Ω \times B . \end{matrix}$ Of course, $u_{1}$ is still given by (2.37) so that taking ${\bar{u}}_{1} = \bar{v} = 0$ in (2.51) and replacing $u_{1}$ and v by their expression (2.37) and (2.52), we obtain the following homogenized problem under the assumption $α_{ε} = o (ε)$ : $\begin{matrix} (2.53) & \{\begin{matrix} u \in H_{0}^{1} (Ω), \\ (1 - | B |) u (x) - div A^{0} (x) \nabla u (x) = (1 - | B |) f (x) & in Ω, \end{matrix} \end{matrix}$ where $A^{0}$ is the matrix defined in (2.40). Remark that if we do not assume $| Y | = 1$ , then $(1 - | B |)$ would be replaced by $(| Y | - | B |)$ in (2.53). The last number being strictly positive, the uniqueness of the solution of (2.53) still holds true.

We now study the last case.

2.4. The limit problem in the case ${lim}_{ε \to 0} \frac{α_{ε}}{ε} = + \infty$

The present case is the one for which the compactness is the best compared with the two previous cases. Indeed, the estimate $‖ α_{ε} \nabla u_{ε} χ_{B_{ε}} ‖_{{(L^{2} (Ω))}^{3}} ⩽ C$ is stronger than the estimate $‖ ε \nabla u_{ε} χ_{B_{ε}} ‖_{{(L^{2} (Ω))}^{3}} ⩽ C$ . Hence one can expect to get an homogenized equation which looks like the one corresponding to the classical case $‖ \nabla u_{ε} χ_{B_{ε}} ‖_{{(L^{2} (Ω))}^{3}} ⩽ C$ . It is actually the case as it will be shown below.

Since the estimate $‖ ε \nabla u_{ε} χ_{B_{ε}} ‖_{{(L^{2} (Ω))}^{3}} ⩽ C$ still holds true as in the critical case, we keep all the results obtained there. In particular all the results related to the stiff part $M_{ε}$ remain true so that we have only to focus on the soft part $B_{ε}$ .

We first prove that the function v arising in (2.20) is now equal to zero. Indeed, from the boundedness in ${(L^{2} (Ω))}^{3}$ of the sequence $α_{ε} \nabla u_{ε} χ_{B_{ε}}$ , we get thanks to the assumption ${lim}_{ε \to 0} \frac{α_{ε}}{ε} = + \infty$ , $\begin{matrix} (2.54) & ε \nabla u_{ε} χ_{B_{ε}} = \frac{ε}{α_{ε}} α_{ε} \nabla u_{ε} χ_{B_{ε}} \to 0 strongly in {(L^{2} (Ω))}^{3} . \end{matrix}$ On the other hand, by virtue of the second convergence in (2.16), $\begin{matrix} (2.55) & ε \nabla u_{ε} χ_{B_{ε}} ⇀ ⇀ \nabla_{y} v (x, y) χ_{B} (y) . \end{matrix}$ Hence, $v = 0$ .

We now want to find the two-scale limit of the sequence $α_{ε} \nabla u_{ε} χ_{B_{ε}}$ which is not straightforward and a refined estimate is needed. We prove it in the following lemma.

Lemma 2.3.

Let $w_{ε}$ be the sequence defined by $\begin{matrix} (2.56) & w_{ε} = \sum_{i \in I_{ε}} \frac{α_{ε}}{ε} (u_{ε} - \frac{1}{| B_{ε}^{i} |} \int_{B_{ε}^{i}} u_{ε} d x) χ_{B_{ε}^{i}} . \end{matrix}$ Then there exist a constant $C > 0$ and $w \in L^{2} (Ω; H^{1} (B))$ such that $\begin{matrix} (2.57) & {‖ w^{ε} ‖}_{L^{2} (Ω)} ⩽ C, \end{matrix}$ and (for a subsequence), $\begin{matrix} (2.58) & w_{ε} ⇀ ⇀ w and α_{ε} \nabla u_{ε} χ_{B_{ε}} ⇀ ⇀ \nabla_{y} w (x, y) χ_{B} (y) . \end{matrix}$

Proof.

We use the Poincaré–Wirtinger inequality in the ball B: there exists a constant C such that: $\begin{matrix} (2.59) & \int_{B} {(ϕ - \frac{1}{| B |} \int_{B} ϕ d y)}^{2} d y ⩽ C \int_{B} | \nabla ϕ |^{2} d y, \forall ϕ \in H^{1} (B) . \end{matrix}$ Taking in (2.59) $ϕ (y) = u_{ε} (ε y + ε i)$ for $y \in B$ and using the change of variable $x = ε y + ε i$ , we are led to the inequality: $\begin{matrix} (2.60) & \int_{Ω} {| u_{ε} - \frac{1}{| B_{ε}^{i} |} \int_{B_{ε}^{i}} u_{ε} d x |}^{2} χ_{B_{ε}^{i}} (x) d x ⩽ C \int_{B_{ε}^{i}} ε^{2} | \nabla u_{ε} |^{2} d x . \end{matrix}$ Inequality (2.60) may be rewritten as $\begin{matrix} (2.61) & \int_{Ω} {| u_{ε} - \frac{1}{| B_{ε}^{i} |} \int_{B_{ε}^{i}} u_{ε} d x |}^{2} χ_{B_{ε}^{i}} (x) d x ⩽ C \frac{ε^{2}}{α_{ε}^{2}} \int_{B_{ε}^{i}} α_{ε}^{2} | \nabla u_{ε} |^{2} d x . \end{matrix}$ Therefore to get estimate (2.57), it suffices to divide the two sides of (2.61) by $\frac{ε^{2}}{α_{ε}^{2}}$ , to sum up over $i \in I_{ε}$ and to take into account estimate (2.1).

Hence, there exists $w \in L^{2} (Ω \times Y)$ such that up to a subsequence, $\begin{matrix} (2.62) & w^{ε} ⇀ ⇀ w, \end{matrix}$ which is the first convergence arising in (2.58). We now prove that $w \in L^{2} (Ω; H^{1} (B))$ and that the two-scale limit G of the sequence $α_{ε} \nabla u_{ε} χ_{B_{ε}}$ is the gradient with respect to y of w so that the second convergence in (2.58) holds true.

Let $ϕ (x, y) \in D {(Ω; D (D))}^{3}$ being $ϕ (x, \cdot)$ extended by zero to Y and then by periodicity to the whole of $R^{3}$ . Bearing in mind (2.57), we get $\begin{array}{rcl} \int_{Ω} α_{ε} \nabla u_{ε} χ_{B_{ε}} ϕ (x, \frac{x}{ε}) d x & = & \sum_{i \in I_{ε}} \int_{B_{ε}^{i}} α_{ε} \nabla u_{ε} ϕ (x, \frac{x}{ε}) d x \\ = & \sum_{i \in I_{ε}} \int_{B_{ε}^{i}} α_{ε} \nabla (u_{ε} - \frac{1}{| B_{ε}^{i} |} \int_{B_{ε}^{i}} u_{ε} d x) ϕ (x, \frac{x}{ε}) d x \\ = & - \sum_{i \in I_{ε}} \int_{B_{ε}^{i}} α_{ε} (u_{ε} - \frac{1}{| B_{ε}^{i} |} \int_{B_{ε}^{i}} u_{ε} d x) ({div}_{x} ϕ + \frac{1}{ε} {div}_{y} ϕ) d x \\ (2.63) & = & - \int_{Ω} (ε w^{ε} {div}_{x} ϕ + w^{ε} {div}_{y} ϕ) d x . \end{array}$

Hence, passing to the limit in (2.63), we get with the help of (2.62) $\begin{array}{l} \int_{Ω \times Y} G (x, y) χ_{B} ϕ (x, y) d y d x \\ (2.64) & = - \int_{Ω \times Y} w (x, y) {div}_{y} ϕ (x, y) d y d x, \forall ϕ \in D {(Ω; D (B))}^{3}, \end{array}$ from which we deduce $\begin{matrix} (2.65) & \nabla_{y} w \in {(L^{2} (Ω \times B))}^{3} and G (x, y) = \nabla_{y} w in Ω \times B . \end{matrix}$ The proof of the lemma is now complete. □

We are now in a position to pass to the limit in (1.9) under the assumption ${lim}_{ε \to 0} \frac{α_{ε}}{ε} = + \infty$ . We choose in (1.9) a test function ϕ in the form $\begin{matrix} (2.66) & ϕ (x) = \bar{u} (x) + ε {\bar{u}}_{1} (x, \frac{x}{ε}) + \frac{ε}{α_{ε}} \bar{w} (x, \frac{x}{ε}), \end{matrix}$ where $(\bar{u}, {\bar{u}}_{1}, \bar{w}) \in H_{0}^{1} (Ω) \times D (Ω; H_{#}^{1} (Y)) \times D (Ω; D (\overline{B}))$ . The function $\bar{w} (x, \cdot)$ is extended by zero outside $\overline{B}$ and then by Y-periodicity to the whole of $R^{3}$ . We get $\begin{array}{l} \int_{Ω} u_{ε} (\bar{u} (x) + ε {\bar{u}}_{1} (x, \frac{x}{ε}) + \frac{ε}{α_{ε}} \bar{w} (x, \frac{x}{ε})) + \nabla u_{ε} {χ_{M_{ε}}}^{t} A^{ε} (x) (\nabla \bar{u} (x) + ε \nabla_{x} {\bar{u}}_{1} (x, \frac{x}{ε}) + \nabla_{y} {\bar{u}}_{1}) \\ + α_{ε}^{2} \nabla u_{ε} {χ_{B_{ε}}}^{t} A^{ε} (x) (\nabla \bar{u} (x) + ε \nabla_{x} {\bar{u}}_{1} (x, \frac{x}{ε}) + \nabla_{y} {\bar{u}}_{1} (x, \frac{x}{ε}) \\ + \frac{ε}{α_{ε}} \nabla_{x} \bar{w} + \frac{1}{α_{ε}} \nabla_{y} \bar{w} (x, \frac{x}{ε})) d x \\ (2.67) & = \int_{Ω} f (x) (\bar{u} (x) + ε {\bar{u}}_{1} (x, \frac{x}{ε}) + \frac{ε}{α_{ε}} \bar{w} (x, \frac{x}{ε})) d x . \end{array}$ Taking the limit $ε \to 0$ in the previous equation and bearing in mind the assumption $lim \frac{ε}{α_{ε}} = 0$ together with (2.65), we get $\begin{array}{l} \int_{Ω \times Y} u (x) \bar{u} (x) d x + \int_{Ω \times Y} (χ_{M} (y) A (x, y) (\nabla u (x) + \nabla_{y} u_{1} (x, y)) (\nabla \bar{u} (x) + \nabla_{y} {\bar{u}}_{1} (x, y)) \\ + χ_{B} (y) A (x, y) \nabla_{y} w (x, y) \nabla_{y} \bar{w} (x, y)) d y d x \\ (2.68) & = \int_{Ω \times Y} f (x) \bar{u} (x) d y d x, \forall \bar{u} \in H_{0}^{1} (Ω) . \end{array}$ Choosing in (2.68) $\bar{u} = {\bar{u}}_{1} = 0$ and $\bar{w} = w$ , we infer thanks to the coerciveness of the matrix A that $w = 0$ . On the other hand, $u_{1}$ is still given in terms of u by formula (2.37) so that the final homogenized equation is $\begin{matrix} (2.69) & \{\begin{matrix} u \in H_{0}^{1} (Ω), \\ u (x) - div A^{0} (x) \nabla u (x) = f (x) & in Ω, \end{matrix} \end{matrix}$ where the matrix $A^{0}$ is given by (2.40). Hence the form of the homogenized equation is very close to the well known equation in the classical case where the sequence $u_{ε}$ is bounded in $H_{0}^{1} (Ω)$ , the only difference being on the values of the entries of $A^{0}$ which are related to the cell equation (2.38) posed in the part M of Y and not longer in Y.

Note that for general values of $| Y |$ , equation (2.69) must be replaced by the equation $\begin{matrix} (2.70) & \{\begin{matrix} u \in H_{0}^{1} (Ω), \\ | Y | u (x) - div A^{0} (x) \nabla u (x) = | Y | f (x) & in Ω, \end{matrix} \end{matrix}$ as seen through equation (2.68).

The proof of Theorem 1.1 is now complete.

References

Allaire , Homogenization and two-scale convergence, SIAM J. Math. Anal. 23(6) (1992), 1482–1518. doi:10.1137/0523084.

Allaire and

Murat , Homogenization of the Neumann problem with nonisolated holes, Asymp. Anal. 7 (1993), 81–95.

Arbogast ,

Douglas and

Hornung , Derivation of the double porosity model of single phase flow via homogenization theory, SIAM J. Math. Anal. 21 (1990), 823–836. doi:10.1137/0521046.

Bensoussan ,

J.-L.

Lions and

Papanicolaou , Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, 1978.

Briane ,

Braides and

Casado-Diaz , Homogenization of non-uniformly bounded periodic diffusion energies in dimension two, Nonlinearity 22(6) (2009), 1459–1480. doi:10.1088/0951-7715/22/6/010.

Briane and

Casado-Diaz , Uniform convergence of sequences of solutions of two-dimensional linear elliptic equations with unbounded coefficients, J. Diff. Equat. 845(8) (2008), 2038–2054. doi:10.1016/j.jde.2008.07.027.

Camar-Eddine and

Seppecher , Closure of the set of diffusion functionals with respect to the Mosco-convergence, Math. Models Methods Appl. Sci. 12(8) (2002), 1153–1176. doi:10.1142/S0218202502002069.

Charef and

Sili , The effective conductivity equation for a highly heterogeneous periodic medium, Ricerche di Matematica 61(2) (2012), 231–244. doi:10.1007/s11587-011-0126-9.

Cherdantsev and

K.D.

Cherednichenko , Two-scale Γ-convergence of integral functionals and its application to homogenization of nonlinear high-contrast periodic composites, Arch. Rat. Mech. Anal. 204 (2012), 445–478. doi:10.1007/s00205-011-0481-4.

10.

Cioranescu ,

Damlamian and

Griso , The periodic unfolding method in homogenization, SIAM J. Math. Anal. 40 (2008), 1585–1620. doi:10.1137/080713148.

11.

Cioranescu and

Saint Jean Paulin , Homogenization in open sets with holes, J. Math. Anal. Appl. 71 (1979), 590–607. doi:10.1016/0022-247X(79)90211-7.

12.

Cioranescu and

Saint Jean Paulin , Homogenization of Reticulated Structures, Springer, 1999.

13.

Lukkassen ,

Nguetseng ,

Nnang and

Wall , Reiterated homogenization of nonlinear monotone operators in a general deterministic setting, J. Funct. Spaces Appl. 7(2) (2009), 121–152. doi:10.1155/2009/102486.

14.

Murat and

Tartar , H-convergence, in: Topics in the Mathematical Modelling of Composite Materials,

Cherkaev and

Kohn , eds, Birkhauser, 1997.

15.

Nguetseng , A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal. 20 (1989), 608–629. doi:10.1137/0520043.

16.

Sili , Homogenization of a nonlinear monotone problem in an anisotropic medium, Math. Models Methods Appl. Sci. 14(3) (2004), 329–353. doi:10.1142/S0218202504003258.

17.

Sili , A diffusion equation through a highly heterogeneous medium, Applic. Analysis 89 (2010), 893–904. doi:10.1080/00036811003717962.

18.

Sili , Diffusion through a composite structure with a high contrasting diffusivity, Asymp. Anal. 89 (2014), 173–187.

19.

Tartar , The General Theory of Homogenization, a Personalized Introduction, Lecture Notes of the Unione Matematica Italiana, Vol. 7, Springer-Verlag, Berlin, UMI, Bologna, 2009.