A remark about the periodic homogenization of certain composite fibered media

1.   Introduction

During the last years, the study of the homogenization of composite heterogeneous media has given rise to an extensive literature and a significant part of that works was devoted to the homogenization of media characterized by high heterogeneities, (see [2], [3]), [4], [5], [6], [8], [14]). The pioneer work for problems of this kind was done in [2] in the study of the double porosity model of single phase flow. Thereafter, the main idea of [2] was taken up in [1] and [12] to give rise to the two-scale convergence method which is a variant of the energy method [17]. Fibered media is an example of composite heterogeneous media with high contrasting properties since usually the material constituting the fibers is very different from the material around it. For instance, in elasticity one can consider rigid fibers immersed in a soft matrix while in the framework of heat conduction one can consider fibers with high conductivity surrounded by a material with a low conductivity. Among the first works devoted to the homogenization of such composite media one can quote [7] where the homogenization process was performed using asymptotic expansions.

From the mathematical point of view, the contrast between the properties of the two materials leads to a degenerate problem in the sense that in general it leads to a lack of compactness. Indeed, the operators under consideration are in general not uniformly bounded (see [5]) or not uniformly coercive with respect to the small parameter (see [16]).

In general, the configuration domain of such media may be described by a domain $\Omega$ of $\mathbb R^3$ which is a periodic replication with a period of size $\varepsilon$ of a set $Y_i^ \varepsilon, \, i \in I_ \varepsilon$. More precisely, $Y_i^ \varepsilon$ is assumed to be the union of a fiber $F_i^ \varepsilon$ with its complement $M_i^ \varepsilon$ (where $M$ stands for "matrix") in $Y_i^ \varepsilon$.

It is well known that the homogenization process in such degenerate problems gives rise to homogenized problems with a different form from the equation at the microscopic level since memory effects, strange terms or nonlocal effects may appear at the limit (see [3], [5], [6], [8], [9], [11], [16]).

The aim of the present work is to show that in the case of fibered media such effects at the limit are not due to the homogenization process itself but to the local structure of the composite media; more precisely, we show that the form of the homogenized problem is already determined by the study of the $3d-1d$ reduction of dimension which occurs locally. To illustrate that, we give here two examples in the framework of the thermal conduction. In the first one we show that one can obtain an extra term (or a strange term, see [11], [3]) in the study of the reduction of dimension $3d-1d$ (see [13]). To prove that result, we consider in section 2 below a thin domain $\Omega^ \varepsilon : = \varepsilon Y \times (0,L)$ as a copy of the local cell arising in a periodic fibered medium for which the configuration domain $\Omega$ is the union of $\frac{1}{ \varepsilon^2}$ such cells, in other words, $\Omega : = \bigcup\limits_{ i \in I_ \varepsilon } \bigl( ( \varepsilon Y + \varepsilon i )\times (0,L) \bigr)$. In fact, in this example the "fibers" play the role of vertical parallel holes and we prove that a strange term already appears at the limit in the $3d-1d$ problem under exactly the same assumptions on the size of the hole as the ones assumed in the corresponding homogenization problem. Furthermore, it is shown that the structure of the homogenized problem is very close to the one of the limit problem obtained in the reduction of dimension $3d-1d$. The homogenization problem will be considered in section 3.

In section 4, we consider another example for which the reduction of dimension $3d-1d$ leads to a limit problem involving a nonlocal effect; the comparison with the corresponding homogenization problem is made in section 5 and once again the similarity between the two limit problems is pointed out.

One can explain the similarity between the homogenization problem and the reduction of dimension problem by the fact that the geometry of the fibered medium is such that the homogenization process implies a local reduction of dimension so that for such media, the homogenization may be viewed as a repetition of local reductions of dimension. More precisely, when we assume that the domain $\Omega$ is made from a single cell of size $\varepsilon$, $\Omega = \Omega^ \varepsilon : = Y^ \varepsilon : = \varepsilon Y \times (0,L)$ where for example $Y$ is the square defined by $Y: = \bigl] -{1\over 2}, {1\over 2} \bigr[^2$ and $L > 0$, one can denote the variable $x$ in $\Omega^ \varepsilon$ by $x = (x', x_3) = ( \varepsilon y, x_3), \, y \in Y, \, x_3 \in (0, L),$ in such a way that $x' = \varepsilon y$. This classical dilation transforms the reduction of dimension problem posed in $\Omega^ \varepsilon$ into a singular perturbation problem posed on the fixed domain $\Omega$ (see [11]) and the limit problem (as $\varepsilon$ goes to zero) is written first in terms of $(y,x_3)$ and then eliminating the variable $y$, one can obtain a limit problem written in terms of $x_3$. When we study the periodic homogenization with $\Omega : = \bigcup\limits_{ i \in I_ \varepsilon } \bigl( ( \varepsilon Y + \varepsilon i )\times (0,L) \bigr)$, the variable $x$ in $\Omega$ writes as $x = (x', x_3)$ with $x' = \varepsilon y + \varepsilon i, \, y \in Y, \, i \in I_ \varepsilon, \, x_3 \in (0, L)$. It is well known (see [1], [10]) that the homogenized problem may be formulated in terms of the variables $(y, x)$. Of course, it is not reasonable to expect a homogenized problem involving only $x_3$ as a macroscopic variable since we have many cells distributed in the plane $x' = (x_1, x_2)$ so that different reductions of dimension occur. For that reason, the role of the part $x' = (x_1,x_2)$ of the macroscopic variable in the homogenized problem is actually the one of a parameter, so that the main variables are still $(y,x_3)$ as in the $1d$- model obtained after the reduction of dimension $3d-1d$. Remarkably, dealing either with only one small cylinder or with $\frac{1}{ \varepsilon^2}$ small cylinders does not affect the form of the limit problem.

For the sake of simplicity and brevity we consider in this work the case of the Laplacian but the results remain valid for more general operators as we will show in forthcoming works.

2.   Strange term in the reduction of dimension

We consider a thin structure described as follows. Let $\varepsilon, \, r_ \varepsilon$ be two decreasing sequences of positive numbers such that $\varepsilon$ tends to zero and $\lim\limits_{ \varepsilon \to 0} \frac{r_ \varepsilon}{ \varepsilon} = 0$, and let $\hat \Omega_ \varepsilon, \, \hat F_ \varepsilon,$ be respectively the parallelepiped and the fiber defined by

We will assume the Dirichlet boundary condition holds on the part

of the boundary of $\hat\Omega_ \varepsilon$ and the Neumann boundary condition holds on the rest

of the boundary of $\hat \Omega_ \varepsilon$. Remark that ${\partial \hat\Omega^D_ \varepsilon}$ is made of the upper and the lower faces of the cylinder $\hat \Omega_ \varepsilon$ together with the lateral boundary of the fiber $\hat F_ \varepsilon$ which plays here the role of a hole. We consider the following equation:

Introduce the change of variables $x' = \varepsilon y , \, u_ \varepsilon (y, x_3) : = \bar u_ \varepsilon( \varepsilon y, x_3)$ and set

Finally, denote by $\nabla'$ the gradient with respect to the two first variables $y = (y_1, y_2)$. Similarly, $\Delta'$ denotes the Laplacian with respect to the same variables. We assume in the sequel that the source term $\hat f_ \varepsilon( \varepsilon y, x_3) = f(y,x_3$ does not depend on $\varepsilon$ with $f \in L^2(\Omega)$ so that equation (4) for $\bar u_ \varepsilon$ is transformed into an equation satisfied by $u_ \varepsilon$ with the following variational formulation

Problem (7) is well-posed. Moreover we extend $u_ \varepsilon$ by zero inside $F_ \varepsilon$ and continue to denote by$u_ \varepsilon$ this extension which belongs to $H^1_D(\Omega): = \lbrace u \in H^1(\Omega), \, \, u(y, 0) = u(y,L) = 0, \, \hbox{almost everywhere in} \, Y \rbrace$. Furthermore, taking $\phi = u_ \varepsilon$ in (7), we obtain easily that there exist $u \in H^1_0(0, L)$ (the subspace of functions in $H^1_D(\Omega)$ depending only on the variable $x_3$) and a subsequence (still denoted by $\varepsilon$) such that:

The limit function $u$ depends on the size of the hole and it is characterized by the following theorem. Indeed, defining $\tilde f(x_3) : = \int_Y f(y, x_3) \ dy$, we have the following:

Theorem 2.1.

Sketch of the proof. Introduce the following function (see [11])

Then: $\frac{\partial w_ \varepsilon }{\partial n } = \frac{1}{ r \ln( \frac{ \varepsilon}{r_ \varepsilon}) } \, \, \hbox{on} \, \, \partial D, \, \, - \Delta' w_ \varepsilon = 0 \, \, \, \hbox{in} \, \, D\setminus \frac{r_ \varepsilon}{ \varepsilon} D, \, \, w_ \varepsilon \rightharpoonup 1 \, \, \hbox{weakly in } H^1(Y)$.

Use as a test function in (7) $\phi(y,x_3) = w_ \varepsilon(y) \bar u(x_3), \, \hbox{for every} \, \bar u \in H_0^1(0,L).$ Then

Integrating by parts with respect to $y$ the first integral on the left, and bearing in mind the property $- \Delta' w_ \varepsilon = 0 \, \, \, \hbox{in} \, \, D\setminus \frac{r_ \varepsilon}{ \varepsilon} D,$ we see that it reduces to the boundary integral: $\int_0^L \int_{\partial D} \frac{1}{ \varepsilon^2} \frac{\partial w_ \varepsilon }{\partial n } u_ \varepsilon d\sigma \bar u dx_3 = \int_0^L \int_{\partial D} \frac{1}{r \varepsilon^2} \frac{1}{ \ln( \frac{ \varepsilon}{r_ \varepsilon}) } u_ \varepsilon d\sigma \bar u dx_3.$ The limit of the last integral is equal to $\frac{1}{r k} \vert \partial D \vert \int_0^L u \bar u \ dx_3 = \frac{2\pi }{k} \int_0^L u \bar u \ dx_3,$ so that we derive easily the two first items of (9).

Finally, let us prove that $u = 0$ under the assumption $k = 0$. It is well known that for given $\varepsilon > 0$, the eigenvalues of the operator $- \frac{1}{ \varepsilon^2} \Delta' - \frac{\partial^2}{\partial x_3^2}$ with eigenvectors in $H^1_D(\Omega_ \varepsilon)$ is an increasing positive sequence of real numbers and that for $f \in L^2(\Omega)$, the solution of (7) is given by $u_ \varepsilon = \sum\limits_{i = 1}^\infty \frac{1}{\lambda^{ \varepsilon}_i} (f, v^ \varepsilon_i) v^ \varepsilon_i$ where $(\lambda^ \varepsilon_i, v^ \varepsilon_i)_{i\geq 1}$ is the sequence of the eigenelements under consideration and where $( , )$ denotes the scalar product of $L^2(\Omega)$. As a consequence, we deduce the estimate

On the other hand, the first eigenvalue is given by

As an immediate consequence, we derive the following inequality

On the other hand, since for $x_3 \in (0 ,L)$ any function $v \in H^1_D(\Omega_ \varepsilon )$ is such that $v(., x_3 ) \in H^1(Y)$ and $v(., x_3) = 0 \, \, \hbox{on} \, \partial( \frac{r_ \varepsilon}{ \varepsilon} D)$, we can use the following estimate proved in [14], pages 44-45:

An easy computation shows that the quantity in the right hand side of (32) goes to $+\infty$ if $r_ \varepsilon \gg e^{ -\frac{k} { \varepsilon^2} }$ and then the result follows from (13) and (12).

3.   Strange term in homogenization

In order to make easier the comparison with the results of the last section, we keep here analogous notations; in particular, $Y\times (0, L)$ and $\bar D(0,r)$ denote respectively the parallelepiped and the disk defined in the previous section. We assume now that $\Omega : = \omega \times (0,L)$ where $\omega$ denotes a domain in $\mathbb R^2$ is the configuration domain of a set made of parallelepipeds $Y_{ \varepsilon}^i : = ( \varepsilon Y + \varepsilon i) \times (0, L)$ distributed with a period $\varepsilon Y$ in the horizontal $x'$-directions; each parallelepiped $Y_{ \varepsilon}^i$ contains a small cylinder playing the role of a hole $F_{ \varepsilon}^i : = (r_ \varepsilon \bar D + \varepsilon i ) \times (0, L)$ (the sequence $r_ \varepsilon$ is the one arising in the previous section). The set of all the holes contained in $\Omega$ will be denoted by $F_ \varepsilon$. We then define

where $I_ \varepsilon$ denotes as usual $I_ \varepsilon: = \lbrace i = (i_1, i_2) \in \mathbb Z^2, \, \, \, Y^i_ \varepsilon \subset \Omega \rbrace$. Note that the analogous configuration domain in the reduction of the dimension problem is $( \varepsilon Y \setminus r_ \varepsilon \bar D) \times (0, L)$ denoted there by $\hat \Omega_ \varepsilon$ (to be distinguished from $\Omega_ \varepsilon$ denoting there the domain obtained after scaling). In other words, the real domain (before scaling) of the $3d-1d$ problem of section 2 is nothing but the representative cell $Y_{ \varepsilon}^i$ of the present homogenization problem.

The equation we want to homogenize is the following

where $H^1_D(\Omega_ \varepsilon) : = \lbrace{ u \in H^1(\Omega_ \varepsilon), \, \, u(x', 0) = u(x',L) = 0, \, \hbox{a.e} \, x' \in \omega, \, u = 0 \, \, \hbox{on} \, \partial F_ \varepsilon \rbrace}$. To give the limit problem, we extend to the whole of $\Omega$ the sequence $u_ \varepsilon$ by zero in $F_ \varepsilon$ and continue to denote this extension by $u_ \varepsilon$. We define

together with the space

We consider a sequence $\delta_ \varepsilon$ of positive numbers such that

For example if $r_ \varepsilon = e^{- \frac{k}{ \varepsilon^2} } , \, \, k > 0$, one can take $\delta_ \varepsilon = e^{- \frac{k}{ \varepsilon} }$. Denote by $C_{\delta_ \varepsilon }$ the circle of radius $\frac{\delta_ \varepsilon}{ \varepsilon}$ centered at the origin and set $C^{i}_{\delta_ \varepsilon } = \varepsilon C_{\delta_ \varepsilon } + \varepsilon i.$ Finally, we introduce the sequence

The following estimate, proved in [3], will be helpful in the sequel.

Lemma 3.1. There exists a constant $C > 0$ such that

We now state the main result of this section through the following theorem.

Theorem 3.2. There exists $u \in H^1_D(\Omega)$ such that the sequence of solutions of (16) is such that

assuming $\lim\limits_{ \varepsilon \to 0} \varepsilon^2 \vert \ln (r_ \varepsilon) \vert = k \in [0, +\infty]$, u is characterized by

Proof. First of all, we remark that the sequence $u_ \varepsilon$ of solutions of (16) extended by zero to the holes is bounded in $H^1_D(\Omega)$; this can be seen using $\phi = u_ \varepsilon$ as test function in (16) and then applying the Cauchy-Schwarz inequality in the right hand side. Therefore, we can assume that there exists $u \in H^1_D(\Omega)$ such that (up to a subsequence) $u_ \varepsilon \rightharpoonup u \,$ weakly in $H^1_D(\Omega)$. We want to find the equation satisfied by $u$.

Let $\phi \in \mathcal {C}^\infty (\Omega) \cap H^1_D(\Omega)$. Define the following sequence of piece wise constant functions

where $D^{i}_{r_ \varepsilon }$ denotes the disk of center $\varepsilon(i_1, i_2)$ and of radius $r_ \varepsilon$. Using the regularity of $\phi$, one can check easily by the use of the mean value Theorem that there exists a constant $C >0$ such that $\vert \phi - \phi_ \varepsilon \vert \leq C \varepsilon$ in $\Omega$.

Define the function $d_ \varepsilon (x') : = \hbox{dist } \bigl( x'; \lbrace \varepsilon i, \, \, i = (i_1, i_2) \in \mathbb Z^2 \rbrace \bigr)$ and denote by $D^{i}_{\delta_ \varepsilon }$ the disk with center $\varepsilon i$ and with radius $\delta_ \varepsilon$, then we will use the following function defined for each $i \in{ \mathbb Z}^2$ such that $Y_ \varepsilon^{i} \subset \Omega$ (i.e., $i \in I_ \varepsilon$) by:

In view of the definition of $w_ \varepsilon$, one can check easily by the use of cylindrical coordinates in the tube $U^i_ \varepsilon : = \lbrace (x', x_3) \in Y^i_ \varepsilon, \, \, r_ \varepsilon < d_ \varepsilon(x') < \delta_ \varepsilon, \, \, x_3 \in (0, L) \rbrace$ the following estimate:

Hence, the second equality arising in (19) allows us to conclude that $w_ \varepsilon$ is bounded in $H^1(\Omega)$ under the hypothesis $r_ \varepsilon \approx e^{- \frac{k} { \varepsilon^2} }$ or the hypothesis $r_ \varepsilon \ll e^{- \frac{k} { \varepsilon^2} }$. Note also that the latest hypothesis which corresponds to "small holes" implies that the sequence $w_ \varepsilon$ which is bounded in $H^1(\Omega)$ strongly converges to $1$ in $H^1(\Omega)$. If we assume only the hypothesis $r_ \varepsilon \approx e^{- \frac{k} { \varepsilon^2} }$, then $w_ \varepsilon$ only converges to $1$ in the weak topology of $H^1(\Omega)$, (see [9]).

We now choose a test function in the form $\phi(x) w_ \varepsilon(x')$ in equation (16) with $\phi \in H^1_D(\Omega)$ and we get

Due to the inequality $\sup\limits_{ x \in \Omega} \vert \phi - \phi_ \varepsilon \vert \leq C \varepsilon$ pointed out above and due to the boundedness in $H^1(\Omega)$ of the sequences $u_ \varepsilon$ and $w_ \varepsilon$, the first integral in the right hand side of (27) clearly tends to zero, while the second one converges to $\int_\Omega \nabla u \nabla \phi \ dx$ since one can assume the strong convergence to $1$ in $L^2(\Omega)$ of the sequence $w_ \varepsilon$ by the Rellich Theorem. Hence, it only remains to compute the limit of the third integral. In view of the definition of $w_ \varepsilon$, that integral reduces to the integral on the tube $U^i_ \varepsilon : = \lbrace (x', x_3) \in Y^i_ \varepsilon, \, \, r_ \varepsilon < d_ \varepsilon(x') < \delta_ \varepsilon, \, \, x_3 \in (0, L) \rbrace$. Using cylindrical coordinates $(r, \theta, x_3) \in ( r_ \varepsilon, \delta_ \varepsilon) \times (0, 2 \pi) \times (0, L)$ and the definition (25) of $w_ \varepsilon$, it is not difficult to check that one can write in terms of $(r, \theta, x_3)$

so that bearing in mind the formula (24) and the fact that $u_ \varepsilon(r_ \varepsilon, \theta, x_3) = 0$, we get:

where $\tilde u_ \varepsilon$ is defined by (20). On the other hand, from Lemma 3.1 and the property $\lim_{ \varepsilon \to 0} \delta_ \varepsilon = 0$ we deduce that $\tilde u_ \varepsilon \to u$ strongly in $L^2(\Omega)$ since $u_ \varepsilon$ converges strongly to $u$ in $L^2(\Omega)$. Finally using the property $\phi_ \varepsilon \to \phi$ strongly in $L^2(\Omega)$, we obtain the first two items of the theorem by passing to the limit in the last term of (29).

We now prove the last part of the theorem using the same idea as in the $3d-1d$ problem showing that the first eigenvalue $\lambda_ \varepsilon^1$ of the operator under consideration tends to infinity when $\varepsilon \to 0$. We establish a lower bound of $\lambda_ \varepsilon^1$ as follows

In each cell $Y_ \varepsilon^i \subset \Omega$, one can write for all $v \in H^1_D(\Omega_ \varepsilon)$ by the use of cylindrical coordinates as above

As a consequence, we infer

Using the following inequality proved in [14],

where $\gamma( \varepsilon) : = \int_{r_ \varepsilon}^ \varepsilon r \ dr \int_{r_ \varepsilon}^ \varepsilon \frac{1}{r} \ dr,$ we get by integrating over $(0, 2 \pi) \times (0, L)$ and summing up over $i \in I_ \varepsilon$,

One can check that the constant $\gamma( \varepsilon)$ goes to zero under the hypothesis $r_ \varepsilon \gg e^{ -\frac{k} { \varepsilon^2} }$ and then the conclusion of the last part of the theorem is a consequence of the inequality (12) which still holds true in the homogenization problem.

4.   Nonlocal effects in the reduction of dimension

We now give the second example regarding nonlocal effects in the limit problem. We begin with the reduction of dimension.

In order to describe the heterogeneities of the medium, we need here to introduce some other notations and to change slightly those of section 2. In this section $D(0, r)$ still denotes the disk defined in the previous sections, $\Omega_ \varepsilon$ is defined by $\Omega_ \varepsilon = \varepsilon Y \times (0, L) = \varepsilon Y \times I$, $Y$ being defined by (1) and $I: = (0, L)$. In addition, we introduce the sets $M_{ \varepsilon} = \Omega_ \varepsilon \setminus { F_{ \varepsilon} }$ where now ${ F_{ \varepsilon} } : = \varepsilon \overline D \times (0, L)$ in such a way that $\Omega_{ \varepsilon} = M_{ \varepsilon} \cup { F_{ \varepsilon} }.$ Similarly, we put $F: = {\overline D} \times I, \quad M: = \Omega \setminus F,$ in such a way that $\Omega = Y \times (0, L) = M \cup F.$ We still denote by $H_D^1(\Omega)$ the space $H^1_D(\Omega) : = \lbrace{ u \in H^1(\Omega), \, \, u(y, 0) = u(y,L) = 0 , \, \forall \, y \in Y \rbrace}$.

We consider the problem

Clearly, problem (35) is the variational formulation in the fixed domain $\Omega$ of the problem satisfied by $u_ \varepsilon(y, x_3) : = \bar u_ \varepsilon( \varepsilon y, x_3)$ and obtained from the similar problem posed in the variable domain $\Omega_ \varepsilon$ and satisfied by $\bar u_ \varepsilon$. Loosely speaking, equation (35) takes into account the contrast between the diffusion in the fiber which is assumed to be of order 1 (before scaling) and the diffusion in the matrix (outside the fiber) which is assumed to be of order $\varepsilon^2$.

As regards the asymptotic behaviour of the sequence $u_ \varepsilon$, the main result is as follows.

Theorem 4.1. The sequence $u_ \varepsilon$ of solutions of (35) is such that:

where the pair $(u_0,v)$ is the unique solution of

Furthermore, one has the convergence of the energies

towards the limit energy

Proof. Taking $\phi = u_ \varepsilon$ in equation (35), we get

Since $u_ \varepsilon(y,L) = u_ \varepsilon(y,0) = 0$ almost everywhere in $Y$, one can apply the one dimensional Poincaré inequality in $(0, L)$ to the function $u_ \varepsilon (y, .)$ for a given $y \in Y$ to obtain

Integrating (41) with respect to $y \in D$ and bearing in mind that $F = \overline D \times (0, L)$, we get

On the other hand, for a given $x_3 \in (0, L)$ the Poincaré-Wirtinger inequality gives the estimate

Integrating (43) with respect to $x_3 \in (0, L)$, we derive with the help of (42) and the fact that $\parallel \nabla' u_ \varepsilon \parallel_{L^2(\Omega)}^2 \leq E_ \varepsilon$ for sufficiently small $\varepsilon$, the estimate

Turning back to equation (35) and applying the Young inequality in the right hand side, we deduce that $E_ \varepsilon \leq C$ which in turn implies by virtue of (44) that

From estimate (45) and the boundedness of the sequence $E_ \varepsilon$, we deduce, up a subsequence still denoted by $\varepsilon$, the existence of $u_{00} \in L^2(I; H^1(Y))$ such that

which together with the estimates (which are consequences of $E_ \varepsilon \leq C$)

easily implies the existence of $v \in H_0^1(I)$ such that

Therefore we obtain all the convergences stated in (36) except the convergence $\frac{1}{ \varepsilon } \nabla'u_ \varepsilon \chi_F \rightharpoonup 0$ replacing the strong convergence by the weak convergence, if we define $u_0$ by

To prove that the weak convergences are actually strong convergences, we first identify the limit problem and then we will prove the convergence of the sequence $E_ \varepsilon$ to $E_0$. The strong convergence of $u_ \varepsilon$ in $L^2(\Omega)$ will be derived from the convergence of the sequence $E_ \varepsilon$ with the help of a kind of Poincaré-Wirtinger inequality as it will be seen below.

The first estimate in (47) amounts to say that the sequence defined by $w_ \varepsilon : = \frac{1}{ \varepsilon} ( u_ \varepsilon - \frac{1}{\vert D \vert}\int_D u_ \varepsilon \ dy )$ is bounded in $L^2(I; H_m^1(D))$ where $H_m^1(D)$ is the subspace of functions in $H^1(D)$ with zero average. Hence, one can assume that for a subsequence at least, it converges weakly in $L^2(I; H_m^1(D))$ to some $w$. Taking a test function $\phi$ in (35) in the form $\phi = \bar u + \bar v + \varepsilon \bar w$ with $\bar u \in \mathcal{D}(\Omega)$ such that $\bar u = 0$ in $F$ and $\bar v \in H^1_0(I)$, $\bar w \in {\mathcal D}(\Omega)$, we can pass to the limit to find the equation

By a density argument we can choose $\bar u = \bar v = 0$ and $\bar w = w$ in (50) so that we get $w = 0$ and equation (37) is obtained. Note that by the same, the convergence $\frac{1}{ \varepsilon } \nabla'u_ \varepsilon \chi_F \rightharpoonup 0$ is proved.

Introduce now the sequence

Choosing $\phi = u_ \varepsilon$ in (35) and $( \bar u, \bar v) = (u_0, v)$ in (37) and thanks to the previous weak convergences, we show that the limit of $X_ \varepsilon$ is zero so that the stated strong convergences take place. To prove the strong convergence in $L^2(\Omega)$ of the sequence $u_ \varepsilon$, we will use the following Poincaré-Wirtinger type inequality: there exists a positive constant C such that

The proof may be done arguing by contradiction. Assume that there exists a sequence $u_n$ in that space such that $\parallel u_n \parallel_{L^2(\Omega)} = 1$ for all $n$ while the sequence $\parallel \nabla' u_n \parallel_{L^2(\Omega)} + \parallel \frac{\partial u_n}{\partial x_3} \parallel_{L^2(F)}$ goes to zero. Then the bidimensional Poincaré-Wirtinger inequality applied to the function $u_n(., x_3)$ for $x_3 \in I$ together with an integration over $I$ of such inequality allows one to get the estimate

On the other hand, the one dimensional Poincaré Inequality applied to the function $u_n(y, .)$ for $y \in D$ and then an integration with respect to $y \in D$ of that inequality lead to the estimate

From (53) and (54), we deduce that $\parallel u_n \parallel_{L^2(\Omega)}$ goes to zero and this is a contradiction with the assumption.

Applying (52) to the sequence $u_ \varepsilon - ( u_0 + v )$, (recall that $u_0 = 0$ in $F$) we get

Applying the previous strong convergences, the right hand side of (55) tends to zero so that the proof of the theorem is now complete.

One can highlight the nonlocal character of the previous equation. Indeed, define $u(x_3) : = \int_Y \bigl( u_0(x) + v(x_3) \bigr) \ dy = \int_{Y \setminus D} u_0(y,x_3) \ dy + v(x_3)$. Let $\hat u$ be the unique solution of

Define $m: = \int_{Y\setminus D} \hat u \ dy > 0.$ Then, we have the following result.

Theorem 4.2. Assume that $f(y,x_3) = f(x_3)$ does not depend on the variable $y$. Then $u_0$ given in (36) may be written as $u_0(x) = f(x_3) \hat u(y)$ and the sequence $\bar u_ \varepsilon$ defined in the variable domain $\Omega_ \varepsilon = \varepsilon Y \times I$ (recall that $u_ \varepsilon(y,x_3) = \bar u_ \varepsilon( \varepsilon y, x_3)$ for $(y, x_3) \in \Omega$) is such that

The pair $(u,v)$ is the unique solution of the nonlocal one-dimensional problem

Proof. Making the change of variable $x' = \varepsilon y$, the strong convergences (57) become immediate consequences of the convergences (36).

The second equation of (58) is obtained from the equation (37) by choosing $\bar u = 0$ and taking into account the fact that $F = D \times I$.

On the other hand, one can check that the function $f(x_3) \hat u(y)$ where $\hat u$ is the solution of (56), solves the same equation as $u_0$, that is the equation obtained from (37) by choosing $\bar v = 0$. By virtue of the uniqueness of $u_0$, we conclude that $u_0(y,x_3) = f(x_3) \hat u(y)$ and then the first equation of (58) is nothing but the equality $u(x_3) : = \int_{Y \setminus D} u_0(y,x_3) \ dy + v(x_3)$ which defines $u$.

5.   Nonlocal effects in homogenization

To describe the geometry of the medium, we need further notations.

Let $D(0,r)$ be the disk defined in the previous sections and let $\omega$ be the square $\omega : = ]-1, 1 [^2$. Assume that $\Omega : = \omega \times (0,L) = \omega \times I$ is now the configuration domain of a set $F_ \varepsilon$ of cylindrical parallel fibers periodically distributed with a period $\varepsilon Y = \varepsilon \bigl( ]- \tfrac{1}{ 2}, \tfrac{1}{ 2}[ \bigr)^2$ in the $x'$-horizontal directions which are surrounded by a poor conductor occupying the matrix $M_ \varepsilon$ in such a way that

Hence, the medium is now a periodic replication of the one arising in the previous section. The equation we want to homogenize is the following

where $H_D^1(\Omega)$ is still the space defined in the beginning of the previous section.

For the sake of brevity, we consider only the case of a source term not depending on the microscopic variable but one can handle also that general case as pointed out in the Remark 5.1 below.

Before stating the main result, we recall the definition of two-scale convergence (see [12], [1]) a well adapted tool for periodic homogenization. A sequence $t_ \varepsilon \in L^2(\Omega)$ two scale converges to a function $t \in L^2(\Omega \times Y)$ if $\int_\Omega t_ \varepsilon(x) \phi(x, \frac{x'}{ \varepsilon} ) \ dx \to \int_\Omega \int_Y t(x,y) \phi(x,y) \ dx dy, \, \, \forall \, \phi \in L^2(\Omega; C_{\#}(Y))$ where $C_{\#}(Y)$ denotes the space of functions which are continuous and $Y$-periodic. It is known that every bounded sequence in $L^2(\Omega)$ admits a two-scale converging subsequence.

In the sequel, the notation $\rightharpoonup\rightharpoonup$ will stand for the two-scale convergence. The main result may be stated as follows.

Theorem 5.1. The sequence $u_ \varepsilon$ of solutions of (61) is such that:

where the pair $(u_0,v) \in \{ \phi \in L^2(\Omega; H_{\#}^1(Y)), \, \, \phi(x,.) = 0 \, \, \hbox{in} \, D \} \times L^2(\omega; H_0^1(I))$ is the unique solution of

Furthermore, by the same approach already used in the $3d-1d$ problem, one can eliminate the microscopic variable $y$ to derive a formulation of the limit problem involving only the macroscopic variable $x$. Indeed, let $\hat u$ be the unique solution of

Define $m: = \int_{Y\setminus D} \hat u \ dy > 0.$

Theorem 5.2. The function $u_0$ given in (62) may be written as $u_0(x,y) = f(x) \hat u(y)$ and the sequence $u_ \varepsilon$ is such that

the pair $(u,v)$ is the unique solution of the nonlocal homogenized problem

In the homogenization setting we cannot reasonably expect strong convergences similar to those obtained in the reduction of dimension due to the oscillations induced by the microscopic variable. However, we have the following result.

Theorem 5.3. Let $u_ \varepsilon$ be the sequence of solutions of (61) and let $(u_0, v)$ be the pair defined by (66), then the sequence

converges to zero.

Remark that this result is a corrector result meaning that $u_ \varepsilon$ may be approximated by $v(x) + u_0(x, \frac{x'}{ \varepsilon})$. Due to the close similarities between the reduction of dimension problem and the homogenization problem and for the sake of brevity, we will give only the outline of the proof of Theorem 5.1.

Proof. As above we take $\phi = u_ \varepsilon$ in the equation (61) and then we prove in the same way that the $L^2$-norm of $u_ \varepsilon$ is dominated up to a positive constant by the energy which is now defined by $E_ \varepsilon: = \int_\Omega \bigl( \chi_{F_ \varepsilon} + \varepsilon^2 \chi_{M_ \varepsilon}\bigr) \bigl( \nabla' u_ \varepsilon \nabla' u_ \varepsilon + \frac{\partial u_ \varepsilon}{\partial x_3}\frac{\partial u_ \varepsilon }{\partial x_3} \bigr) \ dx'dx_3 .$

Once again, we use the Poincaré-Wirtinger inequality in the reference cell $Y$ to obtain the inequality

For given $\varepsilon$, $i \in I_ \varepsilon$ and $x_3 \in(0,L)$, we choose $u(y) : = u_ \varepsilon( \varepsilon y + \varepsilon i, x_3)$ in (71) and then we make the change of variables $x' = \varepsilon y + \varepsilon i$ in such a way we derive the inequality

On the other hand, similarly to the proof in the reduction of dimension problem and due to the Dirichlet boundary condition $u_ \varepsilon(x',0) = u_ \varepsilon(x',L) = 0$, one can apply the one dimensional Poincaré inequality to the sequence $u_ \varepsilon(x',.)$ for almost all $x' \in \omega$ to get after an integration with respect to $x'$ and after summing up over $i \in I_ \varepsilon$,

Summing up over $i \in I_ \varepsilon$ and integrating over $(0,L)$ the inequality (72), we get with the helph of (73)

Using (74) in equation (61) in which we take $\phi = u_ \varepsilon$, we derive easily the estimate $E_ \varepsilon \leq C.$ In particular the last inequality implies that the sequence $\frac{\partial u_ \varepsilon}{\partial x_3} \chi_{F_ \varepsilon}$ is bounded in $L^2(\Omega)$ so that $u_ \varepsilon \chi_{F_ \varepsilon}$ is bounded in $L^2(\omega; H^1_0(0, L))$. Therefore one can extract a subsequence still denoted $\varepsilon$ and find functions $u_{00} \in L^2(\Omega ; H^1_{\#}(Y)), \, K \in (L^2(\Omega \times Y))^2 , \ v \in L^2(\omega; H^1_0(0, L))$ such that the following convergences hold true:

Since $\nabla'u_ \varepsilon \chi_{F_ \varepsilon}$ strongly converges to zero in $L^2(\Omega)$, we deduce from (75) that $u_{00}(x,y) = v(x)$ in $\Omega \times D$ with some $v \in L^2(\Omega)$.

In addition, there exists $\bar v \in L^2(\omega; H_0^1(0,L))$ such that

From the first convergence of (75) we deduce $\bar v(x) : = \int_Y u_{00} (x,y) \chi_D(y) \ dy = \vert D \vert v(x)$. Hence, $v \in L^2(\omega; H^1_0(0, L))$.

On the other hand, the Poincaré-Wirtinger inequality applied in $H^1(D)$ and the same change of variables as in the proof of (72) leads to

Using the boundedness in $L^2(\Omega)$ of the sequence $\frac{1}{ \varepsilon } \nabla' u_ \varepsilon \chi_{F_ \varepsilon}$, we deduce from (77)that the sequence $w_ \varepsilon : = \sum\limits_{ i \in I_ \varepsilon} \frac{1}{ \varepsilon}( u_ \varepsilon - \frac{1}{ \vert D_ \varepsilon^i \vert} \int_{D_ \varepsilon^i} u_ \varepsilon \ dx') \chi_{D_ \varepsilon^i}$ is bounded in $L^2(0,L; H^1_m(D))$ and one can assume possibly by extracting a subsequence that it converges weakly in that space to some $w$. We then prove that the last limit $K$ arising in (75) is equal to $\nabla'_y w$.

The function defined by $u_0(x,y) : = u_{00}(x,y) - v(x)$ in $\Omega \times Y$ satisfies the convergence (62). Taking in (61) a test function in the form $\phi(x) = \bar u_0(x, \frac{x'}{ \varepsilon}) + \bar v(x) + \bar w(x, \frac{x'}{ \varepsilon})$ with regular $\bar u_0, \ \bar w, \ \bar v$ and passing to the limit, we obtain the limit equation(66) by a density argument. Choosing $\bar u_0 = \bar v = 0$ and $\bar w = w$ in that equation, we conclude that $w = 0$ completing the proof of Theorem 5.1.

Remark 1. In order to emphasize the nonlocal effect at the limit, we have assumed in Theorem 4.2 and in Theorem 5.2 that the function $f$ does not depend on the variable $y$. One can handle the general case of a source term $f(y,x)$ depending also on the variable $y$; in this case, setting $\hat f(x) : = \int_Y f(y,x) \ dy$, one can check easily that the solution $u_0(x,y)$ of (62) takes the form $u_0(x,y) = \hat f(x) \hat u(y) + \hat u_0(x,y)$ where $\hat u$ is the solution of (67) and where $\hat u_0(x,y)$ is the solution of (62) but with the right hand side given by $f(y,x) - \int_Y f(y,x) \ dy$. This is due to the simple remark that any right hand side $f(y,x)$ of (62) may be written as $f(y,x) = f(y,x) - \int_Y f(y,x) \ dy + \int_Y f(y,x) \ dy$ so that the solution $u_0$ is the superposition of the two solutions. Of course the same remark holds true in the problem of reduction of dimension. In both cases, the nonlocal effect is due to the term $\hat f(x) \hat u(y)$.