A REMARK ABOUT THE PERIODIC HOMOGENIZATION OF CERTAIN COMPOSITE FIBERED MEDIA

. We explain in this paper the similarity arising in the homogenization process of some composite ﬁbered media with the problem of the reduction of dimension 3 d − 1 d . More precisely, we highlight the fact that when the homogenization process leads to a local reduction of dimension, studying the homogenization problem in the reference conﬁguration domain of the composite amounts to the study of the corresponding reduction of dimension in the reference cell. We give two examples in the framework of the thermal conduction problem: the ﬁrst one concerns the reduction of dimension in a thin parallelepiped of size ε containing another thinner parallelepiped of size r ε (cid:28) ε playing a role of a "hole". As in the homogenization, the one-dimensional limit problem involves a "strange term". In addition both limit problems have the same structure. In the second example, the geometry is similar but now we assume a high contrast between the conductivity (of order 1 ) in the small parallelepiped of size r ε := rε , for some ﬁxed r ( 0 < r < 12 ) and the conductivity (of order ε 2 ) in the big parallelepiped of size ε . We prove that the limit problem is a nonlocal problem and that it has the same structure as the corresponding periodic homogenized problem.

(Communicated by Pierre-Emmanuel Jabin) Abstract. We explain in this paper the similarity arising in the homogenization process of some composite fibered media with the problem of the reduction of dimension 3d − 1d. More precisely, we highlight the fact that when the homogenization process leads to a local reduction of dimension, studying the homogenization problem in the reference configuration domain of the composite amounts to the study of the corresponding reduction of dimension in the reference cell. We give two examples in the framework of the thermal conduction problem: the first one concerns the reduction of dimension in a thin parallelepiped of size ε containing another thinner parallelepiped of size rε ε playing a role of a "hole". As in the homogenization, the one-dimensional limit problem involves a "strange term". In addition both limit problems have the same structure. In the second example, the geometry is similar but now we assume a high contrast between the conductivity (of order 1) in the small parallelepiped of size rε := rε, for some fixed r (0 < r < 1 2 ) and the conductivity (of order ε 2 ) in the big parallelepiped of size ε. We prove that the limit problem is a nonlocal problem and that it has the same structure as the corresponding periodic homogenized problem.
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 126 FRANÇOIS MURAT AND ALI SILI 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 Ω of R 3 which is a periodic replication with a period of size ε of a set Y ε i , i ∈ I ε . More precisely, Y ε i is assumed to be the union of a fiber F ε i with its complement M ε i (where M stands for "matrix") in Y ε i . 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 Ω ε := εY × (0, L) as a copy of the local cell arising in a periodic fibered medium for which the configuration domain Ω is the union of 1 ε 2 such cells, in other words, Ω := i∈Iε (εY + εi) × (0, L) . 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 Ω is made from a single cell of size ε, Ω = Ω ε := Y ε := εY × (0, L) where for example Y is the square defined by Y := − 1 2 , 1 2 2 and L > 0, one can denote the variable x in Ω ε by x = (x , x 3 ) = (εy, x 3 ), y ∈ Y, x 3 ∈ (0, L), in such a way that x = εy. This classical dilation transforms the reduction of dimension problem posed in Ω ε into a singular perturbation problem posed on the fixed domain Ω (see [11]) and the limit problem (as ε 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 Ω := i∈Iε (εY + εi) × (0, L) , the variable x in Ω writes as x = (x , x 3 ) with x = εy + εi, y ∈ Y, i ∈ I ε , x 3 ∈ (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 1 ε 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 ε, r ε be two decreasing sequences of positive numbers such that ε tends to zero and lim ε→0 r ε ε = 0, and letΩ ε ,F ε , be respectively the parallelepiped and the fiber defined by where D(0, r) is the closed disk of radius 0 < r < 1 2 . (1) We will assume the Dirichlet boundary condition holds on the part of the boundary ofΩ ε and the Neumann boundary condition holds on the rest of the boundary ofΩ ε . Remark that ∂Ω D ε is made of the upper and the lower faces of the cylinderΩ ε together with the lateral boundary of the fiberF ε which plays here the role of a hole. We consider the following equation: Introduce the change of variables x = εy, u ε (y, x 3 ) :=ū ε (εy, x 3 ) and set (6) Finally, denote by ∇ the gradient with respect to the two first variables y = (y 1 , y 2 ). Similarly, ∆ denotes the Laplacian with respect to the same variables. We assume in the sequel that the source termf ε (εy, x 3 ) = f (y, x 3 does not depend on ε with f ∈ L 2 (Ω) so that equation (4) forū ε is transformed into an equation satisfied by u ε with the following variational formulation Problem (7) is well-posed. Moreover we extend u ε by zero inside F ε and continue to denote byu ε this extension which belongs to H 1 (7), we obtain easily that there exist u ∈ H 1 0 (0, L) (the subspace of functions in H 1 D (Ω) depending only on the variable x 3 ) and a subsequence (still denoted by ε) such that: The limit function u depends on the size of the hole and it is characterized by the we have the following: Sketch of the proof. Introduce the following function (see [11]) Then: Use as a test function in (7) φ(y, Integrating by parts with respect to y the first integral on the left, and bearing in mind the property −∆ w ε = 0 in D \ rε ε D, we see that it reduces to the boundary integral: uū 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 ε > 0, the eigenvalues of the operator − 1 is an increasing positive sequence of real numbers and that for f ∈ L 2 (Ω), the solution of (7) is given by is the sequence of the eigenelements under consideration and where (, ) denotes the scalar product of L 2 (Ω). 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 , 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 +∞ if r ε e − k ε 2 and then the result follows from (13) and (12).

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 × (0, L) andD(0, r) denote respectively the parallelepiped and the disk defined in the previous section. We assume now that Ω := ω × (0, L) where ω denotes a domain in R 2 is the configuration domain of a set made of parallelepipeds Y i ε := (εY + εi) × (0, L) distributed with a period εY in the horizontal x -directions; each parallelepiped Y i ε contains a small cylinder playing the role of a hole F i ε := (r εD + εi) × (0, L) ( the sequence r ε is the one arising in the previous section). The set of all the holes contained in Ω will be denoted by F ε . We then define where I ε denotes as usual Note that the analogous configuration domain in the reduction of the dimension problem is (εY \ r εD ) × (0, L) denoted there byΩ ε (to be distinguished from Ω ε 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 i ε of the present homogenization problem.
The equation we want to homogenize is the following where To give the limit problem, we extend to the whole of Ω the sequence u ε by zero in F ε and continue to denote this extension by u ε . We define together with the space We consider a sequence δ ε of positive numbers such that For example if r ε = e − k ε 2 , k > 0, one can take δ ε = e − k ε . Denote by C δε the circle of radius δε ε centered at the origin and set C i δε = εC δε + εi. Finally, we introduce the sequenceũ 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 ∈ H 1 D (Ω) such that the sequence of solutions of (16) is such that u ε u weakly in H 1 D (Ω); assuming lim ε→0 ε 2 | ln(r ε )| = k ∈ [0, +∞], u is characterized by if k = +∞, then u is the unique solution of , Proof. First of all, we remark that the sequence u ε of solutions of (16) extended by zero to the holes is bounded in H 1 D (Ω); this can be seen using φ = u ε 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 ∈ H 1 D (Ω) such that ( up to a subsequence) u ε u weakly in H 1 D (Ω). We want to find the equation satisfied by u.
Let φ ∈ C ∞ (Ω) ∩ H 1 D (Ω). Define the following sequence of piece wise constant functions where D i rε denotes the disk of center ε(i 1 , i 2 ) and of radius r ε . Using the regularity of φ, one can check easily by the use of the mean value Theorem that there exists a constant C > 0 such that |φ − φ ε | ≤ Cε in Ω.
Define the function d ε (x ) := dist x ; {εi, i = (i 1 , i 2 ) ∈ Z 2 } and denote by D i δε the disk with center εi and with radius δ ε , then we will use the following function defined for each i ∈ Z 2 such that Y i ε ⊂ Ω (i.e., i ∈ I ε ) by: In view of the definition of w ε , one can check easily by the use of cylindrical coordinates in the tube Hence, the second equality arising in (19) allows us to conclude that w ε is bounded in H 1 (Ω) under the hypothesis r ε ≈ e − k ε 2 or the hypothesis r ε e − k ε 2 . Note also that the latest hypothesis which corresponds to "small holes" implies that the sequence w ε which is bounded in H 1 (Ω) strongly converges to 1 in H 1 (Ω). If we assume only the hypothesis r ε ≈ e − k ε 2 , then w ε only converges to 1 in the weak topology of H 1 (Ω), (see [9]).

FRANÇOIS MURAT AND ALI SILI
We now choose a test function in the form φ(x)w ε (x ) in equation (16) Due to the inequality sup x∈Ω |φ − φ ε | ≤ Cε pointed out above and due to the boundedness in H 1 (Ω) of the sequences u ε and w ε , the first integral in the right hand side of (27) clearly tends to zero, while the second one converges to Ω ∇u∇φ dx since one can assume the strong convergence to 1 in L 2 (Ω) of the sequence w ε by the Rellich Theorem. Hence, it only remains to compute the limit of the third integral. In view of the definition of w ε , that integral reduces to the integral on the tube Using cylindrical coordinates (r, θ, x 3 ) ∈ (r ε , δ ε ) × (0, 2π) × (0, L) and the definition (25) of w ε , it is not difficult to check that one can write in terms of (r, θ, so that bearing in mind the formula (24 ) and the fact that u ε (r ε , θ, x 3 ) = 0, we get: whereũ ε is defined by (20). On the other hand, from Lemma 3.1 and the property lim ε→0 δ ε = 0 we deduce thatũ ε → u strongly in L 2 (Ω) since u ε converges strongly to u in L 2 (Ω). Finally using the property φ ε → φ strongly in L 2 (Ω), 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 λ 1 ε of the operator under consideration tends to infinity when ε → 0. We establish a lower bound of λ 1 ε as follows In each cell Y i ε ⊂ Ω, one can write for all v ∈ H 1 D (Ω ε ) by the use of cylindrical coordinates as above As a consequence, we infer Using the following inequality proved in [14], One can check that the constant γ(ε) goes to zero under the hypothesis r ε e − k ε 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, Ω ε is defined by Ω ε = εY × (0, L) = εY × I, Y being defined by (1) and I := (0, L). In addition, we introduce the sets M ε = Ω ε \ F ε where now F ε := εD × (0, L) in such a way that Ω ε = M ε ∪ F ε . Similarly, we put

FRANÇOIS MURAT AND ALI SILI
We consider the problem Clearly, problem (35) is the variational formulation in the fixed domain Ω of the problem satisfied by u ε (y, x 3 ) :=ū ε (εy, x 3 ) and obtained from the similar problem posed in the variable domain Ω ε and satisfied byū ε . 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 ε 2 .
As regards the asymptotic behaviour of the sequence u ε , the main result is as follows.
Theorem 4.1. The sequence u ε 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 φ = u ε in equation (35), we get Since u ε (y, L) = u ε (y, 0) = 0 almost everywhere in Y , one can apply the one dimensional Poincaré inequality in (0, L) to the function u ε (y, .) for a given y ∈ Y to obtain L 0 |u ε (y, Integrating (41) with respect to y ∈ D and bearing in mind that F = D × (0, L), we get On the other hand, for a given x 3 ∈ (0, L) the Poincaré-Wirtinger inequality gives the estimate Integrating (43) with respect to x 3 ∈ (0, L), we derive with the help of (42) and the fact that ∇ u ε 2 L 2 (Ω) ≤ E ε for sufficiently small ε, the estimate Turning back to equation (35) and applying the Young inequality in the right hand side, we deduce that E ε ≤ C which in turn implies by virtue of (44) that From estimate (45) and the boundedness of the sequence E ε , we deduce, up a subsequence still denoted by ε, the existence of u 00 ∈ L 2 (I; H 1 (Y )) such that u ε u 00 (y, x 3 ) weakly in L 2 (I; H 1 (Y )), which together with the estimates (which are consequences of E ε ≤ C) easily implies the existence of v ∈ H 1 0 (I) such that u 00 (y, Therefore we obtain all the convergences stated in (36) except the convergence 1 ε ∇ u ε χ F 0 replacing the strong convergence by the weak convergence, if we define u 0 by u 0 (y, 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 ε to E 0 . The strong convergence of u ε in L 2 (Ω) will be derived from the convergence of the sequence E ε 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 ε : 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 1 m (D)) to some w. Taking a test function 136 FRANÇOIS MURAT AND ALI SILI φ in (35) in the form φ =ū +v + εw withū ∈ D(Ω) such thatū = 0 in F and v ∈ H 1 0 (I),w ∈ D(Ω), we can pass to the limit to find the equation By a density argument we can chooseū =v = 0 andw = w in (50) so that we get w = 0 and equation (37) is obtained. Note that by the same, the convergence 1 ε ∇ u ε χ F 0 is proved. Introduce now the sequence Choosing φ = u ε in (35) and (ū,v) = (u 0 , v) in (37) and thanks to the previous weak convergences, we show that the limit of X ε is zero so that the stated strong convergences take place. To prove the strong convergence in L 2 (Ω) of the sequence u ε , 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 u n L 2 (Ω) = 1 for all n while the sequence ∇ u n L 2 (Ω) + ∂un ∂x3 L 2 (F ) goes to zero. Then the bidimensional Poincaré-Wirtinger inequality applied to the function u n (., x 3 ) for x 3 ∈ 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 ∈ D and then an integration with respect to y ∈ D of that inequality lead to the estimate 1 |D| D u n dy L 2 (Ω) ≤ C ∂u n ∂x 3 L 2 (Ω) , ∀n.
From (53) and (54), we deduce that u n L 2 (Ω) goes to zero and this is a contradiction with the assumption. Applying (52) to the sequence u ε − (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 Define m := Y \Dû dy > 0. Then, we have the following result.
does not depend on the variable y.
Then u 0 given in (36) may be written as u 0 (x) = f (x 3 )û(y) and the sequenceū ε defined in the variable domain Ω ε = εY × I (recall that u ε (y, The pair (u, v) is the unique solution of the nonlocal one-dimensional problem Proof. Making the change of variable x = εy, the strong convergences (57) become immediate consequences of the convergences (36). The second equation of (58) is obtained from the equation (37) by choosingū = 0 and taking into account the fact that F = D × I.
On the other hand, one can check that the function f (x 3 )û(y) whereû is the solution of (56), solves the same equation as u 0 , that is the equation obtained from (37) by choosingv = 0. By virtue of the uniqueness of u 0 , we conclude that u 0 (y, x 3 ) = f (x 3 )û(y) and then the first equation of (58) is nothing but the equality u(x 3 ) := Y \D u 0 (y, x 3 ) dy + v(x 3 ) which defines u.

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 ω be the square ω :=] − 1, 1[ 2 . Assume that Ω := ω × (0, L) = ω × I is now the configuration domain of a set F ε of cylindrical parallel fibers periodically distributed with a period εY = ε ] − 1 2 , 1 2 [ 2 in the x -horizontal directions which are surrounded by a poor conductor occupying the matrix M ε in such a way that Ω = F ε ∪ M ε , F ε = i∈Iε F i ε , F i ε = (εD(0, r) + εi) × I (59) 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 1 D (Ω) 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 ε ∈ L 2 (Ω) two scale converges to a function t ∈ t(x, y)φ(x, y) dxdy, ∀ φ ∈ L 2 (Ω; 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 (Ω) admits a two-scale converging subsequence.
In the sequel, the notation will stand for the two-scale convergence. The main result may be stated as follows.