On the limit spectrum of a degenerate operator in the framework of periodic homogenization or singular perturbation problems

In this paper we perform the analysis of the spectrum of a degenerate operator $A_\var$ corresponding to the stationary heat equation in a $\var$-periodic composite medium having two components with high contrast diffusivity. We prove that although $ A_\var$ is a bounded self-adjoint operator with compact resolvent, the limits of its eigenvalues when the size $\var$ of the medium tends to zero, make up a part of the spectrum of a unbounded operator $ A_0$, namely the eigenvalues of $ A_0$ located on the left of the first eigenvalue of the bi-dimensional Laplacian with homogeneous Dirichlet condition on the boundary of the representative cell. We also show that the homogenized problem does not differ in any way from the one-dimensional problem obtained in the study of the local reduction of dimension induced by the homogenization.


Introduction, setting of the problem and statement of the results
The purpose of this work is the study of two problems: the first one is concerned by the asymptotic analysis of the eigenelements of a spectral problem in the framework of periodic homogenization; the problem writes as The notations used in (1) are the following. (2) In the sequel, the variable x 3 will play a particular role and it must be distinguished from the horizontal variable x := (x 1 , x 2 ) or y = (y 1 , y 2 ). The gradient and the Laplacian with respect to these variables will be denoted by ∇ and ∆ . We assume that Ω is the reference configuration of a composite medium whose two components are a set F ε of vertical cylindrical fibers and its complement, the matrix M ε . Hence, the projection on thex -plane of the set F ε is made up of a ε-periodic set of disks while the complement of such set represents the projection of M ε , see Figure 1. The characteristic functions of F ε (resp. M ε ) are denoted by χ F ε (resp. χ M ε ). The fibers are distributed in Ω with a period of size ε and the ratio between the conductivity coefficients of the two components is 1 ε 2 . Throughout the paper, for a measurable set B we denote by |B | its Lebesgue measure and by χ B its characteristic function. A generic positive constant the value of which may change from a line to another will be denoted by K .
Let C be a square of R 2 and let D be a disk strictly contained in C . The complement of D in C will be denoted by M in such a way that C = M ∪ D. The geometry of the domain is described as follows.
In Figure 1 we have represented the projection of the domain Ω over the horizontal plane x and the representative cell C = D ∪ M .
The second problem we consider in this work is a special case of the first one. Indeed, we now assume that the composite medium consists of a single fiber F ε := (εD)×(0, L) surrounded by the matrix M ε = εM × (0, L) = ε(C \ D) × (0, L). In other words, the global domain depends now on the small parameter ε and it is defined by Ω ε := (εC ) × (0, L) = F ε ∪ M ε .
In this configuration, the spectral problem (1) takes the form where the space V ε s (the subscript "s" stands for singular perturbation) is now defined by We now introduce the classical scaling u ε (y , x 3 ) = v ε (εy , x 3 ), y ∈ C , (this approach is of course not applicable in the homogenization setting) allowing to transform the problem (4) into the following singular perturbation problem posed on the fixed domain Ω := C × (0, L), V s being the space V ε s corresponding to ε = 1 and defined in (5) and where n C D denotes the outward normal from C \ D towards D and n DC the outward normal from D towards C \ D, see the representative cell in Figure 1.
Note that the study of the asymptotic behavior of (4) is the so-called reduction of dimension problem 3d − 1d since when ε goes to zero the three dimensional domain Ω ε = (εC ) × (0, L) looks like the segment (0, L).
Clearly, the homogenization process induces a local reduction of dimension and one of the main feature of this work is to prove that there is no significant difference between the homogenized problem and the one dimensional problem obtained in the reduction of dimension 3d −1d . The only difference here is that the homogeneous Neumann condition on the boundary of C in the one dimensional model must be replaced by a periodicity condition in the homogenized problem. In other words, studying the behavior of the solutions of (1) is the same as the study of the behavior of the solutions of (4) which in turn is equivalent to the study of (6). This similarity between the two problems has already been explained in [20] in the framework of linear elasticity and in [17] in the framework of the conductivity equation.
Briefly this similarity may be explained by the fact that finding a two-scale limit (see [1,7,18]) taking into account the oscillations induced by the microscopic variable y = x ε of any sequence in Ω ε is simply a matter of finding the weak limit in L 2 (C × (0, L)) of the rescaled sequence φ ε (y, x 3 ) := φ ε (εy, x 3 ) defined on the fixed domain C × (0, L). Note that the change of variable x = εy which implies is the reason for which the two problems (4) and (6) are equivalent. The advantage of dealing with the reduction of dimension instead of the homogenization problem lies on the absence of the oscillations due to the fast variable which needs the use of specific tools from the homogenization theory. This kind of problems was intensively studied during the last years through an abundant literature see [3,4,6,8,10,17,[20][21][22]24] and it is well known that the high contrast between the conductivity coefficients of the components leads in general to a limit problem which may be very different from the equation modeling the composite structure at the scale ε. In particular a nonlocal phenomenon and/or strange term or memory effects may appear highlighting the gap between the properties of the fibers with those of the matrix.
We give here another example by studying the problem of the behavior of the spectrum when the operator is not uniformly bounded below with the respect to the small parameter ε. We choose to perform the study of a simple operator (the Laplacian) in order to focus on the main difficulty and for the sake of brevity and clarity of the presentation.
Unlike the study addressed in [24] where the geometry considered and the choice of the degenerate operator lead to an unbounded spectrum with gaps, we show here that the limit operator is still of a different nature from the original one despite the absence of gaps at the limit (see also [15] and the references therein); the limit spectrum we get in our case is bounded and described by two coupled equations related respectively to the vibrations in the fibers and in the matrix. Other settings have been studied in [2,11,12,16].
Roughly speaking, we show here that a phenomenon similar to that observed in the static case for degenerate operators, see [17,20,21] occurs at the limit: the limiting spectrum is given by solving two equations: the main equation is a one-dimensional equation in (0, L) describing the vibrations in the fibers but this equation alone is not sufficient to describe the phenomenon since it is strongly coupled to another one posed in the matrix. In other words, although the vibrations in the matrix are negligible at the scale ε, compared to those of the fibers, we still have a trace of the matrix vibrations at the limit; in other words, a memory effect appears at the limit.
For uniformly (with respect to the small parameter ε) bounded operators, the structure of the limit spectrum looks like the one of the homogenized problem in the static case, as seen for instance in [11,12,23].
Let us point out that the asymptotic analysis and the techniques leading to the limit problem in the local 3d-1d dimensional reduction, i.e., the singular perturbation problem, are exactly those giving the homogenized problem, except for a few minor differences due to the oscillations of the fast variable. Moreover as in the static case, see [17,20], the homogenized problem is nothing but the exact copy of the one-dimensional model obtained in the local reduction of dimension.
Let us also point out that our main result (see Theorem 3 below) states that the limit spectrum is essentially that of the Laplacian in (0, L) with Dirichlet boundary condition since we prove that all the limit eigenvalues are simple eigenvalues. The latter is true for the 3d − 1d reduction dimension problem as well as for the homogenization problem (see Theorem 5 below); hence one can say that the local reduction of dimension takes precedence over the homogenization process itself.
It is easily seen that for a fixed ε, A ε defined either by (1) or by (6) is a selfadjoint operator with compact resolvent and hence the following classical result takes place.
Taking into account this result, the variational formulation of (1) and of (6) are respectively the following ones where F := D × (0, L) and M := (C \ D) × (0, L). We prove that the spectrum σ 0 of the homogenized problem is bounded: any eigenvalue λ k ∈ σ 0 is such that µ 0 ≤ λ k < µ 1 where the lower bound µ 0 is a positive constant depending both on the first eigenvalue of the operator − d 2 dx 2 3 in H 1 0 (0, L) and the first eigenvalue µ 1 of the two-dimensional Laplacian operator (denoted in the sequel by ∆ y ) with mixed boundary conditions on the boundary of C \ D (periodicity on the sides of the square C and homogeneous Dirichlet condition on the boundary of the disk D) and on the other hand the calculation of these eigenvalues takes into account the vibrations outside the fibers through the mean value C \D u k 0 dy of u k 0 obtained by solving a second order equation posed in C \ D, in such a way that the limit spectral problem is given by the following coupled system of two equations In the case of the singular perturbation problem (6), the limit spectral problem is the following one Note the very close analogy between the two limit problems. The only difference between (10) and (11) lies in the Neumann boundary condition on ∂C of (11) replaced by a periodicity condition in (10) and in the fact that in the homogenized problem (10), the function v depends also on the variable x ; this simply means that the homogenized problem is a duplication of the phenomenon occurring in each cell. In other words, the homogenization process does not affect the vertical variable and does not bring any new features to the limit model.
Note also that the existence and the uniqueness of u k 0 both in (10) and in (11) are ensured by the fact that λ k belongs to the resolvent ρ(−∆ y ) of −∆ y since λ k < µ 1 . We will prove in Section 2 (see (40)) that C \D u k 0 dy = 0 by the use of the property that 1 µ 1 is the best constant in the Poincaré inequality.
Furthermore, the last equation of (10) or (11) shows that λ k 1 in H 1 0 (0, L). From now on and based on the previous comments comparing the homogenization problem with the problem of reduction of the dimension arising locally, we will focus on the asymptotic analysis of the singular perturbation problem (9) (the study of the reduction of dimension occurring in each cell). The reduction of dimension problem is usually encountered in the study of thin structures, see for instance [14] and [19].
We will prove in the next sections that in the case of homogenization, there exists a subsequence of ε still denoted by ε such that (λ k ε ) ε tends to λ k when ε goes to zero and the associate sequence of eigenvectors strongly two-scale converges to u k (x, y) = (λ k u k 0 + 1)v k where (u k 0 (y), v k (x)), λ k ) solves the system (10). For the reduction of dimension problem the associate sequence of eigenvectors u k ε converges strongly in L 2 (Ω) to u k (y, ), λ k ) solves the system (11). Remark 2. One can reasonably and legitimately ask what is the link between the problem (11) (or the problem (10)) and the classical formulation of eigenvalue problems. In fact, (11) is derived from the system (30) which in turn is derived from the equation (29) satisfied by the pair (u k , v k ), see the details of the proof in Section 2 below. If one integrates the first equation of (30) over C \D, we get an equivalent formulation of (30) as follows Another equivalent formulation of (12) is the following Although one can see from the structure of the limit spectrum which is bounded that the limit operator is very different from the starting one, once again we can see from (13) and (14) the sharp difference between the operator A ε and the limit operator A 0 which is not even a selfadjoint operator.
Of course a similar remark applies for the homogenized problem given by (10).
From the technical point of view the main difficulty in the asymptotic analysis comes from the lack of compactness since the sequences of eigenvectors we have to consider are not bounded in H 1 (Ω) so that the strong convergence in L 2 (Ω) (or strong two-scale convergence in the case of homogenization) which allows to conclude that the limit of an eigenvector u k ε is still an eigenvector (i.e. = 0) is not straightforward. To overcome this difficulty, we will use an extension technique (see [9,25]) combined with another slightly more intricate argument. Our main results may be stated as follows.
Theorem 3. For each k = 1, 2, . . . , the sequence of eigenvalues (λ k ε ) ε of (9) is bounded and the associated sequence of eigenvectors (11) with v k = 0 such that for the whole sequence ε, one has For all k, λ k is a simple eigenvalue of the limit operator and λ k ∈ (µ 0 , µ 1 [ where µ 0 depends only on µ 1 and on the first eigenvalue of − d 2 dx 2 3 in H 1 0 (0, L). Conversely, any λ ∈ (µ 0 , µ 1 [ which is an eigenvalue of (11) is a limit of a sequence (λ k ε ) ε of eigenvalues of (9).
The unique accumulation point of the sequence (λ k ) k is the first eigenvalue µ 1 of −∆ y ; hence lim k→+∞ λ k = µ 1 .

Remark 4.
The property v k = 0 may be deduced from the strong convergence (16) of the eigenvectors but we prefer to write it explicitly to highlight the fact that v k is always an eigenvector of − d 2 dx 2 3 with Dirichlet condition.
Regarding the case of homogenization, exactly the same results obtained in the singular perturbation problem remain true but we have to formulate it taking into account the oscillations induced by the homogenization process. We use the notation 2−sc (resp. 2−sc −→) for the two-scale convergence (resp. the strong two-scale convergence), see [1,18,24].
with the following corrector result For all k, λ k is a simple eigenvalue of the limit operator and λ k ∈ (µ 0 , µ 1 [ where the constant µ 0 depends only on the first eigenvalue λ 0 of the operator − d 2 dx 2 3 in H 1 0 (0, L) and the first eigenvalue µ 1 of −∆ y in C \ D with Dirichlet boundary condition on ∂D and periodicity on the sides of C .

Remark 6.
Note that the structure of the limit spectrum is quite complicated because not only the mean value C \D u 0 dy arising in the second equation of the limit system must be calculated by the use of the first equation of the system but the function u 0 itself depends on the corresponding eigenvalue as shown by the first equation. Remark also that in the homogenization setting, the analogous result of the convergence (17) is the convergence Ω (20). However, the latter does not mean that the sequence u k ε χ F ε converges strongly in L 2 (ω; H 1 0 (0, L)) to |D| |C | v k = |D|v k (we have assumed |C | = 1) in which case this convergence would be the exact analogue of (17). Unfortunately, because of the oscillations induced by the homogenization process, such exact analogue of (17) is false. This is one of the few differences between the 3d −1d problem and the homogenization problem.

Remark 7.
Before proceeding to prove the results in the next sections, it should be pointed out that the study can be easily extended to the case of operators in divergence form. In that case, we have to take into account at the limit the contribution of the anisotropy of the fiber (or the fibers in the case of homogenization) as shown in [21]. On the other hand, one can consider other scalings of the form ε γ χ F ε + ε δ χ M ε as in [10,22]. For instance, if one considers coefficients of order ε δ in the matrix M ε instead of ε 2 and 1 in F ε , then loosely speaking the structure of the limit problem depends on the limit of the ratio ε δ−2 as shown in [10,22], the critical case giving rise to a coupled system at the limit is the one corresponding to lim ε δ−2 = l ∈ ]0, +∞[. Here we address the critical case in the framework of the Laplacian operator for the sake of simplicity and brevity.

Remark 8.
In order to highlight the close analogy between the 3d − 1d limit problem and the homogenized problem, the macroscopic variable x will be denoted by x = (y, x 3 ), y ∈ C in the study of the 3d − 1d problem for which Ω := C × (0, L) while in the homogenization problem x will be denoted by Let us remember that in the reduction of dimension problem, Ω is obtained from the original thin domain Ω ε = (εC ) × (0, L) by the scaling x = εy, y ∈ C , which makes our notations homogeneous.
As mentioned above we study in detail the case of a single fiber; this is the purpose of the next section. In the last section, we will indicate the appropriate minor changes we have to do in the case of homogenization.

2.
Proof of the results in the case of a single fiber: the reduction of dimension 3d − 1d

Apriori estimate on the sequence of eigenvalues and eigenvectors
and a subsequence of ε still denoted by ε such that u k ε u k weakly in L 2 (0, L; H 1 (C )) and u k (y, Proof. We first prove an apriori estimate on the sequence of eigenvalues which will play a key role in the sequel. Throughout the paper we refer to λ 0 k as the k-th eigenvalue of − d 2 dx 2 3 in (0, L) with Dirichlet boundary conditions and to µ 1 as the first eigenvalue of−∆ y in C \ D with Dirichlet boundary condition on the boundary of D and Neumann (respectively periodic) boundary condition on the sides of C in the case of a single fiber (respectively homogenization). We claim that Indeed, it is well known that the k-th eigenvalue λ k ε of (9) is given by the Min-Max formula where the space V s is defined by (5) (with ε = 1) and the min runs over all subspaces V k of V s with finite dimension k.
Let φ(y) be an eigenvector associated to µ 1 extended by zero in D. Then φ(y)ψ(x 3 ) belongs to V s for any ψ ∈ H 1 0 (0, L) and φψ = 0 in F . Let V k be the subspace of V s spanned by φv 1  For any u = α 1 φv 1 + · · · + α k φv k ∈ V k , we have u = 0 in F and since v 1 , v 2 , . . . , v k , . . . , is an orthonormal basis in H 1 0 (0, L) we also have Note that the last equality in the second line of (26) is a consequence of the equation Hence, by this choice of V k we obtain estimate (24) by the use of the Min-Max formula combined with (26). We obtain that λ k ∈ (0, µ 1 ) by passing to the limit (for a subsequence of ε) in (24). We will prove later that the value µ 1 cannot be attained by λ k for all k and we will prove also that the sequence (λ k ) k admits a lower bound µ 0 > 0 depending on µ 1 and on the first eigenvalue λ 0 of the onedimensional laplacian operator with Dirichlet boundary conditions in (0, L).
Turning back to (9) and taking u k ε (with u k ε L 2 (Ω) = 1) as a test function, we get The last estimate implies that ∇ u k ε is bounded in L 2 (Ω) and thus u k ε is bounded in L 2 (0, L; H 1 (C )). We then conclude that there exist a sequence of ε and u k ∈ L 2 (C ; H 1 0 (0, L)) such that the convergence (21) holds true.
The sequence ∇ u k ε χ F (y) ∇ u k χ F weakly in L 2 (Ω). But ∇ u k ε χ F which is bounded in L 2 (Ω) by C ε strongly converges to zero in L 2 (Ω). Hence, ∇ u k χ F = 0 which means that u k = v k (x 3 ) a.e. in F . The sequence u k ε χ F (y) (note that the characteristic functions χ F and χ M depend only on the horizontal variable y) is bounded in L 2 (C ; H 1 0 (0, L)) since Hence v k ∈ H 1 0 (0, L) and the convergence (22) holds true.This completes the proof.

The limit problem associated to (9)
We choose a test function in (9) Passing to the limit in this equation, we get Finally a density argument allows to extend (29) to all test functions u ∈ L 2 (0, L; H 1 (C )) such that u = v in F and v ∈ H 1 0 (0, L).
Choosing successively in (29) u ∈ L 2 (0, L; H 1 (C )) such that u = 0 in F and then u ∈ L 2 (0, L; H 1 (C )) such that u = v ∈ H 1 0 (0, L) almost everywhere in Ω and bearing in mind the geometry of Ω := C ×(0, L) = (C \D)∪D ×(0, L), we get that the limit problem (29) may be split into two equations leading to the following equivalent system We are now in a position to show that the values 0 and µ 1 cannot be achieved by the sequence of eigenvalues (λ k ) k .
Proof. If λ k is an eigenvalue corresponding to the eigenvector u k then λ k > 0 as seen through (29) by taking u = u k , v = v k . On the other hand, by construction for each integer k, we have λ k ≤ µ 1 .
Let us prove that µ 1 cannot be achieved. Let φ be an eigenvector associated to the first eigenvalue of −∆ y , i.e., φ is a solution of It is well known that µ 1 is a simple eigenvalue and that φ is regular and keeps a constant sign as well as its normal derivative on ∂D, see [13]. Multiplying (31) by u k ( · , x 3 ) and the first equation of (30) (which holds true for almost all x 3 ∈ (0, L)) by φ and integrating we infer so that for a.e. x 3 ∈ (0, L), If λ k = µ 1 then we get v k = 0 a.e. in (0, L) since the integral ∂D ∂φ ∂n dσ does not vanish. Turning back to the last equation of (30), we get C \D u k dy = 0 since λ k = µ 1 = 0. On the other hand u k is an eigenvector for −∆ y associated to µ 1 = λ k since v k = 0. Therefore there exists η = 0 such that u k = ηφ which implies that C \D φ dy = 0. This is a contradiction since C \D φ dy = 0. Remark 11. Eigenvectors of (30) are pairs (u k , v k ) made up of two inseparable elements. In particular, if v k = 0 then u k = 0 as shown by (30). Indeed, otherwise u k should be an eigenvector of −∆ y associated to the eigenvalue λ k < µ 1 which is a contradiction. Conversely if u k = 0 then v k = 0 since almost everywhere in (0, L), we have v k = u k on the boundary of D.
We now prove that (11) and (30) are equivalent and then we will improve the lower bound of the limit eigenvalues using (11).

Proposition 12.
If (λ k , u k , v k ) solves the system (30) then v k = 0 and u k writes as where (λ k , u k 0 , v k ) solves (11). Furthermore, there exists a positive constant µ 0 depending on the first eigenvalues of −∆ y and − d 2 dx 2 3 such that λ k ≥ µ 0 for all k.
Proof. Assume that (u k , v k ) is a solution of (30). If v k = 0 then u k = 0 otherwise λ k should be an eigenvector of −∆ y by virtue of (30) but this is a contradiction since we have proved that λ k < µ 1 . Hence each solution of (30) is such that u k = 0 and v k = 0 unless u k = v k = 0. We will prove below that the later cannot happen, i.e., we have indeed an eigenvector at the limit, see Proposition 13.
Dividing by v k in the first system of (30), one can check easily that w k := Note that the uniqueness of w k is ensured since λ k < µ 1 belongs to the resolvent of −∆ y . On the other hand, the function λ k u k 0 where u k 0 is defined in (11) is also a solution of (35). Hence, we deduce that w k := u k v k − 1 = λ k u k 0 and therefore (34) follows. Using (34) in (30) we get (11). We now make more precise the lower bound of the sequence of eigenvalues and we prove at the meanwhile that C \D u k 0 (y) dy > 0. Multiplying the first equation of (11) by u 0 and bearing in mind that the best constant in the Poincaré's Inequality is equal to 1 µ 1 , we get On the other hand, the first eigenvalue µ 1 is characterized by Hence the following estimate holds true C \D |∇ y u k 0 (y)| 2 dy ≥ µ 1 C \D |u k 0 (y)| 2 dy.
From (36), we derive with the help of (38) and then from (39) we deduce . Using the second inequality of (40) we get Hence, λ k ≥ µ 0 := φ −1 (λ 0 ) where φ is the continuous increasing function defined on (0, µ 1 ) by So far, we have not yet proved that (u k , v k ) is indeed an eigenvector; this is the purpose of the next subsection.

The strong convergence of the eigenvectors
We prove the following compactness result Proposition 13. For each k, there exists a subsequence of ε such that the sequence of solutions u k ε of (9) converges strongly in L 2 (Ω) to the eigenvector u k of (30).
Proof. One can extend u k ε from F to the whole Ω in such a way that the extension U k ε fulfills Note that the extension only affects the horizontal variable y and the Dirichlet boundary condition on the upper and lower faces of Ω (x 3 = 0 or x 3 = L) is preserved, see for instance [5,9,25]. In addition, one can assume that such extension satisfies in addition the two equations Indeed, if (43) is not true for U k ε , then one can introduce the function W k ε as the unique solution of where V := u ∈ V s , u = 0 on ∂D × (0, L) . Hence, V is the subspace of V s of functions vanishing in F . By the Lax-Milgram Theorem we get the existence and uniqueness for W k ε . Choosing φ ∈ C ∞ 0 (M ), the last equation leads to Turning back to (44) and taking φ ∈ C 1 (M ), we get with the help of (45) On the other hand, using equation (44) with φ = W k ε , we get the following estimate with the help of (42) and (27) We now set u k ε = U k ε − W k ε . Multiplying equation (45) by ε 2 , we see that u k ε fulfills all the required properties including the two equations (43). In the sequel, we denote by U k ε the extension of u k ε satisfying (42) and (43). Consider now the sequence defined in Ω by z k ε = u k ε −U k ε . If we prove that z k ε admits a strongly converging subsequence in L 2 (Ω) then we can deduce the existence of such subsequence for u k ε since U k ε is bounded in H 1 (Ω) by virtue of (42) and (27) and therefore it admits a strongly converging subsequence in L 2 (Ω) according to the Rellich imbedding Theorem.
We first derive the following equation on z k ε by the use of (6) together with (43) Since u k ε and U k ε are bounded respectively in L 2 (0, L; H 1 (C )) and H 1 (Ω), the sequence z k ε is bounded in L 2 (0, L; H 1 (C )). Hence, there exist a subsequence and z k ∈ L 2 (0, L; H 1 (C )) such that z k ε z k weakly in L 2 (0, L; H 1 (C )). Therefore, denoting by U k the weak limit in H 1 (Ω) of the corresponding subsequence U k ε , one can pass easily to the limit in (48) to get the equation Note that by construction, z k ε = 0 in F = D × (0, L) so that the convergence z k ε χ F (y) z k χ D (y) weakly in L 2 (Ω) shows that z k = 0 in F which is equivalently written in the first boundary condition of (49). The Neumann boundary condition may be obtained in a classical way multiplying (48) first by a test function φ ∈ C ∞ 0 (M ) which allows to get equation (49) and then by a test function φ such that φ ∈ V s , φ = 0 in F and φ ∈ C 1 ((0, L) ×C ).
More generally, given a bounded sequence ( f ε ) in L 2 (Ω) and f ∈ L 2 (Ω), we now consider equations of the form Regarding the sequence of solutions of (50), the following lemma holds true.

Lemma 14.
Assume that λ k ε → λ k and that f ε f weakly in L 2 (Ω). Then the sequence w ε is bounded in L 2 (0, L; H 1 (C )) and there exists a subsequence of ε such that w ε w weakly in L 2 (0, L; H 1 (C )) where w is the unique solution of (51).
Proof. We only have to prove that w ε is bounded in L 2 (0, L; H 1 (C )), the limit problem (51) satisfied by w can be established exactly by the same process already used in the proof of (49).
The main ingredient to get that apriori estimate relays on the Poincaré inequality combined with the bound λ k < µ 1 proved in Proposition 10. Note that we have used 1 µ 1 in (52) which is known as the best Poincaré's constant. Multiplying equation (50) by w ε and integrating, we get Choosing in (52) u = w ε ( · , x 3 ) with x 3 ∈ (0, L) and integrating over 0, L), we infer Let δ > 0 be such that 0 < λ k < δ < µ 1 . Turning back to (53) and using (54), we get for ε sufficiently small, Since f ε is bounded in L 2 (Ω), applying once again inequality (54) together with the Young inequality, we derive from (55) the estimate (54) and (56) allow to conclude that w ε is bounded in L 2 (0, L; H 1 (C )).
We continue the proof of the Proposition 13 in the following way.
Multiplying the equations (48) and (50) respectively by w ε and by z k ε and integrating we get Choosing u k ε and (u k , v k ) as test functions respectively in (9) and in (29), we get with the help of the weak convergences proved in Proposition 9 and of the strong convergence proved in Proposition 13, Hence the weak convergences stated in Proposition 9 are in fact strong convergences; in particular, keeping in mind Proposition 13, we get the strong convergences stated in Theorem 3. We now prove that any λ ∈ (µ 0 , µ 1 ) which is an eigenvalue of (11) may be attained as a limit of a sequence (λ k ε ) ε ; By this we can conclude that (11) has no other eigenvalues than those obtained as the limits of the eigenvalues λ k ε and thus we can list all its eigenvalues in increasing order. It is then clear that for a fixed k, we cannot have two subsequences ε and ε with two different limits for λ k ε and λ k ε since this would lead to add a new element to the set of eigenvalues of (11); hence (15) holds for the whole sequence ε.
We argue by contradiction to prove that any λ ∈ (µ 0 , µ 1 [ which is an eigenvalue of (11) may be attained as a limit of a sequence (λ k ε ) ε . If for any sequence, λ k ε does not converge to λ, then there exists a neighborhood of λ which does not contain any λ k ε for all k. In other words, λ belongs to the resolvent of the operator A ε defined by (6). Hence, for any f ∈ L 2 (0, L) ⊂ L 2 (Ω), there exists u ε ∈ D(A ε ) such that (64) Multiplying (64) by φ ∈ V s and integrating we get To get apriori estimates on the sequence u ε , we will use the following Poincaré type inequality Lemma 15. There exists a positive constant K such that , ∀ u ∈ L 2 (0, L; H 1 (C )) ∩ L 2 (D; H 1 0 (0, L)).
Proof. We argue by contradiction. Assuming inequality (66) false, one can find a sequence u n ∈ L 2 (0, L; H 1 (C )) ∩ L 2 (D; H 1 0 (0, L) such that Thanks to the classical Poincaré inequality u − 1 |D| D u dy L 2 (C ) ≤ K ∇ u L 2 (C ) , ∀ u ∈ H 1 (C ) applied to u = u n ( · , x 3 ), x 3 ∈ (0, L), we get after integrating with respect to x 3 , (remember that On the other hand, the Poincaré inequality for functions of H 1 0 (0, L) applied with u(x 3 ) = D u n (y, x 3 ) dy leads to the estimate D u n dy Combining (68) and (69) with (67), we come to a contradiction.
Taking φ = u ε in (65) and applying (66) with u = u ε (note that V s ⊂ L 2 (0, L; H 1 (C )) ∩ L 2 (D; H 1 0 (0, L)), we get the same apriori estimates as those obtained for the sequence u k ε in (27). Indeed all the apriori estimates on the sequence u k ε are based on its L 2 (Ω)-apriori estimate which still holds true for the sequence u ε . Hence we can use the same arguments that led to the system (30) to pass to the limit ε → 0 in (65) and we obtain at the limit (70) Note that v = 0 for g = 0 otherwise the first equation in (70) would implies u = 0 since λ < µ 1 is not an eigenvalue of −∆ y . Hence one can express u in terms of (λ, u 0 , v) according to (34) as u = (λu 0 +1)v. Note that the function u k 0 arising in (11) depends on k only through the associated eigenvalue λ k so that one can drop here the subscript k and denote the associated function depending on λ by u 0 . Therefore (71) takes the form On the other hand, by hypothesis, λ is an eigenvalue of (11) so that the last equation of (11) with the same u 0 which depends only on λ shows that λ 1 + |C \D| |D| + λ |D| C \D u 0 dy is an eigenvalue of − d 2 dx 2 3 . This is a contradiction since equation (72) valid for all g ∈ L 2 (0, L) means that the number We now prove that lim k→+∞ λ k = µ 1 .
Since λ k ∈ (µ 0 , µ 1 ) for any k, the sequence (λ k ) k admits at least an accumulation point and each accumulation point λ is such that µ 0 ≤ λ ≤ µ 1 . Assume that there exists an accumulation point λ such that λ < µ 1 . There exists a subsequence (λ k n , u k n 0 , v k n ) of solutions of (11) such that lim n→+∞ λ k n = λ. Hence the following equation takes place for all n Let δ be a positive number such that λ < δ < µ 1 . For n large enough we have λ k n ≥ δ so that applying the Poincaré inequality after multiplying (73) by u k n 0 , we get for n large enough Applying successively the Cauchy-Schwarz inequality and (74) in the last integral of (75), we infer Therefore, (u k n 0 ) n is bounded in H 1 (C \ D) and one can assume that (possibly for another subsequence) (u k n 0 ) n converges weakly to u 0 in H 1 (C \ D). In particular we have that lim n→+∞ C \D u k n 0 dy = C \D u 0 dy. On the other hand (λ k n , u k n 0 , v k n ) being a solution of (11), the following equation (recall that v k n = 0) shows that the number µ defined by µ := λ 1+ |C \D| |D| + λ |D| C \D u 0 dy is a finite accumulation point of the spectrum of − d 2 dx 2 3 since µ = lim n→+∞ µ n where µ n := λ k n 1+ |C \D| |D| + λ kn |D| C \D u k n 0 dy . This is a contradiction since it is well known that such spectrum is in fact an increasing sequence which tends to +∞.
The last point which remains to prove is that all the limiting eigenvalues are simple and that u k ε converges to u k for the whole sequence ε. Assuming that λ k is a simple eigenvalue, the proof of the convergence of the eigenvectors for the whole sequence ε is known since the work of [23] (see also [9]). We sketch it for the convenience of the reader.
Assume that u k is an eigenvector associated to the simple eigenvalue λ k . Using the fact that the eigenvalues converge for the whole sequence ε, it is easy to check that the multiplicity of λ k is equal or greater than that of λ k ε ; hence λ k ε is simple and there are only two eigenvectors satisfying Ω |u k ε | 2 dx = 1, namely u k ε and −u k ε . Among these two eigenvectors, we choose the one satisfying the inequality Therefore if ε is a subsequence such that u k ε strongly converges in L 2 (Ω) to the eigenvector u associated to λ k , we get by passing to the limit in (78), On the other hand, u k = u or u k = − u since λ k is a simple eigenvalue. The last equality is excluded thanks to (79) so that any subsequence converges to u k . Let us now prove that all the limit eigenvalues are simple eigenvalues. Assume that for some k, (13) holds true for two orthogonal eigenvectors u k v k and ū k v k . By hypothesis, we have We know that u k and u k are given respectively by u k (y, x 3 ) = (λ k u k 0 (y) + 1))v k (x 3 ) and u k (y, where u k 0 (y) given by the first equation of (11) depends only on the eigenvalue λ k .
Turning back to (80), we infer As remarked above v k and v k are always eigenvectors of the operator −

The case of homogenization
In this section we only take up the key points of the previous proofs to highlight the few minor changes to be made, changes due to the oscillations induced by the homogenization process. The points of Theorem 5 whose proofs are identical to the corresponding proofs done in the 3d − 1d case will not be mentioned here for the sake of brevity.

Proof of the estimate (24)
In the spirit of the above section, the natural idea is to choose a test function vanishing over the set F ε of fibers. To that aim, we consider an eigenvector φ(y) corresponding to the first eigenvalue of −∆ y in C \D with the corresponding boundary conditions on the boundary: φ = 0 on ∂D and φ is C -periodic. We extend φ by zero inside D and by periodicity to the whole R 2 . The k-th eigenvalue λ k ε of (8) is now given by the same Min-Max formula as above, namely For each ε, consider the subspace . . , v k as those defined in the previous section.

Proof of the analogous of Proposition 9
To continue the proof in the homogenization case we use the two-scale convergence in order to describe the effect of the oscillations due to the fast variable.
Analogously to the previous section, the use of the estimate (24) in (8) leads to the estimates Hence, by a classical result of the two-scale convergence method, see [1], there exists u k (x, y) ∈ L 2 (Ω; H 1 # (C )) such that for a subsequence of ε, The sequence ε∇ u k ε χ F ε converges strongly (and thus two-scale converges) to zero in L 2 (Ω). On the other hand, one can check that the two-scale limit of ε∇ u k ε χ F ε is also equal to ∇ y u k (x, y)χ D (y), see [21]. Hence we deduce that ∇ y u k (x, y)χ D (y) = 0 in such a way that is bounded in L 2 (Ω) by virtue of (85) so that there exists z ∈ L 2 (ω; H 1 0 (0, L)) such that u k ε χ F ε converges weakly to z in L 2 (ω; H 1 0 (0, L)). Since the weak limit in L 2 (Ω) is the average over the reference cell C of the two-scale limit, we get z( which implies that v k actually belongs to L 2 (ω; H 1 0 (0, L)). Let us remark that since χ F ε (x ) = χ D ( x ε ) 2−sc χ D (y) the natural statement of the convergence (22) is the which implies the weak convergence ∂u k ε ∂x 3 χ F ε |D| ∂v k ∂x 3 weakly in L 2 (Ω).
The convergence (87) is finer and more advantageous. Hence replacing weak convergences by the corresponding two-scale convergences, Proposition 9 still holds true in the homogenization setting.
3.3. The limit problem associated to (8) In contrast to the previous section where it was not necessary to specify the limit of the transversal gradient ∇ u k ε χ F , here we have first to specify the two-scale limit of ∇ u k ε χ F ε in order to pass to the limit in (8). We use the property proved in [21] that the two-scale limit is in fact given by ∇ w(x, y)χ D (y) where w belongs to L 2 (Ω; H 1 (D)/R) and that w = 0 when dealing with isotropic materials which is the case here.
Hence, using this remark and choosing a test function u(x, x ε ) ∈ C ∞ 0 (Ω; H 1 # (C )), u = v(x) in Ω× D with v ∈ C ∞ 0 (Ω), we can pass to the limit in (8) Analogously to the second section, (89) may be written as a system in the following strong form u k (x, y) ∈ L 2 (Ω; H 1 # (C )), −∆ y u k (x, y) = λ k u k in Ω × (C \ D), u k (x, y) = v k (x) on Ω × ∂D, for a.e. x ∈ Ω, u k (x, · ) is C-periodic, v k ∈ L 2 (ω; H 1 0 (0, L)), − The main point now is on whether the new periodic boundary condition on ∂C replacing the homogenous Neumann one is likely to hamper the proof of the Proposition 10.

Proof of the analogous of Proposition 10
After multiplying the first equation in (90) by the eigenvector φ associated to the first eigenvalue µ 1 of −∆ y (with periodic boundary condition on ∂C ) and the equation −∆ y φ = µ 1 φ by u k , we are led to the following equations by integration over C \ D Due to the periodicity of u k and φ, the two boundary integrals on ∂C arising in (91) vanish since the normal derivative has opposite values on opposites faces of C . Except this remark, there is no other change in the rest of the proof.

Proof of the analogous of Proposition 13
The proof of Proposition 12 is based on the use of 1 µ 1 as the best constant in the Poincaré inequality for functions of H 1 (C ) vanishing on ∂D. We have applied that property to the sequence of solutions of (11). In the homogenization setting, we apply the same property to solutions of (10) which also vanish on the boundary ∂D.
The analogous of Proposition 13 will be established in the proof of Theorem 5 below.

Proof of Theorem 5
The approach is identical as that used in the proof of Theorem 3, in particular the extension results are still valid, see [5,9,25]. Despite this, we would like to indicate how to deal with the question of the compactness of the solutions, which is a key point of the proof. More precisely, we now consider the analogous system of (50) which takes the following form in the homogenization setting Remark that the second equality is in fact the Dirichlet boundary condition on the common part of the lateral boundary of M ε with that of F ε . The last boundary condition concerns the rest of the lateral boundary of M ε . The equivalent of system (51) now writes as        w ∈ L 2 (Ω; H 1 # (C )), −∆ y w = λ k w + f in Ω × (C \ D), w = 0 on Ω × ∂D, for a.e. x ∈ Ω, w(x, · ) is C-periodic.
On the other hand, applying the one-dimensional Poincaré inequality to the function Taking the sum over i ∈ I ε , we derive the inequality (99) as a consequence of (101) and (102).
The rest of the proof of Theorem 5 is done exactly as in the previous section for the proof of Theorem 3.