Federica Raimondi
email: federica.raimondi@inria.fr

Homogenization of a class of singular elliptic problems in two-component domains

Keywords: Two-component domains, Homogenization, Periodic Unfolding Method, Quasilinear elliptic equations, Singular equations

This paper deals with the homogenization of a quasilinear elliptic problem having a singular lower order term and posed in a two-component domain with an ε-periodic imperfect interface. We prescribe a Dirichlet condition on the exterior boundary, while we assume that the continuous heat flux is proportional to the jump of the solution on the interface via a function of order ε γ .

We prove an homogenization result for -1 < γ < 1 by means of the periodic unfolding method (see SIAM J. Math. Anal. 40 (2008) 1585-1620 and The Periodic Unfolding Method (2018) Springer), adapted to two-component domains in (J. Math. Sci. 176 (2011) 891-927).

One of the main tools in the homogenization process is a convergence result for a suitable auxiliary linear problem, associated with the weak limit of the sequence {u ε } of the solutions, as ε → 0. More precisely, our result shows that the gradient of u ε behaves like that of the solution of the auxiliary problem, which allows us to pass to the limit in the quasilinear term, and to study the singular term near its singularity, via an accurate a priori estimate.

Introduction

In this paper we study the asymptotic behavior of a class of quasilinear elliptic problems presenting singular lower order terms and posed in two-component domains.

More precisely, the two-component domain is a bounded open subset of R N which is the union of two disjoint open subsets, ε 1 and ε 1 , and their common boundary Ŵ ε . The disconnected component ε 2 is the union of the ε-periodic translated sets of εY 2 , where Y 2 is well contained in the reference cell Y and Ŵ = ∂Y 2 . The connected component ε 1 is obtained by removing from the closure of ε 2 such that the interface Ŵ ε = ∂ ε 2 . We deal with the homogenization, as ε goes to zero, of the following problem:

         -div(A ε (x, u ε )∇u ε ) = f ζ (u ε ) in \ Ŵ ε , u ε 1 = 0 on ∂ , (A ε (x, u ε 1 )∇u ε 1 )ν ε = (A ε (x, u ε 2 )∇u ε 2 )ν ε on Ŵ ε , (A ε (x, u ε 1 )∇u ε 1 )ν ε = -ε γ h ε (u ε 1 -u ε 2 ) on Ŵ ε ,
where ν ε is the unit external normal vector to ε 1 . We prescribe a Dirichlet condition on the exterior boundary, while we assume that the continuous heat flux is proportional to the jump of the solution on the interface by means of a function of order ε γ . The quasilinear diffusion matrix field is defined by A ε (x, t) = A( x ε , t), where the matrix field A is uniformly elliptic, bounded, periodic in the first variable and Carathéodory. The nonlinear real function ζ (s) is nonnegative and singular at s = 0, while f is a nonnegative datum whose summability depends on the growth of ζ near its singularity. Concerning the boundary condition, h ε (x) = h( x ε ) where h is assumed to be a periodic, nonnegative and bounded function, and γ ∈] -1, 1[. This problem describes the stationary heat diffusion in a two-component composite with an ε-periodic imperfect interface. In particular, quasilinear diffusion terms describe the behavior of materials like glass or wood, in which the heat diffusion depends on the range of the temperature (see for instance [START_REF] Bever | Encyclopedia of Material Science and Engineering[END_REF]). We refer to [START_REF] Donato | Existence and homogenization for a singular problem through rough surfaces[END_REF]Section 3] for more details about source terms depending on the solution itself and becoming infinite when the solutions vanish. These kind of source terms can model, for instance, electrical conductors where each point becomes a source of heat as a current flows inside. The boundary condition on Ŵ ε models a jump of the solution through a rough interface and we refer to [START_REF] Carslaw | Conduction of Heat in Solids[END_REF] for a physical justification of this model (see also [START_REF] Monsurrò | Homogenization of a two-component composite with interfacial thermal barrier[END_REF]).

In view of a counterexample suggested by H.C. Hummel in [START_REF] Hummel | Homogenization for heat transfer in polycristals with interfacial resistances[END_REF], one cannot expect bounded a priori estimates for the solution when γ > 1. For this case we refer to [START_REF] Donato | Homogenization of two heat conductors with an interfacial contact resistance[END_REF] where different a priori estimates needed. As for our problem, existence and uniqueness results has been proved in [START_REF] Donato | Uniqueness result for a class of singular elliptic problems in two-component domains[END_REF][START_REF] Donato | Existence and uniqueness results for a class of singular elliptic problems in two-component domains[END_REF] for γ 1.

Here we prove an homogenization result for γ ∈] -1, 1[ by means of the periodic unfolding method. This method was originally introduced by D. Cioranescu, A. Damlamian and G. Griso in [START_REF] Cioranescu | Periodic unfolding and homogenization[END_REF] and [START_REF] Cioranescu | The periodic unfolding method in homogenization[END_REF] for fixed domains and extended to perforated ones in [START_REF] Cioranescu | Periodic unfolding and Robin problems in perforated domains[END_REF] and [START_REF] Cioranescu | The periodic unfolding method in perforated domains[END_REF]. Then, it has been adapted to twocomponent domains by P. Donato, K.H. Le Nguyen and R. Tardieu in [START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF] (see also [START_REF] Donato | Homogenization of diffusion problems with a nonlinear interfacial resistance[END_REF]).

This work has been partially inspired by the works [START_REF] Donato | Existence and homogenization for a singular problem through rough surfaces[END_REF] and [START_REF] Donato | Homogenization of a class of singular elliptic problems in perforated domains[END_REF]. Let us point out the main differences with respect to them and the additional difficulties. In [START_REF] Donato | Existence and homogenization for a singular problem through rough surfaces[END_REF] the authors treat the same singularity when A is linear and in a different geometrical framework where the domain has two connected components separated by an oscillating interface. This interface tends to a flat one, so that the integral on Ŵ ε goes to the one on Ŵ, roughly speaking. But here Ŵ ε is the union of disconnected sets (of order 1 ε ) so that its measures goes to inifnity, whence the integral on Ŵ ε needs particular care.

Let us mention that in [START_REF] Donato | Homogenization of a class of singular elliptic problems in perforated domains[END_REF] the authors consider the same singular problem with A quasilinear but posed in a periodically perforated domain. Here the holes are replaced by a second material so that we have to treat in addition all the integrals on the second component and the boundary term with jump.

In Section 2 we present the setting of the problem and we state the main results.

In Section 3 we give a short presentation of the periodic unfolding method, adapted to two-component domains.

In Section 4 we prove two a priori estimates uniform with respect to ε and we give an estimate of the integral of the singular term close to the singular set {u ε = 0}, in terms of the quasilinear one (cf. Proposition 4.3).

In Section 5 we prove a crucial convergence result, Theorem 5.4. It is one of the main tools when proving our homogenization result and it shows that the gradient of u ε is equivalent to the gradient of the solution of a suitable auxiliary linear problem, associated with a weak cluster point of the sequence {u ε }, as ε → 0. This idea was originally introduced in [START_REF] Bensoussan | H-convergence for quasi-linear elliptic equations with quadratic growth[END_REF] (see also [START_REF] Bensoussan | H-Convergence for Quasilinear Elliptic Equations Under Natural Hypotheses on the Correctors, Composite Media and Homogenization Theory[END_REF]) where some nonlinear problems with quadratic growth are considered. We refer to [START_REF] Chourabi | Homogenization and correctors of a class of elliptic problems in perforated domains[END_REF] and [START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF] for the study of the auxiliary problem, while for the proof of the convergence result we adapt some techniques from [START_REF] Donato | Existence and homogenization for a singular problem through rough surfaces[END_REF] and [START_REF] Donato | Homogenization of a class of singular elliptic problems in perforated domains[END_REF].

Section 6 is devoted to the proofs of Proposition 2.5 and the homogenization theorem. For these results Theorem 5.4 plays an important rule not only in the study of the quasilinear term but also in that of the singular one. Actually, as done in [START_REF] Donato | Existence and homogenization for a singular problem through rough surfaces[END_REF] and [START_REF] Donato | Homogenization of a class of singular elliptic problems in perforated domains[END_REF], we split the integral of the singular term into the sum of two integrals: one on the set where the solution is close to the singularity and one where is it far from it, Finally, let us point out that the homogenization result we prove here shows that the conductivity of the first material is the same obtained when there is no material occupying ε 2 . On the other hand, since in the limit problem it appears f instead of θ 1 f (being θ 1 the proportion of the material occupying ε 1 ), one has anyway to take into account even the source term in the second component.

The first paper on this subject is due to [START_REF] Auriault | Macroscopic modelling of heat transfer in composites with interfacial thermal barrier[END_REF] for the linear (nonsingular) case by multiple scale method. In [START_REF] Donato | Homogenization of two heat conductors with an interfacial contact resistance[END_REF] and [START_REF] Monsurrò | Homogenization of a two-component composite with interfacial thermal barrier[END_REF] the authors also studied the linear case by using the Tartar method. For similar homogenization elliptic problems we refer to [START_REF] Canon | Homogenization of diffusion in composite media with interfacial barrier[END_REF][START_REF] Donato | Existence and homogenization for a singular problem through rough surfaces[END_REF][START_REF] Donato | Homogenization of a class of singular elliptic problems in perforated domains[END_REF][START_REF] Hummel | Homogenization for heat transfer in polycristals with interfacial resistances[END_REF] and [START_REF] Pernin | Homogénéisation d'un problème de diffusion en milieu composite à deux composantes[END_REF]. In [START_REF] Donato | Homogenization of diffusion problems with a nonlinear interfacial resistance[END_REF][START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF] the authors study the linear case in presence of linear and nonlinear boundary conditions, respectively.

Setting of the problem and main results

Throughout the paper, we use the same notation as in [START_REF] Donato | Homogenization of diffusion problems with a nonlinear interfacial resistance[END_REF][START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF] for the periodic unfolding method in two-component domains.

The domain. For

N ∈ N, N 2, let be a bounded open set in R N with a Lipschitz-continuous boundary ∂ . Also let Y . = N i=1 [0, l i [ be a reference cell, with l i > 0, i = 1, . . . , N. We suppose that Y 1 and Y 2 are two disjoint connected open subsets of Y such that Y 2 = ∅, Y 2 ⊂ Y and Y = Y 1 ∪ Y 2 , with a common boundary Ŵ = ∂Y 2 Lipschitz-continuous.
Let {ε} ε>0 be a positive parameter taking values in a sequence converging to zero and set:

• for any k ∈ Z N , k l = (k 1 l 1 , . . . , k N l N )
and

Y k = k l + Y, Y k i = k l + Y i , i = 1, 2, • K ε = {k ∈ Z N | εY k 2 ⊂ } and ε 2 = k∈K ε εY k 2 , ε 1 = \ ε 2 , Ŵ ε = ∂ ε 2 .
By construction, results the union of the two disjoint components and their common boundary, i.e. = ε 1 ∪ ε 2 ∪ Ŵ ε (see Fig. 1). Also we introduce the following sets:

K ε = k ∈ Z N | εY k ⊂ , ε = interior k∈ K ε ε(k l + Y ) , ε = \ ε ,
and

ε i = k∈ K ε εY k i , ε i = ε i \ ε i , i = 1, 2, Ŵ ε = ∂ ε 2 .
Also, there exists ε 0 > 0 such that ∀ε ε 0 , ε 2 = ∅.

(2.1) Let us note that here, for the sake of simplicity, we do not allow the second component to meet the boundary of the domain and also we suppose Y 2 connected. Actually the cases where the holes meet the boundary of and Y 2 has a finite number of connected components can be treated as done in [START_REF] Donato | Homogenization of diffusion problems with a nonlinear interfacial resistance[END_REF] and [START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF], respectively.

In the sequel we denote by

• M ω (v) . = 1 |ω| ω v dx the average of any function v ∈ L 1 (ω), for any open set ω of R N , • χ ω the characteristic function of any open set ω of R N ,
• ∼ the zero extension to the whole of of functions defined on ε 1 or ε 2 ,

• θ i . = |Y i | |Y | , i = 1, 2, • c different positive constants independent on ε, • M(α, β, Y ) the set of matrix fields A = (a i,j ) 1 i,j N ∈ (L ∞ (Y )) N ×N such that A(x)λ, λ α|λ| 2 and A(x)λ β|λ|, ∀λ ∈ R N and a.e. in Y, with α, β ∈ R, 0 < α β, • {e 1 , . . . , e N } the canonical basis of R N , • v i = v | ε i the restriction to ε i of functions v defined in , i = 1, 2.

Moreover let us recall the classical decomposition for every real function

v v = v + -v -, v + . = max{v, 0} and v -. = -min{v, 0} a.e. in , (2.2)
where v + and v -are both nonnegative. In particular, one has The problem. The aim of the paper is to study the asymptotic behavior, as ε goes to zero, of the following problem:

(v 1 -v 2 ) v - 1 -v - 2 = v + 1 -v + 2 v - 1 -v - 2 -v - 1 -v - 2 2 = -v + 1 v - 2 -v + 2 v - 1 -v - 1 -v -
         -div(A ε (x, u ε )∇u ε ) = f ζ (u ε ) in \ Ŵ ε , u ε 1 = 0 on ∂ , (A ε (x, u ε 1 )∇u ε 1 )ν ε = (A ε (x, u ε 2 )∇u ε 2 )ν ε on Ŵ ε , (A ε (x, u ε 1 )∇u ε 1 )ν ε = -ε γ h ε (u ε 1 -u ε 2 ) on Ŵ ε , (2.4)
where ν ε is the unit external normal vector to ε 1 , and we prescribe a Dirichlet condition on the exterior boundary and a jump of the solution on the interface Ŵ ε , in the case γ ∈] -1, 1[.

Assumptions on the data.

H 1 ) The real N × N matrix field A : (y, t) ∈ Y × R → A(y, t) = (a i,j (y, t)) i,j =1,...,N ∈ R N 2 satisfies the following conditions:

                   i) A is a Carathéodory function s.t. A(•, t
) is Y -periodic and in M(α, β, Y ), for every t ∈ R; ii) there exists a real function ω : R → R satisfying the following conditions:

ω is continuous and non decreasing, with ω(t

) > 0 ∀t > 0, -|A(y, t 1 ) -A(y, t 2 )| ω(|t 1 -t 2 |) for a.e. y ∈ Y, ∀ t 1 = t 2 , -∀s > 0, lim y→0 + s y dt ω(t) = +∞. H 2 ) The functions ζ and f verify      i) ζ : [0, +∞[→ [0, +∞] is a function such that ζ ∈ C 0 ([0, +∞[), 0 ζ (s) 1 s k for every s ∈]0, +∞[, with 0 < k 1; ii) ζ is non increasing; iii) f 0 a.e. in , f ≡ 0, with f ∈ L l ( ), for l 2 1 + k ( 1).
H 3 ) -1 < γ < 1, and h is a Y -periodic function in L ∞ (Ŵ) such that there exists h 0 ∈ R : 0 < h 0 < h(y) a.e. on Ŵ.

Under the above assumptions we set, for every t ∈ R,

A ε (x, t) . = A x ε , t for a.e. x ∈ , h ε (x) . = h x ε for a.e. x ∈ Ŵ ε . (2.5)
The functional framework. We now introduce the functional spaces used in the literature to handle (2.4)-type problems.

Let V ε . = {v ∈ H 1 ( ε 1 )|v = 0 on ∂ } endowed with the norm v V ε = ∇v L 2 ( ε 1 )
. Remark 2.1. It is known (see for instance [18, Lemma 1], [START_REF] Cioranescu | Homogenization of Reticulated Structures[END_REF]) that a Poincaré inequality in V ε holds with a constant c P independent on ε, that is

v L 2 ( ε 1 ) c P ∇v L 2 ( ε 1 ) ∀v ∈ V ε . (2.6)
Consequently, the norm in V ε is equivalent to that in H 1 ( ε 1 ) via a constant independent on ε. For every γ ∈ R, let H ε γ be the space defined by

H ε γ . = v ∈ L 2 ( )| v 1 ∈ V ε , v 2 ∈ H 1 ε 2 ,
which, after the identification ∇v := ∇v 1 + ∇v 2 , is equipped by the norm

v 2 H ε γ . = ∇v 2 L 2 ( \Ŵ ε ) + ε γ v 1 -v 2 2 L 2 (Ŵ ε ) .
Proposition 2.2 ([22,29]). Let γ 1. There exist some positive constants c 1 , c 2 and C, independent of ε, such that

∀v ∈ H ε γ , c 1 v 2 V ε ×H 1 ( ε 2 ) v 2 H ε γ c 2 1 + ε γ -1 v 2 V ε ×H 1 ( ε 2 ) . In addition, if v ε = (v ε 1 , v ε 2 ) is a bounded sequence in H ε γ , then v ε 1 H 1 ( ε 1 ) C, v ε 2 H 1 ( ε 2 ) C, v ε 1 -v ε 2 L 2 (Ŵ ε ) Cε -γ 2 .
The variational formulation associated with problem (2.4) reads

         Find u ε ∈ H ε γ such that u ε > 0 a.e. in , f ζ (u ε )ϕ dx < +∞, and \Ŵ ε A ε (x, u ε )∇u ε ∇ϕ dx + Ŵ ε ε γ h ε (u ε 1 -u ε 2 )(ϕ 1 -ϕ 2 )dσ = f ζ (u ε )ϕ dx, ∀ϕ ∈ H ε γ .
(2.7)

In [START_REF] Donato | Uniqueness result for a class of singular elliptic problems in two-component domains[END_REF][START_REF] Donato | Existence and uniqueness results for a class of singular elliptic problems in two-component domains[END_REF] it is proved that, under assumptions H 1 )-H 3 ), problem (2.7) admits a unique solution.

Let us introduce here the homogenized matrix

A 0 (t), t ∈ R, corresponding to our case γ ∈] -1, 1[. It is defined by A 0 (t)λ . = 1 |Y | Y 1 A(y, t) λ -∇ y χ λ (y, t) dy ∀λ ∈ R N , (2.8)
where, for every

λ ∈ R N , χ λ (•, t) ∈ H 1 (Y 1 ) are unique solutions of the cell problems      -div(A(•, t)∇ y χ λ (•, t) = -div(A(•, t)λ) in Y 1 , A(•, t)(λ -∇ y χ λ (•, t))ν 1 = 0 on Ŵ, χ λ (•, t) Y-periodic and M Y 1 (χ λ ) = 0.
(2.9) The homogenized matrix A 0 is actually the one obtained in the framework of perforated domains. It has been originally introduced in [START_REF] Cioranescu | Homogenization in open sets with holes[END_REF] for linear problems with Neumann conditions in perforated domain, successively extended to quasilinear ones in [START_REF] Artola | Homogénéisation d'une classe de problèmes non linéaires[END_REF] and [START_REF] Artola | Un résultat d'homogénéisation pour une classe de problèmes de diffusion non linéaires stationnaires[END_REF].

We recall (see [START_REF] Cabarrubias | Homogenization of a quasilinear elliptic problem with nonlinear Robin boundary comditions[END_REF] and [START_REF] Cioranescu | Homogenization in open sets with holes[END_REF]) that the matrix A 0 satisfies the following properties:

i) A 0 is continuous and A 0 (t) ∈ M α, β 2 α , for every t ∈ R;
ii) there exists a positive constant C, depending only on α, β, Y and T s.t.

A 0 (t 1 ) -A 0 (t 2 ) Cω |t 1 -t 2 | for every t 1 , t 2 ∈ R, with t 1 = t 2 ,
where ω is the function given in H 1 ).

(2.10)

Remark 2.3. Observe that assumptions H 1 ) ii and H 2 ) ii are only needed for the uniqueness of the solution of problem (2.7). If they do not hold true, the homogenized problem is still the same but all the convergences remain valid only for a subsequence.

The main results. We now state the main results of this work, which will be proved in Section 6.

Theorem 2.4. Under assumptions H 1 )-H 3 ), let u ε ∈ H ε γ be the unique solution of problem (2.7). Then, there exist a subsequence (still denoted by ε),

u 1 ∈ H 1 0 ( ), u 1 ∈ L 2 ( , H 1 per (Y 1 )) with M Ŵ ( u 1 ) = 0 for almost every x ∈ and u 2 ∈ L 2 ( , H 1 (Y 2 )) with M Ŵ (u 2 ) = 0 for almost every x ∈ such that                i) T ε i (u ε i ) → u 1 strongly in L 2 ( , H 1 (Y i )), i = 1, 2, ii) T ε (ζ (u ε )) → ζ (u 1 ) a.e. in × Y, iii) u ε 1 ⇀ θ 1 u 1 weakly in L 2 ( ), iv) T ε 1 (∇u ε 1 ) ⇀ ∇u 1 + ∇ y u 1 weakly in L 2 ( × Y 1 ) v) T ε 2 (∇u ε 2 ) ⇀ ∇ y u 2 weakly in L 2 ( × Y 2 ), (2.11)
and

u 1 0 a.e. in and f ζ (u 1 )ϕ dx < +∞, ∀ϕ ∈ H 1 0 ( ).
(2.12)

Moreover, the pair (u 1 , u 1 ) is the unique solution of the unfolded limit equation

∀ϕ ∈ H 1 0 ( ) and ∀ψ ∈ L 2 ( ; H 1 per (Y 1 )) ×Y 1 A(y, u 1 )(∇u 1 + ∇ y u 1 )(∇ϕ + ∇ y ψ) dx dy = |Y | f ζ (u 1 )ϕ dx.
(2.13) Proposition 2.5. Under assumptions H 1 )-H 3 ), let (u 1 , u 1 , ū2 ) be given by Theorem 2.4. Then

u 1 (y, x) = -N i=1 χ e i (y, u 1 (x)) ∂u 1 ∂x i (x) ∈ L 2 ( ; H 1 per (Y 1 )), ∇ y ū2 ≡ 0 a.e. in × Y 2 ,
where χ e i (•, u 1 ), i = 1, . . . , N are the solutions of the cell problems (2.9), written for λ = e i . Then the homogenization result for problem (2.7) is Theorem 2.6. Under assumptions H 1 )-H 3 ), let u ε ∈ H ε γ be the unique solution of problem (2.7) and u 1 given by Theorem 2.4. Then u 1 > 0 almost everywhere in and u 1 is the unique solution of the following singular limit problem:

-div(A 0 (u 1 )∇u 1 ) = f ζ (u 1 ) in , u 1 = 0 on ∂ , (2.14)
where the homogenized matrix A 0 (t) is given by (2.8) and verifies

A 0 (u 1 )∇u 1 = 1 |Y | Y 1 A(y, u 1 )(∇u 1 + ∇ y u 1 ) dy. (2.15)
Consequently, convergences (2.11) (with ∇ y ū2 ≡ 0) hold for the whole sequence.

The periodic unfolding method

In this section we give a short presentation of the periodic unfolding method adapted to twocomponent domains by P. Donato, K.H. Le Nguyen and R. Tardieu in [START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF]. This method was originally introduced by D. Cioranescu, A. Damlamian and G. Griso in [START_REF] Cioranescu | Periodic unfolding and homogenization[END_REF] and [START_REF] Cioranescu | The periodic unfolding method in homogenization[END_REF] for fixed domains and extended to perforated ones in [START_REF] Cioranescu | Periodic unfolding and Robin problems in perforated domains[END_REF] and [START_REF] Cioranescu | The periodic unfolding method in perforated domains[END_REF].

To this aim, we recall the unfolding operators T ε 1 and T ε 2 and the boundary unfolding operator T b ε : T ε 1 and T b ε are exactly those ones introduced in [START_REF] Cioranescu | The periodic unfolding method in perforated domains[END_REF] for perforated domains, while T ε 2 has been introduced for the two-component domain in [START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF].

Let z ∈ R N , we denote by [z] Y its integer part such that z -[z] Y belongs to Y and set {z} Y . = z -[z] Y . Then, for every positive ε,

x = ε x ε Y + x ε Y ∀x ∈ R N .
Definition 3.1. For any Lebesgue-measurable function φ on ε i , the unfolding operators T ε i , i = 1, 2, are defined as follows:

T ε i (φ)(x, y) . = φ(ε[ x ε ] Y + εy) a.e. for (x, y) ∈ ε × Y i , 0 a.e. for (x, y) ∈ ε × Y i . (3.1)
For any Lebesgue-measurable function φ on Ŵ ε , the boundary unfolding operator T b ε is defined as follows:

T b ε (φ)(x, y) . = φ(ε[ x ε ] Y + εy) a.e. for (x, y) ∈ ε × Ŵ, 0 a.e. for (x, y) ∈ ε × Ŵ. (3.2)
T ε i (φ) instead of T ε i (φ i ), i = 1, 2
, for the sake of semplicity. Also we define T ε (ϕ) as follows:

T ε (φ)(x, y) . = T ε 1 (φ) in × Y 1 , T ε 2 (φ) in × Y 2 .
We now recall the main properties of the unfolding operators.

Proposition 3.3 ([17,22]). Let p ∈ [1, +∞[ and i = 1, 2.
1. T ε i is a linear and continuous operator from

L p ( ε i ) to L p ( × Y ). 2. T ε i (φψ) = T ε i (φ)T ε i (ψ) for every φ, ψ ∈ L p ( ε i ). 3. Let φ ∈ L p (Y i ) be a Y -periodic function and set φ ε (x) = φ( x ε ). Then T ε i (φ ε ) → φ strongly in L p ( × Y i ).
4. For all φ ∈ L 1 ( ε i ), one has

1 |Y | ×Y i T ε i (φ)(x, y) dx dy = ε i φ(x) dx = ε i φ(x) dx - ε i φ(x) dx.
5. For all φ ∈ L 1 (Ŵ ε ), one has

Ŵ ε φ(x) dσ x = 1 ε|Y | ×Ŵ T b ε (φ)(x, y) dx dσ y . 6. T ε i (φ) L p ( ×Y i ) |Y | 1 p φ L p ( ε i ) for every φ ∈ L p ( ε i ). 7. T b ε (φ) L p ( ×Ŵ) ε 1 p |Y | 1 p φ L p (Ŵ ε ) for every φ ∈ L p (Ŵ ε ). 8. For φ ∈ L p ( ), T ε i (φ) → φ strongly in L p ( × Y i ). 9. Let {φ ε } be a sequence in L p ( ) such that φ ε → φ strongly in L p ( ). Then T ε i (φ ε ) → φ strongly in L p ( × Y i ). 10. Let {φ ε } be a sequence in L p ( ε i ) such that φ ε L p ( ε i ) c. If T ε i (φ ε ) ⇀ φ weakly in L p ( × Y i ), then φ ε ⇀ θ i M Y i ( φ) weakly in L p ( ). 11. If φ ∈ W 1,p ( ε i ), then ∇ y [T ε i (φ)] = εT ε i (∇φ) and T ε i (φ) ∈ L p ( , W 1,p (Y i )). 12. If φ ∈ L s (Ŵ ε ) for s ∈ [1, +∞[, then T b ε (φ) L s ( ×Ŵ) |Y | 1 s ε 1 s φ L s (Ŵ ε ) .
We state below the main propositions proved in [START_REF] Cioranescu | The periodic unfolding method in domains with holes[END_REF] and [START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF] concerning the jump on the interface and some convergence results, under the same notations as in [START_REF] Donato | Homogenization of diffusion problems with a nonlinear interfacial resistance[END_REF].

ε Ŵ ε h ε u ε 1 -u ε 2 ϕ dσ x = 1 |Y | ×Ŵ h(y) T ε 1 u ε 1 -T ε 2 u ε 2 T ε 1 (ϕ) dx dσ y .
Theorem 3.5 ([12]). [START_REF] Cioranescu | The periodic unfolding method in perforated domains[END_REF][START_REF] Donato | Homogenization of diffusion problems with a nonlinear interfacial resistance[END_REF][START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF]Let γ ∈ R and u ε be a bounded sequence in H ε γ . Then,

T ε 1 ∇u ε 1 L 2 ( ×Y 1 ) c, T ε 2 ∇u ε 2 L 2 ( ×Y 2 ) c, T ε 1 u ε 1 -T ε 2 u ε 2 L 2 ( ×Ŵ) cε 1-γ 2 , (3.3)
and there exist a subsequence (still denoted by ε), u 1 ∈ H 1 0 ( ) and

u 1 ∈ L 2 ( , H 1 per (Y 1 )) with M Ŵ ( u 1 ) = 0 for almost every x ∈ such that T ε 1 (u ε 1 ) → u 1 strongly in L 2 ( , H 1 (Y 1 )), T ε 1 (∇u ε 1 ) ⇀ ∇u 1 + ∇ y u 1 weakly in L 2 ( × Y 1 ). (3.4) Moreover, if γ 1, there exist a subsequence (still denoted by ε), u 2 ∈ L 2 ( ) and u 2 ∈ L 2 ( , H 1 (Y 2 )) with M Ŵ (u 2 ) = 0 for almost every x ∈ such that T ε 2 (u ε 2 ) ⇀ u 2 weakly in L 2 ( , H 1 (Y 2 )), T ε 2 (∇u ε 2 ) ⇀ ∇ y u 2 weakly in L 2 ( × Y 2 ). (3.5)
Furthermore, if γ < 1, then u 1 = u 2 and also

T ε 2 u ε 2 → u 1 strongly in L 2 , H 1 (Y 2 ) . (3.6)

A priori estimates

In this section we prove two uniform a priori estimates (with respect to ε) for a solution of problem (2.7). In addition, we recall Proposition 4.3 which gives a bound to the integral of the singular term close to the singularity.

These estimates allow us to prove the first part of Theorem 2.4 at the end of this section.

Proposition 4.1. Under assumptions H 1 )-H 3 ), let u ε ∈ H ε γ be the solution of problem (2.7). The following a priori estimate holds true:

∇u ε L 2 ( \Ŵ ε ) C 1 f 1 1+k L 2 1+k ( ) , (4.1)
where C 1 depends on α, c p and c 1 . Also

u ε 1 -u ε 2 L 2 (Ŵ ε ) ε -γ 2 C 2 f 1 1+k L 2 1+k ( ) , (4.2)
where C 2 depends on α, h 0 , c p , c 1 .

Proof. Let u ε ∈ H ε γ be the solution of problem (2.7) and let us choose u ε as test function. By H 1 )-H 3 ) and applying the Young inequality with exponents 2 1-k and 2 1+k , we get for every η > 0

α ∇u ε 2 L 2 ( \Ŵ ε ) + ε γ h 0 Ŵ ε u ε 1 -u ε 2 2 dσ \Ŵ ε A ε x, u ε ∇u ε ∇u ε dx + Ŵ ε ε γ h ε u ε 1 -u ε 2 2 dσ = f ζ u ε u ε dx f u 1-k dx η u ε 2 L 2 ( \Ŵ ε ) + c(η) f 2 1+k L 2 1+k ( ) .
In view of Remark 2.1, the above inequality leads to

α ∇u ε 2 L 2 ( \Ŵ ε ) + ε γ h 0 Ŵ ε u ε 1 -u ε 2 2 dσ c(η, c p ) u ε 2 V ε ×H 1 ( ε 2 ) + c(η) f 2 1+k L 2 1+k ( ) .
Thanks to Proposition 2.2 we get

α ∇u ε 2 L 2 ( \Ŵ ε ) + ε γ h 0 Ŵ ε u ε 1 -u ε 2 2 dσ c(η, c p , c 1 ) u ε 2 H ε γ + c(η) f 2 1+k L 2 1+k ( ) . So that, α -c(η, c p , c 1 ) ∇u ε 2 L 2 ( \Ŵ ε ) + ε γ h 0 -c(η, c p , c 1 ) u ε 1 -u ε 2 2 L 2 (Ŵ ε ) c(η) f 2 1+k L 2 1+k ( ) .
Whence, choosing η sufficiently small so that αc(η, c p , c 1 ) 0 and h 0c(η, c p , c 1 ) 0, we deduce the result from the previous estimate.

Proposition 4.2. Under assumptions H 1 )-H 3 ), let u ε ∈ H ε

γ be the solution of problem (2.7). Then

f ζ u ε ϕ L 1 ( ) c, for every nonnegative ϕ ∈ H ε γ with c depending on α, β, c p , c 1 , ∇ϕ L 2 ( \Ŵ ε ) and f L l ( ) .
Proof. Let u ε ∈ H ε γ be the solution of problem (2.7) and let us choose a nonnegative ϕ ∈ H 1 0 ( ) as test function. Since ϕ has no jump on Ŵ ε , the boundary term vanishes. Hence, by using the Hölder inequality and nonnegativity of f , ζ and ϕ, we have

0 f ζ u ε ϕ dx = \Ŵ ε A ε x, u ε ∇u ε ∇ϕ dx β ∇u ε L 2 ( \Ŵ ε ) ∇ϕ L 2 ( ) , 1
If now ϕ = (ϕ ε 1 , ϕ ε 2 ) ∈ H ε γ is nonnegative, since Ŵ ε is Lipschitz continuous, there exist nonnegative ψ 1 and ψ 2 in H 1 0 ( ) such that ϕ = (ϕ ε 1 , ϕ ε 2 ) = (ψ 1 | ε 1 , ψ 2 | ε 2 ). Then 0 f ζ u ε ϕ dx = ε 1 f ζ u ε ϕ ε 1 dx + ε 2 f ζ u ε ϕ ε 2 dx f ζ u ε ψ 1 dx + f ζ u ε ψ 2 dx c.
We also recall the following result from [START_REF] Donato | Existence and uniqueness results for a class of singular elliptic problems in two-component domains[END_REF] (written for h = ε γ h ε and λ = 0). It gives an estimate of the integral of the singular term close to the singular set {u ε = 0}.

Proposition 4.3 ([26]

). Under assumptions H 1 )-H 3 ), let u ε ∈ H ε γ be the solution of problem (2.7) and δ a fixed positive real number. Then

{0<u ε δ} f ζ u ε ϕ dx \Ŵ ε A ε x, u ε ∇u ε ∇ϕZ δ u ε dx + ε γ Ŵ ε h ε u ε 1 -u ε 2 Z δ u ε 1 ϕ 1 -Z δ u ε 2 ϕ 2 dσ,
for every ϕ ∈ H ε γ , ϕ 0, where Z δ is an auxiliary function defined by Under assumptions H 1 )-H 3 ), let u ε ∈ H ε γ be the unique solution of problem (2.7). Then, there exist a subsequence (still denoted by ε), (2.11) and (2.12) hold.

Z δ (s) =      1, if 0 s δ, -s δ + 2, if δ s 2δ, 0, if s 2δ.
u 1 ∈ H 1 0 ( ), u 1 ∈ L 2 ( , H 1 per (Y 1 )) with M Ŵ ( u 1 ) = 0 for almost every x ∈ and u 2 ∈ L 2 ( , H 1 (Y 2 )) with M Ŵ (u 2 ) = 0 for almost every x ∈ such that
Proof. Let u ε be the solution of problem (2.7). Proposition 4.1 allows us to apply Theorem 3.5. It provides the existence of

u 1 ∈ H 1 0 ( ), u 1 ∈ L 2 ( , H 1 per (Y 1
)) with M Ŵ ( u 1 ) = 0 for almost every x ∈ and u 2 ∈ L 2 ( , H 1 (Y 2 )) with M Ŵ (u 2 ) = 0 for almost every x ∈ such that, up to a subsequence, one has convergences (2.11) i,iv,v , since γ < 1. Now observe that, by construction, for every x ∈ there exists ε x > 0 such that

x ∈ ε , ∀ε ε x .
Also, by defintion of unfolding one has, for i = 1, 2 Consequently, for almost every (x, y) ∈ × Y , there exists ε x > 0 such that

T ε i ζ (φ) (x, y) = ζ (T ε i (φ)(x, y)) a.e. in ε × Y i , 0 a.e. in ε × Y i .
T ε ζ u ε = ζ T ε u ε , ∀ε ε x . (4.5)
On the other hand, using the continuity of ζ and (2.11) i , we have

ζ T ε u ε → ζ (u 1 ) a.e. in × Y. (4.6)
This, together with (4.5), gives the convergence (2.11) ii . Also, from Proposition 2.2 and convergence (2.11) i , Proposition 3.3 10 gives

u ε 1 ⇀ θ 1 M Y 1 (u 1 ) weakly in L 2 ( ),
which reads as (2.11) iii being u 1 independent on y.

To show that u 1 is nonnegative almost everywhere in , we note that every solution u ε is positive almost everywhere in . Then, the definition of the unfolding operator implies T ε (u ε ) 0 almost everywhere in × Y so that, in view of (2.11) i , u 1 0 a.e. in .

It remains to prove the second condition in (2.12). Let us choose first a nonnegative ϕ ∈ H 1 0 ( ). Propositions 3.3 2,4 and 4.2, for the subsequence mentioned before, lead to lim inf

ε→0 1 |Y | ×Y T ε (f )T ε ζ u ε T ε (ϕ) dx dy lim inf ε→0 f ζ u ε ϕ dx < +∞. (4.7)
Now, from Proposition 3.3 8 , T ε (f ) and T ε (ϕ) converge to f and ϕ, respectively, almost everywhere in × Y , up to a subsequence. Thus, by (2.11) ii ,

T ε (f )T ε ζ u ε T ε (ϕ) → f ζ (u 1 )ϕ a.e. in × Y.
Since T ε (f ), T ε (ζ (u ε )) and T ε (ϕ) are nonnegative functions, we can use Fatou's lemma and (4.7) to obtain

1 |Y | ×Y f ζ (u 1 )ϕ dx dy lim inf ε→0 1 |Y | ×Y T ε (f )T ε ζ u ε T ε (ϕ) dx dy < +∞.
Being the functions f and u 1 independent on y, this implies in particular that f ζ (u 1 )ϕ dx < +∞ and ends the proof for ϕ 0. For ϕ with any sign, it suffices to decompose ϕ as ϕ +ϕ -.

A convergence result for an auxiliary problem

This section is devoted to the study of a suitable auxiliary problem and a crucial convergence result, which is one of the main tools needed for proving Theorem 2.6. In the same spirit of what we have done in [START_REF] Donato | Homogenization of a class of singular elliptic problems in perforated domains[END_REF] in the case of a periodically perforated domain, we first consider the auxiliary linear problem (5.2) below and state existence and homogenization results for it. Then, we prove the convergence result given in Theorem 5. [START_REF] Bensoussan | H-convergence for quasi-linear elliptic equations with quadratic growth[END_REF]. It shows that the gradient of u ε is equivalent to the gradient of the solution of the auxiliary linear problem, associated with a weak cluster point of the sequence {u ε }, as ε → 0.

We refer to [START_REF] Chourabi | Homogenization and correctors of a class of elliptic problems in perforated domains[END_REF] and [START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF] for the study of the auxiliary problem. For the proof of the convergence result we adapt some techniques from [START_REF] Donato | Existence and homogenization for a singular problem through rough surfaces[END_REF] and [START_REF] Donato | Homogenization of a class of singular elliptic problems in perforated domains[END_REF], which inspired this work.

First of all, as in [START_REF] Chourabi | Homogenization and correctors of a class of elliptic problems in perforated domains[END_REF] we introduce a linear operator L ε from H -1 ( ) to (V ε ) ′ verifying the following assumption:

H 4 ) If {ϕ ε } is a sequence such that ϕ ε V ε c and ϕ ε ⇀ θ 1 ϕ 0 weakly in L 2 ( ), (5.1)
then lim ε→0 L ε (Z), ϕ ε (V ε ) ′ ,V ε = Z, ϕ 0 H -1 ( ),H 1 0 ( ) .
Let us point out that assumption H 4 ) is satisfied, for example, by the adjoint of the linear operator P ε 1 introduced by Cioranescu and Saint Jean Paulin in [START_REF] Cioranescu | Homogenization in open sets with holes[END_REF]. Then, see [START_REF] Chourabi | Homogenization of elliptic problems with quadratic growth and nonhomogenous Robin conditions in perforated domains[END_REF]Remark 3.1], if H 4 ) holds, one has ϕ 0 ∈ H 1 0 ( ).

The auxiliary problem

The suitable auxiliary problem we are interested in is the following one:

               -div(A ε (x, u ε 1 )∇v ε 1 ) = L ε (Z) in ε 1 , -div(A ε (x, u ε 2 )∇v ε 2 ) = 0 in ε 2 , v ε 1 = 0 on ∂ , (A ε (x, u ε 1 )∇v ε 1 )ν ε = (A ε (x, u ε 2 )∇v ε 2 )ν ε on Ŵ ε , (A ε (x, u ε 1 )∇v ε 1 )ν ε = -ε γ h ε (v ε 1 -v ε 2 ) on Ŵ ε , (5.2)
under assumptions H 1 ), H 3 )-H 4 ), and Z ∈ H -1 ( ).

The variational formulation of problem (5.2) is

     Find v ε ∈ H ε γ such that \Ŵ ε A ε (x, u ε )∇v ε ∇ϕ dx + Ŵ ε ε γ h ε (v ε 1 -v ε 2 )(ϕ 1 -ϕ 2 ) dσ = L ε (Z), ϕ 1 (V ε ) ′ ,V ε , ∀ϕ ∈ H ε γ .
(5.

3)

The existence and the uniqueness of a solution of problem (5.3) is a straightforward consequence of the Lax-Milgram theorem. Observe now that in view of Proposition 4.1 and (2.11) iii , the sequence {u ε } satisfies conditions (5.1), i.e.

u ε 1 V ε c and u ε 1 ⇀ θ 1 u 1 weakly in L 2 ( ).
(5.4)

The homogenization result below extends Theorem 3.3 of [START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF] to the case where the matrix field depends on x. We omit its proof, which follows along the line of [START_REF] Donato | The periodic unfolding method for a class of imperfect transmission problems[END_REF]Theorem 3.3]. The terms containing the functional Z and the quasilinearity of A can be treated as in [START_REF] Chourabi | Homogenization and correctors of a class of elliptic problems in perforated domains[END_REF] and [START_REF] Cabarrubias | Homogenization of a quasilinear elliptic problem with nonlinear Robin boundary comditions[END_REF], respectively.

Theorem 5.1. Under assumptions H 1 ), H 3 )-H 4 ) and Z ∈ H -1 ( ), let v ε ∈ H ε γ be the unique solution of problem (5.3). Then, there exist

v 1 ∈ H 1 0 ( ) and v 1 ∈ L 2 ( , H 1 per (Y 1 )) with M Ŵ ( v 1 ) = 0 such that          i) v ε i ⇀ θ i v 1 weakly in L 2 ( ), i = 1, 2, ii) T ε i (v ε i ) → v 1 strongly in L 2 ( ; H 1 (Y i )), i = 1, 2 iii) T ε 1 (∇v ε 1 ) ⇀ ∇v 1 + ∇ y v 1 weakly in L 2 ( × Y 1 ), iv) T ε 2 (∇v ε 2 ) ⇀ 0 weakly in L 2 ( × Y 2 ), (5.5)
and the pair (v 1 , v 1 ) is the unique solution of the limit equation

∀ϕ ∈ H 1 0 ( ) and ∀ψ ∈ L 2 ( ; H 1 per (Y 1 )), ×Y 1 A(y, u 1 )(∇v 1 + ∇ y v 1 )(∇ϕ + ∇ y ψ) dx dy = |Y | Z, ϕ H -1 ( ),H 1 0 ( ) .
Finally, v 1 is the unique solution of the following limit problem:

div(A 0 (u 1 )∇v 1 ) = Z in , v 1 = 0 on ∂ , (5.6) where the homogenized matrix A 0 is given by (2.8).

Let us consider problem (5.2) for Z = -div(A 0 (u 1 )∇u 1 ). Its variational formulation is

\Ŵ ε A ε x, u ε ∇v ε ∇ϕ dx + Ŵ ε ε γ h ε v ε 1 -v ε 2 (ϕ 1 -ϕ 2 ) dσ = L ε -div A 0 (u 1 )∇u 1 , ϕ 1 (V ε ) ′ ,V ε , ∀ϕ ∈ H ε γ , (5.7)
which admits a unique solution v ε ∈ H ε γ . Theorem 5.1 (written for Z = -div(A 0 (u 1 )∇u 1 )) gives

-div(A 0 (u 1 )∇v 1 ) = -div(A 0 (u 1 )∇u 1 ) in , v 1 = 0 on ∂ .
Thanks to the assumptions on A, the uniqueness of this problem ensures that

v 1 = u 1 .
(5.8) Moreover, the same arguments used to prove (2.11) iii give

v ε H ε γ c and v ε 1 ⇀ θ 1 u 1 weakly in L 2 ( ).
(5.9)

As in [START_REF] Donato | Existence and homogenization for a singular problem through rough surfaces[END_REF] we define now the following auxiliary functions u m :

∀m ∈ N, m 1, u m . = T m (u 1 ),
where T m is the usual truncation function at level m, so that 0 u m u 1 and u m → u 1 strongly in H 1 0 ( ), as m → +∞.

(5.10)

Then, we denote v m ε ∈ H 1 0 ( ) the solution of the following problem:

     -div(A ε (x, u ε 1 )∇v m ε,1 ) = L ε (-div(A 0 (u 1 )∇u m )) in ε 1 , -div(A ε (x, u ε 2 )∇v m ε,2 ) = 0 in ε 1 , v m ε = 0 on ∂ .

Its variational formulation is

A ε x, u ε ∇v m ε ∇ϕ dx = L ε -div A 0 (u 1 )∇u m , ϕ 1 (V ε ) ′ ,V ε , ∀ϕ ∈ H ε γ .
(5.11)

Again, the existence and the uniqueness of a solution of problem (5.11) is straightforward proved by using the Lax-Milgram theorem.

Remark 5.2. Let us notice that v m ε ∈ H 1 0 ( ), and consequently v m ε,1 = v m ε,2 on Ŵ ε , for every ε.

Thus, in particular, Theorem 5.1 applies to this case with no jump (for Z = -div(A 0 (u 1 )∇u m )) and, again by uniqueness, we obtain

v m ε H ε γ c and v m ε,1 ⇀ θ 1 u m weakly in L 2 ( ).
(5.12) Also the following convergence holds true:

T ε v m ε → u m strongly in L 2 ; H 1 (Y ) . (5.13)
Moreover, by classical results from [START_REF] Stampacchia | Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus[END_REF], we have that for every fixed m v m ε L ∞ ( ) c m , for every ε.

(5.14)

Remark 5.3. The sequence {(v m ε,1 ) -} satisfies conditions (5.1). Indeed, in view of the estimate in (5.12) and convergence (5.13), one has

v m ε,1 - V ε v m ε,1 V ε c,
T ε 1 v m ε,1 -→ u - m = 0 strongly in L 2 ( × Y 1 ),
since u m 0 by construction. Also, by using Proposition 3.3 10 , one has

v m ε,1 -⇀ θ 1 M Y 1 u - m = 0 weakly in L 2 ( ).
Let us also point out that, from (5.13), one has even

T ε 2 v m ε,2 -→ u - m = 0 strongly in L 2 ( × Y 2 ).
This is needed in the next subsection.

A convergence result

We are now able to prove the main result of this section. Here we adapt the arguments we used in [START_REF] Donato | Homogenization of a class of singular elliptic problems in perforated domains[END_REF] for the quasilinear singular case in periodically perforated domains to the two-component case, where the holes are replaced by the second component. This is why we analize in detail only the terms differing from the previous work, namely the boundary term and the quasilinear diffusion term in the second component.

The proof follows the same steps introduced in that of [START_REF] Donato | Existence and homogenization for a singular problem through rough surfaces[END_REF]Theorem 8.5], which concerns the singular case when A is linear and the domain is made up of two connected components separated by an oscillating interface.

Theorem 5.4. Under assumptions H 1 )-H 4 ), let u ε and v ε be solutions of problems (2.7) and (5.7), respectively. Then, up to a subsequence,

lim ε→0 ∇u ε -∇v ε L 2 ( \Ŵ ε ) = 0.
Proof. The proof is done in 3 steps.

Step 1. Let us prove that lim

ε→0 \Ŵ ε ∇ v m ε -2 dx = 0 ∀m 1. (5.15)
Let us choose -(v m ε ) -∈ H ε γ as test function in (5.11). From the ellipticity of A, taking into account that From Remark 5.3 and H 4 ), we obtain

∇v m ε = ∇(v m ε ) + -∇(v m ε ) -, and (2.3) we get 0 α ∇ v m ε -2 L 2 ( \Ŵ ε ) - \Ŵ ε A ε x, u ε ∇v m ε ∇ v m ε -dx = L ε -div A 0 (u 1 )∇u m , -v m ε,1 - (V ε ) ′ ,V ε .
lim ε→0 L ε -div A 0 (u 1 )∇u m , -v m ε,1 - (V ε ) ′ ,V ε = -div A 0 (u 1 )∇u m , 0 H -1 ( ),H 1 0 ( ) = 0,
which concludes the first step.

Step 2. In this step we show that

lim m→+∞ lim ε→0 \Ŵ ε ∇ u ε -v m ε 2 dx = 0. (5.16)
To do that, let us choose u εv m ε ∈ H ε γ as test function in (2.7) and (5.11). By subtraction, H 1 ) i and the nonnegativity of the boundary term, one has

α ∇ u ε -v m ε 2 L 2 ( \Ŵ ε ) \Ŵ ε A ε x, u ε ∇ u ε -v m ε ∇ u ε -v m ε dx + ε γ Ŵ ε h ε u ε 1 -u ε 2 2 dσ = f ζ u ε u ε -v m ε dx -L ε -div A 0 (u 1 )∇u m , u ε -v m ε 1 V ′ ε ,V ε . (5.17)
Let us prove that, as ε → 0, we have

α lim sup ε→0 ∇ u ε -v m ε 2 L 2 ( \Ŵ ε ) f ζ (u 1 )(u 1 -u m )χ {u 1 >0} dx --div A 0 (u 1 )∇u m , u 1 -u m H -1 ( ),H 1 0 ( ) , (5.18)
so that we obtain the result (5.16) as m → +∞, via convergence (5.10) and the Lebesgue theorem.

Concerning the second term in the right-hand side of the previous inequality, let us observe that u ε -v m ε satisfies (5.1), in view of (5.4) and (5.12). So that, by H 4 )

lim ε→0 L ε (-div A 0 (u 1 )∇u m , u ε -v m ε 1 V ′ ε ,V ε = -div A 0 (u 1 )∇u m , u 1 -u m H -1 ( ),H 1 0 ( ) . (5.19)
Let δ > 0. We split the integral of the singular term in (5.17) in two terms as follows: where, for i = 1, 2,

f ζ u ε u ε -v m ε dx = f ζ u ε u ε -v m ε + dx + f ζ u ε v m ε -dx 2 i=1 I δ ε,i + J δ ε,i + K ε , i = 1, 2, (5.20)
I δ ε,i . = ε i ∩{0<u ε i δ} f ζ u ε i u ε i dx, J δ ε,i . = ε i ∩{u ε i >δ} f ζ u ε i u ε i -v m ε,i
+ dx, and

K ε . = f ζ u ε v m ε -dx.
The terms corresponding to i = 1 are exactly the ones considered in [START_REF] Donato | Homogenization of a class of singular elliptic problems in perforated domains[END_REF]Theorem 5.5] for the case of periodically perforated domains. It is easy to check that the additional terms I δ ε,2 and J δ ε,2 can be treated exactly in the same way. Indeed, all the convergences used there for the first component hold true also here in the second component, thanks to (2.1) and convergence (3.6).

Hence,

     lim δ→0 lim ε→0 (I δ ε,1 + I δ ε,2 ) = 0, lim δ→0 lim ε→0 (J δ ε,1 + J δ ε,2 ) = (θ 1 + θ 2 ) f ζ (u 1 )(u 1 -u m )χ {u 1 >0} dx = f ζ (u 1 )(u 1 -u m )χ {u 1 >0} dx.
(5.21)

On the contrary, the term K ε needs to be treated specifically, since its computation gives rise to a different boundary term. To this aim, let us observe that

∩{u ε >δ} f ζ u ε v m ε -χ {u 1 =δ} dx = 0,
for every δ ∈ R + \D, where D is a countable set of values (see for instance [START_REF] Donato | Existence and homogenization for a singular problem through rough surfaces[END_REF][START_REF] Giachetti | An elliptic problem with a lower order term having singular behaviour[END_REF] and [START_REF] Donato | Homogenization of a class of singular elliptic problems in perforated domains[END_REF]). Hence, for δ 0 ∈ R + \ D, we can write

K ε = ∩{0<u ε δ 0 } f ζ u ε v m ε -dx + ∩{u ε >δ 0 } f ζ u ε v m ε -χ {u 1 =δ 0 } dx . = A ε + B ε . (5.22)
From Proposition 4.3 written with δ = δ 0 we get

0 A ε \Ŵ ε A ε x, u ε ∇u ε ∇ v m ε -Z δ 0 u ε dx + ε γ Ŵ ε h ε u ε 1 -u ε 2 Z δ 0 u ε 1 v m ε,1 --Z δ 0 u ε 2 v m ε,2 -dσ.
The first term in the right-hand side of the previous inequality goes to zero as ε goes to zero, because of H 1 ) i , the Hölder inequality, (4.1) and (5.15). On the other hand, since v m ε ∈ H 1 0 ( ), also (v m ε ) -belongs to H 1 0 ( ), so that (v m ε,1 ) -= (v m ε,2 ) -on Ŵ ε and, since Z δ is nonincreasing (see definition (4.3)), we have As done for the previous terms, in order to handle B ε we split it as follows:

ε γ Ŵ ε h ε u ε 1 -u ε 2 Z δ 0 u ε 1 v m ε,1 --Z δ 0 u ε 2 v m ε,2 -dσ = ε γ Ŵ ε h ε v m ε,1 -u ε 1 -u ε 2 Z δ 0 u ε 1 -Z δ 0 u ε 2 dσ 0.
B ε = B ε,1 + B ε,2 with B ε,i . = ε i ∩{u ε i >δ 0 } f ζ u ε i v m ε,i -χ {u 1 =δ 0 } dx, i = 1, 2.
The same arguments used to prove that lim ε→0 B ε,1 = 0 in the proof of [START_REF] Donato | Homogenization of a class of singular elliptic problems in perforated domains[END_REF]Theorem 5.5] apply also to the second component. So, using again (2. By collecting (5.17), (5.19), (5.20), (5.21), (5.24), we finally obtain the validity of (5.18).

Step 3. In this last step, we show that

lim m→+∞ lim ε→0 \Ŵ ε ∇ v m ε -v ε 2 dx = 0. (5.25)
We take v m εv ε as test function in the variational formulations (5.11) and (5.7). By subtraction, we have

\Ŵ ε A ε x, u ε ∇ v m ε -v ε ∇ v m ε -v ε dx -ε γ Ŵ ε h ε v ε 1 -v ε 2 v m ε -v ε 1 -v m ε -v ε 2 dσ = L ε -div A 0 (u 1 )∇(u m -u 1 ) , v m ε -v ε 1 V ′ ε ,V ε .
In view of Remark 5.2, it results

-ε γ Ŵ ε h ε v ε 1 -v ε 2 v m ε -v ε 1 -v m ε -v ε 2 dσ = ε γ Ŵ ε h ε v ε 1 -v ε 2 2 dσ 0.
Consequently, by passing to the limit on ε, for H 1 ) and H 4 )(whose assumptions are satisfied both by v ε and v m ε thanks to (5.9) and (5.12)), we get This together with (5.10) gives 0 lim

0 α lim ε→0 ∇ v m ε -v ε 2 L 2 ( \Ŵ ε ) lim ε→0 L ε -div A 0 (u 1 )∇(u m -u 1 ) , v m ε -v ε 1 V ′ ε ,V ε = -div A 0 (u 1 )∇(u m -u 1 ) , u m -u 1 H -1 ( ),H 1 0 ( ) .
m→+∞ lim ε→0 α ∇ v m ε -v ε 2 L 2 ( \Ŵ ε ) lim m→+∞ A 0 (u 1 )∇(u m -u 1 )∇(u m -u 1 ) dx = 0.
At last, coupling (5.16) and (5.25) we obtain the desired result.

Proof of the homogenization result

In this last section we prove the second part of Theorem 2.4, Proposition 2.5 and Theorem 2.6. First, we treat Proposition 2.5 which is a consequence of the convergence result given in Theorem 5.4.

Proof of Proposition 2.5. Let v ε be the solution of problem (5.7). The homogenization result given in Theorem 5.1 (written for Z = -div(A 0 (u 1 )∇u 1 ) and v 1 = u 1 , see (5.8)) gives, for the subsequence verifying (2.4),

T ε 1 (∇v ε 1 ) ⇀ ∇u 1 + ∇ y v 1 weakly in L 2 ( × Y 1 ), T ε 2 (∇v ε 2 ) ⇀ 0 weakly in L 2 ( × Y 2 ), (6.1)
where

v 1 is a function in L 2 ( ; H 1 per (Y 1 )) with M Ŵ ( v 1 ) = 0, such that v 1 (x, y) = - N i=1 χ e i y, u 1 (x) ∂u 1 ∂x i (x).
Now let us observe that, from Theorem 5.4 and Proposition 3.3 9 , we get

T ε ∇u ε -∇v ε → 0 strongly in L 2 ( × Y ).
This, together with (2.11) iv,v and (6.1), leads to

∇ y v 1 = ∇ y u 1 a.e. in × Y 1 , ∇ y ū2 ≡ 0 a.e. in × Y 2 , (6.2)
which implies v 1 = u 1 + w(x), for some function w only depending on x.

Since M Ŵ ( v 1 ) = M Ŵ ( u 1 ) = 0 and M Ŵ ( v 1 ) = M Ŵ ( u 1 ) + M Ŵ (w),
we derive w = 0 and Whence we obtain the convergence

v 1 = u 1 .
T ε 1 ∇v ε 1 ⇀ ∇u 1 + ∇ y u 1 weakly in L 2 ( × Y 1 ), (6.3)
and the claimed expression of u 1 .

We are now able to prove the homogenization theorems.

Proof of Theorem 2.4. Let us first remark that, under our assumptions, Proposition 4.4 holds true. In particular, by collecting (2.11) iv and (6.2) ii , one gets

T ε 2 ∇u ε 2 ⇀ 0 weakly in L 2 ( × Y 2 ). (6.4)
Notice that the function u 1 ∈ H 1 0 ( ) is nonnegative due to (2.12). Moreover, conditions (5.4) ensures the validity of H 4 ) and, as a consequence, Theorem 5.4 holds true. Now, to identify the limit problem satisfied by (u 1 , u 1 ), we take ϕ, φ ∈ D( ) and ξ ∈ H 1 per (Y ), and use

ψ ε (x) = ϕ(x) + εφ(x)ξ x ε ∈ H ε γ
as test function in (2.7). We obtain

\Ŵ ε A ε x, u ε ∇u ε ∇ψ ε dx + ε γ Ŵ ε h ε u ε 1 -u ε 2 (ψ ε,1 -ψ ε,2 ) dσ = f ζ u ε ψ ε dx. (6.5)
Let us consider the solution v ε of problem (5.3) and write

lim ε→0 \Ŵ ε A ε x, u ε ∇u ε ∇ψ ε dx = lim ε→0 \Ŵ ε A ε x, u ε ∇v ε ∇ψ ε dx + lim ε→0 \Ŵ ε A ε x, u ε ∇ u ε -v ε ∇ψ ε dx. (6.6)
By using assumption H 1 ), the Hölder inequality and Theorem 5.4, one has

lim ε→0 \Ŵ ε A ε x, u ε ∇ u ε -v ε ∇ψ ε dx = 0, (6.7)
taking into account that the norm of ψ ε is bounded in H ε γ . By Theorem 5.1 (written for Z = -div(A 0 (u 1 )∇u 1 )), we obtain

lim ε→0 \Ŵ ε A ε x, u ε ∇v ε ∇ψ ε dx = 1 |Y | ×Y 1 A(y, u 1 )(∇u 1 + ∇ y u 1 )(∇ϕ + φ∇ y ξ ) dx dy, (6.8)
in view of (6.3). Concerning the boundary term in (6.5), by unfolding, using Lemma 3.4 and (3.3) we get

ε γ Ŵ ε h ε u ε 1 -u ε 2 (ψ ε,1 -ψ ε,2 ) dσ = ε γ +1 Ŵ ε h ε u ε 1 -u ε 2 (φ 1 ξ 1 -φ 2 ξ 2 ) dσ = ε γ |Y | ×Ŵ h(y) T ε 1 u ε 1 -T ε 2 u ε 2 ξ 1 (y)T ε 1 (ϕ 1 ) -ξ 2 (y)T ε 2 (ϕ 2 ) dx dσ y cε γ T ε 1 u ε 1 -T ε 2 u ε 2 L 2 ( ×Ŵ) T ε 1 (ϕ 1 ) ξ 1 (y) -ξ 2 (y) L 2 ( ×Ŵ) cε γ ε 1-γ 2 = cε 1+γ 2 .
Since -1 < γ < 1, passing to the limit as ε → 0, we obtain

lim ε→0 ε γ Ŵ ε h ε u ε 1 -u ε 2 (ψ ε,1 -ψ ε,2 ) dσ = 0. ( 6.9)
In order to pass to the limit in the singular term of (6.5), we define

µ ε (x) . = εφ(x)ξ x ε , that is ψ ε = ϕ + µ ε . (6.10)
Then one has

T ε (µ ε ) = εT ε (φ)ξ and ∇µ ε = ε∇φξ • ε + φ∇ y ξ • ε , and
i) T ε (µ ε ) → 0 strongly in L 2 ( × Y ), ii) T ε (∇µ ε ) → φ∇ y ξ strongly in L 2 ( × Y ). (6.11)
From now on, without loss of generality we assume ϕ 0 and µ ε 0 in (6.10). Indeed we can decompose these functions in their positive and negative parts as in (2.2). Now let us split the singular integral into two terms: one near the singularity and one far from it. For every positive δ we write

0 f ζ u ε ψ ε dx = {0<u ε δ} f ζ u ε ψ ε dx + {u ε >δ} f ζ u ε ψ ε dx . = I δ ε + J δ ε . (6.12)
In view of Proposition 4.3 written for ϕ, µ ε 0 we have Taking into account the decomposition given in (6.10), and the fact that since ϕ, φ ∈ D( ) then ϕ 1 = ϕ 2 and φ 1 = φ 2 on Ŵ ε , we have

0 lim sup ε→0 I δ ε lim sup ε→0 \Ŵ ε A ε x, u ε ∇u ε ∇ψ ε Z δ u ε dx + ε γ Ŵ ε h ε u ε 1 -u ε 2 Z δ u ε 1 ψ ε,1 -Z δ u ε 2 ψ ε,2 dσ . ( 6
ε γ Ŵ ε h ε u ε 1 -u ε 2 ϕ 1 Z δ u ε 1 -Z δ u ε 2 + εφ 1 Z δ u ε 1 ξ 1 -Z δ u ε 2 ξ 2 dσ = ε γ Ŵ ε h ε u ε 1 -u ε 2 ϕ 1 (Z δ u ε 1 -Z δ u ε 2 dσ + ε γ +1 Ŵ ε h ε u ε 1 -u ε 2 φ 1 Z δ u ε 1 ξ 1 -Z δ u ε 2 ξ 2 dσ ε γ +1 Ŵ ε h ε u ε 1 -u ε 2 φ 1 Z δ u ε 1 ξ 1 -Z δ u ε 2 ξ 2 dσ,
where we used H 3 ), ϕ 1 0 and the growth condition of Z δ .

From the properties of the unfolding operators (Lemma 3.4 and estimate (3.3)), the fact that Z δ 1 by definition and γ ∈] -1, 1[, we get

ε γ +1 Ŵ ε h ε u ε 1 -u ε 2 φ 1 Z δ u ε 1 ξ 1 -Z δ u ε 2 ξ 2 dσ cε γ T ε 1 (φ 1 ) T ε 1 u ε 1 -T ε 2 u ε 2 L 2 ( ×Ŵ) ξ 1 (Z δ T ε 1 u ε 1 -ξ 2 Z δ T ε 2 u ε 2 ] L 2 ( ×Ŵ) cε γ ε 1-γ 2 = c ε γ +1 2
→ 0, which implies (6.14). Now, we write

\Ŵ ε A ε x, u ε ∇u ε ∇ψ ε Z δ u ε dx = ε 1 A ε x, u ε 1 ∇u ε 1 ∇ψ ε,1 Z δ u ε 1 dx + ε 2 A ε x, u ε 2 ∇u ε 2 ∇ψ ε,2 Z δ u ε 2 dx.
The arguments used in Theorem 2.8 of [START_REF] Donato | Homogenization of a class of singular elliptic problems in perforated domains[END_REF] to handle the term related to the first component apply also to the second one. So that, in view of (6.4) and (6. since, due to the expression of u 1 given by Proposition 2.5, the functions ∇u 1 and ∇ y u 1 vanish where u 1 is equal to 0. In order to study the limit behavior of J δ ε (defined in (6.12)), once again we split it as follows:

J δ ε = {u ε 1 >δ} f ζ u ε 1 ψ ε,1 dx + {u ε 2 >δ} f ζ u ε 2 ψ ε,2 dx.
The integral on the first component is the same considered in the proof of [START_REF] Donato | Homogenization of a class of singular elliptic problems in perforated domains[END_REF]Theorem 2.8]. By using those same computations also on the second component, we get Then, when passing to the limit in (6.5), we combine (6.6)-(6.9), (6.12), (6.16), (6.17 Proof of Theorem 2.6. From the expression of u 1 given in Proposition 2.5, a standard computation shows that u 1 is a solution of the following problem:

div(A 0 (u 1 )∇u 1 ) = f ζ (u 1 )χ {u 1 >0} in , u 1 = 0 on ∂ . (6.19) Also, since the conditions given in (2.10) are satisfied by A 0 , Theorem 6 from [START_REF] Donato | Uniqueness result for a class of singular elliptic problems in two-component domains[END_REF] shows that u 1 is the unique solution of this problem. This implies the uniqueness of u 1 under the condition M Ŵ ( u 1 ) = 0, in view of Proposition 2.5. Hence convergence (2.11) holds for the whole sequence, as well as (6.4). We lastly prove that u 1 > 0 almost everywhere in . From the strong maximum principle, by contradiction, we derive u 1 ≡ 0 in . This imply f ζ (u 1 )χ {u 1 >0} ϕ dx = 0 for every ϕ ∈ D( ). This means f ≡ 0 on which contradicts assumption H 2 ) iii . Consequently u 1 > 0 almost everywhere in and χ {u 1 >0} ≡ 1. Then u 1 satisfies the limit equation (2.14).

  singular. Near the singularity, making use of the estimate given in Proposition 4.3, we shift the study of the singular term to that of the quasilinear one, for which we can use Theorem 5.4.

Fig. 1 .

 1 Fig. 1. The two-component domain and the reference cell Y .

Remark 3 . 2 .

 32 If φ is defined in , we simply write

  4 ([22]). Let ϕ ∈ D( ) and u ε ∈ H ε γ . For ε small enough one has

  result for ϕ ∈ H 1 0 ( ), via the estimate (4.1).

(4. 3 )

 3 Proposition 4.4.

1 )

 1 and convergence (3.6), we get lim ε→0 B ε,2 = 0. This implies, together with (5.22)-(5.23), lim ε→0 K ε = 0.(5.24)

  , u ε ∇u ε ∇ψ ε Z δ u ε dx.(6.15)

  1 A(y, u 1 )(∇u 1 + ∇ y u 1 )(∇ϕ + φ∇ y ξ )χ {u 1 =0} dx dy = 0

(θ 1 +

 1 θ 2 ) f ζ (u 1 )ϕχ {u 1 >0} dx = f ζ (u 1 )ϕχ {u 1 >0} dx.(6.17

  )

) and have ×Y 1 A 1 A

 11 (y, u 1 )(∇u 1 + ∇ y u 1 )(∇ϕ+ φ∇ y ξ ) dx dy = |Y | f ζ (u 1 )ϕχ {u 1 >0} dx, for every ϕ, φ ∈ D( ) and ξ ∈ H 1 per (Y ). By density we obtain×Y (y, u 1 )(∇u 1 + ∇ y u 1 )(∇ϕ + ∇ y ψ) dx dy = |Y | f ζ (u 1 )ϕχ {u 1 >0} dx(6.18)for every ϕ ∈ H 1 0 ( ) and ψ ∈ L 2 ( ; H 1 per (Y )).

Acknowledgements

The author wishes to express her deep gratitude to Patrizia Donato for helpful discussions and valuable suggestions.


domain: phys spi

text:
Nadine Gerges
email: nadine.gerges@cea.fr

Silvère Gousset

Camille Petit-Etienne

Marie Panabière

Etienne Le Coarer

Jumana Boussey

Yann Ferrec

Cécile Gourgon

SILICON MICRO-INTERFEROMETERS FOR IMAGING SPECTROMETER FOR GREENHOUSE GAS GLOBAL SURVEY

Keywords: microelectronics, silicon, hyperspectral imaging

NanoCarb is a miniature Fourier transform imaging spectrometer dedicated to the measurement of greenhouse gases (CO2 and CH4) from a constellation of small satellites. The key element of NanoCarb is an array of Fabry-Perot microinterferometers having a stepcase shape. Lateral dimensions and height of each step depend on the used material, the focusing lenses and the targetted optical path difference to be measured. In this paper, we describe a grayscale lithography process for a large surface patterning with high vertical resolution. This process is combined with plasma etching. Therefore, this two processes leads to a 20 nm step height resolution in resist and 50 nm in silicon. In addition to the manufacturing of this plate, we also investigated the deposit of a narrow-band filter on this plate used to select the spectral band of CO2 (1608 nm), with promising preliminary results.

INTRODUCTION

The H2020 SCARBO (Space CARBon Observatory) is a European project led by Airbus Defense and Space, involving several partners including ONERA and IPAG. The project aims at designing and assessing the feasibility of a constellation of small satellites at an affordable cost for the measurement of GreenHouse Gases (GHG), CO2 and CH4. NanoCarb is one of the main components of the project, a Fourier transform imaging spectrometer dedicated to GHG measurement [START_REF] Gousset | NANOCARB-21: a miniature Fourier-transform spectroimaging concept for a daily monitoring of greenhouse gas concentration on the Earth surface[END_REF] . The NanoCarb concept is based on two main features. The first one is consisting in a very compact static imaging interferometer, whose principle is presented by Ferrec et al [START_REF] Ferrec | NanoCarb part 1: compact snapshot imaging interferometer for CO2 monitoring from space International Conference on Space Optics-ICSO[END_REF] : the spectro-imager is divided into two optical subsystems, the front optics, which consists of an afocal system with a field stop, and the interferometric core which consists of a combination of an interferometer array with a micro-lens (µlens) array [START_REF] Gousset | NanoCarb hyperspectral sensor: on performance optimization and analysis for greenhouse gas monitoring from a constellation of small satellites[END_REF] . The interferometric plate has a stepcase shape fabricated of silicon, and as the surfaces are semireflective, each step behaves like a Fabry-Perot interferometer. We thus obtain a set of small images on the focal plane array (FPA), each image being associated to one thickness. The second feature of NanoCarb is to measure only a partial interferogram (as shown by Gousset et al. [START_REF] Gousset | NanoCarb hyperspectral sensor: on performance optimization and analysis for greenhouse gas monitoring from a constellation of small satellites[END_REF] ), that is a few optical path differences (OPD) well chosen, depending on the spectral signature of the target species. An interferential filter is used to select the spectral band (spectral filtering), in front of the μ-interferometer array (interferometric filtering). This principle is well suited for CO2 and CH4, thanks to a periodic spectral pattern over specific bands. For instance, for CO2 around 1.6 µm of wavelength, this OPD is about 5.2 mm [START_REF] Ehrhardt | Characterization by OCT of a new kind of micro-interferometric components for the NanoCarb miniature imaging spectrometer International Conference on Space Optics-ICSO[END_REF] . This work represents a collaboration between the LTM (Laboratoire des Technologies de la Microélectronique) and ONERA, and has several objectives (Figure 1) : -Develop fabrication processes for the realization of 2D micro-interferometers array in Si whose thicknesses must be perfectly controlled at 50 nm. -Develop optical filters located on the back side of the interferometric plate which aim to let through only the light that is in the spectral band of interest (1608 nm for CO2).

Figure 1 -Schematic representation of the objectives of our work

EXPERIMENTAL

The shape of the interferometric plate is quite uncommon: indeed, the transverse size of the 10 steps is about 4 mm, but their depths must be as small as 50 nm. There are several methods to fabricate perfectly controlled steps in resist and transfer them into silicon from a series of lithography/etching sequences, but in case of a large number of steps, this process is time consuming and expensive. Moreover, there are various techniques of grayscale patterning, such as special optical masks [START_REF] Waits | MEMS-based gray-scale lithography[END_REF] , LED-projection [START_REF] Eckstein | Direct write grayscale lithography for arbitrary shaped microoptical surfaces[END_REF] , direct-write lasers [START_REF] Mckenna | Maskless direct write grayscale lithography for MEMS applications[END_REF] [8] , and electron-beam lithography [START_REF] Kim | Controlling resist thickness and etch depth for fabrication of 3D structures in electron-beam grayscale lithography[END_REF] . The latter is not effective due to long exposure times. The challenge in our work is to fabricate such a plate using only one grayscale lithography and plasma etching step. Thus, we developed a UV grayscale lithography process for a large surface patterning with high vertical resolution. This process is combined with plasma etching to transfer the as-obtained resist patterns into the silicon substrate. This method is an efficient and quick way for the realization of such arrays into silicon. Figure 2 material. For the high index material, filters based on Ta2O5 (n=2.1) and Nb2O5 (n=2.2) are demonstrated [10] [11] . It is also possible to control the transmission wavelength in the near infrared by combining Si/SiOx with air gaps whose thickness allows to adjust the central wavelength [START_REF] Silva | Large area silicon-airsilicon DBRs for infrared filter applications[END_REF] [13] .

We did not chose solutions because the architectures are complex and these high index materials are not available at LTM. Silicon has a high index of 3.4 and can be deposited on the technological platform together with SiO2. We optimized by simulations a filter that transmits at 1608 nm and consists of 12 layers of Si and SiO2. In addition to the Fabry-Perot function, the interferometric plate can also be used to host the spectral filter, in order to limit the number of optical components and thus unwanted multiple reflections. This interference filter is a narrowband filter, for instance, the spectral band for the CO2 camera is centered at 1608 nm with a FWHM of 6 nm. This poses two difficulties of realization. First, it requires a very high rejection rate over a very wide spectral range. The second difficulty is that the variation of the transmitted spectral band with the angle of incidence has to be small. In the literature, it is possible to find a number of publications on the manufacture of interference filters. The objective is to use a pair of materials with a large index gap. The SiO2 (n=1.44) is generally used as a low index This study was completed by experimental realizations of filters by deposition on a PECVD CORIAL tool. This equipment allows to obtain very uniform films thanks to the uniform distribution of gases. The optical characterization at IPAG and ONERA using a Bruker FTIR and a PerkinElmer grating spectrometer respectively, shows that the filter transmits at 1614 nm with a transmission of 45.9% and a FWHM of 6.5 nm. Figure 3 shows the results obtained with the PerkinElmer spectrometer at ONERA as a function of the angle of incidence. We can notice that the spectral shift of this filter is 1 nm, 3 nm and 12 nm at 5°, 8° and 15° respectively. We deduce that the effective index of the filter is equal to 2.2. These results show that this filter corresponds to the expected specifications in terms of angle of incidence, but it is essential to use an in-situ control during the deposition in order to obtain the accuracy of the peak centering.

CONCLUSIONS

In this study, we addressed the issue of realizing a perfectly controlled staircase structure in only one lithography step and subsequent pattern transfer into silicon by plasma etching. We thus demonstrated the possibility to manufacture a Si plate with 50 nm steps to be used as an array of Fabry-Perot interferometers for NanoCarb. In addition to the manufacturing of this plate, we also investigated the deposit of a narrow-band filter on this plate, with promising preliminary results.

Figure 2 -

 2 Figure 2 -(a): heights and optical microscopy image of 10 strips patterned in the resist with various power exposure ratios, (b) the profile obtained by etching into silicon combines two measurements; the upper one is optical microscopy image of 10 strips (400×4000 µm per strip) patterned in the resist with various power exposure ratios and the corresponding developed resist thickness, and the lower one is the profile obtained by stripping those resist after etching the patterns into silicon. As shown in this figure, the depth of the 5 th and the 6 th step is 118 nm and 98 nm, respectively, which results in a step height of 20 nm in the resist and around 50 nm in silicon, as required by the NanoCarb design.In addition to the Fabry-Perot function, the interferometric plate can also be used to host the spectral filter, in order to limit the number of optical components and thus unwanted multiple reflections. This interference filter is a narrowband filter, for instance, the spectral band for the CO2 camera is centered at 1608 nm with a FWHM of 6 nm. This poses two difficulties of realization. First, it requires a very high rejection rate over a very wide spectral range. The second difficulty is that the variation of the transmitted spectral band with the angle of incidence has to be small. In the literature, it is possible to find a number of publications on the manufacture of interference filters. The objective is to use a pair of materials with a large index gap. The SiO2 (n=1.44) is generally used as a low index

Figure 3 -

 3 Figure 3 -Optical characterization of the filter composed of 12 Si/SiO2 layers for several angles of incidence