Xavier Claeys

Bérangère Delourme
email: delourme@caltech.edu

High order asymptotics for wave propagation across thin periodic interfaces

Keywords: periodic thin interfaces, matched asymptotic expansions, homogenization, Helmholtz equation

This work deals with the scattering of acoustic waves by a thin ring that contains many regularly-spaced heterogeneities. We provide and justify a complete description of the solution with respect to the period and the thickness of the heterogeneities. Our approach mixes matched asymptotic expansions and homogenization theory.

Introduction

This work is dedicated to the study of asymptotic models associated with electromagnetic wave scattering from thin rings that contain regularly spaced heterogeneities. We use asymptotic analysis techniques to study a case where the thickness of the ring and the distance between two consecutive heterogeneities (of size δ) are very small compared to the wavelength of the incident wave and the diameter of the ring, which yields a precise description of the behavior of the solution as the thickness of the ring goes to 0.

Without being exhaustive, let us indicate some works from the mathematical literature that share similarities with our problem. Many papers deal with the construction of approximate boundary conditions for the scattering by a perfect conductor coated by a periodic layer or by a periodic rough boundary. The construction of such approximate conditions relies on an asymptotic expansion of the solution with respect to the thickness and the period of the layer. Concerning electromagnetic scattering by impenetrable objects with rough boundaries, the first two terms of the expansion have been derived by Y.Achdou [START_REF] Achdou | Etude de la réflexion d'une onde électromagnétique par un métal recouvert d'un revêtement métallisé[END_REF] and M. Artola and M. Cessenat [START_REF] Artola | Diffraction d'une onde électromagnétique par une couche composite mince accolée à un corps conducteur épais. I. Cas des inclusions fortement conductrices[END_REF] for the Maxwell equations in planar geometries. This work has been extended to the case of a circular geometry by A.Zebic [START_REF] Zebic | Conditions de frontière équivalentes en électromagnétisme[END_REF] and to the case of general smooth geometries by T. Abboud and H. Ammari [START_REF] Abboud | Diffraction at a curved grating: TM and TE cases, homogenization[END_REF] for the Helmholtz equation. Higher order terms have been derived by A.L Madureira and F.Valentin [START_REF] Madureira | Asymptotics of the Poisson problem in domains with curved rough boundaries[END_REF] for the Laplace problem and by A.Bendali and J.R. Poirier [START_REF] Poirier | High order asymptotic expansion for the scattering of fast oscillating periodic surfaces[END_REF] for the Helmholtz equation. In addition, M. Dauge, E. Faou, V. Péron [START_REF] Dauge | Comportement asymptotique à haute conductivité de l'épaisseur de peau en électromagnétisme[END_REF] and H. Haddar, P. Joly and H.M. Nguyen [START_REF] Haddar | Generalized impedance boundary conditions for scattering by strongly absorbing obstacles: the scalar case[END_REF] have derived asymptotic expansions in the case of strongly absorbing (non-rough) obstacles. Similar methods have been applied to the case of an impenetrable object coated by a thin dielectric layer, see [START_REF] Ammari | Conditions aux limites approchées pour les couches minces périodiques[END_REF][START_REF] Tlemcani | A Two-Scale Asymptotic Analysis of a Time-Harmonic Scattering Problem with a Multi Layered Thin Periodic Domain[END_REF].

In the present article we are particularly interested in problems involving wave transmission through a rough thin dielectric layer. Although this case has already been studied in [START_REF] Drissi | Simulation des silencieux d'échappement par une méthode d'éléments finis homogénéisés[END_REF][START_REF] Ciuperca | Approximate transmission conditions through a rough thin layer: the case of periodic roughness[END_REF][START_REF] Bresch | High order multi-scale wall-laws, part I: The periodic case[END_REF][START_REF] Bonnetier | A priori convergence estimates for a rough Poisson-Dirichlet problem with natural vertical boundary conditions[END_REF] for thin perforated plates and periodic rough thin dielectric layers, only low order asymptotics have been derived so far (expansions up to 2nd order). Note that Reference [START_REF] Delourme | Approximate models for wave propagation across thin periodic interfaces[END_REF] deals with the same configuration as the present paper. However, the purpose is different since it mainly focuses on the construction of stable approximate transmission conditions up to order 2 -only the first two terms of the asymptotic expansion are derived. Note that high order asymptotics for wave transmission through thin dielectric layer have already been derived by K. Schmidt and S. Tordeux in [START_REF] Schmidt | Asymptotic modelling of conductive thin sheets[END_REF], but this work did not consider rapidly oscillating material characteristics.

Our problem is similar to the one considered in [START_REF] Schmidt | Asymptotic modelling of conductive thin sheets[END_REF] (i.e. a 2-D wave transmission through a thin dielectric layer) with, in addition, fast oscillations in the material characteristics of the layer. We derive an asymptotic expansion up to any order, we provide error estimates for it, and obtain explicit approximate transmission conditions at any order (which is different from [START_REF] Delourme | Approximate models for wave propagation across thin periodic interfaces[END_REF] that only considers order 2 asymptotics). More precisely, denoting u δ (r, θ) the solution to our problem (with |rr * | the distance to the layer, and θ the longitudinal coordinate) we propose an expansion of the form

u δ (r, θ) = +∞ n=0 δ n u n (r, θ)
away from the layer,

u δ (r, θ) = +∞ n=0 δ n U n r-r * δ r * θ δ , θ
in the neighbourhood of the layer.

and provide error estimates for these expansions: there are the main results of the present article, see Theorem 10.1 and Theorem 10.2.

The expansion above is not the same in the vicinity of the layer, and away from it: this is a boundary layer phenomenon. Moreover, the expansion close to the layer involves multiple scales inherited from the fast oscillating structure of the equations. Actually, we mix homogeneization techniques with matched asymptotics -we believe that this is a remarkable feature of the present work. Homogeneization theory was developed by A. Bensoussan, J.L. Lions, G. Papanicolaou and E. Sanchez-Palencia [START_REF] Bensoussan | Asymptotic analysis for periodic structures[END_REF][START_REF] Sánchez-Palencia | Un problème d' écoulement lent d'une fluide visqueux incompressible au travers d'une paroi finement perforée[END_REF][START_REF] Sánchez-Palencia | Nonhomogeneous media and vibration theory[END_REF] to analyze materials with fast oscillating characteristics. Matched asymptotics mainly originated from the work of Van Dyke [START_REF] Van Dyke | Perturbation methods in fluid mechanics[END_REF] to treat boundary layer problems arising in fluid mechanics. While the analysis proposed by Van Dyke was formal, rigorous theory for the method of matched expansions was developed later on by the Russian school. Main general references [START_REF] Maz'ya | Asymptotic theory of elliptic boundary value problems in singularly perturbed domains[END_REF][START_REF] Il ′ In | Application of the method of matching asymptotic expansions to the solution of boundary value problems[END_REF][START_REF] Il ′ In | Matching of asymptotic expansions of solutions of boundary value problems[END_REF][START_REF] Gadyl ′ Shin | The method of matching asymptotic expansions in a singularly perturbed boundary value problem for the Laplace operator[END_REF][START_REF] Joly | Matching of asymptotic expansions for wave propagation in media with thin slots. I. The asymptotic expansion[END_REF][START_REF] Joly | Matching of asymptotic expansions for waves propagation in media with thin slots. II. The error estimates[END_REF] provide rigorous matched asymptotics for a wide variety of singularly perturbed elliptic problems (wave transmission through thin dielectric layer is not considered though) with particular attention to perturbations centered around points with all kinds of boundary conditions: domains containing small inclusions, or with rounded corners on the boundary are typical examples.

The remainder of this report is organized as follows. In Section 1, we describe the setting of the problem and introduce the "Ansatz" of the asymptotic expansion. In Section 2 and 3, we derive the equations satisfied by the far field and near field terms of the matched asymptotic expansion of the solution. Section 4 is devoted to the study of Laplace problems in a normalized strip which allows us to explicit the behavior of the near field terms at infinity in Section 5 and then to write the matching conditions in Section 6. In Section 7, we prove the existence and uniqueness of the far and near field terms of the asymptotic expansion. In particular, we obtain in Section 8 semi-explicit formulas which uncouple far and near field problems (Theorem 8.3 is also a remarkable result of the present paper). The last two sections are devoted to the justification of the asymptotic expansion by constructing first a global expansion and then by establishing error estimates.

Setting of the problem

As a model problem we consider the propagation in harmonic regime of an electromagnetic wave in a medium that is invariant in one direction of space. We focus on the TE mode of the field that is ruled by the following 2-D Helmholtz equation,

div(ǫ -1 δ ∇u δ ) + ω 2 µ δ u δ = -f in R 2 and lim r→∞ √ r(∂ r u δ + iω u δ ) = 0, (1)
where ω refers to the pulsation of the time variation and f ∈ L 2 (R 2 ) refers to a source function for which there exists r e > 0 such that f (x) = 0 for |x| > r e . Besides (r, θ) denote the polar coordinates in R 2 . The characteristics of the medium of propagation are defined in a very specific manner, as follows.

The medium of propagation The functions ǫ δ (x) and µ δ (x) are assumed to be non-negative and defined over R 2 , referring respectively to the permittivity and permeability. Both are assumed to be constant except inside a ring of thickness δ centered around the origin (see Figure 1): there exist r * ∈ (0, r e ) and ǫ ∞ , µ ∞ , c ∞ > 0 independent of δ such that

ǫ ∞ µ ∞ = 1 c 2 ∞ and ǫ δ (r, θ) = ǫ ∞ , µ δ (r, θ) = µ ∞ for |r -r * | > 2πδ.
In this problem, the geometric parameter δ > 0 is assumed to be much smaller than the wavelength far from the heterogeneities: δ ω/c ∞ → 0. Inside the ring {|rr * | ≤ δ/2} the functions ǫ δ (x) and µ δ (x) are assumed to have a periodic structure: the ring contains many small heterogeneities regularly placed according to the azimuthal θ-direction. We suppose that the number N of heterogeneities is related to the thickness of the ring as follows

δ = r * N where N ∈ N and N → ∞ . (2)
Such a relation implies that δ only takes discrete values, so that the expression "δ → 0" should be understood in the sense of [START_REF] Achdou | Etude de la réflexion d'une onde électromagnétique par un métal recouvert d'un revêtement métallisé[END_REF]. Inside the thin ring, we assume that the permeability and permittivity are then given by

ǫ δ (r, θ) = ǫ r -r * δ , θ r * δ and µ δ (r, θ) = µ r -r * δ , θ r * δ , (3)
where ǫ, µ ∈ L 2 loc (R 2 ) are independent of δ. Since (r, θ) are polar coordinates, for such a definition to be meaningful, it is necessary that ǫ and µ satisfy some periodicity assumption, see Fig. 1 (b). We have to assume in addition that ǫ(ν, s + 2π) = ǫ(ν, s), µ(ν, s + 2π) = µ(ν, s),

and ǫ(ν, s) = ǫ ∞ µ(ν, s) = µ ∞ if |ν| > π, if |ν| > π, where s = θ r * δ and ν = r -r * δ . (4)
We also make standard assumptions on the bounds for material properties

∃α > 0 such that α < ǫ(ν, s) < 1 α and α < µ(ν, s) < 1 α ∀ ν, s ∈ R.
To sum up, with the preceding definitions, the medium of propagation is everywhere homogeneous except inside a thin ring. Besides, as δ → 0, this ring of heterogeneities gets close to the following limit circle,

Γ := x ∈ R 2 , |x| = r * .
Reformulation as a problem posed in a bounded domain In order to analyze Problem (1), we will need to reformulate it as a problem posed in a bounded domain denoted Ω defined as follows

Ω = Ω -∪ Γ ∪ Ω + with Ω + := { x ∈ Ω s.t. r * < |x| < r e } and Ω -:= { x ∈ Ω s.t. |x| < r * } so that ǫ δ = ǫ ∞ and µ δ = µ ∞ outside Ω.
We restate the Sommerfeld radiation condition as a condition set on ∂Ω by means of a Dirichlet-to-Neumann map. Such an operator can be defined by the formula

T u = - +∞ p=-∞ u p k H (1) ′ |p| (kr e ) H (1)
|p| (kr e )

e ipθ where u p = 1 2π 2π 0 u(r e , θ)e -ipθ dθ .

It is well known that T is continuous as an operator from H 1 2 (∂Ω) to H -1 2 (∂Ω). Moreover the Sommerfeld radiation condition lim r→∞ √ r(∂ r u δ + iω u δ ) = 0 is then equivalent to ∂ r u δ + T u δ = 0 on ∂Ω. Such a formulation of the radiation condition allows to rewrite (1) as a variational problem set on the bounded domain Ω, as follows

Find u δ ∈ H 1 (Ω) such that a δ (u δ , v) = Ω f v dx ∀v ∈ H 1 (Ω),
where

a δ (u, v) = Ω ǫ -1 δ ∇u • ∇v -ω 2 µ δ uv dx + ∂Ω v T u dσ. (5)
It is well-known that this variational formulation is well posed, see for example [START_REF] Colton | Inverse acoustic and electromagnetic scattering theory[END_REF]. The bilinear form a δ (•, •) actually satisfies infsup conditions uniformly with respect to δ as will be shown in Proposition 10.3.

General methodology and main results

Our purpose is to describe the terms of the expansion of u δ for δ → 0. In our case, due to the fast variations of ǫ δ , µ δ with respect to the angular coordinate, it does not seem possible to write a uniform expansion of the solution in the whole domain Ω. Roughly speaking, the solution u δ oscillates rapidly in a region confined to the vicinity of the periodic ring: this is a boundary layer phenomenon. We use matched asymptotics to cope with this. We first give a brief review on how to apply this method in the present context which, we hope, will help the reader understanding our approach. We follow five steps.

Step I: Far field ansatz (Section 2): we start from a guess of the general form (called "ansatz") of the expansion of u δ both in Ω -and Ω + (see Fig. 2). For the present case we choose

u δ (r, θ) = +∞ n=0 δ n u + n (r, θ) in Ω + , and u δ (r, θ) = +∞ n=0 δ n u - n (r, θ) in Ω -. (6)
The motivation for choosing such an ansatz comes from the already known asymptotic expansion in the analysis of problems involving a thin layer with a geometry that share similarities with ours, see in particular [START_REF] Schmidt | Asymptotic modelling of conductive thin sheets[END_REF]. We plug (6) into (1) in order to formally derive equations that the terms u ± n should satisfy. Unfortunately this will not yield a characterization of the u ± n as these equations will not be well posed: transmission conditions will be missing at the interface Γ.

Step II: Near field ansatz (Sections 3-4-5): the lack of conditions at the interface will be a motivation for studying the expansion of the field close to the periodic ring. Indeed, in this region, it cannot be expected that an expansion of the form (6) still holds because of the rapidly oscillating structure of the geometry. As a consequence we consider a different form for the asymptotic expansion, namely

u δ (r, θ) = +∞ n=0 δ n U n r -r * δ , r * θ δ ; θ , with U n (ν, s + 2kπ, τ + 2lπ) = U n (ν, s, τ ) ∀k, l ∈ Z . (7)
Such an ansatz is inspired by the theory of homogenization, see for example [START_REF] Allaire | Shape optimization by the homogenization method[END_REF][START_REF] Achdou | Etude de la réflexion d'une onde électromagnétique par un métal recouvert d'un revêtement métallisé[END_REF][START_REF] Madureira | Asymptotics of the Poisson problem in domains with curved rough boundaries[END_REF]. According to the periodicity conditions that we impose on U n , it suffices to describe these functions over the infinite periodicity cell B × S 1 represented in figure 2 (b)

B := R × S 1 with S 1 = R/2πZ (unit circle of R 2 ). ( 8
)
Once again we plug the ansatz (7) inside (1) after the change of variables (4). This will yield equations that should be satisfied by the terms U n (ν, s, τ ), at least formally. However, once again, this set of equations will not be well posed because of a lack of conditions at infinity (for |ν| → ∞).

Step III: Matching principle (Section 6): in order to obtain well posed problems that would yield a characterization of the far and near field terms u ± n , U n , the equations derived previously will have to be completed with conditions: transmission conditions for u ± n and conditions at infinity for U n . The method of matched asymptotic expansions provides a procedure called "matching principle" for deriving conditions coupling the behavior of U n (ν, s, τ ) for |ν| → ∞ with the behavior of u ± n close to Γ. During this step we will apply this procedure.

Step IV: Well-posedness of recurrent problems (Sections 7-8): the final output of Step III above will be a system of recurrent equations. We will show that this system is well posed, so that the terms u ± n , U n can be taken as the unique solution to these equations (Theorem 7.1). Semi explicit formula of the far and near field terms are also provided (Theorem 8.3) This step of the analysis will be completely rigorous.

Step V: Error estimate (Section 9): as Step I,II and III of the asymptotic construction partly rely on formal calculus, the purpose of Step V will be an a posteriori justification of the definition of u ± n , U n by means of error estimates.

The first main outcome of this analysis will be an explicit recurrent construction of the terms of the expansions ( 6) and ( 7) up to any order (see Theorem 7.1 and Problem (59) ). The second outcome of this analysis will be a global error estimate for these expansions (Theorem 10.1), from which we deduce an optimal error estimate for the far field (see Theorem 10.2):

Theorem. Let γ ∈ (0, γ * ) and O = { x(r, θ) ∈ Ω | |r -r * | > γ }. Then, for any N ≥ 0, there exists C N > 0 independent of δ such that u δ - N n=0 δ n u n H 1 (O) ≤ C N δ N +1
We will provide a completely rigorous justification for this error estimate as well.

Admittedly

Step I,II and III of this construction contain a formal part. On the other hand, if we started directly from Step IV which would yield an analysis that is completely rigorous from a mathematical point of view, it would be impossible for the reader to understand where our equations come from. This is what motivated the general structure of the present article.

Notation: periodic functions In the sequel, we will often refer to functions v(α) that are 2π-periodic with respect to the variable α = θ, s or τ . This is our motivation for introducing the following space: for any Banach space V we consider

H k # ( S 1 , V ) = v ∈ H k loc ( R, V ) such that v(α + 2π) = v(α) .
This also provides a definition for

C ∞ # ( S 1 , V ) = ∩ k≥0 H k # ( S 1 , V ). Note that H k # ( S 1 , V ) ⊂ H k ( (0, 2π), V ) but H k # ( S 1 , V ) = H k ( (0, 2π), V ), simply because compatibility conditions relating v(0), ∂ α v(0), . . . to v(2π), ∂ α v(2π), . . . may hold for elements of H k # ( S 1 , V ).

Far field terms (Step I)

The first step of our work consists in deriving equations for the far field terms. To do so, we substitute u δ by its far field expansion [START_REF] Bensoussan | Asymptotic analysis for periodic structures[END_REF] in the Helmholtz equation [START_REF] Abboud | Diffraction at a curved grating: TM and TE cases, homogenization[END_REF] and formally separate the different powers of δ. This yields the equations satisfied by the far field terms: Find

u ± n ∈ H 2 (Ω ± ) such that ǫ -1 ∞ △u ± n + ω 2 µ ∞ u ± n = f • δ 0 n in Ω ± , ∂ r u + n + T u + n = 0 on ∂Ω. (9)
where δ 0 n is the kronecker symbol (that has nothing to do with the small parameter δ), i.e, δ 0 0 = 1 and δ 0 n = 0 for n = 0. We emphasize that u ± n are not entirely defined since we have not prescribed yet any boundary condition on Γ: we have to find transmission conditions between u + n and u - n through Γ. For our asymptotic construction, we impose in addition that

u n | ± Γ ∈ C ∞ # (S 1 ) and ∂ r u n | ± Γ ∈ C ∞ # (S 1 ) . (10)
Admittedly [START_REF] Claeys | Analyse asymptotique et numérique de la diffraction d'ondes par des fils minces[END_REF] is slightly abusive as we should write C ∞ (Γ) instead of C ∞ # (S 1 ). We assume indeed that Γ is parameterized by the coordinate θ, so that we consider u n | ± Γ and ∂ r u n | ± Γ as periodic functions of θ.

Radial expansion According to classical elliptic regularity results, see for example Theorem 4.18 in [START_REF] Mclean | Strongly elliptic systems and boundary integral equations[END_REF], u ± n are smooth in a neighborhood of Γ, so they admit a radial expansion with respect to r up to any order. More generally, for a function v ∈ H 2 (Ω + ) and v k ∈ C ∞ (S 1 ), we shall say that k 0

(r -r * ) k v k (θ) is the radial expansion of v for r -r * → 0 + if ∀n ≥ 0, ∃v n ∈ H 2 (Ω + ) s.t. v(r, θ) - n k=0 (r -r * ) k v k (θ) = (r -r * ) n+1 v n (r, θ). ( 11
)
Decomposition of the Helmholtz operator We would like to describe in detail the terms of the radial expansion of u ± n (r, θ). To do so we introduce a particular decomposition of the Helmholtz operator. Our approach is an application of a more general method proposed in [START_REF] Claeys | Analyse asymptotique et numérique de la diffraction d'ondes par des fils minces[END_REF]. Using the expression of the Laplace operator in the polar coordinates, we can decompose the operator r

2 (ǫ -1 ∞ ∆+ω 2 µ ∞ ) according to ρ = r-r * A u = r 2 (ǫ -1 ∞ ∆ + ω 2 µ ∞ )u = 1 ρ 2 4 j=0 ρ j A j ( ρ∂ ρ , ∂ θ )u, (12)
where A j ρ∂ ρ , ∂ θ , (j = 0 • • • 4), are given by

A 0 (ρ∂ ρ , ∂ θ )u = r 2 * ǫ -1 ∞ (ρ∂ ρ ) 2 -ρ∂ ρ u, A 1 (ρ∂ ρ , ∂ θ )u = r * ǫ -1 ∞ 2(ρ∂ ρ ) 2 -ρ∂ ρ u, A 2 (ρ∂ ρ , ∂ θ )u = ǫ -1 ∞ (ρ∂ ρ ) 2 + ǫ -1 ∞ ∂ 2 θ + µ ∞ ω 2 r 2 * u, A 3 (ρ∂ ρ , ∂ θ )u = 2ω 2 µ ∞ r * u, A 4 (ρ∂ ρ , ∂ θ )u = ω 2 µ ∞ u. (13)
The operators A j satisfy some kind of homogeneity property: for any u(r, θ)

= (r-r * ) k u k (θ) = ρ k u k (θ),
there exists a function v k (θ) that only depends on θ such that A j u = (rr * ) k v k (θ). This remark leads us to introduce, for all k ∈ R, the operators A j (k) :

A j (k, ∂ θ ) • v = ρ -k A j ρ∂ ρ , ∂ θ .{ ρ k v(θ) } (14)
The A j (k, ∂ θ ) are differential operators with respect to θ. We shall see that A 0 (k, ∂ θ ) plays a particular role in the forthcoming analysis. Note that

A 0 (k, ∂ θ ) = A 0 (k) = r 2 * ǫ -1 ∞ (k 2 -k) so, as a simple number, A 0 (k) is invertible except for k = 0, 1 and A 0 (k) -1 v(θ) = v(θ)/ r 2 * ǫ -1 ∞ (k 2 -k) .
Far field behavior close to the interface In this paragraph we will describe the general form of the asymptotics of solutions to the Helmholtz equation in the vicinity of the interface Γ. We consider here that r > r * , but the same analysis could be carried out mutatis mutandis for r < r * taking the same notations.

Assume that v ∈ H 2 (Ω + ) satisfies ǫ -1 ∞ ∆v + ω 2 µ ∞ v = 0 in a neighborhood of the interface Γ, and that it admits a radial expansion of the form [START_REF] Colton | Inverse acoustic and electromagnetic scattering theory[END_REF] up to any order. What does Helmholtz Equation imply on the terms v k (θ) of its asymptotic series? Inserting this series in the decomposition (12) of the operator A yields 4 k=0 j∈N ρ k A k ρ j v j = 0 which yields, after reordering,

k 0 ρ k 4 j=0 A j (k -j, ∂ θ ) • v k-j (θ) = 0 =⇒ 4 j=0 A j (k -j, ∂ θ ) • v k-j (θ) = 0 , ∀k 0,
where we take the convention that v j = 0 for j < 0. The equations above yield an iterative process for determining v k (θ), k 2 provided that v 0 (θ), v 1 (θ) are already known: this is a straightforward consequence of

v k (θ) = -A 0 (k) -1 4 j=1 A j (k -j, ∂ θ ) • v k-j (θ) , ∀ k 2. ( 15
)
that makes sense only for j 2 since A 0 (0) = A 0 (1) = 0. In this construction process, the first two terms of the asymptotics v 0 (θ), v 1 (θ) play the role of initial conditions. Note that they are simply the Dirichlet and Neumann trace of

v, v 0 (θ) = v| + Γ and v 1 (θ) = ∂ r v| + Γ . ( 16
)
As a consequence, if ǫ ∞ ∆v+ω 2 µ ∞ v = 0 in the vicinity of Γ and v| + Γ and ∂ r v| + Γ are known, then the whole expansion of v can be explicitly constructed by means of ( 15)-( 16). This conclusion can be formalized as follows. Introduce two families of differential operators with respect to θ that are denoted ( s 0 k (∂ θ ) ) k≥0 and ( s 1 k (∂ θ ) ) k≥0 . These operators are defined by an iterative procedure that mimics the construction above.

s 0 k (∂ θ ) = 0 ∀k < 0, s 0 0 (∂ θ ) = Id, s 0 1 = 0, s 0 k (∂ θ ) = -A 0 (k) -1 4 j=1 A j (k -j, ∂ θ ) • s 0 k-j (∂ θ ) k ≥ 2, s 1 k (∂ θ ) = 0 ∀k < 0, s 1 0 (∂ θ ) = 0, s 1 1 (∂ θ ) = Id, s 1 k (∂ θ ) = -A 0 (k) -1 4 j=1 A j (k -j, ∂ θ ) • s 1 k-j (∂ θ ) k ≥ 2. ( 17
) Proposition 2.1. Let v ∈ H 2 (Ω + ) satisfy ǫ -1 ∞ ∆v + ω 2 µ ∞ v = 0
in the vicinity of the interface Γ, and assume that it admits a radial expansion up to any order. Then ∀n ≥ 0 there exists

v n ∈ H 2 (Ω + ) such that v(r, θ) = n k=0 (r -r * ) k s 0 k (∂ θ )v| + Γ + s 1 k (∂ θ )∂ r v| + Γ + (r -r * ) n+1 v n (r, θ) ∀n ≥ 0. ( 18
)
Proof. By assumption we have v(r, θ) = k≥0 (rr * ) k v k (θ). To prove [START_REF] Il ′ In | Application of the method of matching asymptotic expansions to the solution of boundary value problems[END_REF], we prove that

v k (θ) = s 0 k (θ)v 0 (θ) + s 1 k (∂ θ )v 1 (θ) by induction. This is immediate for k = 0 and k = 1. Now assume that v j (θ) = s 0 j (∂ θ )v 0 (θ) + s 1 j (∂ θ )v 1 (θ)
holds for any j ≤ k. Then according to [START_REF] Gadyl ′ Shin | The method of matching asymptotic expansions in a singularly perturbed boundary value problem for the Laplace operator[END_REF] and [START_REF] Il ′ In | Matching of asymptotic expansions of solutions of boundary value problems[END_REF] we have

v k+1 (θ) = -A 0 (k + 1) -1 4 j=1 A j (k + 1 -j, ∂ θ ) • s 0 k+1-j (∂ θ )v 0 (θ) + s 1 k+1-j (∂ θ )v 1 (θ) , = s 0 k+1 (∂ θ )v 0 (θ) + s 1 k+1 (∂ θ )v 1 (θ).
Proposition 2.1 also holds for functions defined in Ω -. Since we impose Equations ( 9) and (10), Proposition 2.1 can be applied to the far field terms: ∀n, p ≥ 0, there exists u n,p ∈ L 2 (Ω) such that u n,p | Ω± ∈ H 2 (Ω ± ) and that satisfies

u ± n (r, θ) = p k=0 (r -r * ) k s 0 k (∂ θ )u n | ± Γ + s 1 k (∂ θ )∂ r u n | ± Γ + (r -r * ) p+1 u n,p (r, θ) . ( 19
)
and

v Γ = v| + Γ + v| - Γ 2 , ∂ r v Γ = ∂ r v| + Γ + ∂ r v| - Γ 2 .

Near field terms (Step II)

This paragraph is dedicated to the derivation of equations that should be satisfied by the terms of the near field expansion. This will be a much more involved task than for the far field terms: these issues were predictable because of the scaling appearing in [START_REF] Bonnetier | A priori convergence estimates for a rough Poisson-Dirichlet problem with natural vertical boundary conditions[END_REF]. Let us rewrite [START_REF] Bonnetier | A priori convergence estimates for a rough Poisson-Dirichlet problem with natural vertical boundary conditions[END_REF] as follows:

u δ (r, θ) = ∞ n=0 δ n U δ n (r, θ) where U δ n (r, θ) := U n r -r * δ , r * θ δ , θ ∀ n ∈ N. (20)
In this ansatz the terms U n are functions of variables denoted (ν, s, τ ). Although the identity τ = θ will always hold throughout our analysis, it seemed to us that distinguishing both variables would help remove any ambiguity in the equations.

Derivation of the equations satisfied by the near field terms

In this paragraph, we will formally derive the equations that the terms of the expansion (20) should satisfy. We shall use the formulas

∂U δ n ∂r (r, θ) = 1 δ ∂U n ∂ν r -r * δ , r * θ δ ; θ , ∂U δ n ∂θ (r, θ) = r * δ ∂U n ∂s r -r * δ , r * θ δ ; θ + ∂U n ∂τ r -r * δ , r * θ δ ; θ . (21)
Using the expression of the laplacian in polar coordinates and replacing r by r * +δν and taking into account that ǫ -1 does not depend on θ, easy calculation yields

r 2 div( ǫ -1 δ ∇U δ n ) + µ δ ω 2 U δ n = 1 δ 2 r 2 * ∂ ν ǫ -1 ∂ ν U n + ∂ s ǫ -1 ∂ s U n + 1 δ r * 2ν∂ ν ǫ -1 ∂ ν U n + ǫ -1 ∂ ν U n + ∂ s ǫ -1 ∂ τ U n + ∂ τ ǫ -1 ∂ s U n + ν∂ ν ǫ -1 ν∂ ν U n + ǫ -1 ∂ 2 τ U n + r 2 * µ ω 2 U n + δ 2νr * µ ω 2 U n + δ 2 ν 2 µ ω 2 U n . (22)
Since u δ solves the homogeneous Helmholtz equation in the vicinity of the periodic ring, we have formally [START_REF] Madureira | Asymptotics of the Poisson problem in domains with curved rough boundaries[END_REF] in this equation, collecting the terms in δ n , and setting as a convention that U n = 0 for n ≤ 0, we obtain the following equations

n∈N δ n r 2 ( div( ǫ -1 δ ∇U δ n ) + µ δ ω 2 U δ n ) = 0. Plugging (
ν -2 A 0 (∂ ν , ∂ s ) U n = -ν -2 4 j=1 ν j A j U n-j in B × S 1 . (23)
Admittedly, we should impose the equations of (23) only for s = r * τ /δ. However following a typical homogeneization methodology, we deliberately choose to "relax" the constraint s = r * τ /δ and to impose the equations of (23) for any s and τ , which is stronger. This choice will be justified a posteriori by the error estimate of Section 10. The differential operator A 0 = A 0 (∂ ν , ∂ s ) is defined by

A 0 (∂ ν , ∂ s )U := r 2 * ν 2 ∂ ν (ǫ -1 ∂ ν U ) + ∂ s (ǫ -1 ∂ s U ) . (24)
Observe that A 0 does not contain any dependency nor any partial derivative with respect to τ which will be a key feature later on. The other operators A j , 1 ≤ j ≤ 4 are differential operators in (ν, s, τ ) defined by

A 1 U := 2ν 2 ∂ ν ǫ -1 ∂ ν U + ǫ -1 ν∂ ν U + ν∂ s ǫ -1 ∂ τ U + ν∂ τ ǫ -1 ∂ s U A 2 U := ν∂ ν ǫ -1 ν∂ ν U + ǫ -1 ∂ 2 τ U + r 2 * µ ω 2 U A 3 U := 2ω 2 µr * U, A 4 U := ω 2 µ U. (25)
Note that, according to these definitions, for any function U (ν, s, τ ) = U (ν, τ ) that does not depend on the variable s we have

A k (∂ ν , ∂ s , ∂ τ )U (ν, τ ) = A k (ν∂ ν , ∂ τ )U (ν, τ
) for k = 0, . . . , 4 and |ν| > 2π, where A k (ν∂ ν , ∂ τ ) has been defined in (13).

Precise statement of the near field equations

The material of subsection 3.1 was formal. In the present paragraph we will give a rigorous mathematical sense to Equations [START_REF] Maz'ya | Asymptotic theory of elliptic boundary value problems in singularly perturbed domains[END_REF]. We need to introduce an adapted functional framework. First of all, we will assume that the functions U n admit a smooth dependency with respect to τ and that U n (ν, s, τ +2π) = U n (ν, s, τ ).

Weighted spaces

In accordance with our choice of ansatz for the near field, we also have to impose the periodicity condition U n (ν, s + 2π, τ ) = U n (ν, s, τ ). Moreover, following the usual procedure of matched asymptotics, we also have to discard any possibility for the terms U n to blow up exponentially for |ν| → ∞, which can be justified a posteriori. In the present case, this condition will be enforced by imposing that, for any n and any θ, the function

U n (•, τ ) ∈ V 1 + (B)
where we set, for σ = ±,

V k σ (B) = U ∈ H k loc (R 2 ) U 2 V k σ = α+β≤k B |∂ α ν ∂ β s U (ν, s)| 2 e -σ|ν| dνds < +∞ and U (ν, s + 2π) = U (ν, s) . (26)
We emphasize that the definition of V k σ (B) encompasses a periodicity assumption with respect to s. We will use the above definition for k = 0 and k = 1. The spaces V k ± (B) equipped with the norm

V k ± are Banach spaces. Besides we have the obvious inclusion V k -(B) ⊂ V k + (B). The elements of V 1 -(B)
may be interpreted as "evanescent at infinity". Finally we impose that the functions U n should belong to the following space

U n ∈ C ∞ S 1 , V 1 + (B) ∀n ∈ Z . (27)

Dual spaces

In the sequel, we may write equations in the space V 1 + (B) ′ the topological dual space of V 1 + (B) i.e. the space of linear functionals that are continuous over V 1 + for the norm

V 1
+ . The duality pairing on

(V 1 + ) ′ × V 1 + will be denoted < ; > + .
As usual, we equip this space with the continuity norm associated to this pairing,

g (V 1 + ) ′ = sup U∈V 1 + \{0} g, U + U V 1 + .
The following lemma gives a detailed description of V 1 + (B) ′ by showing that it can be identified with a space of distributions over R 2 that are periodic in s.

Lemma 3.1. Set (T # ϕ)(ν, s) = k∈Z ϕ(ν, s + 2kπ), ∀ϕ ∈ D(R 2 ). For any g ∈ V 1 + (B) ′ set T ⋆ # g : ϕ → g, T # ϕ + that is an element of D ′ (R 2 ). The image of V 1 + (B) ′ under T ⋆ # is exactly the space of distributions h ∈ D ′ (R 2 ) of the form h = h 1 -div(h 2 ) where h 1 , h 2 ∈ L 2 loc (R 2 ) such that h k (ν, s + 2lπ) = h k (ν, s), ∀s ∈ R, ∀l ∈ Z and h k | B ∈ V 0 -(B) for k = 1, 2. Remark 3.2. Assume that g ∈ V 1 + (B) ′ is in C ∞ # (B). Then, (T ⋆ # g) is the 2π periodic function (in s) which satisfies (T ⋆ # g) |B = g.
In other words T ⋆ # g is the extension of g by periodicity. Note also that one practical interest of the previous notation is the justification of the Green formula [START_REF] Sánchez-Palencia | Nonhomogeneous media and vibration theory[END_REF].

Proof. Take a g ∈ V 1 + (B) ′ . Observe that (u, v) → B (∇u • ∇v + uv)e -|ν| dνds is a scalar product over V 1 + (B).
As a consequence according to Riesz representation theorem, there exists

u g ∈ V 1 + (B) such that g, v + = B e -|ν| ∇u g • ∇v + e -|ν| u g vdνds ∀v ∈ V 1 + (B) .
Set g 1 = e -|ν| u g and g 2 = e -|ν| ∇u g , and observe that

g k ∈ V 0 -(B). For k = 1, 2 consider h k ∈ L 2 loc (R 2 ) such that h k (ν, s + 2lπ), ∀l ∈ Z and h k | B = g k . Observe that B g 1 T # ϕdx = R 2 h 1 ϕdx, ∀ϕ ∈ D(R 2 )
and the same remark holds for g 2 , h 2 . As a consequence

T ⋆ # g = h 1 -div(h 2 ) since T ⋆ # g, ϕ D ′ ,D = B g 1 T # ϕdνds + B g 2 • ∇(T # ϕ)dνds, = B g 1 T # ϕdνds + B g 2 • T # (∇ϕ)dνds, = R 2 h 1 ϕ dνds + R 2 h 2 • ∇ϕ dνds ∀ϕ ∈ D(R 2 ), where < •, • > D ′ ,D denotes the duality product between D ′ (R 2 ) and D(R 2 ). Conversely consider h ∈ D ′ (R 2 ) of the form h = h 1 -div(h 2 )
where h 1 , h 2 satisfy the conditions stated in the lemma. Consider

g k = h k | B , k = 1, 2 and define g ∈ V 1 + (B) ′ by g, ϕ + = g 1 , v + -div(g 2 ), v + = B g 1 ϕ + g 2 • ∇ϕdνds. (28)
A calculus similar to what precedes shows that T ⋆ # g = h.

Convention of notation The elements of

V 1 + (B) ′ can be interpreted as periodic distributions of D ′ (R 2 ). As a consequence, whenever h ∈ D ′ (R 2 ) we shall adopt the following convention of notation " g = h in V 1 + (B) ′ " ⇐⇒ " g ∈ V 1 + (B) ′ and T ⋆ # g = h in D ′ (R 2 ) ". ( 29
)
For a distribution h belonging a priori to D ′ (R 2 ) we shall also write

h ∈ V 1 + (B) ′ which would mean precisely: "∃g ∈ V 1 + (B) ′ such that h = T # g in D ′ (R 2 )". Concerning V 1 -(B)
′ , a lemma similar to Lemma 3.1 holds, so we set mutatis mutandis the same conventions of notation for

V 1 -(B) ′ . Besides V 1 + (B) ′ ⊂ V 1 -(B) ′ since V 1 -(B) ⊂ V 1 + (B). And indeed careful verifications show that, for h ∈ D ′ (R 2 ), we have " g = h in V 1 + (B) ′ " =⇒ " g = h in V 1 -(B) ′ " .

Equations of the near field

Now we come back to the equations that we collected previously for the near field terms, imposing smooth dependency with respect to τ and discarding any exponential growth for |ν| → ∞. We obtain the following recurrent system of equations, where U n is the unknown and U n-1 , U n-2 , U n-3 and U n-4 are considered already known

U n ∈ C ∞ S 1 , V 1 + (B) A 0 (∂ ν , ∂ s ) U n = - 4 j=1 ν j A j (∂ ν , ∂ s , ∂ τ ) U n-j in V 1 -(B) ′ , ∀τ ∈ S 1 . (30)
In these equations the operators A j , j = 1 . . . 4 involve ∂ ν , ∂ s , ∂ τ and admit some dependency with respect to ν and s, as can be checked from [START_REF] Nédélec | Acoustic and electromagnetic equations[END_REF]. Besides note that writing equations ( 30) in V 1 -(B) ′ allows the right hand side of the equation to admit a polynomial growth for |ν| → ∞. We will use this system of equations for defining each U n . However [START_REF] Schmidt | Asymptotic modelling of conductive thin sheets[END_REF] is not well posed as the conditions at infinity are not constraining enough.

Study of the Laplace equation in the normalized strip

The equations that we have just derived for the terms of the near field asymptotics raise some difficulty at least for two reasons. First of all, (30) has a recurrent structure. In addition, there is no particular reason for assuming that the right-hand side would remain bounded at infinity. As a consequence, (30) appears as a complicated problem that does not fit any particular standard framework.

Consequently, we stop for a while our asymptotic procedure, and we dedicate the present section to the study of equations that take the same form as [START_REF] Schmidt | Asymptotic modelling of conductive thin sheets[END_REF]. We will also describe precisely the behavior at infinity of the solutions to such equations and we will give important results of well-posedness (Propositions 4.5, 4.7 and 4.8) . Note that the present section only contains completely rigorous analysis.

In this paragraph, we only consider functions of ν, s, and (ν, s) will be thought as cartesian coordinates over R 2 , so that the operator ν -2 A 0 then simply rewrites ν

-2 A 0 • U = r 2 * div(ǫ -1 ∇U ). Suppose given g ∈ V 1
+ (B) ′ and consider the problem

Find U ∈ V 1 + (B) such that -div(ǫ -1 ∇U ) = g in V 1 + (B) ′ . (31)
Here the left hand side div(ǫ -1 ∇U ) makes sense as an element of D ′ (R 2 ) so that (31) should be understood in the sense of [START_REF] Sánchez-Palencia | Un problème d' écoulement lent d'une fluide visqueux incompressible au travers d'une paroi finement perforée[END_REF]. Let D # (B) refer to the space of functions ϕ ∈ C ∞ (R 2 ) such that ϕ(ν, s + 2kπ) = ϕ(ν, s)∀k ∈ Z, ∀ν, s ∈ R and there exists ν 0 > 0 such that ϕ(ν, s) = 0 for |ν| > ν 0 . In particular we have

T # ( D(R 2 ) ) = D # (B)
according to Lemma 11.2 so that Equation ( 31) can be reformulated variationally as follows

B ǫ -1 ∇U • ∇ϕdx = g, ϕ + ∀ϕ ∈ D # (B) . (32)

Standard result of well-posedness : variational framework

According to standard techniques for the Laplace equation, existence of a solution to [START_REF] Tlemcani | A Two-Scale Asymptotic Analysis of a Time-Harmonic Scattering Problem with a Multi Layered Thin Periodic Domain[END_REF] is at hand in a slightly different functional framework. Indeed, it is natural to rely on a variational approach in order to determine whether this problem is well posed. As a variational framework, we consider the following spaces

W(B) = V ∈ H 1 loc (R 2 ) B |∇V | 2 + |V | 2 1 + ν 2 dνds < ∞ and V (ν, s + 2π) = V (ν, s) .
Note that we have the inclusions

V 1 -(B) ⊂ W(B) ⊂ V 1 + (B). This implies in particular V 1 + (B) ′ ⊂ W(B) ′
where W(B) ′ is the topological dual to the space W(B). Observe that 1 ∈ W(B) and ν / ∈ W(B). Besides a Poincaré inequality holds for this space. Indeed set Γ ± := {±π} × (0, 2π). Then there exists a constant C > 0 such that, for any U ∈ W(B), we have

U 2 W = B |∇U | 2 + |U | 2 1 + ν 2 dνds ≤ C B |∇U | 2 dνds + C Γ- U ds + Γ+ U ds 2 .
This is a consequence of Hardy's inequality, see for instance Lemma 2.5.7 in [START_REF] Nédélec | Acoustic and electromagnetic equations[END_REF]. The next proposition is a byproduct of this inequality. The proof follows directly from Lax-Milgram's Lemma.

Proposition 4.1.

Let g ∈ V 1 + (B) ′ .
Then there exists at most one function

U ∈ W(B) satisfying both -div(ǫ -1 ∇U ) = g in V 1
+ (B) ′ and the condition

Γ- U ds + Γ+ U ds = 0 with Γ ± := {±π} × (0, 2π) . ( 33
)
This solution exists if and only if the following compatibility condition is satisfied: g, 1 + = 0. Besides, there is continuous dependency of U with respect to g i.e. there exists a constant C > 0 independent of g such that

U W C g (V 1 + ) ′ . ( 34
)
The only technical point for proving Proposition 4.1 consists in establishing that (32) still holds with ϕ chosen arbitrarily in W(B). But this is a consequence of the density of D # (B) in W(B) for the norm • W , see Lemma 11.3.

Asymptotic behavior at infinity

To go further into the analysis of Problem [START_REF] Tlemcani | A Two-Scale Asymptotic Analysis of a Time-Harmonic Scattering Problem with a Multi Layered Thin Periodic Domain[END_REF], we need to study functions U (ν, s) admitting a behavior at infinity of the form "polynomial + evanescent at infinity". This is our motivation for introducing the following notation.

Definition 4.2 (Property P ∞ ).

A function U ∈ V 1 + (B) will be said to satisfy property (P ∞ ) if and only if there exists a function

U ∈ V 1 -(B)
, and two functions p ± (ν, s) that admit polynomial dependency with respect to ν, such that

U (ν, s) = p ± (ν, s) + U(ν, s) for ± ν > 2π and s ∈ S 1 . (35)
Besides, when U satisfies such a property, we set

ℓ ± D (U ) = p ± (0, s) and ℓ ± N (U ) = ∂ ν p ± (0, s), as well as ℓ D (U ) = 2π ℓ + D (U ) + ℓ - D (U ) , [ ℓ D (U ) ] = ℓ + D (U ) -ℓ - D (U ) , ℓ N (U ) = 1 2 ℓ + N (U ) + ℓ - N (U ) , [ ℓ N (U ) ] = ℓ + N (U ) -ℓ - N (U ) . ( 36
)
Such a property will be commonly encountered during the rest of our study. In particular, solutions to a Laplace equation with a right-hand side that is evanescent at infinity satisfy such a property. The following result is classical from the point of view of Kondratiev's theory, see for example chapter 5 of [START_REF] Kozlov | Elliptic boundary value problems in domains with point singularities[END_REF]. However, for the sake of completeness, and since we do not expect our reader to be familiar with such a result, we give a detailed proof.

Proposition 4.3.

Any U ∈ V 1 + (B) such that div(ǫ -1 ∇U ) ∈ V 1 + (B) ′ satisfies Property (P ∞ ). Besides, for any such U there exist a ± , b ± ∈ C and U ∈ V 1 -(B) such that U (ν, s) = a ± + b ± ν + U(ν, s) for ± ν > 2 π , s ∈ S 1 . ( 37
)
Proof. We only prove this result for ν > 2π, since the proof for ν < -2π follows the same lines. Moreover, using a suitable cut-off function if necessary, we assume that U vanishes for ν ≤ 2π. Take a C ∞ cut-off function χ : B → [0, 1] such that χ(ν, s) = χ(ν), and χ(ν) = 0 for ν < π and χ(ν

) = 1 for ν ≥ 2π. Set g = -div(ǫ -1 ∇U ) ∈ V 1 + (B) ′ .
We apply Laplace transform in ν and Fourier decomposition in s to the equation -div(ǫ

-1 ∇U ) = g. Set ûk (λ) = 1 2π B U (ν, s)e -λν-iks dν ds for ℜe{λ} > +1/2 , k ∈ Z, ĝk (λ) = 1 2π
g, e -λν-iks

+ for ℜe{λ} > -1/2 , k ∈ Z.
The functions χ(ν) exp(-λν-iks) and exp(-λν-iks) coincide on the support of g so, since χ(ν) exp(-λν-

iks) ∈ V 1 + (B)
for ℜe{λ} > -1/2, the function ĝk (λ) is well defined. Adapting for example the proof of Theorem 7.23 (a) in [START_REF] Rudin | Functional analysis[END_REF] that relies on Morera's Theorem, it is easy to show that that ĝk (λ) is analytic for ℜe{λ} > -1/2. Finally, Parseval Identity and Riesz Theorem prove that there exists

C 1 > 0 such +∞ k=-∞ +∞ -∞ | ĝk (-1/2 + iξ) | 2 1 + k 2 + ξ 2 dξ C 1 g 2 (V 1 + ) ′ < +∞. ( 38
)
Now let us take χ(ν) exp(-λνiks) as a test function that we plug into the variational equation [START_REF] Tordeux | Méthode asymptotiques pour la propagation des ondes dans des milieux comportant des fentes[END_REF]. Once again, since χ(ν) exp(-λνiks) and exp(-λνiks) coincide on the support of U and g, this yields

ǫ -1 ∞ (k 2 -λ 2 )û k (λ) = ĝk (λ) for ℜe{λ} > 1/2 . This identity shows that ûk (λ) = ǫ -1 ∞ (k 2 -λ 2 ) -1 ĝk (λ) for ℜe{λ} > -1/2, λ = k.
As a consequence the functions ûk (λ) can be extended so that: 1) ûk (λ) is analytical for ℜe{λ} > -1/2 if k = 0, and 2) û0 (λ) is analytic over {λ ∈ C | ℜe{λ} > -1/2, λ = 0 } with a pole of order 2 at 0. Applying inversion formula for the Fourier-Laplace transform, we get

U (ν, s) = k∈Z 1 2iπ ℜe{λ}= 1 2 ûk (λ) e λν+iks dλ .
where the contours are parameterized by λ = 1 2 + iξ and oriented toward ξ → +∞. The series converges in V 1 + (B) according to Parseval Theorem. Moreover, as the functions ûk (λ) have been meromorphically extended for ℜe{λ} > -1/2, we can shift the integration contour to λ = -1 2 + iξ, ξ ∈ R (oriented toward ξ → +∞), applying the residue formula for k = 0. For a justification of this classical operation, see for example Theorem 5.4.1 in [START_REF] Kozlov | Elliptic boundary value problems in domains with point singularities[END_REF]. This yields

U (ν, s) = dĝ 0 dλ (0) + ĝ0 (0) ν + V(ν, s), with V(ν, s) = k∈Z 1 2iπ ℜe{λ}=-1 2 ĝk (λ) k 2 -λ 2 e λν+iks dλ. (39) Let us show that the function U(ν, s) = χ(ν)V(ν, s) belongs to V 1 -(B). To prove this, it is sufficient to show that V(ν, s) exp(ν/2) ∈ H 1 (B). Using the parameterization λ = -1 2 + iξ, ξ ∈ R for the integral in the definition of V(ν, s), we have e ν 2 V(ν, s) = 1 2π k∈Z +∞ -∞ ĝk (-1 2 + iξ) k 2 -(-1 2 + iξ) 2 e iξν+iks dξ .
It appears that e ν we obtain

e ν 2 V 2 L 2 (B) + ∇( e ν 2 V ) 2 L 2 (B) = 1 2π k∈Z +∞ -∞ (1 + k 2 + ξ 2 ) ĝk (-1 2 + iξ) k 2 -(-1 2 + iξ) 2 2 dξ, ≤ C 2 2 2π k∈Z +∞ -∞ | ĝk (-1 2 + iξ) | 2 1 + k 2 + ξ 2 dξ ≤ C 1 C 2 2 2π g 2 (V 1 + ) ′ < +∞. (40)
Straightforward application of Definition 4.2 shows that when U satisfies the assumptions of Proposition 4.3, then a ± = ℓ ± D (U ) and b ± = ℓ ± N (U ). In the case where U satisfies the assumptions of Proposition 4.3 and in addition

U ∈ W(B), then ℓ ± N (U ) = 0 since ν / ∈ W(B).
Now we introduce a particular non-trivial solution to [START_REF] Tlemcani | A Two-Scale Asymptotic Analysis of a Time-Harmonic Scattering Problem with a Multi Layered Thin Periodic Domain[END_REF] with g = 0. The following proposition also proves that ( 31) is ill posed with a kernel of dimension 2 at least.

Proposition 4.4.

There exists a unique

N ∈ V 1 + (B) such that N (ν, s) -ν ∈ W(B) and that satisfies div(ǫ -1 ∇N ) = 0 in V 1 + (B) ′ and ℓ D (N ) = 0. Proof. Note that div(ǫ -1 ∇ν) = ∂ ν (ǫ -1 -ǫ -1 ∞ ) ∈ V 1 + (B) ′ since ǫ -1 -ǫ -1 ∞ = 0 for |ν| > π.
As a consequence, according to Proposition 4.1, there exists a unique N * ∈ W(B) satisfying [START_REF] Van Dyke | Perturbation methods in fluid mechanics[END_REF] and such that div(ǫ

-1 ∇N * ) = -∂ ν (ǫ -1 -ǫ -1 ∞ ) in V 1 + (B) ′ . Then it suffices to set N = ν + N * -ℓ D (N * )
. Uniqueness of such a function is a consequence of the uniqueness part of Proposition 4.1.

Both 1 and N verify the assumptions of Proposition 4.3. Thus, successively identifying a ± and b ± with ℓ ± D (U ) and ℓ ± N (U ) for ±ν > 2π and both for 1 and N yields the following simple identities

ℓ N (1) = 0 , [ ℓ N (1) ] = 0 , ℓ D (1) = 1 and [ ℓ D (1) ] = 0, ℓ N (N ) = 1 , [ ℓ N (N ) ] = 0 , ℓ D (N ) = 0 and [ ℓ D (N ) ] = 0 (a priori) . ( 41
)

General results of well posedness

Now we show that it is possible to add conditions formulated in terms of the mean and jump operators (36) to Problem (31) so as to obtain a well posed problem where no compatibility condition on the right hand side is required anymore.

Proposition 4.5.

Consider g ∈ V 1 + (B) ′ and α, β ∈ C arbitrary.
There exists a unique solution to the problem

Find U ∈ V 1 + (B) such that -div(ǫ -1 ∇U ) = g in V 1 + (B) ′ , ℓ D (U ) = α and ℓ N (U ) = β . ( 42
)
Moreover there is continuous dependency of the solution with respect to the data, i.e, there exists C > 0 independent of α, β, g such that

U V 1 + C |α| + |β| + g (V 1 + ) ′ . ( 43
)
Proof.

Uniqueness

U ∈ V 1 -(B) such that U (ν, s) = ±( ℓ + D (U ) + ℓ + N (U )ν ) + U(ν, s) for ±ν > 2π. Set B + = {(ν, s) ∈ B | ν > 0}. Applying Lemma 11.1 to U yields 0 = lim R→∞ B ǫ -1 ∇U • ∇ψ R dsdν = lim R→∞ B ǫ -1 ∇ ℓ + N (U )ν + ℓ + D (U ) + U(ν, s) • ∇ψ R dsdν, = 2 ℓ + N (U ) B+ ǫ -1 ∂ ν ψ R dsdν + lim R→∞ B ǫ -1 ∇U • ∇ψ R dsdν = 4π ǫ -1 ∞ ℓ + N (U ) .
As a consequence ℓ + N (U ) = 0 and this shows that U ∈ W(B). Take C = Γ-U ds + Γ+ U ds. Applying Proposition 4.1 to U -C we see that U = C. Since U is a constant, we have U = ℓ D (U ) = 0 which concludes the uniqueness part of the proof.

Existence Now consider the function χ(ν, s) = 1ψ(ν) where ψ is the cut-off function introduced in the uniqueness part above. This function χ satisfies

B div ǫ -1 ∇ χ(ν) |ν| dν ds = 2πǫ -1 ∞ +2π -2π ∂ 2 ν { χ(ν) |ν| } dν = 4πǫ -1 ∞ . Set g ∞ = g, 1 + and h = g ∞ • (4πǫ -1 ∞ ) -1 div{ǫ -1 ∇ χ(ν) |ν| }. We have h, 1 + = g, 1 + . Set g * = g -h that satisfies the compatibility condition g * , 1 + = 0. Besides h ∈ V 1 + (B) ′ so g * ∈ V 1 + (
B) ′ as well. We can apply the existence part of Proposition 4.1 with g * as right-hand side: there exists a function

U * ∈ W(B) ⊂ V 1 + (B) satisfying (33) such that -div(ǫ -1 ∇U * ) = g * in V 1 + (B) ′ .
Taking into account (41), we finally construct a solution to (42) by setting

U = U * -ℓ D (U * ) + g ∞ • (4πǫ -1 ∞ ) -1 χ(ν) |ν| + α + βN . (44)
Continuity To prove (46) for the solution that we have just constructed, it is sufficient to prove that the first three terms in the right hand side of (44) continuously depend on α, β, g. Clearly g ∞ satisfies such an inequality since | g,

1 + | 1 V 1 + g (V 1 + ) ′ .
Besides U * also satisfies an inequality similar to (43). Indeed direct estimation shows that there exists C > 0 such that

V V 1 + ≤ C V W ∀ V ∈ W(B).
Combining this with (34) then proves that there exists C > 0 independent of g such that

U * V 1 + ≤ C g (V 1 + ) ′ .
There only remain to provide an estimate for ℓ D (U * ) . Consider the function

V * (ν, s) = V * (ν) such that V * (ν) = 0 if |ν| ≤ π and V * (ν) = π -|ν| if |ν| > π. Set B ± R = { (ν, s) ∈ B | πR ≤ ±ν ≤ 2πR }. Observe that supp( ∂ ν ψ R ) ⊂ B + R ∪ B - R .
As a consequence, for R > π we have

0 = Γ+ U * ds + Γ- U * ds = B ∇(ψ R U * ) • ∇V * dx, = B ψ R ∇U * • ∇V * dx - B + R U * ∂ ν ψ R dx + B - R U * ∂ ν ψ R dx,
where Γ ± have been defined in [START_REF] Van Dyke | Perturbation methods in fluid mechanics[END_REF]. We can apply Proposition 4.3 to U * and since U * ∈ W(B), there exists

U * ∈ V 1 -(B) such that U * (ν, s) = ℓ ± D (U * ) + U * (ν, s) for ±ν > 2π.
As a consequence, using Lemma 11.1, we see that

lim R→+∞ - B + R U * ∂ ν ψ R dx + B - R U * ∂ ν ψ R dx = 4π ℓ D (U * ) . Also, the decomposition U * (ν, s) = ℓ ± D (U * ) + U * (ν, s) for ±ν > 2π shows that ∇U * ∈ V 0 -(B). As a consequence, since W(B) is dense in V 1 + (B) for V 1 + according to Lemma 11.3 in appendix, we have B ǫ -1 ∇U * ∇V dx = g * , V + , ∀V ∈ V 1 + (B).
Hence by the dominated convergence theorem

lim R→+∞ B ψ R ∇U * • ∇V * dx = ǫ ∞ B ǫ -1 ∇U * • ∇V * dx = ǫ ∞ g * , V * + .
Finally we see that ℓ D (U * ) = -ǫ ∞ g * , V * + /4π. Since there exists a constant independent of g such that g

* (V 1 + ) ′ ≤ C g (V 1
+ ) ′ , this clearly proves the continuous dependency of ℓ D (U * ) with respect to g. In conclusion U depends continuously on α, β, g with respect to the norm • V 1 + . Remark 4.6. Using Proposition 4.5, it is easy to check that if g(ν, s) = ν k , k ∈ N then the following problem is well-posed,

Find U ∈ V 1 + (B) such that -div(ǫ -1 ∇U ) = g in V 1 -(B) ′ , ℓ D (U ) = α and ℓ N (U ) = β . ( 45
)
Moreover there is continuous dependency of the solution with respect to the data i.e there exists C > 0 independent of α, β, g such that

U V 1 + C |α| + |β| + g (V 1 + ) ′ . ( 46
)
Whereas ℓ D (U ) and ℓ N (U ) should be considered as data in Problem (42), we can provide explicit formula relating directly [ℓ D (U )] and [ℓ N (U )] to div(ǫ -1 ∇U ), ℓ D (U ) and ℓ N (U ) . We postpone the proof of the following proposition in appendix since it involves tedious calculus.

Proposition 4.7.

Assume that U ∈ V 1 + (B) satisfies -div(ǫ -1 ∇U ) = g in V 1 + (B) ′ for some g ∈ V 1 + (B) ′ . Then we have [ ℓ N (U ) ] = - ǫ ∞ 2π g, 1 + and [ ℓ D (U ) ] = ℓ N (U ) • [ ℓ D (N ) ] - ǫ ∞ 2π g, N + .
When U is solution to (42), a byproduct of Proposition 4.7 above is the continuous dependency of ℓ ± D (U ) and ℓ ± N (U ) with respect to α, β, g.

A natural question is then: is there continuous dependency of the remainder in (4.3) with respect to α, β, g?

The answer is positive and it is easy to see it. Indeed according to (46) and Proposition 4.7 there exists a constant C > 0 independent of α, β, g such that

U V 1 + ≤ C |α| + |β| + g (V 1 + ) ′ . ( 47
)
However this estimate is not optimal. A sharper one would involve

U V 1 -instead of U V 1 + .
Such an estimate can be established as is stated by the following proposition. The proof is postponed to appendix. Proposition 4.8.

For g ∈ V 1 + (B) ′ and α, β ∈ C, let U ∈ V 1 + (B) be the unique solution to Problem (42). Let χ : R → [0, 1] be a C ∞ cut-off function that satisfies χ(ν) = 0 for ν < π and χ(ν) = 1 for ν > 2π. Finally let U ∈ V 1 -(B) be defined by U(ν, s) = U (ν, s) -P (ν),
where

P (ν) = χ(ν) [ ℓ + D (U ) + ℓ + N (U ) ν ] + χ(-ν) [ ℓ - D (U ) + ℓ - N (U ) ν ].
Then there is continuous dependency of U with respect to α, β, g: there exists a constant C > 0 independent of α, β, g such that

U V 1 -≤ C |α| + |β| + g (V 1 + ) ′ . ( 48
)
5 Expansion of the near field terms at infinity

In the previous section we considered functions which only depend on the variables ν and s. We are now coming back to the more complex situation of equations [START_REF] Schmidt | Asymptotic modelling of conductive thin sheets[END_REF], where the near field terms U n also depend on the third variable τ . Let us rewrite this system of equations:

U n ∈ C ∞ S 1 , V 1 + (B) A 0 (∂ ν , ∂ s ) U n = - 4 j=1 ν j A j (∂ ν , ∂ s , ∂ τ ) U n-j in V 1 -(B) ′ , ∀τ ∈ S 1 . (49)
These equations take, at fixed n, a form very close to [START_REF] Tlemcani | A Two-Scale Asymptotic Analysis of a Time-Harmonic Scattering Problem with a Multi Layered Thin Periodic Domain[END_REF], except that it is parameterized by the third variable τ . Besides, in [START_REF] Schmidt | Asymptotic modelling of conductive thin sheets[END_REF], no condition is prescribed on < ℓ D (U n ) > and < ℓ N (U n ) > (this precise point will be part of the focus of next section).

Remark 5.1. In equation ( 49), the variable τ plays the role of a parameter: more precisely, for any τ ∈ S 1 , U n (., ., τ

) ∈ V 1 + (B).
Note that, according to the estimates (43, 46, 48), if the right hand side of equation ( 49) admits C ∞ dependency with respect to τ , then the solution to Equation (49) admits C ∞ dependency with respect to τ as well. As we shall see, in the remainder of this paper, near field terms U n will always have C ∞ dependency with respect to τ .

At this point of our work, it is interesting to prove that the structure of (49) implies a particular form for the behavior of each U n as |ν| → ∞. [START_REF] Schmidt | Asymptotic modelling of conductive thin sheets[END_REF]. Then each U p , p = 0, . . . , n satisfies Property (P ∞ ) and there exist functions

Proposition 5.2. Consider a family U 0 , U 1 , . . . U n ∈ C ∞ S 1 , V 1 + (B) satisfying
U p ∈ C ∞ S 1 , V 1 -(B)
such that for p = 0, . . . , n, and for ±ν ≥ 2π there holds

U p (ν, s, τ ) = p+1 k=0 ν k • s 0 k (∂ τ ) • ℓ ± D U p-k (•, τ ) + p+1 k=0 ν k • s 1 k (∂ τ ) • ℓ ± N U p+1-k (•, τ ) + U p (ν, s, τ ). (50)
First of all, despite appearances, the coefficients ℓ ± N (U p+1 ) do not come into play in Formula (50) since s 1 0 = 0 according to [START_REF] Il ′ In | Matching of asymptotic expansions of solutions of boundary value problems[END_REF]. Besides, Proposition 5.2 shows that, provided that System (30) is satisfied, the function U n satisfy Property

(P ∞ ) (although ∇ • (ǫ -1 ∇U n ) / ∈ V 1 + (B) ′ in general)
and their behavior for |ν| → ∞ consists in a polynomial part that does not depend on s and an exponentially decreasing part. In addition the polynomial part only depends on a set of functions ℓ ± D ( U n (•, τ ) ) and ℓ ± N ( U n (•, τ ) ) that admit a dependency only with respect to τ .

Proof. We proceed by induction. For the first term U 0 the decomposition (50) is a direct consequence of Proposition 4.3 and the definition of s 1 0 , s 1 1 , s 0 0 , s 0 1 given by [START_REF] Il ′ In | Matching of asymptotic expansions of solutions of boundary value problems[END_REF]. Assume now that (50) holds for p = 0, 1, . . . n -1 and let us prove it for p = n. Consider a cut-off function χ ∈ C ∞ (B) such that χ(ν, s) = χ(ν) = 1 for ν > 2π, and χ(ν, s) = χ(ν) = 0 for ν < π. For each q = 0 . . . n -1 set

p ± k (ν, τ ) = k+1 q=0 ν q • s 0 q (∂ τ ) • ℓ ± D U k-q (•, τ ) + k+1 q=0 ν q • s 1 q (∂ τ ) • ℓ ± N U k+1-q (•, τ ) , P k (ν, τ ) = χ(ν) p + k (ν, τ ) + χ(-ν) p - k (ν, τ ),
so that U k (ν, s, τ ) = P k (ν, τ ) + U k (ν, s, τ ) for k = 0, . . . n -1 and for |ν| > 2π. Besides, in order to make notations shorter, we set c D,± q (τ ) = ℓ ± D ( U q (•, τ ) ) and c N,± q (τ ) = ℓ ± N ( U q (•, τ ) ), q = 0 . . . n -1.

Using definitions ( 14) and [START_REF] Il ′ In | Matching of asymptotic expansions of solutions of boundary value problems[END_REF] we see that, for ν > π, we have

4 k=1 ν k A k (∂ ν , ∂ s , ∂ τ )p + n-k (ν, τ ) = +∞ k=1 ν k A k (∂ ν , ∂ τ )p + n-k (ν, τ ) = +∞ k=1 +∞ q=0 ν k+q A k (q, ∂ τ ) • s 0 q (∂ τ )c D,+ n-k-q (τ ) + s 1 q (∂ τ )c N,+ n+1-k-q (τ ) . (51)
In the equation above we used the conventions that c q (τ ) = 0 for q < 0, and A k (q, ∂ τ ) = 0 for k > 4 and for any q, so that the sum contains only finitely many terms. Let us apply the change of index j = k + q and keep the index k, which yields

4 k=1 ν k A k (∂ ν , ∂ s , ∂ τ )p + n-k (ν, τ ) = +∞ j=1 ν j +∞ k=1 A k (j -k, ∂ τ ) • s 0 j-k (∂ τ )c D,+ n-j (τ ) + s 1 j-k (∂ τ )c N,+ n+1-j (τ ) , = - +∞ j=1 ν j A 0 (j) • s 0 j (∂ τ )c D,+ n-j (τ ) + s 1 j (∂ τ )c N,+ n+1-j (τ ) = -A 0 (∂ ν , ∂ s ) p + n , with p + n (ν, τ ) = n+1 q=2 ν q • s 0 q (∂ τ )c D,+ n-q (τ ) + s 1 q (∂ τ )c N,+ n+1-q (τ ) . (52)
In the last line of this calculus we used the fact that A 0 (0) = A 0 (1) = 0. The same calculus holds replacing + byand introducing a functions p - n (ν, τ ). Define P n (ν, s, τ ) = χ(ν) p + n (ν, τ )+χ(-ν) p - n (ν, τ ). Since this function is polynomial for |ν| large enough, explicit calculus shows that P n satisfies (P ∞ ) and that ℓ ± D ( P n (•, τ )) = ℓ ± N ( P n (•, τ )) = 0 as p ± n does not contain any monomial of order 0 or 1. According to (51)-(52) the function P n satisfies

Q n := A 0 (∂ ν , ∂ s ) P n + 4 k=1 ν k A k (∂ ν , ∂ s , ∂ τ )P n-k ∈ C ∞ S 1 , V 1 + (B) ′ .
Use this remark, and take into account that

U k ∈ C ∞ S 1 , V 1 + (B) ′ for k = 0, . . . , n -1 to see that U n (ν, s, τ ) = U n (ν, s, τ ) -P n (ν, τ ) solves, for each τ ∈ S 1 , a Laplace problem with a right hand side that belongs to C ∞ S 1 , V 1 + (B) ′ , namely A 0 (∂ ν , ∂ s ) U n = Q n + 4 k=1 ν k A k (∂ ν , ∂ s , ∂ τ )U n-k in R 2 .
Clearly, for each τ ∈ S 1 , we can apply Proposition 4.3 to U n which yields the existence of

U n ∈ C ∞ S 1 , V 1 -(B) and c D,± n , c N,± n ∈ C ∞ (S 1 ) such that U n (ν, s, τ ) = c D,± n (τ ) + c N,± n (τ ) ν + U n (ν, s, τ ) for ± ν > π .
Each term of this decomposition has C ∞ dependency with respect to τ according to (46)-(48) and Proposition 4.7. There only remains to recall that U n = U n + P n , by construction. Moreover, according to the definition (17), we have s 0 1 = 0 and s 1 0 = 0. As a consequence, setting

P n (ν, s, τ ) = χ(ν) p + n (ν, τ ) + χ(-ν) p - n (ν, τ ) with p ± n (ν, τ ) = p ± n (ν, τ ) + c D,± n (τ ) + c N,± n (τ ) ν = n+1 q=0 ν q • s 0 q (∂ τ )c D,+ n-q (τ ) + s 1 q (∂ τ )c N,+ n+1-q (τ ) , we have U n -P n ∈ C ∞ S 1 , V 1 -(R 2
) . Finally since p ± n has no monomial of order less than 2, we have

ℓ ± D ( U n (•, τ ) ) = ℓ ± D (P n (•, τ )) = c D,± n (τ ) and similarly ℓ ± N ( U n (•, τ ) ) = ℓ ± N (P n (•, τ )) = c N,± n (τ ).
As a consequence, Formula (50) holds for p = n.

Matching conditions (Step III)

In Section 2 we derived formally equations ( 9) that should hold in order for the functions u n to be the terms of the expansion of u δ with respect to δ. However, the set of equations ( 9) is not a well posed problem so it cannot, for the moment, characterize the u n 's. Besides, Adding equations to (9) may lead to a well posed problem, which would yield a definition of the u n as solution of this modified set of equations. A similar remark can be formulated for the U n and the set of equations [START_REF] Schmidt | Asymptotic modelling of conductive thin sheets[END_REF].

The method of matched asymptotics provides such additional equations by means of the matching principle. This principle will yield exactly the proper amount of additional equations needed. Consequently it will lead to a definition of u n , and thus to a construction of the expansion of u δ . In spite of several valuable attempts, for example in [START_REF] Van Dyke | Perturbation methods in fluid mechanics[END_REF] , [START_REF] Joly | Matching of asymptotic expansions for wave propagation in media with thin slots. I. The asymptotic expansion[END_REF] and [START_REF] Joly | Matching of asymptotic expansions for waves propagation in media with thin slots. II. The error estimates[END_REF], there does not exist any generic theory of matched asymptotics at present. As a consequence, this principle can only be justified a posteriori by error estimates that will validate our construction of the expansion u δ .

Recall that we have set ρ = rr * and ν = ρ/δ. In the present situation, the matching principle consists in imposing that the expansion of the near field terms for ν → +∞ (resp. ν → -∞) should coincide with the expansion of the far field terms for ρ → 0 + (resp. ρ → 0 -). Let us write it formally for ν → +∞ and ρ → 0

+ +∞ n=0 δ n U n (ν, s, τ ) ≃ +∞ n=0 δ n n+1 k=0 ν k s 0 k (∂ τ )ℓ + D (U n-k ) + s 1 k (∂ τ )ℓ + N (U n+1-k ) , ≃ +∞ n=0 δ n n+1 k=0 ρ δ k s 0 k (∂ τ )ℓ + D (U n-k ) + s 1 k (∂ τ ), ℓ + N (U n+1-k ) ≃ +∞ n=-1 +∞ k=0 δ n ρ k s 0 k (∂ τ )ℓ + D (U n ) + s 1 k (∂ τ )ℓ + N (U n+1 ) . +∞ n=0 δ n u n (ρ, θ) ≃ +∞ n=-1 +∞ k=0 δ n ρ k s 0 k (∂ θ )u n | + Γ + s 1 k (∂ θ )∂ r u n | + Γ . (53)
Here we took the convention that u -1 = 0. Comparing both expansions in (53), we see that they coincide provided that ℓ

+ D (U n (•, τ = θ)) = u n | + Γ and ℓ + N (U n (•, τ = θ)) = ∂ r u n-1 | + Γ ∀n 0.
Similar conditions can be derived in the same manner for ν → -∞ and ρ → 0 -, so that finally matching conditions take the form

ℓ ± D U n (•, τ = θ) = u n | ± Γ (θ) and ℓ ± N U n (•, τ = θ) ) = ∂ r u n-1 | ± Γ (θ) ∀n 0 . ( 54
)
Indeed we have to take into account the equation τ = θ when writing the matching equations above. According to the definitions (36), conditions (54) can be rewritten in terms of jump and mean value operators

∀n 0 ,    [u n ] Γ = [ ℓ D (U n ) ], [∂ r u n-1 ] Γ = [ ℓ N (U n ) ], ℓ D (U n ) = u n Γ , ℓ N (U n ) = ∂ r u n-1 Γ . (55)
Implicit in the writing of identities (55) is the equation τ = θ. Recall also that we take the convention u -1 = 0. This is the final form of matching conditions that we will keep for the remaining of this analysis. Note that Equations (55) take the form of transmission conditions.

Recurrent system of equations (Step IV) : existence and uniqueness of the terms of the asymptotic expansion

Let us summarize the equations that we derived formally, gathering ( 9), ( 30) and (55). This yields, for each n ≥ 0, a system of equations coupling u ± n (r, θ), U n (ν, s, τ = θ) that writes at each step as follows:

Find

u n ∈ H 1 (Ω + ) × H 1 (Ω -), and U n ∈ C ∞ # S 1 , V 1 + (B) satisfying          ǫ -1 ∞ △u n + ω 2 µ ∞ u n = f δ 0 n in Ω ± , ∂ r u n + T u n = 0 on ∂Ω, [u n ] Γ = [ ℓ D (U n ) ] and [∂ r u n-1 ] Γ = [ ℓ N (U n ) ]    A 0 (∂ ν , ∂ s ) U n = - 4 j=1 ν j A j U n-j in V 1 -(B) ′ , ∀τ ∈ (0, 2π), ℓ D (U n ) = u n Γ and ℓ N (U n ) = ∂ r u n-1 Γ . ( 56
)
Note that, from now on, we will implicitly assume that τ = θ. To fully understand the system of equations above recall that V 1 + (B) has been defined by ( 26) and the operators A j have been defined by ( 24)-( 25). The main goal of this section is to prove the following result of well-posedness: Theorem 7.1. There exist a unique sequence of far field terms

(u n ) n∈N , u n ∈ H 1 (Ω + ) × H 1 (Ω -) and a unique sequence of near field terms (U n ) n∈N , U n ∈ C ∞ # S 1 , V 1 + (B)
, such that recurrent system (56) is satisfied.

Unfortunately it does not seem possible to prove directly Theorem 7.1: indeed, Formulation (56) is not appropriate for a direct recursion proof. More precisely, a careful inspection shows that Problem (56) is not well posed to prove that Problem (56) is well-posed (and completely defines u n+1 and U n+1 ) under the assumption that U k and u k exist for any k ≤ n. Roughly speaking, the difficulty arises from the fact that [ℓ N (U n+2 )] has to be known to define [∂ r u n+1 ]. This issue seems to be a characteristic of thin transmission layer problems as this was already pointed out at the beginning of Section 3 in [START_REF] Schmidt | Asymptotic modelling of conductive thin sheets[END_REF].

Reformulation of recurrent problems

Well posedness turns out to be much easier to prove with another formulation of Problem (56). Set

U D n = ℓ D (U n ) and U N n = U n -ℓ D (U n
) so that we have the following decomposition

U n = U D n + U N n with ℓ D (U N n ) = 0 and U D n = U D n (τ = θ) . ( 57
) Note that [ℓ D (U D n )] = [ℓ N (U D n )] = 0 so that [ℓ D (U n )] = [ℓ D (U N n )] and [ℓ N (U n )] = [ℓ N (U N n )]. Indeed U D n is assumed to be a function of θ alone such that U D n ∈ C ∞ # (S 1
). We also consider a decomposition of the far field terms in two parts:

u n = u D n + u N n where          ǫ -1 ∞ △u D n + ω 2 µ ∞ u D n = f • δ 0 n in Ω \ Γ, ∂ r u D n + T u D n = 0 on ∂Ω, [u D n ] Γ = [u n ] Γ , [∂ r u D n ] Γ = 0.          ǫ -1 ∞ △u N n + ω 2 µ ∞ u N n = 0 in Ω \ Γ, ∂ r u N n + T u N n = 0 on ∂Ω, [u N n ] Γ = 0 , [∂ r u N n ] Γ = [∂ r u n ] Γ . ( 58
)
Still we propose to construct (u n , U n ), n ≥ 0 in a recursive manner, but we do not use Problem (56) solving it at each step n assuming that (u 0 , U 0 ), . . . , (u n-1 , U n-1 ) are known in order to define (u n , U n ). Instead we consider the change of unknown given by ( 57)-( 58) and rewrite (56) as follows

Find u D n ∈ H 1 (Ω + ) × H 1 (Ω -), u N n-1 ∈ H 1 (Ω + ) × H 1 (Ω -) and U D n-1 ∈ C ∞ # (S 1 ), U N n ∈ C ∞ # S 1 , V 1 + (B) satisfying          ǫ -1 ∞ △u D n + ω 2 µ ∞ u D n = f δ 0 n in Ω ± , ∂ r u D n + T u D n = 0 on ∂Ω, [u D n ] Γ -[ ℓ D (U N n ) ] = 0 and [∂ r u D n ] Γ = 0,          ǫ -1 ∞ △u N n-1 + ω 2 µ ∞ u N n-1 = 0 in Ω ± , ∂ r u N n-1 + T u N n-1 = 0 on ∂Ω, [u N n-1 ] Γ = 0 and [∂ r u N n-1 ] Γ -[ ℓ N (U N n ) ] = 0,            A 0 (∂ ν , ∂ s ) U N n + ν A 1 • U D n-1 = -νA 1 U N n-1 - 4 j=2 ν j A j U n-j in V 1 -(B) ′ , ∀τ ∈ [0, 2π], ℓ D (U N n ) = 0 and ℓ N (U N n ) -∂ r u N n-1 Γ = ∂ r u D n-1 Γ U D n-1 -u N n-1 Γ = u D n-1 Γ . ( 59
)
Remark 7.2. The two sequences of problems ( 56) and ( 59) are equivalent: a sequence (u n , U n ) n≥0 satisfies (56) if and only if it satisfies (56).

Well posedness of recurrent problems

Our effective recursive procedure for defining u n , U n can be described as follows. At step n, assuming that u D k , u N k-1 , U N k and U D k-1 for k = 0 . . . n -1 are known data, we solve Problem (59). This procedure of definition can be considered valid if and only if Problem (59) is well posed.

Proposition 7.3. [Uniqueness] Assume that u D k , u N k-1 , U N k and U D k-1 for k = 0 . . . n -1 are known data.
Then Problem (59) admits at most one solution.

Proof. Let v D , v N ∈ H 1 (Ω + ) × H 1 (Ω -), V N ∈ C ∞ # ( S 1 , V 1 + (B) ) and V D ∈ C ∞ # (S 1
) be a solution to (59) with zero right hand sides. We have to show that v D = v N = 0, V N = 0 and V D = 0 necessarily. First of all, looking at the definition of A 1 given by ( 25), we see that -ν

-1 A 1 • V D = r * ( ∂ τ V D )∂ s ( ǫ -1 ) since V D is constant with respect to ν and s. Setting g = r * ( ∂ τ V D )∂ s ( ǫ -1 ), we have g(•, τ ), 1 + = 0. Besides V N satisfies div(ǫ -1 ∇V N ) = g in V 1 + (B) ′ which implies [ℓ N (V N
)] = 0 according to Proposition 4.7. As a consequence v N is solution to a Helmholtz problem with homogeneous transmission conditions so that v N = 0. This directly implies V D = 0 hence g = 0 according to the last equation of (59). Now we see that V N has to be the solution to a problem that takes the form of (42) with zeros data. This implies V N = 0 according to Proposition 4.5. Finally v D has to satisfy a Helmholtz problem with homogeneous transmission conditions across Γ. As a consequence v D = 0 as well. Proof. We focus only on the construction of u N n-1 , U N n and U D n-1 since, once U N n has been defined, the definition of u D n amounts to solving a classical transmission problem which does not raise any technical difficulty. Observe that the equation ruling U N n fits the framework of Proposition 5.2, and in particular U 0 , . . . , U n-1 take the form (50). Introduce

Q n (ν, τ ) = χ(ν) p + n (ν, τ )+χ(-ν) p - n (ν, τ ) with p ± n = n+1 k=2 ν k s 0 k { ℓ ± D (U n-k ) }+s 1 k { ℓ ± N (U n+1-k ) } .
Since p + n do not contain any monomial of degree less than 2, we have

ℓ ± D (Q n ) = 0 and ℓ ± N (Q n ) = 0.
Moreover, according to Proposition 5.2 and Equation ( 51)-( 52), the function Q n (ν, τ ) solves the recurrent equations ruling U N n up to a right hand side that exponentially decreases at infinity. More precisely

g := ν -2 A 0 (∂ ν , ∂ s )Q n + ν -1 A 1 U N n-1 + 4 j=2 ν j-2 A j U n-j ∈ C ∞ ( S 1 , V 1 + (B) ′ ).
According to Proposition 4.5, there exists a unique

R n ∈ C ∞ # ( S 1 , V 1 + (B)) satisfying ν -2 A 0 (∂ ν , ∂ s )R n = -g in V 1 + (B) ′ , and < ℓ D (R n ) >= 0, and < ℓ N (R n ) >= 0. Finally setting V n = R n + Q n , since ℓ ± D (V n ) = ℓ ± D (R n ) and ℓ ± N (V n ) = ℓ ± N (R n ) by construction, we have A 0 (∂ ν , ∂ s )V n = -νA 1 U N n-1 - 4 j=2 ν j A j U n-j in R 2 and ℓ D (V n ) = 0 , ℓ N (V n ) = 0 .
Note however that, a priori, we have

[ℓ D (V n )] = 0 and [ℓ N (V n )] = 0. Now we define u N n-1 ∈ H 1 (Ω + ) × H 1 (Ω -)
as the unique function satisfying the following system of equations

         ǫ -1 ∞ △u N n-1 + ω 2 µ ∞ u N n-1 = 0 in Ω ± , ∂ r u N n-1 + T u N n-1 = 0 on ∂Ω, [u N n-1 ] Γ = 0 and [∂ r u N n-1 ] Γ = [ℓ N (V n )].
Of course, we will have to verify a posteriori that such a u N n-1 satisfies the first system of equations of (59). Now that u N n-1 is constructed, we directly set

U D n-1 = u N n-1 Γ + u D n-1 Γ .
There only remains to construct U N n . For this purpose, we introduce a last intermediate function, denoted W n , that is defined as the unique function satisfying

W n ∈ C ∞ S 1 , V 1 + (B) ν -2 A 0 (∂ ν , ∂ s )W n = -ν -1 A 1 U D n-1 in V 1 -(B) ′ , ℓ D (W n ) = 0 and ℓ N (W n ) = ∂ r u N n-1 Γ + ∂ r u D n-1 Γ . ( 60
)
This problem is well posed. Indeed according to the definition of A 0 and A 1 given by ( 24) and ( 25)

we have ν -2 A 0 (∂ ν , ∂ s )W n = div(ǫ -1 ∇W n ) on the one hand, and ν -1 A 1 U D n-1 = r * ∂ θ (U D n-1 ) ∂ s (ǫ -1 ) (recall that U D n-1 depends only on θ by definition). Setting h = ν -1 A 1 U D n-1 , this implies in particular that h ∈ V 1 + (B) ′ and that < h, 1 > + = 0. As a consequence, applying Proposition 4.7 to W n yields [ℓ N (W n )] = 0. We finally set U N n = V n + W n . The function U N
n satisfies the third system of equations in (59). Besides we have

[∂ r u N n-1 ] = [ℓ N (V n )] = [ℓ N (U n )] which confirms that u N
n-1 properly satisfies also the equations of (59). This concludes the construction of a solution, hence the existence.

As usual, we took as a convention in the above definitions, that U n (ξ) = N n (ξ) = 0 if n ≤ -1. We may consider that these definitions are valid only if Problems (61) and (62) are well posed. Such a verification is achieved using Proposition 4.5 and Remark 4.6. Besides the linearity of Problems (61) and (62) ensures that the dependency of U n (ξ) and N n (ξ) is polynomial. A brief induction shows that the degree of U n (ξ) and N n (ξ) as polynomials in ξ cannot be greater than n.

It is clear from (61), by a recurrence argument, that U n (ξ) satisfies Property (P ∞ ) for any n ≥ 0. Similarly < ℓ D (U n (ξ)) > and < ℓ N (U n (ξ)) > are polynomials in ξ, and the conditions stated in (61) that hold for any ξ ∈ C, prove that < ℓ D (U k n ) >=< ℓ N (U k n ) >= 0 for any k ≥ 0 and any n ≥ 1. Note however that, by construction, we have < ℓ D (U 0 ) >= 1 and < ℓ N (U 0 ) >= 0. Similar remarks may be formulated, mutatis mutandis, about (N n (ξ)) n≥0 .

The functions (U n (ξ)) n≥0 and (N n (ξ)) n≥0 may be used to construct solutions to equations (30): we first naturally associate with U n (ξ) and N n (ξ), the differential operators in τ

U n (ν, s, ∂ τ ) = n k=0 U k n (ν, s)∂ k τ , N n (ν, s, ∂ τ ) = n k=0 N k n (ν, s)∂ k τ , as well as ℓ ± D (U n (∂ τ )), ℓ ± N (U n (∂ τ )), ℓ ± D (N n (∂ τ ) and ℓ N (D n (∂ τ )): ℓ ± D (U n (∂ τ )) = n k=0 ℓ ± D (U k n ) ∂ k τ , ℓ ± N (U n (∂ τ )) = n k=0 ℓ ± N (U k n ) ∂ k τ , ℓ ± D (U n (∂ τ )) = n k=0 ℓ ± D (U k n )] ∂ k τ , ℓ n N (U n (∂ τ )) = n k=0 ℓ N (U k n ), ∂ k τ .
Now, take any function c(τ ) ∈ C ∞ # (S 1 ), and consider the functions

C n (ν, s, τ ) = U n (ν, s, ∂ τ )c(τ ) = n k=0 U k n (ν, s) ∂ k τ c(τ ) n ≥ 0,
where U k n (ν, s) are the coefficients of U n (ν, s, ξ) as polynomial in ξ. It is easy to check that the functions C n (ν, s, τ ) satisfy recurrence equations of the form [START_REF] Schmidt | Asymptotic modelling of conductive thin sheets[END_REF]. Indeed we have

             4 j=0 ν j A j (∂ ν , ∂ s , ∂ τ ) C n-j (ν, s, τ ) = 4 j=0 ν j A j (∂ ν , ∂ s , ∂ τ ) U n-j (ν, s, ∂ τ ) c(τ ) = 0, ℓ N C n (•, τ ) = 0 ∀n ≥ 0, ℓ D C n (•, τ ) = 0 ∀n ≥ 1 and ℓ D C 0 (•, τ ) = c(τ ). (63)
Similar remarks can be formulated, replacing U n (ξ) by N n (ξ), and setting

C n (ν, s, τ ) = N n (ν, s, ∂ τ )c(τ ).
In this case we have

ℓ D C n (•, τ ) = 0 for all n ≥ 0, ℓ N C n (•, τ ) = 0 for n ≥ 1, and ℓ N C 0 (•, τ ) = c(τ )
. Now we state a result about the terms of the asymptotic expansion that we defined as the solutions to (59).

Theorem 8.3.

Let (u n , U n ) n≥0 be the sequence that were recursively constructed as the solution to (59). These functions also satisfy, on the one hand,

                                 ε -1 ∞ ∆u n + ω 2 µ ∞ u n = δ 0 n f in Ω + ∪ Ω -, ∂ r u n-1 + iωu n-1 = 0 on ∂Ω, [u n ] Γ = n j=1 j k=0 [ℓ D (U k j )] ∂ k θ u n-j Γ + n-1 j=0 j k=0 [ℓ D (N k j )] ∂ k θ ∂ r u n-1-j Γ (a), [∂ r u n ] Γ = n+1 j=2 j k=0 [ℓ N (U k j )] ∂ k θ u n+1-j Γ + n j=1 j k=0 [ℓ N (N k j )] ∂ k θ ∂ r u n-1-j Γ (b), (64)
and on the other hand,

U n (ν, s, τ ) = n j=0 U j (ν, s, ∂ τ ) u n-j Γ (τ ) + n-1 j=0 N j (ν, s, ∂ τ ) ∂ r u n-1-j Γ (τ ) . (65)
Proof. First, let us remark that for any n ∈ N, problem (64) is well posed, since the right hand side only depends on u n-1 , u n-2 , . . . u 0 . To prove the proposition, it is thus sufficient to show that the sequence (u n , U n ) n∈N defined by ( 64)-( 65) is solution to (56). The jump relations in (64) state that we have

[ u n ] Γ = [ ℓ D (U n ) ] = 0, and [ ∂ r u n ] Γ = [ ℓ N (U n+1 ) ].
Recall that ℓ D (N k j ) = 0 for all k, n ≥ 0, and that ℓ D (U k j ) = 0 if j = 0, and ℓ D (U 0 0 ) = 1 according to the construction of U n (ν, s, ξ) and N n (ν, s, ξ). As a consequence we have

ℓ D U n (•, τ ) = n j=0 j k=0 ℓ D (U k j ) ∂ k τ u n-j Γ + n-1 j=0 j k=0 ℓ D (N k j ) ∂ r u n-1-j Γ = u n Γ + 0.
Similarly, recall that ℓ N (U k j ) = 0 for all k, n ≥ 0, and that ℓ N (N k j ) = 0 if j = 0, and ℓ N (N 0 0 ) = 1 . This yields

ℓ N U n (•, τ ) = n j=0 j k=0 ℓ N (U k j ) ∂ k τ u n-j Γ + n-1 j=0 j k=0 ℓ N (N k j ) ∂ r u n-1-j Γ = 0 + ∂ r u n-1 Γ .
To sum up we have just verified that the matching conditions (55) are satisfied by functions (u n , U n ) that would be defined by ( 64)-(65). There only remains to verify that the sequence U 0 , U 1 , . . . defined by (65) satisfies [START_REF] Schmidt | Asymptotic modelling of conductive thin sheets[END_REF].

4 j=0 ν j A j (∂ ν , ∂ s , ∂ τ ) U n-j (ν, s, τ ) = 4 j=0 ν j A j (∂ ν , ∂ s , ∂ τ ) n-j k=0 U k (ν, s, ∂ τ ) u n-j-k Γ (τ ) + 4 j=0 ν j A j (∂ ν , ∂ s , ∂ τ ) n-1-j k=0 N k (ν, s, ∂ τ ) u n-j-k Γ (τ ). (66)
We show that both sums above are equal to 0. We prove it only for the first sum, as the second sum may be treated exactly in the same manner. Once again, let us take the convention that A j = 0 for j < 0 or j > 4, and u n = 0 for n < 0. We have

4 j=0 ν j A j (∂ ν , ∂ s , ∂ τ ) n-j k=0 U k (ν, s, ∂ τ ) u n-j-k Γ (τ ) (with p = j + k) = 4 j=0 ν j A j (∂ ν , ∂ s , ∂ τ ) n p=0 U p-j (ν, s, ∂ τ ) u n-p Γ (τ ), = n p=0 4 j=0 ν j A j (∂ ν , ∂ s , ∂ τ )U p-j (ν, s, ∂ τ ) u n-p Γ (τ ) = 0.
In the last line of the above calculus, we used (63). The same calculus can be achieved with the second sum in the right hand side of (66). This proves that we have

4 j=0 ν j A j (∂ ν , ∂ s , ∂ τ ) U n-j (ν, s, τ ) = 0.
To sum up, we have verified that the functions (u n , U n ) that would be defined by means of (64)-(65) are also solution to (59). We conclude the proof by using Proposition 7.3.

The previous proposition states that the definition of u n for a fixed n ≥ 0 only requires the computation of auxiliary coefficients

[ℓ N (N k j )], [ℓ N (U k j )], [ℓ D (U k j )] and [ℓ D (N k j )
] which represent the effect of the periodic ring on the far field terms. This is interesting because it suggests to consider the following approximate transmission conditions

[ũ δ ] Γ = n j=1 δ j j k=0 [ℓ D (U k j )] ∂ k θ ũδ Γ + n-1 j=0 δ j+1 j k=0 [ℓ D (N k j )] ∂ k θ ∂ r ũδ Γ , [∂ r ũδ ] Γ = n+1 j=2 δ j-1 j k=0 [ℓ N (U k j )] ∂ k θ ũδ Γ + n j=1 δ j j k=0 [ℓ N (N k j )] ∂ k θ ∂ r ũδ Γ .
where ũδ would be solution to the Helmholtz problem with such transmission conditions across the interface. It is not clear whether such a problem would be well posed though ; see [START_REF] Delourme | Approximate models for wave propagation across thin periodic interfaces[END_REF] for the investigation of approximate transmission conditions associated with our problem.

Definition of the global expansion

The procedure of asymptotic analysis that we have been applying so far led to the definition of functions u n , U n that provide the terms of the expansion of u δ either "close" to the thin ring of inhomogeneities, or "far" from it. A global expansion can be constructed by interpolating those two expansions by means of a cut-off functions as follows (see Figure 3):

u N,δ (r, θ) = 1 -χ r -r * η(δ) N n=0 δ n u n (r, θ) + χ r -r * η(δ) N n=0 δ n U n r -r * δ , r * θ δ , θ , with η(δ) = δ 1-1 N and χ ∈ C ∞ (R + ) , and χ(ρ) = 1 for ρ ≤ 1 , χ(ρ) = 0 for ρ ≥ 2.
(67) We have to formulate two comments about such a formula. First of all, observe that according to the definition of χ we have u N,δ = N n=0 δ n u n for |rr * | > 2η(δ), so Formula (67) takes the usual form of an expansion in δ except in a narrow region around the thin ring.

The second interesting comment concerns the near field terms in (67): we have to give a precise meaning to the truncated near field expansion

N n=0 δ n U n r-r * δ , r * θ δ , θ
. Indeed, regarding the regularity a priori of the functions U n that are only assumed to belong to

C ∞ S 1 , V 1 + (B)
, it is not clear whether the expression "U n ( (rr * )/δ, r * θ/δ, θ )" makes sense, since U n (ν, s, τ ) has been defined considering that s and τ are separated variables. The following proposition shows the a priori regularity of U n actually allows to consider such a function.

* δ : C ∞ # ( S 1 , H k loc,# (B) ) → H k loc (Ω\{0}) for k = 0, 1 characterized by φ * δ ϕ (r, θ) = ϕ φ δ (r, θ) = ϕ r -r * δ , r * θ δ , θ ∀ϕ ∈ D(B × S 1 ).
Moreover, set a > 0, K a := { x(r, θ) | |rr * | < a } and B δ a :=]a/δ, a/δ[×S 1 : there exists a constant C > 0 such that for any a < r * /2 we have

∀ϕ ∈ C ∞ ( S 1 , L 2 loc (B) ) , ∀δ ∈ (0, 1), φ * δ ϕ 2 L 2 (Ka) ≤ Cδ. (68)
Proof. Note that D # (B × S 1 ) is dense in H 1 # ( S 1 , L 2 loc (B) ) (the proof is very close to the proof of Lemma 11.3 so it is left to the reader). Relying on this density argument, it suffices to prove that for any 0 < a < r * , there exists a constant C a > 0 such that

φ * δ ϕ 2 L 2 (Ka) ≤ C a ϕ 2 H 1 ( S 1 ,L 2 (B δ a ) )
∀ϕ ∈ D # (B × S 1 ) .

For any ϕ ∈ D(B × S 1 ) we have

φ * δ ϕ (r, θ) = ϕ r -r * δ , r * θ δ , 0 + θ 0 ∂ τ ϕ r -r * δ , r * θ δ , τ dτ . ( 69
)
Consequently, introducing the usual change of variables ν = (rr * )/δ, s = r * θ/δ and using the periodicity of ϕ with respect to s, we get

φ * δ ϕ 2 L 2 (Ka)) ≤ 2(a + r * ) δ 2 r * N δ ϕ(•, 0) 2 L 2 (B δ a ) + 2(a + r * ) δ 2 r * N δ 2π 0 ∂ τ ϕ(•, τ ) 2 L 2 (B δ a ) dτ ,
where N δ = 2πr * /δ is the number of periodicity cells in the periodic ring (recall that, according to the general assumptions stated in Section 1, we have N δ ∈ N). There only remains to estimate ϕ(•, 0

) L 2 (B δ a )
in order to conclude. As τ → ϕ(•, τ

) L 2 (B δ a ) is in H 1 (S 1
), it suffices to apply the trace theorem for H 1 (S 1 ). The continuity estimates (68) is then obtained. Note in addition that this estimate holds as well if a depends on δ in such a way that a δ → 0 when δ → 0. Later on during the error analysis, we shall be particularly interested in the case where a δ = η(δ).

This proposition shows that (67) indeed makes sense and that the near field terms should be understood in the sense of "φ * δ U n ". Moreover, using density arguments, the usual formula for the derivatives of a composition of functions holds, namely ∀ϕ ∈ C ∞ # ( S 1 , H 1 loc (B) )

φ * δ 1 δ ∂ ν g := ∂ r φ * δ g and φ * δ r * δ ∂ s g + ∂ τ g := ∂ θ φ * δ g .
10 Error estimates (Step V)

Now we have defined the terms u n , U n solution to Problem (59), and constructed a global expansion through (67). Note however that, for the moment, the expansion (67) has only the status of a "good candidate" for being a sharp approximation to u δ . In this section, we are going to prove that, it is indeed a strong approximation by providing error estimates. To be more precise we are going to prove the following result Theorem 10.1. Let u δ be the solution of Problem ( 5), and u N,δ be defined by Formula (67). For any N ≥ 0, there exists C N > 0 independent of δ (but a priori depending on N ) such that

u δ -u N,δ H 1 (Ω) C N δ N -2 ∀δ ∈ (0, 1). ( 70
)
As a byproduct, such an estimate yields optimal error bounds for the far field expansion in any region that excludes a neighborhood of the thin ring:

Theorem 10.2. Let γ ∈ (0, γ * ) and O = { x(r, θ) ∈ Ω | |r -r * | > γ }.
Then, for any N ≥ 0, there exists C N > 0 independent of δ such that

u δ - N n=0 δ n u n H 1 (O) ≤ C N δ N +1
Proof. Theorem 10.2 results from a direct application of Theorem 10.1 and the triangular inequality:

u δ - N n=0 δ n u n H 1 (O) ≤ u δ -u N +3,δ H 1 (O) + δ N +1 2 k=0 δ k u N +k+1 H 1 (O) = O(δ N +1 ) .

Stability

As is standard in asymptotic analysis, the first step in the proof of Theorem 10.1 consists in establishing that the solution to Problem (5) continuously depends on the data, with a continuity constant that remains bounded as δ → 0.

Proposition 10.3. Problem ( 5) is well-posed. Moreover, there exist two positive constants C and δ 0 independent of δ such that,

u H 1 (Ω) ≤ C sup v∈H 1 (Ω)\{0} |a δ (u, v)| v H 1 (Ω) ∀u ∈ H 1 (Ω), ∀δ ∈ (0, δ 0 ). (71)
Proof. The well posedness of Problem ( 5) is a classical result, see for example [START_REF] Mclean | Strongly elliptic systems and boundary integral equations[END_REF]. In order to prove estimate (71), we proceed by contradiction which is a standard approach (see for instance theorem 2.1 of [START_REF] Haddar | Generalized impedance boundary conditions for scattering by strongly absorbing obstacles: the scalar case[END_REF], lemma 12.14 of [START_REF] Tordeux | Méthode asymptotiques pour la propagation des ondes dans des milieux comportant des fentes[END_REF] or theorem 1.2.1 of [START_REF] Claeys | Analyse asymptotique et numérique de la diffraction d'ondes par des fils minces[END_REF] for similar proofs). Assume that (71) does not hold.

Then there exists a sequence δ n which tends to 0 and a sequence u n ∈ H 1 (Ω), n ∈ N such that

u n H 1 (Ω) = 1 (a) and lim n→∞ sup v∈H 1 (Ω),v =0 |a δn (u n , v)| v H 1 (Ω) = 0 (b).
The sequence u n is bounded in H1(Ω) so, introducing a subsequence if necessary, we may assume that it converges toward a function u ∈ H 1 (Ω) weakly in H1(Ω) and strongly in L 2 (Ω). Moreover, for any v ∈ H 1 (Ω), lim n→+∞ (ǫ -1 δnǫ -1 ∞ )∇v L 2 (Ω) = 0 and lim n→+∞ (µ δnµ ∞ )v L 2 (Ω) = 0, since ǫ δn and µ δn actually converge to ǫ ∞ and µ ∞ almost everywhere in Ω. As a consequence

lim n→+∞ a δn (u n , v) = a 0 (u, v), (72)
where a 0 (u, v) = Ω ǫ -1 ∞ ∇u • ∇vω 2 µ ∞ uv dx + ∂Ω vT udσ. Combining (72) and the assumption (b), we have a 0 (u, v) = 0 for any v ∈ H 1 (Ω). It exactly means that u is an outgoing wave which solves the homogeneous Helmholtz equation in Ω. We thus get u = 0 (see for instance [START_REF] Colton | Inverse acoustic and electromagnetic scattering theory[END_REF] or [START_REF] Nédélec | Acoustic and electromagnetic equations[END_REF]) and consequently lim n→+∞ u n L 2 (Ω) = 0. Also, since u n satisfies the Helmholtz equation, it is easy to check that

Re{a δn (u n , u n )} = Ω ǫ δn |∇u n | 2 dx + Re ∂Ω u n T u n dσ -ω 2 Ω µ δn |u n | 2 dx.
But, it is well known (see for instance [START_REF] Abboud | Diffraction at a curved grating: TM and TE cases, homogenization[END_REF]) that

Re{ ∂Ω u n T u n dσ} ≥ 1 2re ∂Ω |u n | 2 . Then, ∇u n 2 L 2 (Ω) + 1 2r e ∂Ω |u n | 2 dσ ≤ C |a δn (u n , u n )| + u n 2 L 2 (Ω) ∀n ≥ 0 .
for a constant C > 0 independent of n, chosen larger than ω 2 sup R 2 |µ|/2 Letting n → +∞, we obtain lim n→∞ u n H 1 (Ω) = 0, which contradicts the assumption (a).

Consistency

The second step in the error analysis consists in applying the stability estimate to the difference u δu N,δ . As a consequence, according to Proposition 10.3, it suffices to provide estimates for expressions of the As a consequence ν → +∞ when x ∈ T + δ . According to (59), Proposition 5.2 can be applied to U 0 , . . . , U N . As a consequence there exists a family U 0 , . . . ,

U N belonging to C ∞ S 1 , V 1 -(B) such that for x(r, θ) ∈ T + δ we have N n=0 δ n φ * δ U n = N n=0 δ n φ * δ n+1 k=0 ν k s 0 k (∂ τ ) • ℓ + D (U n-k ) + s 1 k (∂ τ ) • ℓ + N (U n+1-k ) + U n , = N n=0 n+1 k=0 δ n r -r * δ k s 0 k (∂ θ ) • ℓ + D (U n-k ) + s 1 k (∂ θ ) • ℓ + N (U n+1-k ) + N n=0 δ n φ * δ U n . (77)
In the first line ℓ + D (U n-k ) should be understood as ℓ + D ( U n-k (•, τ ) ), whereas it should be understood as ℓ + D ( U n-k (•, θ) ) in the second line. Let us rearrange the first sum in the right-hand side above, using the change of indices p = nk, q = k and taking the matching conditions (54) into account:

N n=0 n+1 k=0 δ n r -r * δ k s 0 k (∂ θ ) • ℓ + D (U n-k ) + s 1 k (∂ θ ) • ℓ + N (U n+1-k ) = N p=-1 N -p q=0 δ p (r -r * ) q s 0 k (∂ θ ) • ℓ + D (U p ) + s 1 k (∂ θ ) • ℓ + N (U p+1 ) , = N p=0 N -p q=0 δ p (r -r * ) q s 0 k (∂ θ ) • u p | + Γ + s 1 k (∂ θ ) • ∂ r u p | + Γ . (78)
In the calculus above we used our convention that U n = 0 for n ≤ 0, and we also used that s 1 0 (∂ θ ) = 0 by definition. Now according to the definition of u n given by (59), Proposition 2.1 can be applied to each u n and yields [START_REF] Joly | Matching of asymptotic expansions for wave propagation in media with thin slots. I. The asymptotic expansion[END_REF], which can be plugged into the last sum in (78). This gives

N p=0 N -p q=0 δ p (r -r * ) q s 0 k (∂ θ ) • u p | + Γ + s 1 k (∂ θ ) • ∂ r u p | + Γ = N p=0 δ n u n - N n=0 δ n (r -r * ) N +1-n u n,N -n (r, θ), (79)
where u 0,N , u 1,N -1 , . . . , u N,0 ∈ H 1 (Ω ± ) do not depend on δ. The same calculus as for T + δ can be achieved for T - δ . Combining equations (77),(78), (79) both for T + δ and T - δ , and taking into account that η(δ) < |rr * | < 2 η(δ) inside T δ , we obtain

N n=0 δ n u n -φ * δ U n L 2 (T δ ) = N n=0 δ n (r -r * ) N +1-n u n,N -n - N n=0 δ n φ * δ U n L 2 (T δ ) , ≤ 2 N +1 N n=0 δ n η(δ) N +1-n u n,N -n L 2 (Ω) + N n=0 δ n φ * δ U n L 2 (T δ ) . (80)
To treat the last terms in the inequality above, recall that each U n is "evanescent at infinity". Observe that

T δ ⊂ K a δ = { x(r, θ) | |r -r * | < a δ } for a δ = 2η(δ)
, so we can apply Estimate (68): there exists a constant C > 0 independent of δ such that, ∀δ ∈ (0, 1), we have

φ * δ U n 2 L 2 (T δ ) e -1/η(δ) φ * δ e |ν| 2 U n 2 L 2 (T δ ) C δ e -1/η(δ) e |ν| 2 U n 2 H 1 ( S 1 ,L 2 (B δ a ) ) , C δ e -1/η(δ) 2π 0 U n (•, τ ) 2 V 0 -(B) + ∂ τ U n (•, τ ) 2 V 0 -(B) dτ. (81)
Besides δe -1/η(δ) = O(δ 2N ) since η(δ) = δ 1-1/N . Let us plug (81) into (80). Taking into account in addition that u n,N -n L 2 (Ω) is independent of δ, and that δ n η(δ

) N +1-n = O(δ N ) since η(δ) = δ 1-1/N ,
we obtain the existence of a constant C > 0 independent of δ such that

N p=0 δ n u n -φ * δ U n L 2 (T δ ) ≤ C δ N ∀δ ∈ (0, 1). (82)
Of course the constant C depends on N and on u n , U n , n = 0, . . . , N . It also depends on r * . Coming back to (76), we see that Inequality (82) leads to an error bound for the first term in the right-hand side of (74). A similar bound can be derived for the second term using the same approach. We summarize both estimates in the following proposition.

Proposition 10.4 (Estimate for the matching error).

For a given N ∈ N, let (u 0 , U 0 ), . . . (u n , U n ) be the unique solutions to Problems (57)-( 58)-(59). Let χ δ (x) = χ( (rr * )/η(δ) ) where χ and η(δ) are defined in (67). Then there exists a constant C, δ 0 > 0 independent of δ (depending on N though) such that

N n=0 δ n Ω ∇v • ∇χ δ u n -φ * δ U n dx - N n=0 δ n Ω v ∇χ δ • ∇ u n -φ * δ U n dx ≤ C v H 1 (Ω) δ N -1 ∀v ∈ H 1 (Ω) , ∀δ ∈ (0, δ 0 ).

Near field error

In order to prove Theorem 10.1, the only remaining step consists in estimating the last term of (74), i.e,

N n=0 δ n a δ φ * δ U n , χ δ v .
As we have already said, this term does not vanish since the near field truncated expansion does not satisfy exactly the original homogeneous Helmholtz equation (see 30) . To obtain an upper-bound, we would like to use the near field terms equations [START_REF] Schmidt | Asymptotic modelling of conductive thin sheets[END_REF] which are contained (in a variational way) in (59). However, it is not possible to make direct use of these equations since the quantity which appears in (74) is not U n but φ * δ U n . The next proposition will be used to bypass this difficulty and can be seen as an extension of Proposition 9.1.

Proposition 10.5.

The map φ

* δ : C ∞ ( S 1 , L 2 loc (B) ) → L 2 loc (Ω \ {0}
) defined in Proposition 9.1 can be extended in a unique manner as a continuous map φ

* δ : C ∞ ( S 1 , H 1 loc (B) ′ ) → H 1 loc (Ω \ {0}) ′ characterized by the following identities: ∀g ∈ C ∞ ( S 1 , L 2 loc (B) ) we have φ * δ 1 δ ∂ ν g := ∂ r φ * δ g and φ * δ r * δ ∂ s g + ∂ τ g := ∂ θ φ * δ g . (83)
As an extension, the map φ * δ of Proposition 10.5 coincides over C ∞ ( S 1 , L 2 loc (B) ) with the map defined by Proposition 9.1. Besides the relations (83) should be understood in the weak sense, namely (for g ∈

C ∞ ( S 1 , L 2 loc (B) )) φ * δ δ -1 ∂ ν g , v Ω = - Ω φ * δ g (x) ∂ r (vr(x))/r dx, φ * δ r * δ -1 ∂ s g + ∂ τ g , v Ω = - Ω φ * δ g (x) ∂ θ v(x) dx ∀v ∈ D(Ω \ {0}), (84)
where < , > Ω refers to the duality pairing between D(Ω \ {0}) and D ′ (Ω \ {0}). Identities (84) are a weak counterpart of Identities [START_REF] Kozlov | Elliptic boundary value problems in domains with point singularities[END_REF]. These properties of the map φ * δ can be used for studying the third term in (74). Observe that, in (74), we have supp{χ δ v} ⊂ Ω \ {0}, so we have

N n=0 δ n a δ φ * δ U n , χ δ v = N n=0 δ n Ω ǫ -1 δ ∂ r φ * δ U n ∂ r (χ δ v) dx + N n=0 δ n Ω ǫ -1 δ r -2 ∂ θ φ * δ U n ∂ θ (χ δ v) dx - N n=0 δ n Ω ω 2 µ δ φ * δ U n (χ δ v) dx. (85)
Let us study the first integral in the right hand side above. Note that ǫ

-1 δ = φ * δ (ǫ -1 ), and that U n ∈ C ∞ # ( S 1 , V 1 + (B)
) by construction. Using the identities (83), we obtain

Ω ǫ -1 δ ∂ r φ * δ U n ∂ r (χ δ v) dx = δ -2 φ * δ (r * + δν)∂ ν ǫ -1 (r * + δν) ∂ ν U n , χ δ v r 2 Ω .
A similar identity can be derived for the second and third integral in the right hand side of (85). Gathering all three expressions obtained in this manner leads to a calculus similar to [START_REF] Madureira | Asymptotics of the Poisson problem in domains with curved rough boundaries[END_REF]. Indeed, according to the definition of A j given by ( 24)-( 25), we obtain the following equality

N n=0 δ n a δ φ * δ U n , χ δ v = N n=0 δ n φ * δ (ν δ) -2 4 j=0 (δ ν) j A j U n , χ δ v r 2 Ω , = φ * δ (ν δ) -2 N n=0 4 j=0 δ n+j ν j A j U n , χ δ v r 2 Ω . (86)
Thanks to Equation (30) contained in (59), many terms in the right hand side of (86) cancel. Indeed, using the change of index p = n + j we obtain

N n=0 4 j=0 δ n+j ν j A j U n = N p=0 4 j=0 δ p ν j A j U p-j =0 according to (30) + N n=N -3 4 j=N +1-n δ n+j ν j A j U n .
Plugging this into Equation (86) yields

N n=0 δ n a δ φ * δ U n , χ δ v = N n=N -3 4 j=N +1-n δ n+j-2 φ * δ ν j-2 A j U n , χ δ v r 2 Ω .
Note that in the sum of the right-hand side above, we have j

+ n ≥ N + 1. As a consequence N n=0 δ n a δ φ * δ U n , χ δ v ≤ δ N -1 N n=N -3 4 j=N +1-n φ * δ ν j-2 A j U n , χ δ v r 2 Ω . (87)
To obtain a suitable estimate we only need to show that, for any j = 1 . . . 4 and any n = 0 . . . N there exists a constant C n,j > 0 independent of δ such that ∀v ∈ H 1 (Ω), ∀δ ∈ (0, 1) we have

φ * δ ν j-2 A j U n , χ δ v r 2 Ω ≤ C n,j δ -1 v H 1 (Ω) .
We show how to derive such a bound for j = 1 (and so n = N ). The case j = 2, 3 and 4 can be treated in a similar way, so there are left to the reader.

We start by writing the decomposition of U n in the form "polynomial + evanescent": according to Proposition 5.2, there exist two functions p ± (ν, τ ) that admit polynomial dependency with respect to ν with coefficients that depend on τ and belong to C ∞ # (S 1 ), and there exists a function

U ∈ C ∞ ( S 1 , V 1 -(B) ) such that U n (ν, s, τ ) = P (ν, τ ) + U(ν, s, τ ) with P (ν, τ ) = χ(ν)p + (ν, τ ) + χ(-ν)p -(ν, τ ).
In this decomposition χ : R → [0, 1] is a C ∞ (R) function such that χ = χ(ν), χ(ν) = 1 for ν > 2π and χ(ν) = 0 for ν < π. Plugging this decomposition into the term that we need to bound from above, we get

φ * δ ν -1 A 1 U n , χ δ v r 2 Ω = φ * δ ν -1 A 1 P , χ δ v r 2 Ω + φ * δ ν -1 A 1 U , χ δ v r 2 Ω . (88)
Also, we will denote by C δ the support of the truncation function χ δ :

C δ := supp(χ δ ) = x ∈ R 2 , r * -2δ 1-1/N < r < r * + 2δ 1-1/N .
Using the small size of C δ we can prove the two following estimates (see for instance lemma 3.10 of [START_REF] Tordeux | Méthode asymptotiques pour la propagation des ondes dans des milieux comportant des fentes[END_REF]) : there exist two positive constants C 1 and C 2 such that

u L 1 (C δ ) ≤ C 1 δ (N -1)/2N u L 2 (Cη) , u L 2 (C δ ) ≤ C 2 δ (N -1)/2N u H 1 (Ω) . (89)
Estimate of the first term We will successively derive an upper bound for each of the terms in the right hand side of (88). We focus first on the term containing P . Observe that (ν -1 A 1 P )(ν, τ ) is a smooth function, both with respect to ν and τ . Looking at the precise statement of Proposition 5.2 we see that the degree of p ± (ν, τ ), as a polynomial in ν, does not exceed N . In addition P admits a smooth dependency with respect to τ . As a consequence, we see that there exists a constant C > 0 independent of δ such that

| (ν -1 A 1 • P )(ν, τ ) | ≤ C (1 + |ν| N -1 ) ∀ν ∈ R , ∀τ ∈ [0, 2π] .
To establish the previous estimate we have used the fact that A 1 = A 1 when |ν| > π and the homogeneity of A 1 . Now observe that |rr * |/δ ≤ 2 η(δ) δ ≤ 2δ -1 N over supp(χ δ v) according to the definition of χ δ given by (67). Since the change of variables induced by φ * δ enforces ν = (rr * )/δ and θ = τ , there exist constants C 1 , C 2 > 0 independent of δ such that ∀v ∈ H 1 (Ω) and ∀δ ∈ (0, 1) we have

φ * δ ν -1 A 1 P , χ δ v r 2 Ω ≤ C 1 (1 + δ -N -1 N ) v L 1 (C δ ) ≤ C 2 v H 1 (Ω) . (90)
Estimate of the second term Now we focus on the second term in (88) that contains φ * δ ν -1 A 1 U . Let us rewrite this function in a different form. Using the definition of A 1 given by [START_REF] Nédélec | Acoustic and electromagnetic equations[END_REF] and the relations (83), we have

φ * δ ν -1 A 1 U = 2r ∂ r φ * δ ǫ -1 ∂ ν U + ∂ θ φ * δ ǫ -1 ∂ τ U + φ * δ ǫ -1 ∂ ν U + ǫ -1 ∂ s ∂ τ U - δ r * ǫ -1 ∂ 2 τ U . Since U ∈ C ∞ # ( S 1 , V 1 -(B) ) we have g 1 := ǫ -1 ∂ ν U ∈ C ∞ # ( S 1 , V 0 -(B) ) and g 2 := ǫ -1 ∂ τ U ∈ C ∞ # ( S 1 , V 0 -(B)
) on the one hand, and

g 3 := ǫ -1 ∂ ν U + ǫ -1 ∂ s ∂ τ U, g 4 := -δ r -1 * ǫ -1 ∂ 2 τ U ∈ C ∞ ( S 1 , V 0 -(B)
) on the other hand. Applying the definition of the weak derivative, the Cauchy-Schwarz inequality and the estimate (89), we obtain the existence of constants C 1 , C 2 > 0 independent of δ such that

φ * δ ν -1 A 1 U , χ δ v r 2 Ω ≤ C 1 χ δ v r 2 H 1 (C δ ) 4 i=1 φ * δ g i L 2 (C δ ) , ≤ C 2 δ -(N +1)/2N v H 1 (Ω) 4 i=1 φ * δ g i L 2 (C δ ). (91)
In order to obtain a proper error estimate, there only remain to bound from above each φ * δ g i L 2 (C δ ) . To do so, we use Estimate (68). Since C δ has the same form as the set K a of this inequality with a = 2δ N -1/N , this yields the existence of a constant C > 0 independent of δ such that

φ * δ g i 2 L 2 (C δ ) ≤ C √ δ g i 2 L 2 (S 1 ,V 0 -(B))
∀δ ∈ (0, 1), i = 1, . . . , 4.

Now it suffices to plug (92) into (91). This provides a bound for φ * δ ν -1 A 1 U , χ δ v r 2 Ω , which finally leads to the following estimate that we wanted to establish: there exists C > 0 independent of δ such that

φ * δ ν -1 A 1 U n , χ δ v r 2 Ω ≤ C n,j δ -1/2N v H 1 (Ω)
∀v ∈ H 1 (Ω), ∀δ ∈ (0, 1) .

Summary Recall that φ * δ ν j-2 A j U n , χ δ v Ω can be estimated using the same approach as the one we have just presented. As a consequence, coming back to Equation (87), this proves the following proposition .

Proposition 10.6 (Estimate for the near field error). For a given N ∈ N, let (u 0 , U 0 ), . . . , (u N , U N ) be the unique solutions to Problems (57)-( 58)-(59). Let χ δ (x) = χ( (rr * )/η(δ) ) where χ and η(δ) are defined in (67). Then there exists a constant C, δ 0 > 0 independent of δ (depending on N though) such that

N n=0 δ n a δ φ * δ U n , χ δ v ≤ C v H 1 (Ω) δ N -2 ∀v ∈ H 1 (Ω)
, ∀δ ∈ (0, δ 0 ).

Conclusion of the error analysis

Now we have at hand all the ingredients required for proving a global error estimate. Indeed this error estimate can be obtained by means of Proposition 10.3, as well as Equation (74) where we take into account Proposition 10.4 and 10.6. This proves Theorem 10.1.

Conclusion

Our study comes to an end. To conclude, let us remind the main results we obtained. First of all we described a complete asymptotic expansion of the solution of a scattering problem by a thin periodic layer. The approach is based on the well-known method of matched asymptotic expansion. We pay particular attention to write the matching conditions in an optimal way and to avoid the double recursion issue (that naturally appears in the transmission problems) by introducing new fields called Dirichlet and Neumann fields. These two main ingredients as well as a detailed analysis of the near field periodic problems allows us to prove quite easily the existence and uniqueness of the terms of the expansion. Besides, we also derived explicit formulas for the far and near field terms which can be very useful to construct approximate models up to any order. Finally, we proved the convergence of our expansion to the exact one with an optimal convergence rate. It would now be interesting to extend the previous approach to the more general case of a smooth thin and periodic interface. Whereas the calculations are more involved in this configuration, it seems that the mains issues of the analysis remain the same. Another interesting extension would be to generalize the method to the 3D Maxwell case, the analysis becoming much more complex due to the classical Maxwell Equations compactness issues.

Lemma 11.2.

The operator T # defined by (T # ϕ)(ν, s) = k∈Z ϕ(ν, s + 2kπ) maps D(R 2 ) in D # (B) and is onto.

Proof. We have to prove that for any ϕ ∈ D # (B) there exists v ∈ D(R 2 ) such that T # v = ϕ. Assume first that ϕ ∈ D((0, 2π) × R), so that ϕ(ν, s) = 0 if 0 ≤ s < ε or 2πε < s ≤ 2π for some ε > 0.

We extend ϕ by 0 outside (0, 2π) × R, so that ϕ ∈ D(R 2 ). Denote (τ α ϕ)(ν, s) = ϕ(ν, α + s). Note that each α k (ν) belongs to H 1 (R) and satisfies α k (ν) = 0 whenever |ν| > ν 0 for any k. Besides the series above converges in H 1 (B). Indeed set ϕ N (ν, s) = |k|≤N α k (ν)e iks . We have lim N →∞ ϕϕ N H 1 (B) = 0.

Finally take an arbitrary ε > 0. There exists N ≥ 1 such that ϕϕ N H 1 (B) ≤ ε. Besides, since

D(R) = { v ∈ C ∞ (R) | supp(v) is compact } is dense in H 1 (R),

  Assume that α = β = 0 and g = 0. Let us show that [ ℓ N (U ) ] = 0. Consider ψ R (ν) = ψ(ν/R) where ψ is a C ∞ function such that ψ(ν, s) = ψ(-ν, s) = ψ(ν), and ψ(ν) = 0 for |ν| > 2π, and ψ(ν) = 1 for |ν| < π. Note that since ℓ D (U ) = ℓ N (U ) = 0, according to Proposition 4.3, there exists a function

Proposition 7 . 4 (

 74 Existence). Assume that u D k , u N k-1 , U N k and U D k-1 for k = 0 . . . n-1 are known data. Then Problem (59) admits at least one solution.

Proposition 9. 1 .

 1 Consider the immersion map φ δ : B → B × S 1 such that φ δ (r, θ) = (ν, s, τ ) where ν = (rr * )/δ, s = r * θ/δ mod 2π and τ = θ. It induces a unique continuous map φ

  We have supp(τ 2kπ ϕ) ∩ supp(ϕ) = ∅ except if k = 0. As a consequence, defining v(ν, s) by v(ν, s) = ϕ(ν, s) if 0 ≤ s ≤ 2π, and v(ν, s) = 0 if s < 0 or s > 2π, we have v ∈ D(R 2 ) and T # v = ϕ. Now take any ϕ ∈ D # (B). Using a cut-off function and the special case that we have just examined, we may consider in addition that ϕ(ν, s) = 0 if π/3 < s < 5π/3. Introduce a C ∞ cut-off function χ : R → R + such that χ(s) = 1 if 0 < s < π/3 and χ(s) = 0 if 2π/3 < s, and define v(ν, s) in the following manner:v(ν, s) = 0 if s > 2π/3 or s < -2π/3, and v(ν, s) = χ(s)ϕ(ν, s) if 0 ≤ s ≤ 2π/3, and v(ν, s) = χ(-s)ϕ(ν, 2π + s) if -2π/3 ≤ s ≤ 0. With such a definition for v, we have T # v = ϕ once again. Lemma 11.3. The space D # (B) is dense in W(B) for the norm W , and it is dense in V 1 + (B)for the norm(V 1 + ) . Proof. Consider any ϕ ∈ V 1 + (B) chosen arbitrarily. Take a C ∞ cut-off function χ : R → R + such that χ(ν) = 1 if |ν| < 1 and χ(ν) = 0 if |ν| > 2. Set χ k (ν) = χ(ν/k). The dominated convergence theorem shows that lim k→∞ ϕχ k ϕ V 1 + = 0.On the other hand, if ϕ ∈ W(B), and if χ k refers to the same cutoff function as above, dominated convergence theorem shows once again that lim k→∞ ϕ -χ k ϕ W = 0. Let H 1 #,c (B) = { ϕ ∈ H 1 # (B) | ∃ν 0 > 0 such that ϕ(ν) = 0 for |ν| > ν 0 }. Observe that V 1 + ,W and H 1 (B) all induce the same topology over H 1 #,c (B). As a consequence, to prove the lemma, it is sufficient to show that any element of H 1 #,c (B) can be approximated by functions chosen in D # (B). From now on, let us consider a function ϕ ∈ H 1 #,c (B). Since ϕ has compact support, it suffices to prove that there exists a sequence v n ∈ D # (B), n ∈ N such that lim n→∞ ϕv n H 1 (B) = 0. Let us decompose ϕ in a Fourier series with respect to the variable s. ϕ(ν, s) = +∞ k=-∞ α k (ν)e iks with α k (ν) = 1 2π 2π 0 ϕ(ν, s)e -iks ds, k ∈ Z .

-β k 2 H 1 ( 1 N ε 2 ≤ 4 ε 2 .

 21122 there existsβ k ∈ D(R) such that α k R) < ε 2 /(4 N 3 ). Clearly β k (ν)e iks ∈ D # (B) for each k = -N, . . . , +N . Besides, an easy calculus yields ( α k (ν)β k (ν) )e iks 2 H 1 (B) ≤ ε 2 /N for any k = -N, . . . , +N . Setting v = |k|≤N β k (ν)e iks , we obtain ϕ Nv 2 H 1 (B) = N k=-N ( α k (ν)β k (ν) )e iks 2 H 1 (B) ≤ 2N + Clearly v ∈ D # (B).As a consequence, to sum up, for an arbitrary ε > 0, we have constructed a functionv ∈ D # (B) such that ϕv H 1 (B) ≤ ϕϕ N H 1 (B) + ϕ Nv H 1 (B) ≤ 3 ε.This concludes the proof of the density of D # (B) in V 1 + (B).

2 V 2 H 1 (B) ≤ C 3 +∞

 2213 We conclude the proof by using estimates provided by (38) and (40): there exist constants C 3 , C 4 > 0 independent of ℓ + D (U + ), ℓ + N (U + ), g + such that e ν Domain of the near field

Figure 2 :Figure 3 :

 23 Figure 2: Far and near fields domains

Notation: mean and jump operators For the sake of brevity, from now on, we will write s 0 j {v} and s 1 j {v} instead of s 0 j (∂ θ )v and s 1 j (∂ θ )v. In the sequel we will need notations for Dirichlet and Neumann jump and mean values on Γ for a function v ∈ H 1 (Ω ± ) such that ∆v ∈ L 2 (Ω ± ) so we set[v] Γ = v| + Γ -v| - Γ , [∂ r v] Γ = ∂ r v| + Γ -∂ r v| - Γ

V(ν, s) is exactly the inverse Fourier transform of the function(ξ, k) → ( k 2 -(-1 2 + iξ) 2 ) -1 ĝk (-1 2 +iξ). Note that there exists a constant C 2 > 0 such that (1+ξ 2 +k 2 ) ≤ C 2 |k 2 -(-1 2 +iξ) 2 |, ∀k ∈ Z, ∀ξ ∈ R.As a consequence, applying Parseval Theorem once more and taking into account (38),

Approximate transmission conditions

In the previous part, we proved that the near and far field terms are well defined. We may provide a refined analysis by trying to uncouple the equations ruling the far field on the one hand, and the near field on the other hand. The main motivation for this will be the derivation of approximate transmission conditions for Problem [START_REF] Abboud | Diffraction at a curved grating: TM and TE cases, homogenization[END_REF] up to any order.

We would like to reformulate equations (56) as a system involving only far field terms, where transmission conditions would not contain any direct dependency on the near field terms. To do this, we need to make the dependency of the U n 's with respect to the variable τ as explicit as possible. Actually, a careful examination of equations (56) should convince the reader that such a dependency is inherited from the terms u k Γ and ∂ r u k Γ , k ≤ n (note the linearity dependence of the near field equations [START_REF] Maz'ya | Asymptotic theory of elliptic boundary value problems in singularly perturbed domains[END_REF][START_REF] Mclean | Strongly elliptic systems and boundary integral equations[END_REF][START_REF] Nédélec | Acoustic and electromagnetic equations[END_REF] in term of partial derivatives in the variable τ ).

Keeping in mind the previous comments, let us introduce intermediate notation. For ξ ∈ C, define Âj (ξ) as the operator A j where ∂ τ has been replaced by ξ. In other words, we have Â0 (ξ) = A 0 , Â3 (ξ) = A 3 and Â4 (ξ) = A 4 , whereas

Now we construct two families of functions ( U n (ξ) ) n≥0 and ( N n (ξ) ) n≥0 . Each U n (ξ) and N n (ξ) is assumed to be polynomial with respect to ξ, with coefficients in V + 1 (B) namely

We define both families ( U n (ξ) ) n≥0 and ( N n (ξ) ) n≥0 by a recursive procedure formulated by means of the operators Âj (ξ), j = 0 . . . 4 as follows Definition 8.1.

-U 0 (ν, s, ξ) = 1,

Definition 8.2.

-N 0 (ν, s, ξ) = N (ν, s),

form a δ (u δu N,δ , v) in order to obtain a bound for u δu N,δ H 1 (Ω) . Let us introduce the function χ δ (x) = χ( (rr * )/η(δ) ) and observe that

(73) Now take an arbitrary v ∈ H 1 (Ω), and plug Formula (67) into a δ (u δu N,δ , v). Taking into account (73) we obtain

The first two terms in the right hand side above compensate each other. Indeed note that, since χ δ v vanishes in the neighborhood of Γ, we have

Besides, according to the definition of u n given by (59), we have

The first two terms in the right-hand side above will be called matching error: it measures the mismatch between the far and near field expansions in the overlapping areas. The last term will be called near field error. It measures how much the near field truncated expansion fails to satisfy the original Helmholtz expansion. In the two following technical parts we shall provide a suitable upper bound for each of these three terms.

Matching error

In this paragraph we provide a bound for the first two terms in the right-hand side of (74). We present the derivation of such a bound only for the first term, since the derivation of a bound for the second is nearly the same. Let us introduce the set

Observe that ∇χ δ L ∞ (R) ≤ δ -1 ∇χ L ∞ (R) . We want to obtain an upper bound for

, and ψ(ν) = 0 for |ν| > 2π, and ψ(ν) = 1 for |ν| < π. We have

As a consequence, by Cauchy-Schwarz inequality, there exists C > 0 depending only on

Proof. Take a cut-off function ψ(ν, s) = ψ(-ν, s) = ψ(ν), and ψ(ν) = 0 for |ν| > 2π, and ψ(ν) = 1 for |ν| < π. Define ψ α (ν) = ψ(ν/α). The dominated convergence theorem shows that ψ α converges toward

As a consequence we have in particular lim α→∞ < g,

Plugging this decomposition into the first integral in the right hand side of (93), and using Lemma 11.1, yields

Using similar arguments, we also prove that lim α→∞ B - α ∇U • ∇ψ α dνds = 2π ℓ - N (U ). Gathering these two asymptotic results proves the first identity.

In the same manner we can prove the second identity. First, by the dominated convergence theorem it is easily seen that lim

Moreover, according to Proposition 4.3 we can see that

Reminding that ℓ D (N ) = 0, we obtain the desired formula:

where

Then there is continuous dependency of U with respect to α, β, g: there exists a constant C > 0 independent of α, β, g such that

′ and, using (46), it is easy to establish that there exists C > 0 independent of g such that

Besides, by linearity, we have g = g 0 + g + + g -. Denote by U ± , U 0 ∈ V 1 -(B) the remainder terms associated respectively with U + , U -and U 0 following the statement of the proposition we want to establish. Clearly we have U = U + + U -+ U 0 . All these preliminary remarks show that, to prove the proposition, it suffices to prove that there exists C > 0 independent of α, β, g such that

The derivation of an estimate for U -would follow the same lines as the proof of ii) so we skip it. Note also that ℓ + D (U + ) = ℓ + D (U ) and ℓ + N (U + ) = ℓ + N (U ) so that according to Proposition 4.7 there exists C > 0 independent of α, β, g such that |ℓ +

Proof of ii) Let φ(ν) = χ + (ν/2), so that φ(ν) = 0 for ν < 2π and φ(ν) = 1 for ν > 4π. Since U + (ν, s) = 0 for ν ≤ π, according to (46) and (47), there exists a constant C > 0 independent of ℓ + D (U + ), ℓ + N (U + ), g + such that the following inequalities hold

As a consequence, it is sufficient to provide an upper bound for φ U + V 1 -in order to prove ii). Now let us come back to the proof of Proposition 4. However U + (ν, s) = V + (ν, s) a priori for ν < 2π. Elementary estimates then show that there exist constants C 1 , C 2 > 0 independent of ℓ + D (U + ), ℓ + N (U + ), g + such that

List of Figures