On the homogenization of thin perforated walls of finite length

Abstract

The present work deals with the resolution of the Poisson equation in a bounded domain made of a thin and periodic layer of finite length placed into a homogeneous medium. We provide and justify a high order asymptotic expansion which takes into account the boundary layer effect occurring in the vicinity of the periodic layer as well as the corner singularities appearing in the neighborhood of the extremities of the layer. Our approach combines the method of matched asymptotic expansions and the method of periodic surface homogenization.

Keywords

asymptotic analysis periodic surface homogenization singular asymptotic expansions

Introduction

The present work is dedicated to the construction of a high order asymptotic expansion of the solution to a Poisson problem posed in a polygonal domain which excludes a set of similar small obstacles equi-spaced along the line between two re-entrant corners. The distance between two consecutive obstacles, which appear to be holes in the domain, and the diameter of the obstacles are of the same order of magnitude δ, which is supposed to be small compared to the dimensions of the domain. The presence of this thin periodic layer of holes is responsible for the appearance of two different kinds of singular behaviors. First, a highly oscillatory boundary layer appears in the vicinity of the periodic layer. Strongly localized, it decays exponentially fast as the distance to the periodic layer increases. Additionally, since the thin periodic layer has a finite length and ends in corners of the boundary, corners singularities come up in the neighborhood of its extremities. The objective of this work is to provide a sophisticated asymptotic expansion that takes into account these two types of singular behaviors.

The boundary layer effect occurring in the vicinity of the periodic layer is well-known. It can be described using a two-scale asymptotic expansion (inspired by the periodic homogenization theory) that superposes slowly varying macroscopic terms and periodic correctors that have a two-scale behavior: these functions are the combination of highly oscillatory and decaying functions (periodic of period δ with respect to the tangential direction of the periodic interface and exponentially decaying with respect to $d / δ$ , d denoting the distance to the periodic interface) multiplied by slowly varying functions. This boundary layer effect has been widely investigated since the work of Sanchez-Palencia [36,37], Achdou [2,3] and Artola–Cessenat [5,6]. In particular, high order asymptotics have been derived in [4,9,12,27] for the Laplace equation and in [34,35] for the Helmholtz equation.

On the other hand, corner singularities appearing when dealing with singularly perturbed boundaries have also been widely investigated. Among the numerous examples of such singularly perturbed problems, we can mention the cases of small inclusions (see [29, Chapter 2] for the case of one inclusion and [8] for the case of several inclusions), perturbed corners [15], propagation of waves in thin slots [23,24], the diffraction by wires [13], or the mathematical investigation of patched antennas [7]. Again, this effect can be depicted using two-scale asymptotic expansion methods that are the method of multiscale expansion (sometimes called compound method) and the method of matched asymptotic expansions (see [22,29,38]). Following these methods, the solution of the perturbed problem may be seen as the superposition of slowly varying macroscopic terms that do not see directly the perturbation and microscopic terms that take into account the local perturbation.

Recently, Vial and co-authors [10,39] investigated a Poisson problem in a polygonal domain surrounded by a thin and homogeneous layer, while Nazarov [31] studied the resolution of a general elliptic problem in a polygonal domain with periodically changing boundary. In their studies they have combined the two different kinds of asymptotic expansions mentioned above in order to deal with both corner singularities and the boundary layer effect. Based on the multiscale method, the authors of [10,39] constructed and justified a complete asymptotic expansion for the case of the homogeneous layer. For the periodic boundary in [31] the first terms of the asymptotic expansion have been constructed and error estimates have been carried out. This asymptotic expansion relies on a sophisticated analysis of solution behavior at infinity for the Poisson problem in an infinite cone with oscillating boundary with Dirichlet boundary conditions by Nazarov [30], where he published an analysis for Neumann boundary conditions in [32]. In the present paper, we are going to extend the work for the homogeneous layer and the periodic boundary by constructing explicitely and rigorously justifying asymptotic expansion for the above mentioned periodic layer transmission problem to any order (with Neumann boundary conditions on the perforations of the layer).

The remainder of the paper is organized as follows. In Section 1 we are going to define the problem and show the main ingredients of the asymptotic expansion following the method of matched asymptotic expansions. The asymptotic expansion for the solution away from the corners, the far field, is given in Section 2, where the problems for the terms of the far field expansion and their behaviour when approaching the corners is analysed in Section 3. The terms of this expansion take into account the boundary layer effect due to the thin layer with small perforations and satisfy transmission conditions. The asymptotic expansion of the solution close to the re-entrant corners in stretched coordinates, the near field, is derived in Section 4. Then, the matching of the far field and near field expansions and the iterative construction of the terms of the asymptotic expansions are conducted in Section 5. Finally, in Section 6 the asymptotic expansion is justified with an error analysis. Not to stress the main line of the justification too much we have released some details in the appendices.

1. Description of the problem and main results

1.1. Description of the problem

In this section we are going to define the domain of interest $Ω^{δ} \in R^{2}$ , its limit when $δ \to 0$ and the problem considered. With the coordinates $x = (x_{1}, x_{2})$ of $R^{2}$ let $Ω_{B}$ and $Ω_{T}$ be the two adjacent rectangular domains defined by $\begin{matrix} Ω_{B} = (- L, L) \times (- H_{B}, 0), Ω_{T} = (- L^{'}, L^{'}) \times (0, H_{T}), \end{matrix}$ where $L^{'} > L$ , $H_{B}$ and $H_{T}$ are positive numbers. We denote by Γ the common interface of $Ω_{B}$ and $Ω_{T}$ , i.e., $\begin{matrix} \overline{Γ} = \overline{\partial Ω_{B}} \cap \overline{\partial Ω_{T}} and Γ = (- L, L) \times {0} \end{matrix}$ and we consider the (non-convex) polygonal domain (see Fig. 1(a)) $\begin{matrix} Ω = Ω_{B} \cup Ω_{T} \cup Γ, \end{matrix}$ which has two reentrant corners at $x_{O}^{\pm} = (\pm L, 0)$ with both an angle of $\frac{3 π}{2}$ .

Fig. 1.

Illustration of the polygonal domain Ω and the domain of interest $Ω^{δ}$ . (a) The domain $Ω = Ω_{T} \cap Ω_{B} \cap Γ$ . (b) The domain of interest $Ω^{δ} = Ω ∖ \overline{Ω_{hole}^{δ}}$ .

Besides, let ${\hat{Ω}}_{hole} \in R^{2}$ be a smooth canonical bounded open set (not necessarily connected) strictly included in the domain $(0, 1) \times (- 1, 1)$ . Then, let $N^{*} : = N ∖ {0}$ denote the set of positive integers and let δ be a positive real number (that is supposed to be small) such that $\begin{matrix} (1.1) & \frac{2 L}{δ} = q \in N^{*} . \end{matrix}$

Now, let $Ω_{hole}^{δ}$ be a thin (periodic) layer consisting of q equi-spaced similar obstacles which can be defined by scaling and shifting the canonical obstacle ${\hat{Ω}}_{hole}$ (see Fig. 1(b)): $\begin{matrix} (1.2) & Ω_{hole}^{δ} = ⋃_{ℓ = 1}^{q} {- L e_{1} + δ {{\hat{Ω}}_{hole} + (ℓ - 1) e_{1}}} . \end{matrix}$ Here, $e_{1}$ and $e_{2}$ denote the unit vectors of $R^{2}$ and δ is assumed to be smaller than $H_{T}$ and $H_{B}$ such that $Ω_{hole}^{δ}$ does not touch the top or bottom boundaries of Ω. Finally, we define our domain of interest as $\begin{array}{rcl} Ω^{δ} = (Ω_{B} \cup Ω_{T} \cup Γ) ∖ \overline{Ω_{hole}^{δ}} . \end{array}$ Its boundary $\partial Ω^{δ}$ consists of the boundary of the set of holes $Γ^{δ} = \partial Ω_{hole}^{δ}$ and $Γ_{D} = \partial Ω^{δ} ∖ Γ^{δ} = \partial Ω$ , the boundary of Ω. Here and in what follows, we denote by n the outward unit normal vector of $\partial Ω^{δ}$ . Note, that in the limit $δ \to 0$ the repetition of holes degenerates to the interface Γ, the domain $Ω^{δ}$ to the domain $Ω_{T} \cup Ω_{B}$ and its boundary $\partial Ω^{δ}$ to $\partial Ω \cup Γ$ .

The domain $Ω^{δ}$ being defined, we can introduce the problem to be considered in this article: Seek $u^{δ}$ solution to $\begin{matrix} (1.3) & \{\begin{matrix} - Δ u^{δ} = f & in Ω^{δ}, \\ \nabla u^{δ} \cdot n = 0 & on Γ^{δ}, \\ u^{δ} = 0 & on Γ_{D}, \end{matrix} \end{matrix}$ where $f \in L^{2} (Ω^{δ})$ . It is natural to search for $u^{δ} \in H_{Γ_{D}}^{1} (Ω^{δ})$ where $\begin{array}{rcl} (1.4) & H_{Γ_{D}}^{1} (Ω^{δ}) = {u \in H^{1} (Ω^{δ}) such that u = 0 on Γ_{D}} . \end{array}$ The well-posedness of problem (1.3) in $H_{Γ_{D}}^{1} (Ω^{δ})$ directly follows from Lax–Milgram theorem:

Proposition 1.1 (Existence, uniqueness and stability).

Let $f \in L^{2} (Ω^{δ})$ . Then, for any $δ > 0$ there exists a unique solution $u^{δ}$ of problem (1.3) in $H_{Γ_{D}}^{1} (Ω^{δ})$ , and with a constant C (independent of δ) it holds $\begin{matrix} (1.5) & {‖ u^{δ} ‖}_{H^{1} (Ω^{δ})} ⩽ C ‖ f ‖_{L^{2} (Ω^{δ})} . \end{matrix}$

The objective of this paper is to describe the behavior of $u^{δ}$ as δ tends to 0. For the sake of simplicity, we shall assume that f has a compact support in a subset of $Ω_{T}$ with distance $δ_{0} > 0$ to Γ. Our work relies on a construction of an asymptotic expansion of $u^{δ}$ as δ tends to 0.

Remark 1.2.
The construction is for simplicity for the specific geometrical setting, where Γ is a straight line ending in two corners of the polygonal boundary $\partial Ω$ , where the angles between Γ and $\partial Ω$ are both ends at angles $\frac{π}{2}$ or π, respectively. Nevertheless, the study may be extended to a polygon Ω of different angles.
Remark 1.3.
It is worth noting that the choice of the boundary condition imposed on the small obstacles $Γ^{δ}$ constituting the periodic layer, here homogeneous Neumann boundary conditions, has a strong impact on the asymptotic expansion. A homogeneous Dirichlet condition would yield to a completely different asymptotic expansion (see for instance Appendix A in [16], or [11]).
Remark 1.4.
The smoothness of ${\hat{Ω}}_{hole}$ is not required for the existence of $u^{δ} \in H^{1} (Ω^{δ})$ . The well-posedness result remains valid if ${\hat{Ω}}_{hole}$ is a Lipschitz domain. However, we use this assumption in the forthcoming analysis (in particular in Proposition 4.5 and Section 6).
1.2. Ansatz of the asymptotic expansion

As mentioned in the introduction, due to the periodic layer, it seems not possible to write a simple asymptotic expansion valid in the whole domain. We have to take into account both the boundary layer effect in the vicinity of Γ and the additional corner singularities appearing in the neighborhood of the two reentrant corners. To do so, we shall distinguish a far field area located ‘far’ from the reentrant corners $x_{O}^{\pm}$ and two near field zones located in the vicinity of them (see Fig. 2).

Fig. 2.

Schematic representation of the overlapping subdomains for the asymptotic expansion. The far field area (hatched) away from the corners $x_{O}^{\pm}$ is overlapping the near field area (light grey) in the matching zone (dark grey).

1.2.1. Far field expansion

Far from the two corners $x_{O}^{\pm}$ (hatched area in Fig. 2), we shall see that $u^{δ}$ is the superposition of a macroscopic part (that is not oscillatory) and a boundary layer localized in the neighborhood of the thin periodic layer. More precisely, we choose the following ansatz: $\begin{matrix} (1.6) & u^{δ} (x) = \sum_{(n, q) \in N^{2}} δ^{\frac{2}{3} n + q} u_{F F, n, q}^{δ} (x), \end{matrix}$ where $x = (x_{1}, x_{2})$ , and for $(n, q) \in N^{2}$ $\begin{matrix} (1.7) & u_{F F, n, q}^{δ} (x) = \{\begin{matrix} u_{n, q}^{δ} (x) & if | x_{1} | > L, \\ χ (\frac{x_{2}}{δ}) u_{n, q}^{δ} (x) + Π_{n, q}^{δ} (x_{1}, \frac{x}{δ}) & if | x_{1} | < L . \end{matrix} \end{matrix}$ Here $χ : R \mapsto (0, 1)$ denotes a smooth cut-off function satisfying $\begin{matrix} (1.8) & χ (t) = \{\begin{matrix} 1 & if | t | > 2, \\ 0 & if | t | < 1 . \end{matrix} \end{matrix}$ The macroscopic terms $u_{n, q}^{δ}$ are defined in the limit domain $Ω_{T} \cup Ω_{B}$ . A priori, they are not continuous across Γ. As for the boundary layer correctors $Π_{n, q}^{δ} (x_{1}, X_{1}, X_{2})$ (also sometimes denoted periodic correctors), and as usual in the periodic homogenization theory, there are 1-periodic with respect to the scaled tangential variable $X_{1}$ . Consequently, they are defined in $(- L, L) \times B$ , where $B$ is the infinite periodicity cell (see Fig. 3(a)): $\begin{matrix} (1.9) & B = {(0, 1) \times R} ∖ \overline{{\hat{Ω}}_{hole}} . \end{matrix}$ Moreover, the periodic correctors are super-algebraically decaying as the scaled variable $X_{2}$ tends to $\pm \infty$ (they decay faster than any power of $X_{2}$ ), more precisely, for any $(k, ℓ) \in N^{2}$ , $\begin{matrix} (1.10) & lim_{| X_{2} | \to + \infty} X_{2}^{k} \partial_{X_{2}}^{ℓ} Π_{n, q}^{δ} = 0 . \end{matrix}$ The macroscopic terms as well as the boundary layer corrector terms might have a polynomial dependence with respect to $ln δ$ : there is $N (n, q) \in N$ such that $\begin{matrix} u_{n, q}^{δ} = \sum_{s = 0}^{N (n, q)} {(ln δ)}^{s} u_{n, q, s} and Π_{n, q}^{δ} = \sum_{s = 0}^{N (n, q)} {(ln δ)}^{s} Π_{n, q, s}, \end{matrix}$ where $u_{n, q, s}$ and $Π_{n, q, s}$ do not depend on δ.

Remark 1.5.
Here and in what follows, although it might be surprising at first glance, we call far field expansion the expansion (1.6), i.e., the superposition of the macroscopic terms and the boundary layer correctors. Besides, it should also be noted that, for any $k \in N$ , we consider $δ^{\frac{2 (n + 3 k)}{3} + q}$ and $δ^{\frac{2 n}{3} + (q + 2 k)}$ as different scales as they would be different powers of δ. In fact, we shall see that n and q play a different role in the asymptotic procedure. Finally, following Remark 1.2, the consideration of the more general case of two angles of measure α, would yield to an expansion of the form (1.6) substituting $δ^{\frac{2 n}{3} + q}$ for $δ^{\frac{π n}{α} + q}$ (see [10]).

Fig. 3.
The periodicity cell $B$ and the normalized domains ${\hat{Ω}}^{\pm}$ . (a) The periodicity cell $B$ . (b) The domain ${\hat{Ω}}^{-}$ . (c) The domain ${\hat{Ω}}^{+}$ .
1.2.2. Near field expansions

In the vicinity of the two corners $x_{O}^{\pm}$ (light grey areas in Fig. 2), the solution varies rapidly in all directions. Therefore, we shall see that $\begin{matrix} (1.11) & u^{δ} (x) = \sum_{(n, q) \in N^{2}} δ^{\frac{2 n}{3} + q} U_{n, q, \pm}^{δ} (\frac{x - x_{O}^{\pm}}{δ}), \end{matrix}$ for some near field terms $U_{n, q, \pm}^{δ}$ defined in the fixed unbounded domains $\begin{matrix} (1.12) & {\hat{Ω}}^{-} = K^{-} ∖ ⋃_{ℓ \in N} {\overline{{\hat{Ω}}_{hole}} + ℓ e_{1}}, {\hat{Ω}}^{+} = K^{+} ∖ ⋃_{ℓ \in N^{*}} {\overline{{\hat{Ω}}_{hole}} - ℓ e_{1}} \end{matrix}$ shown in Fig. 3(b) and (c), where $K^{\pm}$ are the unbounded angular domains $\begin{matrix} K^{\pm} = {X = R^{\pm} (cos θ^{\pm}, sin θ^{\pm}), R^{\pm} \in R_{+}^{*}, θ^{\pm} \in I^{\pm}} \in R^{2} \end{matrix}$ of angular sectors $I^{+} = (0, \frac{3 π}{2})$ and $I^{-} = (- \frac{π}{2}, π)$ . If the domain ${\hat{Ω}}_{hole}$ is symmetric with respect to the axis $X_{1} = 1 / 2$ , then the domain ${\hat{Ω}}^{-}$ is nothing but the domain ${\hat{Ω}}^{+}$ mirrored with respect to the axis $X_{1} = 0$ . However, this is not the case in general. Similarly to the far field terms the near field terms might also have a polynomial dependence with respect to $ln δ$ , i.e., for all $(n, q) \in N^{2}$ , there is $N (n, q) \in N$ such that $\begin{matrix} U_{n, q, \pm}^{δ} = \sum_{s = 0}^{N (n, q)} {(ln δ)}^{s} U_{n, q, \pm, s}, \end{matrix}$ where the functions $U_{n, q, \pm, s}$ do not depend on δ.

1.2.3. Matching principle

To link the two different expansions, we assume that they are both valid in two intermediate areas $Ω_{M}^{δ, \pm}$ (dark shaded in Fig. 2) of the following form: $\begin{matrix} Ω_{M}^{δ, \pm} = {x = (x_{1}, x_{2}) \in Ω, \sqrt{δ} ⩽ d (x, x_{O}^{\pm}) ⩽ 2 \sqrt{δ}}, \end{matrix}$ where d denotes the usual Euclidean distance. The precise definition of the matching areas is not important. The reader might just keep in mind that they correspond to a neighborhood of the corners $x_{O}^{\pm}$ of the reentrant corners for the far field terms (macroscopic and boundary layer correctors) and to $R^{\pm}$ going to $+ \infty$ for the near field terms (expressed in the scaled variables).

1.3. Far and near field equations

The ‘ansatz’ being assumed, the next objective is to construct the terms $u_{n, q}^{δ}$ , $Π_{n, q}^{δ}$ and $U_{n, q, \pm}^{δ}$ of the far and near field expansions (the asymptotic expansions are justified later, by proving error estimates). This is by far the longest part of the work (Sections 1 to 5). The usual starting point of this construction consists in the formal derivation of the near field and far field problems, that is to say problems satisfied by the near and far field terms.

1.3.1. Far field equations: Macroscopic and boundary layer correctors equations

Inserting the far field expansion into the initial problem (1.3) and separating the different powers of δ (the complete procedure, based on the separation of the scales, is explained in [18] (Appendix)) gives a collection of equations for the macroscopic terms and the boundary layer terms:

Macroscopic equations. The macroscopic terms $u_{n, q}^{δ}$ satisfy $\begin{matrix} (1.13) & - Δ u_{n, q}^{δ} = \{\begin{matrix} f & if n = q = 0, \\ 0 & otherwise, \end{matrix} in Ω_{T} \cup Ω_{B}, \end{matrix}$ together with homogeneous Dirichlet boundary conditions on $Γ_{D}$ $\begin{matrix} (1.14) & u_{n, q}^{δ} = 0 on Γ_{D} . \end{matrix}$

Boundary layer corrector equations. The boundary layer correctors satisfy $\begin{matrix} (1.15) & \{\begin{matrix} - Δ_{X} Π_{n, q}^{δ} (x_{1}, X) = G_{n, q}^{δ} & in B, \\ \partial_{n} Π_{n, q}^{δ} = 0 & on \partial {\hat{Ω}}_{hole}, \end{matrix} \end{matrix}$ where, for any $p \in N$ , $\begin{array}{rcl} G_{n, q}^{δ} & = & \sum_{p = 0}^{q} \frac{\partial^{p} u_{n, q - p}^{δ}}{{\partial x_{2}}^{p}} (x_{1}, 0^{+}) [Δ, χ_{+}] (\frac{{(X_{2})}^{p}}{p!}) + \sum_{p = 0}^{q} \frac{\partial^{p} u_{n, q - p}^{δ}}{{\partial x_{2}}^{p}} (x_{1}, 0^{-}) [Δ, χ_{-}] (\frac{{(X_{2})}^{p}}{p!}) \\ (1.16) & + \partial_{x_{1}}^{2} Π_{n, q - 2}^{δ} (x_{1}, X) + 2 \partial_{x_{1}} \partial_{X_{1}} Π_{n, q - 1}^{δ} (x_{1}, X) . \end{array}$ Here, for any sufficiently smooth function v, $[Δ, v]$ denotes the commutator between Δ and v, that is to say $\begin{matrix} [Δ, v] u = Δ (v u) - v Δ u = \nabla v \cdot \nabla u + u Δ v . \end{matrix}$ Moreover, the smooth truncation functions $χ_{\pm}$ are defined by $\begin{matrix} (1.17) & χ_{\pm} (X_{2}) = \{\begin{matrix} χ (X_{2}), & if \pm X_{2} > 0, \\ 0 & otherwise, \end{matrix} (χ_{\pm} (X_{2}) = 1_{(\pm X_{2} > 0)} χ (X_{2})), \end{matrix}$ and, for $p \in N$ , $\begin{matrix} \frac{\partial^{p} u_{n, q - p}^{δ}}{{\partial x_{2}}^{p}} (x_{1}, 0^{\pm}) = lim_{h \to 0^{\pm}} \frac{\partial^{p} u_{n, q - p}^{δ}}{{\partial x_{2}}^{p}} (x_{1}, h) . \end{matrix}$ Note that equations (1.13)–(1.14), posed in the domains $Ω_{T}$ and $Ω_{B}$ , do not define entirely the macroscopic terms. Indeed, we first have to prescribe transmission conditions across the interface Γ (for instance the jump of their trace and the jump of their normal trace across Γ). This information will appear to be a consequence of the boundary layer equations (Section 2). Then, we also have to prescribe the behavior of the macroscopic terms in the vicinity of the two corner points $x_{O}^{\pm}$ . This information will be provided by the matching conditions (Section 5).

1.3.2. Near field equations

The near field equations are obtained in a much more direct way. Inserting the near field ansatz (1.11) into the Laplace equation (1.3) and separating formally the different powers of δ, it is easily seen that the near field term $U_{n, q}^{δ}$ satisfy $\begin{matrix} (1.18) & \{\begin{matrix} - Δ_{X} U_{n, q}^{δ} = 0 & in {\hat{Ω}}^{\pm}, \\ U_{n, q}^{δ} = 0 & on \partial K^{\pm}, \\ \partial_{n} U_{n, q}^{δ} = 0 & on \partial {\hat{Ω}}_{hole}^{\pm} = \partial {\hat{Ω}}^{\pm} ∖ \partial K^{\pm} . \end{matrix} \end{matrix}$

1.4. Outlook of the paper and main result

The remainder of the paper is organized as follows. In Section 2, we investigate the boundary layer problems. We derive transmission condition for the macroscopic term $u_{n, q}^{δ}$ up to any order (Proposition 2.4). We also obtain an explicit formula for the periodic correctors $Π_{n, q}^{δ}$ (see (2.30)). In particular, we shall see that the periodic corrector $Π_{n, q}^{δ}$ is completely determined providing that the macroscopic terms $u_{n, p}^{δ}$ are defined for $p ⩽ q$ .

Then, Section 3 is dedicated to the analysis of the far field problems (consisting of the far field equations (1.13) together with the transmission conditions (2.29a), (2.29b)). We first introduce two families of so-called macroscopic singularities $s_{m, q}^{\pm}$ (Proposition 3.5 and Proposition 3.8). These functions are particular solutions of the homogeneous Poisson equations (with prescribed jump conditions across the interface Γ) that blow up in the vicinity of the reentrant corners. These two families are then used to derive a quasi-explicit formula for the far field terms (Proposition 3.11). This quasi-explicit formula defines the macroscopic terms $u_{n, q}^{δ} \in H_{loc}^{1} (Ω_{T} \cup Ω_{B})$ up to the prescription of $2 n$ constants called $ℓ_{- m}^{\pm} (u_{n, q}^{δ})$ , $1 ⩽ m ⩽ n$ .

Section 4 deals with the resolution of the near field problems (1.18). As done for the macroscopic terms, we define two families of near field singularities $S_{m}^{\pm}$ , that are particular solutions of the homogeneous Poisson problem posed in ${\hat{Ω}}^{\pm}$ that blow up at infinity (Proposition 4.8). Based on these near field singularities, we then derive a quasi-explicit formula for the near field terms (4.21). Here again, this quasi-explicit formula defines the near field terms $U_{n, q, \pm}^{δ}$ up to the prescription of n constants called $L_{m} (U_{n, q, \pm}^{δ})$ , $1 ⩽ m ⩽ n$ .

Section 5 is dedicated to the derivation of the matching conditions and the definition of the terms of the asymptotic expansions. Based on an asymptotic representation of the far field terms close to the reentrant corners and of the near field terms at infinity, we obtain a collection of matching conditions (5.11), (5.13), (5.14) and (5.15) that permit to determine the constants $L_{m} (U_{n, q, \pm}^{δ})$ for the near fields and the constants $ℓ_{- m}^{\pm} (u_{n, q}^{δ})$ for the macroscopic fields. As a consequence, all the terms of the asymptotic expansion are then constructed (through an iterative procedure).

Finally, Section 6 deals with the justification of the asymptotic. We prove the following macroscopic error estimate:

Theorem 1.6.
Let $N_{0} > 0$ such that $3 N_{0}$ is an integer and let $D_{N_{0}}$ denote the set of couples $(n, q) \in N^{2}$ such that $\frac{2}{3} n + q ⩽ N_{0}$ . Furthermore, for a given number $α > 0$ , let $\begin{matrix} Ω_{α} = Ω^{δ} ∖ (- L - α, L + α) \times (- α, α) . \end{matrix}$ Then, there exist a constant $δ_{0} > 0$ , a constant $C = C (α, δ_{0}) > 0$ , and a constant $k = k (N_{0}) ⩾ 0$ such that for any $δ \in (0, δ_{0})$ $\begin{matrix} (1.19) & {‖ u^{δ} - \sum_{(n, q) \in D_{N_{0}}} δ^{\frac{2}{3} n + q} u_{n, q}^{δ} ‖}_{H^{1} (Ω_{α})} ⩽ C δ^{N_{0} + \frac{1}{3}} {(ln δ)}^{k} . \end{matrix}$

2. Analysis of the boundary layer problems: Transmission conditions

This section is dedicated to the analysis of the boundary layers problems (1.18). It permits us to derive (necessary) transmission conditions for the macroscopic fields $u_{n, q}^{δ}$ across Γ (Proposition 2.4). For a given $n \in N$ , we shall propose a recursive procedure to write the jump of the trace and of the normal trace of $u_{n, q}^{δ}$ as linear combinations of the mean values of the macroscopic fields of lower order $u_{n, k}^{δ}$ ( $k < q$ ) and their tangential derivatives. This procedure is done by induction on the index q and is completely independent of the index n and of the superscript δ (of $u_{n, q}^{δ}$ ). That is why we shall omit the index n and the superscript δ in this section.

For any sufficiently smooth function u defined in $Ω_{T} \cup Ω_{B}$ , we denote by ${[u (x_{1}, 0)]}_{Γ}$ and ${⟨ u (x_{1}, 0) ⟩}_{Γ}$ its jump and mean values across Γ: for $| x_{1} | < L$ , $\begin{matrix} (2.1) & \begin{matrix} {[u (x_{1}, 0)]}_{Γ} = lim_{h \to 0^{+}} (u (x_{1}, h) - u (x_{1}, - h)), \\ {⟨ u (x_{1}, 0) ⟩}_{Γ} = \frac{1}{2} lim_{h \to 0^{+}} (u (x_{1}, h) + u (x_{1}, - h)) . \end{matrix} \end{matrix}$

Let ${(f_{q})}_{q \in N}$ be a sequence of functions belonging to $L^{2} (Ω_{T}) \cap L^{2} (Ω_{B})$ that are compactly supported in $Ω_{T}$ . We consider the following sequence of coupled problems (obtained by rewriting (1.13), (1.14), (1.15) and omitting the index n): $\begin{matrix} (2.2) & \{\begin{matrix} - Δ u_{q} = f_{q} & in Ω_{T} \cup Ω_{B}, \\ u_{q} = 0 & on Γ_{D}, \end{matrix} \{\begin{matrix} - Δ_{X} Π_{q} (x_{1}, X) = G_{q} (x_{1}, X) & in B, \\ \partial_{n} Π_{q} = 0 & on \partial {\hat{Ω}}_{hole}, \end{matrix} \end{matrix}$ where $\begin{array}{rcl} G_{q} (x_{1}, X) & = & \sum_{p = 0}^{q} (2 ⟨ g_{p} (X) ⟩ {⟨ \partial_{x_{2}}^{p} u_{q - p} (x_{1}, 0) ⟩}_{Γ} + \frac{1}{2} [g_{p} (X)] {[\partial_{x_{2}}^{p} u_{q - p} (x_{1}, 0)]}_{Γ}) \\ (2.3) & + 2 \partial_{x_{1}} \partial_{X_{1}} Π_{q - 1} + \partial_{x_{1}}^{2} Π_{q - 2} . \end{array}$ Here, we use $\begin{matrix} (2.4) & ⟨ g_{p} (X) ⟩ : = \frac{1}{2} [Δ, χ_{+} + χ_{-}] (\frac{X_{2}^{p}}{p!}), [g_{p} (X)] : = [Δ, χ_{+} - χ_{-}] (\frac{X_{2}^{p}}{p!}), p \in N, \end{matrix}$ and later also $⟨ g_{p}^{\pm} (X) ⟩ : = [Δ, χ^{\pm}] (\frac{X_{2}^{p}}{p!})$ will be needed. As previously, we impose $Π_{q}$ to be 1-periodic with respect to $X_{1}$ and to be super-algebraically decaying as $| X_{2} |$ tends to $+ \infty$ : for any $(k, ℓ) \in N^{2}$ , $\begin{matrix} (2.5) & lim_{| X_{2} | \to + \infty} X_{2}^{k} \partial_{X_{2}}^{ℓ} Π_{q} = 0 . \end{matrix}$ Note that the right-hand side $G_{q}$ in (2.3) corresponds to the right-hand side $G_{n, q}^{δ}$ of problem (1.15) (for a given n). The problems for $Π_{q}$ , $q \in N$ are coupled to the others by the source terms. In difference, the problems for $u_{q}$ , $q \in N$ are not complete and their coupling to other problems will be exposed in following.

The present section is organized as follows: in Section 2.1, we give a standard existence and uniqueness result (Proposition 2.2), which shows that under two compatibility conditions the boundary layer problems for $Π_{q}$ in (2.2) have a unique decaying solution. In Section 2.2, we use Proposition 2.2 to derive transmission conditions for the first two terms $u_{0}$ and $u_{1}$ (see (2.12), (2.18), (2.19), (2.27)), and we obtain an explicit tensorial representation for the associated boundary layer correctors (cf. (2.13)–(2.20)). Finally the approach is extended in Section 2.3 to obtain transmission conditions up to any order for the macroscopic fields $u_{q}$ .

Remark 2.1.
The asymptotic construction described in this section is entirely similar to the construction of a multi-scale expansion for an infinite periodic thin layer (without corner singularity). A complete description of this case may be found in [1,3,35,37] and references therein.

2.1. Preliminary step: Existence result for the boundary layer problem

In this subsection, we give a standard result of existence for the boundary layer corrector problems for $Π_{q}$ in (2.2). It will be subsequently used to construct exponentially decaying boundary layer correctors. Let us introduce the two weighted Sobolev spaces $\begin{matrix} (2.6) & V^{\pm} (B) = {Π \in H_{loc}^{1} (B), Π (0, X_{2}) = Π (1, X_{2}), and (Π w_{e}^{\pm}) \in H^{1} (B)}, \end{matrix}$ where the weighting functions $w_{e}^{\pm} (X_{1}, X_{2}) = χ (X_{2}) exp (\pm \frac{| X_{2} |}{2})$ . The functions of $V^{-} (B)$ correspond to the periodic (w.r.t. $X_{1}$ ) functions of $H_{loc}^{1} (B)$ that grow slower than $exp (\frac{| X_{2} |}{2})$ as $X_{2}$ tends to $\pm \infty$ . By contrast, the functions of $V^{+} (B)$ correspond to the periodic functions of $H_{loc}^{1} (B)$ decaying faster than $exp (- \frac{| X_{2} |}{2})$ as $X_{2}$ tends to $\pm \infty$ . As a consequence, they are super-algebraically decaying, means they satisfy (2.5) for $ℓ ⩽ 1$ . Note also that $V^{+} (B) \subset V^{-} (B)$ .

Based on this functional framework, we consider the following problem: for given $g \in {(V^{-} (B))}^{'}$ find $Π \in V^{-} (B)$ such that $\begin{matrix} (2.7) & \{\begin{matrix} - Δ_{X} Π = g & in B, \\ \partial_{n} Π = 0 & on \partial {\hat{Ω}}_{hole}, \\ \partial_{X_{1}} Π (0, X_{2}) = \partial_{X_{1}} Π (1, X_{2}), & X_{2} \in R . \end{matrix} \end{matrix}$

Proposition 2.2.

Problem (2.7) has a finite dimensional kernel of dimension 2, spanned by the functions $N = 1_{B}$ and $D$ , where $D$ is the unique harmonic function of $V^{-} (B)$ such that there exists $D_{\infty} \in R$ $\begin{matrix} \tilde{D} (X_{1}, X_{2}) = D (X_{1}, X_{2}) - χ_{+} (X_{2}) (X_{2} + D_{\infty}) - χ_{-} (X_{2}) (X_{2} - D_{\infty}) \end{matrix}$ belongs to $V^{+} (B)$ . The constant $D_{\infty}$ only depends on the geometry of the periodicity cell $B$ .

If f is orthogonal to $D$ and $N$ in the $L^{2} (B)$ sense, meaning that $\begin{array}{l} (C_{D}) & \int_{B} g (X) D (X) d X = 0, \\ (C_{N}) & \int_{B} g (X) N (X) d X = 0 \end{array}$ then, there exists a unique solution $Π \in V^{+} (B)$ .

Conversely, if problems (2.7) admits a solution $Π \in V^{+} (B)$ , then it satisfies the compatibility conditions ( C D ) , ( C N ) .

For the proof of the previous proposition we refer the reader to [32, Proposition 2.2] and [14, Section 5]. General results on the elliptic problems in infinite cylinder can be found in [26] (Chapter 5). Note that all these results remain the same with a different exponential growth or decay constant in the definition of $w_{e}^{\pm}$ unless it does not exceed an upper bound which is determined by the least exponentially decaying or growing functions in the kernel of $- Δ$ in $B$ .

Based on the previous proposition, we shall construct $Π_{q} (x_{1}, X)$ in $C ((- L, L), V^{+} (B))$ . Transmission conditions for the macroscopic terms ${[u_{q}]}_{Γ}$ and ${[\partial_{x_{2}} u_{q}]}_{Γ}$ will directly follow from the compatibility conditions ( C D ), ( C N ) applied to problem (2.2) for $Π_{q}$ , $q \in N$ . It will guarantee that the boundary layer correctors $Π_{q}$ are exponentially decaying. Let us give a couple of useful relations, which are easy to obtain by direct calculations (noting that $g_{0}^{\pm} = [Δ, χ_{\pm}] 1$ ), and will be extensively used in the next subsections: Lemma 2.3.
The following relations hold: $\begin{matrix} (2.8) & \begin{matrix} \int_{B} ⟨ g_{0} (X) ⟩ D (X) d X = 0, \int_{B} [g_{0} (X)] D (X) d X = - 2, \\ \int_{B} ⟨ g_{0} (X) ⟩ N (X) d X = 0, \int_{B} [g_{0} (X)] N (X) d X = 0, \\ \int_{B} ⟨ g_{1} (X) ⟩ D (X) d X = D_{\infty}, \int_{B} [g_{1} (X)] D (X) d X = 0, \\ \int_{B} ⟨ g_{1} (X) ⟩ N (X) d X = 0, \int_{B} [g_{1} (X)] N (X) d X = 2 . \end{matrix} \end{matrix}$

2.2. Derivation of the first terms

We can now turn to the formal computation of the first solutions of the sequence of problems (2.2). We emphasize that the upcoming iterative procedure is formal in the sense that we shall provide necessary transmission conditions for the macroscopic terms $u_{q}$ but we shall not address the question of their existence in this part (this question will be investigated in Section 3). Throughout this section, we assume that the macroscopic terms exist and are smooth above and below the interface Γ.

2.2.1. Step 0: ${[u_{0}]}_{Γ}$ and $Π_{0}$

The limit boundary layer term (or periodic corrector) $Π_{0}$ is solution of $\begin{matrix} (2.9) & \{\begin{matrix} - Δ_{X} Π_{0} (x_{1}, X) = G_{0} (x_{1}, X) & in B, \\ \partial_{n} Π_{0} = 0 & on \partial {\hat{Ω}}_{hole}, \end{matrix} \end{matrix}$ where $G_{0} (x_{1}, X) = 2 ⟨ g_{0} (X) ⟩ {⟨ u_{0} (x_{1}, 0) ⟩}_{Γ} + \frac{1}{2} [g_{0} (X)] {[u_{0} (x_{1}, 0)]}_{Γ}$ . Problem (2.9) is a partial differential equation with respect to the microscopic variables $X_{1}$ and $X_{2}$ , wherein the macroscopic variable $x_{1}$ plays the role of a parameter. For a fixed $x_{1}$ in $(- L, L)$ (considered as a parameter), $G_{0} (x_{1}, \cdot)$ belongs to ${(V^{-} (B))}^{'}$ since it is compactly supported. Then, in view of Proposition 2.2, there exists an exponentially decaying solution $Π_{0} (x_{1}, \cdot) \in V^{+} (B)$ if and only if the two compatibility conditions ( C D ), ( C N ) (Proposition 2.2) are satisfied. Thanks to the second line of Lemma 2.3, $\begin{matrix} (2.10) & \int_{B} G_{0} (x_{1}, X) N (X) d X = 0, \end{matrix}$ which means that ( C N ) is always satisfied. Besides, in view of the first line of Lemma 2.3, $\begin{matrix} (2.11) & \int_{B} G_{0} (x_{1}, X) D (X) d X = - {[u_{0} (x_{1}, 0)]}_{Γ} . \end{matrix}$ As a consequence, we obtain a necessary and sufficient condition for $Π_{0}$ to be exponentially decaying: $\begin{matrix} (2.12) & {[u_{0} (x_{1}, 0)]}_{Γ} = 0 . \end{matrix}$ This condition provides a first transmission condition for the limit macroscopic term $u_{0}$ . Under the previous condition, $G_{0} (x_{1}, X) = 2 ⟨ g_{0} (X) ⟩ {⟨ u_{0} (x_{1}, 0) ⟩}_{Γ}$ , and, using the linearity of problem (2.9), we can obtain a tensorial representation of $Π_{0} \in C ((- L, L), V^{+} (B))$ , in which macroscopic and microscopic variables are separated: $\begin{matrix} (2.13) & Π_{0} (x_{1}, X) = {⟨ u_{0} (x_{1}, 0) ⟩}_{Γ} W_{0}^{t} (X) . \end{matrix}$ Here the profile function $W_{0}^{t} (X)$ is the unique function of $V^{+} (B)$ satisfying $\begin{matrix} (2.14) & \{\begin{matrix} - Δ_{X} W_{0}^{t} (X) = F_{0}^{t} (X) & in B, \\ \partial_{n} W_{0}^{t} = 0 & on \partial {\hat{Ω}}_{hole}, \\ \partial_{X_{1}} W_{0}^{t} (0, X_{2}) = \partial_{X_{1}} W_{0}^{t} (1, X_{2}), & X_{2} \in R, \end{matrix} F_{0}^{t} (X) = 2 ⟨ g_{0} (X) ⟩ . \end{matrix}$ A direct calculation shows that $\begin{matrix} (2.15) & W_{0}^{t} (X) = (1 - χ (X_{2})) . \end{matrix}$ Note that the continuity of $Π_{0}$ with respect to $x_{1}$ is a consequence of the continuity of $G_{0}$ with respect to $x_{1}$ .

2.2.2. Step 1: ${[\partial_{x_{2}} u_{0}]}_{Γ}$ , ${[u_{1}]}_{Γ}$ , and $Π_{1}$

In view of the general sequence of problems (2.2), the second boundary layer (or periodic corrector) $Π_{1}$ satisfies $\begin{matrix} (2.16) & \{\begin{matrix} - Δ_{X} Π_{1} (x_{1}, X) = G_{1} (x_{1}, X) & in B, \\ \partial_{n} Π_{1} = 0 & on \partial {\hat{Ω}}_{hole} . \end{matrix} \end{matrix}$ where, thanks to (2.15) ( $\partial_{X_{1}} W_{0}^{t} = 0$ ), $\begin{array}{rcl} G_{1} (x_{1}, X) & = & \frac{1}{2} [g_{1} (X)] {[\partial_{x_{2}} u_{0} (x_{1}, 0)]}_{Γ} + \frac{1}{2} [g_{0} (X)] {[u_{1} (x_{1}, 0)]}_{Γ} \\ (2.17) & + F_{0}^{t} (X) {⟨ u_{1} (x_{1}, 0) ⟩}_{Γ} + 2 ⟨ g_{1} (X) ⟩ {⟨ \partial_{x_{2}} u_{0} (x_{1}, 0) ⟩}_{Γ} . \end{array}$ As for $Π_{0}$ , problem (2.16) is a partial differential equation with respect to the microscopic variables $X_{1}$ and $X_{2}$ , where the macroscopic variable $x_{1}$ plays the role of a parameter. For a fixed $x_{1}$ in $(- L, L)$ , $G_{1} (x_{1}, \cdot)$ is compactly supported in $B$ , and, consequently, belongs to ${(V^{-} (B))}^{'}$ . Then, thanks to Proposition 2.2, there exists an exponentially decaying solution $Π_{1} (x_{1}, \cdot) \in V^{+} (B)$ if and only if the two compatibility conditions ( C D ), ( C N ) are satisfied. In view of Lemma 2.3, $F_{0}^{t} (X)$ , $[g_{0} (X)]$ , $⟨ g_{1} (X) ⟩$ are orthogonal to $N$ . Then, the second formula of the fourth line of Lemma 2.3 gives $\begin{matrix} \int_{B} G_{1} (x_{1}, X) N (X) d X = {[\partial_{x_{2}} u_{0} (x_{1}, 0)]}_{Γ} . \end{matrix}$ Therefore, the compatibility condition ( C N ) is fulfilled if and only if $\begin{matrix} (2.18) & {[\partial_{x_{2}} u_{0} (x_{1}, 0)]}_{Γ} = 0 . \end{matrix}$ Next, using the first and third lines of Lemma 2.3, we obtain $\begin{matrix} \int_{B} G_{1} (x_{1}, X) D (X) d X = - {[u_{1} (x_{1}, 0)]}_{Γ} + 2 D_{\infty} {⟨ \partial_{x_{2}} u_{0} (x_{1}, 0) ⟩}_{Γ} . \end{matrix}$ Therefore, the compatibility condition ( C D ) is fulfilled if and only if $\begin{matrix} (2.19) & {[u_{1} (x_{1}, 0)]}_{Γ} = 2 D_{\infty} {⟨ \partial_{x_{2}} u_{0} (x_{1}, 0) ⟩}_{Γ} . \end{matrix}$ Under the two conditions (2.18)–(2.19), problem (2.16) has a unique solution $Π_{1}$ in $C ((- L, L), V^{+} (B))$ (the continuity of $Π_{1}$ with respect to $x_{1}$ results from the continuity of $G_{1}$ with respect to $x_{1}$ ). Using (here again) the linearity of problem (2.16), we can write $Π_{1}$ as a tensorial product between profile functions that only depend on the microscopic variables $X_{1}$ and $X_{2}$ , and functions that only depend on the macroscopic variable $x_{1}$ (more precisely, the latter functions consist of the average traces of the macroscopic terms of order 0 and 1 on Γ): $\begin{matrix} (2.20) & Π_{1} (x_{1}, X) = {⟨ u_{1} (x_{1}, 0) ⟩}_{Γ} W_{0}^{t} (X) + {⟨ \partial_{x_{2}} u_{0} (x_{1}, 0) ⟩}_{Γ} W_{1}^{n} (X), \end{matrix}$ where $W_{0}^{t}$ is defined by (2.14) and $W_{1}^{n} \in V^{+} (B)$ is the unique decaying solution to the following problem: $\begin{matrix} (2.21) & \{\begin{matrix} - Δ_{X} W_{1}^{n} (X) = F_{1}^{n} (X) + \frac{D_{1}^{n}}{2} [g_{0} (X)] & in B, \\ \partial_{n} W_{1}^{n} = 0 & on \partial {\hat{Ω}}_{hole}, \\ \partial_{X_{1}} W_{1}^{n} (0, X_{2}) = \partial_{X_{1}} W_{1}^{n} (1, X_{2}), & X_{2} \in R, \end{matrix} \end{matrix}$ where, $\begin{matrix} (2.22) & F_{1}^{n} (X) = 2 {⟨ g_{1} (X) ⟩}_{Γ} and D_{1}^{n} = \int_{B} F_{1}^{n} (X) D (X) d X = 2 D_{\infty} . \end{matrix}$ It is easily seen that the right-hand side of (2.21) is orthogonal to both $N$ and $D$ . A direct computation shows that $\begin{matrix} (2.23) & W_{1}^{n} (X) = \tilde{D} (X), \end{matrix}$ the function $\tilde{D}$ being defined in the first point of Proposition 2.2.

2.2.3. Step 2: ${[\partial_{x_{2}} u_{1}]}_{Γ}$ ( ${[u_{2}]}_{Γ}$ and $Π_{2}$ )

We can continue the iterative procedure started in the two previous steps as follows. The periodic corrector $Π_{2}$ satisfies the following equation $\begin{matrix} (2.24) & \{\begin{matrix} - Δ_{X} Π_{2} (x_{1}, X) = G_{2} (x_{1}, X) & in B, \\ \partial_{n} Π_{2} 0 & on \partial {\hat{Ω}}_{hole} . \end{matrix} \end{matrix}$ Here, $\begin{array}{rcl} G_{2} (x_{1}, X) & = & \frac{1}{2} [g_{0} (X)] {[u_{2}]}_{Γ} + \frac{1}{2} [g_{1} (X)] {[\partial_{x_{2}} u_{1}]}_{Γ} + F_{0}^{t} (X) {⟨ u_{2} ⟩}_{Γ} \\ (2.25) & + F_{1}^{n} (X) {⟨ \partial_{x_{2}} u_{1} ⟩}_{Γ} + F_{2}^{t} (X) \partial_{x_{1}}^{2} {⟨ u_{0} ⟩}_{Γ} + F_{2}^{n} (X) \partial_{x_{1}} {⟨ \partial_{x_{2}} u_{0} ⟩}_{Γ} . \end{array}$ $F_{0}^{t} (X)$ and $F_{1}^{n} (X)$ are given by (2.14)–(2.22), and, $\begin{matrix} (2.26) & F_{2}^{t} (X) = - 2 ⟨ g_{2} (X) ⟩ + W_{0}^{t} (X), F_{2}^{n} (X) = 2 \partial_{X_{1}} W_{1}^{n} (X) . \end{matrix}$ In formula (2.25), for the sake of concision, we have omitted the dependence on $x_{1}$ of the macroscopic terms. To obtain this formula, we have replaced $Π_{0}$ and $Π_{1}$ with their tensorial representations (2.13), (2.20), we have substituted $- \partial_{x_{2}}^{2} u_{0} (x, 0^{\pm})$ by $\partial_{x_{1}}^{2} u_{0} (x, 0^{\pm})$ using the macroscopic equation (2.2) ( $Δ u_{0} = 0$ in the vicinity of Γ) and we have taken into account the jump conditions (2.12), (2.18) for $u_{0}$ .

For a fixed $x_{1} \in (- L, L)$ , it is easily verified that $G_{2} (x_{1}, \cdot)$ belongs to ${(V^{-} (B))}^{'}$ : indeed, the first five terms of (2.25) are compactly supported and the last one is exponentially decaying (more precisely, $w_{e}^{+} F_{2}^{t}$ and $w_{e}^{+} F_{2}^{n}$ belong to $L^{2} (B)$ ). Then again, the existence of an exponentially decaying corrector $Π_{2} (x_{1}, \cdot) \in V^{+} (B)$ results from the orthogonality of $G_{2} (x_{1}, \cdot)$ with $N$ and $D$ . As previously, enforcing the compatibility condition ( C N ) provides the transmission condition for the jump of the normal trace of $u_{1}$ across Γ: $\begin{matrix} (2.27) & {[\partial_{x_{2}} u_{1}]}_{Γ} = N_{2}^{t} \partial_{x_{1}}^{2} {⟨ u_{0} ⟩}_{Γ} + N_{2}^{n} \partial_{x_{1}} {⟨ \partial_{x_{2}} u_{0} ⟩}_{Γ}, \end{matrix}$ where $\begin{matrix} (2.28) & N_{2}^{t} = - \int_{B} F_{2}^{t} (X) N (X) d X, N_{2}^{n} = - \int_{B} F_{2}^{n} (X) N (X) d X . \end{matrix}$ Then, enforcing the compatibility condition ( C N ) provides the jump ${[u_{2}]}_{Γ}$ , and the existence of $Π_{2}$ is proved. Naturally an explicit expression of ${[u_{2}]}_{Γ}$ and a tensorial representation of $Π_{2}$ can be written (see the upcoming formulas (2.29a)–(2.30)), but, for the sake of concision, we do not write it here.

2.3. Transmission conditions up to any order

We are now in a position to extend the previous approach up to any order. For each $q \in N$ , similarly to the first steps, our global iterative approach relies on the following procedure:

We compute the right-hand side $G_{q} (x_{1}, X)$ of the periodic corrector problem (2.2) of order q: we write $G_{q}$ as a tensorial product between functions that only depend on the microscopic variables $X_{1}$ and $X_{2}$ and functions that only depend on the macroscopic variable $x_{1}$ . More specifically, the latter functions consist of the trace and normal trace of the macroscopic terms of order lower than q and their tangential derivatives (see (2.9), (2.17), (2.25)).

We compute the normal jump ${[\partial_{x_{2}} u_{q - 1} (x_{1}, 0)]}_{Γ}$ by enforcing $G_{q}$ to be orthogonal to $N$ , i.e., to satisfy the compatibility condition ( C N ) (see (2.12), (2.19)).

We compute the jump ${[u_{q} (x_{1}, 0)]}_{Γ}$ by imposing $G_{q}$ to satisfy the compatibility condition ( C D ) ensuring that $G_{q}$ is orthogonal to $D$ (see (2.18), (2.27)).

We write a tensorial representation of the periodic corrector $Π_{q}$ introducing at most two new profile functions (see (2.13), (2.20)).

Applying this general scheme, we can prove the following proposition, whose proof can be found in [18] (Proposition 2.4, Appendix B.1).

Proposition 2.4.
Assume that the macroscopic terms $u_{q}$ satisfying (2.2) exist. Then, there exists four sequences of real constants $N_{p}^{n}$ , $N_{p}^{t}$ , $D_{p}^{n}$ , $D_{p}^{t}$ such that $\begin{array}{rcl} (2.29a) & {[u_{q} (x_{1}, 0)]}_{Γ} = \sum_{p = 1}^{q} D_{p}^{t} \partial_{x_{1}}^{p} {⟨ u_{q - p} (x_{1}, 0) ⟩}_{Γ} + \sum_{p = 1}^{q} D_{p}^{n} \partial_{x_{1}}^{p - 1} {⟨ \partial_{x_{2}} u_{q - p} (x_{1}, 0) ⟩}_{Γ}, \\ (2.29b) & {[\partial_{x_{2}} u_{q} (x_{1}, 0)]}_{Γ} = \sum_{p = 1}^{q} N_{p + 1}^{t} \partial_{x_{1}}^{p + 1} {⟨ u_{q - p} (x_{1}, 0) ⟩}_{Γ} + \sum_{p = 1}^{q} N_{p + 1}^{n} \partial_{x_{1}}^{p} {⟨ \partial_{x_{2}} u_{q - p} (x_{1}, 0) ⟩}_{Γ} . \end{array}$ Moreover, there exist two families of decaying profile functions $W_{p}^{t}$ and $W_{p}^{n}$ belonging to $V^{+} (B)$ such that the periodic corrector $Π_{q} \in C ((- L, L), V^{+} (B))$ admits the following representation: $\begin{matrix} (2.30) & Π_{q} (x_{1}, X) = \sum_{p = 0}^{q} \partial_{x_{1}}^{p} {⟨ u_{q - p} (x_{1}, 0) ⟩}_{Γ} W_{p}^{t} (X) + \sum_{p = 1}^{q} \partial_{x_{1}}^{p - 1} {⟨ \partial_{x_{2}} u_{q - p} (x_{1}, 0) ⟩}_{Γ} W_{p}^{n} (X) . \end{matrix}$

In the previous definition,we have used the superscript $t$ (in $D_{p}^{t}$ , $N_{p}^{t}$ ) to refer to some constants associated with tangential derivatives of the average trace of the macroscopic terms. Similarly, the superscript $n$ (in $D_{p}^{n}$ , $N_{p}^{n}$ ) is used for the constants associated with tangential derivatives of the average of the normal trace of the macroscopic terms. The definitions of the functions $W_{p}^{t}$ and $W_{p}^{n}$ and of the constants $D_{q}^{t}$ , $N_{q}^{t}$ , $D_{q}^{n}$ , $N_{q}^{n}$ are given explicitely in (A.1), (A.3), (A.4), and (A.6).

We point out that the periodic correctors $Π_{q}$ do not appear (explicitly) in (2.29b): they have been eliminated. In other words, the resolution of macroscopic and boundary layer problems are decoupled and the construction of $Π_{q}$ can be made a posteriori.
3. Analysis of the macroscopic problems (macroscopic singularities)

Thanks to the previous section (see in particular Proposition 2.4, reminding that the index n and the superscript δ have been deliberately omitted in the previous section), we can see that if the macroscopic terms $u_{n, q}^{δ}$ (solution to (1.13)) exist, they satisfy the following transmission problems: for any $(n, q) \in N^{2}$ , $\begin{matrix} (3.1a) & \{\begin{matrix} - Δ u_{n, q}^{δ} = f_{n, q} in Ω_{T} \cup Ω_{B}, \\ [u_{n, q}^{δ} {(x_{1}, 0)]}_{Γ} = g_{n, q}^{δ}, \\ [\partial_{x_{2}} u_{n, q}^{δ} {(x_{1}, 0)]}_{Γ} = h_{n, q}^{δ}, \\ u_{n, q}^{δ} = 0 on Γ_{D}, \end{matrix} f_{n, q} = \{\begin{matrix} f & if n = q = 0, \\ 0 & otherwise, \end{matrix} \end{matrix}$ where $\begin{array}{rcl} (3.1b) & g_{n, q}^{δ} (x_{1}) = \sum_{p = 1}^{q} D_{p}^{t} \partial_{x_{1}}^{p} {⟨ u_{n, q - p}^{δ} (x_{1}, 0) ⟩}_{Γ} + \sum_{p = 1}^{q} D_{p}^{n} \partial_{x_{1}}^{p - 1} {⟨ \partial_{x_{2}} u_{n, q - p}^{δ} (x_{1}, 0) ⟩}_{Γ}, \\ (3.1c) & h_{n, q}^{δ} (x_{1}) = \sum_{p = 1}^{q} N_{p + 1}^{t} \partial_{x_{1}}^{p + 1} {⟨ u_{n, q - p}^{δ} (x_{1}, 0) ⟩}_{Γ} + \sum_{p = 1}^{q} N_{p + 1}^{n} \partial_{x_{1}}^{p} {⟨ \partial_{x_{2}} u_{n, q - p}^{δ} (x_{1}, 0) ⟩}_{Γ} . \end{array}$ As previously mentioned, the constants $D_{q}^{t}$ , $N_{q}^{t}$ , $D_{q}^{n}$ , $N_{q}^{n}$ , which only depend on the geometry of the periodicity cell $B$ , are defined in (A.3)–(A.6).

The present section is dedicated to the analysis of problems (3.1). In Section 3.1, we give general results of well-posedness for transmission problems: we first introduce a variational framework, then we present an alternative functional framework based on weighted Sobolev spaces. In Section 3.2, we explain the reason why the variational framework is not adapted for the resolution of problem (3.1) for $q = 1$ (and higher). This leads us to consider singular (extra-variational) macroscopic terms that may blow up in the vicinity of the two corners. In Section 3.3, we construct several sequences of singular functions that are used in Section 3.4 to write a general formula for the macroscopic terms (Proposition 3.11).

3.1. General results of existence for transmission problem

The problems under consideration can be investigated using the general framework for transmission problems posed in polygonal domains developed in [33]. In the present paper, we first recall a classical well-posedness result based on a variational form of the problem. Then, based on weighted Sobolev spaces, we describe the behavior of the solutions close to the two reentrant corners.

3.1.1. Variational framework

Let us introduce the classical Hilbert spaces associated with our problems $\begin{matrix} H_{Γ_{D}}^{1} (Ω_{T} \cup Ω_{B}) = {u \in H^{1} (Ω_{T} \cup Ω_{B}), s.t. u = 0 on Γ_{D}}, \end{matrix}$ which incorporates discontinuous functions over Γ (see Fig. 1(a)). Its restrictions to $Ω_{T}$ and $Ω_{B}$ are denoted by $H_{Γ_{D}}^{1} (Ω_{T})$ and $H_{Γ_{D}}^{1} (Ω_{B})$ . We denote by $H_{00}^{1 / 2} (Γ)$ the restriction of the trace of the function $H_{Γ_{D}}^{1} (Ω_{T})$ to Γ (for a complete description of the trace of functions, we refer the reader to [19]), i.e., $\begin{matrix} H_{00}^{1 / 2} (Γ) = {μ \in H^{1 / 2} (Γ), s.t. \exists v \in H_{Γ_{D}}^{1} (Ω_{T}) : v = μ on Γ} . \end{matrix}$ Naturally, the space $H_{00}^{1 / 2} (Γ)$ is also the restriction of the trace of the functions of $H_{Γ_{D}}^{1} (Ω_{B})$ to Γ. Based on a variational formulation, and thanks to the Lax–Milgram lemma, we can prove the following well-posedness result:

Proposition 3.1.
Let $f \in L^{2} (Ω)$ , $g \in H_{00}^{1 / 2} (Γ)$ , and $h \in L^{2} (Γ)$ . Then, the following problem has a unique solution u belonging to $H_{Γ_{D}}^{1} (Ω_{T} \cup Ω_{B})$ : $\begin{matrix} (3.2) & \{\begin{matrix} - Δ u = f & in Ω_{T} \cup Ω_{B}, \\ {[u]}_{Γ} = g & on Γ, \\ {[\partial_{x_{2}} u]}_{Γ} = h & on Γ . \end{matrix} \end{matrix}$

3.1.2. Weighted Sobolev spaces and asymptotic behaviour

In the next subsections, we shall study the behavior of the macroscopic terms in the neighborhood of the two reentrant corners. It is well-known that the Hilbert spaces $H^{m} (Ω_{B})$ (resp. $H^{m} (Ω_{T})$ ) are not well-adapted to this investigation. By contrast, the weighted Sobolev spaces provide a more convenient functional framework. We refer the reader to the Kondrat’ev theory (see [25], [26, Chapter 5 and Chapter 6] for a complete presentation of these spaces and their applications). In this part, we introduce the weighted Sobolev spaces associated with our problem following the presentation of [26, Chapter 6]. Let us first define the polar coordinates $(r^{\pm}, θ^{\pm})$ centered at the vertex $x_{O}^{\pm}$ , i.e., $\begin{matrix} (3.3) & x_{1} - {(x_{O}^{\pm})}_{1} = r^{\pm} cos (θ^{\pm}), x_{2} - {(x_{O}^{\pm})}_{2} = r^{\pm} sin (θ^{\pm}) . \end{matrix}$ Next, we consider the two infinite angular (or conical) domains $K_{x_{O}^{\pm}}$ centered at $x_{O}^{\pm}$ of opening $\frac{3 π}{2}$ $\begin{matrix} (3.4) & \begin{matrix} K_{x_{O}^{\pm}} = {(r^{\pm} cos θ^{\pm}, r^{\pm} sin θ^{\pm}) \in R^{2}, r^{\pm} > 0, θ^{\pm} \in I^{\pm}}, \\ I^{+} = (0, \frac{3 π}{2}), I^{-} = (- \frac{π}{2}, π), \end{matrix} \end{matrix}$ and, for $ℓ \in {0, 1, 2}$ , we define the space $V_{2, β}^{ℓ} (K_{x_{O}^{\pm}})$ as the closure of $C_{0}^{\infty} (\overline{K_{x_{O}^{\pm}}} ∖ {0})$ with respect to the norm $\begin{array}{rcl} ‖ u ‖_{V_{2, β}^{ℓ} (K_{x_{O}^{\pm}})} = {(\int_{K_{x_{O}^{\pm}}} \sum_{| α | ⩽ ℓ} {(r^{\pm})}^{2 (β - ℓ + | α |)} {| \partial_{x_{1}}^{α_{1}} \partial_{x_{2}}^{α_{2}} u |}^{2} d x)}^{1 / 2}, \\ (3.5) & α = (α_{1}, α_{2}) \in N^{2}, | α | = α_{1} + α_{2} . \end{array}$ Then, let $\begin{matrix} (3.6) & χ_{L}^{\pm} (x) = (1 - χ (2 r^{\pm} / L)) \end{matrix}$ be the cut-off function equal to one in the vicinity of $x_{O}^{\pm}$ and vanishing in the vicinity of $x_{O}^{\mp}$ (the support of $χ_{L}^{\pm}$ is localized in the neighborhood of the vertice $x_{O}^{\pm}$ ), and let $χ_{L}^{0} = 1 - χ_{L}^{+} - χ_{L}^{-}$ . We remind that the truncation function χ is defined by (1.8). For $ℓ \in {0, 1, 2}$ , we introduce the space $V_{2, β}^{ℓ} (Ω)$ $\begin{matrix} (3.7) & V_{2, β}^{ℓ} (Ω) = {u \in H_{loc}^{ℓ} (Ω), {‖ χ_{L}^{-} u ‖}_{V_{2, β}^{ℓ} (K_{x_{O}^{-}})} + {‖ χ_{L}^{+} u ‖}_{V_{2, β}^{ℓ} (K_{x_{O}^{+}})} + {‖ χ_{L}^{0} u ‖}_{H^{ℓ} (Ω)} < + \infty}, \end{matrix}$ equipped with the following norm $\begin{matrix} (3.8) & ‖ u ‖_{V_{2, β}^{ℓ} (Ω)} = {‖ χ_{L}^{-} u ‖}_{V_{2, β}^{ℓ} (K_{x_{O}^{-}})} + {‖ χ_{L}^{+} u ‖}_{V_{2, β}^{ℓ} (K_{x_{O}^{+}})} + {‖ χ_{L}^{0} u ‖}_{H^{ℓ} (Ω)} . \end{matrix}$ Here, we have used the convention $H^{0} (Ω) = L^{2} (Ω)$ . Note, that the space $V_{2, β}^{ℓ} (Ω)$ is independent of the exact choice of χ and so the truncation functions $χ_{L}^{\pm}$ and that $\begin{array}{rcl} (3.9) & V_{2, β^{'}}^{ℓ} (Ω) \subset V_{2, β}^{ℓ} (Ω) for any β^{'} < β . \end{array}$ In the same way, we also define $V_{2, β}^{ℓ} (Ω_{T})$ (resp. $V_{2, β}^{ℓ} (Ω_{B})$ ) as well as their associated norm $‖ \cdot ‖_{V_{2, β}^{ℓ} (Ω_{T})}$ (resp. $‖ \cdot ‖_{V_{2, β}^{ℓ} (Ω_{B})}$ ) replacing Ω with $Ω_{T}$ (resp. $Ω_{B}$ ) in the definitions (3.7) and (3.8). Finally, for $ℓ \in {1, 2}$ , we introduce the space $V_{2, β}^{ℓ - 1 / 2} (Γ)$ of the trace of the functions in $V_{2, β}^{ℓ} (Ω_{T})$ on the interface Γ. As norm in $V_{2, β}^{ℓ - 1 / 2} (Γ)$ we take $\begin{matrix} (3.10) & ‖ u ‖_{V_{2, β}^{1 / 2} (Γ)} = inf {‖ v ‖_{V_{2, β}^{1} (Ω_{T})} : v \in V_{2, β}^{1}, v_{| Γ} = u} . \end{matrix}$

When studying the behavior of the far field terms close to the reentrant corners, the set $\begin{matrix} (3.11) & Λ = {λ_{m} \in R, such that λ_{m} = \frac{2 m}{3}, m \in Z ∖ {0}} \end{matrix}$ of singular exponents will play a crucial role (see [19, Chapters 1–4]). It consists of the real numbers λ whose square $λ^{2}$ is an eigenvalue of the unbounded operator $A$ which maps $D (A) = H_{0}^{1} (0, \frac{3 π}{2}) \cap H^{2} (0, \frac{3 π}{2})$ into $L^{2} (0, \frac{3 π}{2})$ such that $A u = - u^{″}$ . Note that the associated eigenvectors are given by $\begin{matrix} (3.12) & w_{m} (t) = sin (λ_{m} t), m \in Z ∖ {0} . \end{matrix}$ The following proposition, which is a standard result in the literature on elliptic problems in angular domains (cf. [33, Theorem 3.6 and Corollary 4.4] for the proof), provides an explicit asymptotic representation of the solution of the transmission problems in a neighbourhood of the corners (see also [26, Chapter 6] for a complete and detailed explanation of the overall approach):

Proposition 3.2.
Let $β < 0$ be a real number such that $1 - β \notin Λ$ . Assume that $f \in V_{2, β}^{0} (Ω_{T}) \cap V_{2, β}^{0} (Ω_{B}) \subset L^{2} (Ω)$ , $g \in V_{2, β}^{3 / 2} (Γ) \subset H^{3 / 2} (Γ)$ and $h \in V_{2, β}^{1 / 2} (Γ) \subset H^{1 / 2} (Γ)$ . Then, the unique solution $u \in H_{Γ_{D}}^{1} (Ω_{T}) \cap H_{Γ_{D}}^{1} (Ω_{B})$ of problem (3.2) admits the following decomposition: $\begin{matrix} (3.13) & u = \sum_{1 ⩽ q < \frac{3}{2} (1 - β)} c_{q}^{\pm} {(r^{\pm})}^{λ_{q}} w_{q, 0, \pm} (θ^{\pm}) + w^{\pm}, \end{matrix}$ where $w^{\pm} \in V_{2, β}^{2} (Ω_{T}) \cap V_{2, β}^{2} (Ω_{B})$ , $w_{q, 0, +} (t) = w_{q} (t)$ and $w_{q, 0, -} (t) = w_{q} (t - \frac{π}{2})$ (where $w_{q}$ were given in (3.12)). Moreover, there exists a constant C independent of u such that $\begin{matrix} (3.14) & {‖ w^{\pm} ‖}_{V_{2, β}^{2} (Ω_{T})} + {‖ w^{\pm} ‖}_{V_{2, β}^{2} (Ω_{B})} + \sum_{1 ⩽ q < \frac{3 (1 - β)}{2}} | c_{q} |^{\pm} ⩽ C (‖ f ‖_{V_{2, β}^{0} (Ω_{T})} + ‖ f ‖_{V_{2, β}^{0} (Ω_{B})}) . \end{matrix}$

The expansion (3.13) is nothing but a modal expansion of the solution u is the vicinity of the two corners. Without doubt a similar expansion could be obtained using the technique of separation of variables (see [20, Chapter 2]). The sum $\sum_{1 ⩽ q < \frac{3}{2} (1 - β)} c_{q}^{\pm} {(r^{\pm})}^{λ_{q}} w_{q, 0, \pm} (θ^{\pm})$ is an asymptotic expansion for $r^{\pm} \to 0$ whose remainder $w^{\pm}$ decays faster to zero as any term in the sum. Obviously, due the embedding (3.9) asymptotic expansions of higher order in $r^{\pm}$ are obtained when β is decreased (or $| β |$ increased).
3.2. The necessary introduction of singular macroscopic terms

3.2.1. The limit macroscopic term and its behavior in the vicinity of the corners

The limit macroscopic term $u_{0, 0}^{δ}$ satisfies problem (3.2) with $f = f_{0, 0} = f \in L^{2} (Ω_{T}) \cap L^{2} (Ω_{B})$ , $g = 0$ and $h = 0$ . In view of Proposition 3.1, there exists a unique solution $u_{0, 0}^{δ}$ belonging to $H_{0, Γ_{D}}^{1} (Ω_{T}) \cap H_{Γ_{D}}^{1} (Ω_{B})$ . Indeed, $u_{0, 0}^{δ}$ is independent of δ (it will be denoted by $u_{0, 0}$ ) and belongs to $H_{0}^{1} (Ω)$ , since its trace does not jump across Γ.

The existence and uniqueness of $u_{0, 0}$ being granted, we can investigate its behavior in the neighborhood of the two reentrant corners. Since we have assumed that f is compactly supported in $Ω_{T}$ , $f \in V_{2, β}^{0} (Ω_{T}) \cap V_{2, β}^{0} (Ω_{B})$ for any $β \in R$ . Then, in view of Proposition 3.2, $u_{0, 0}$ has the following asymptotic expansion in the vicinity of the two corners vertices $x_{O}^{\pm}$ : for any $k \in N$ , there exists $u_{0, 0, k} \in V_{2, β}^{2} (Ω_{T}) \cap V_{2, β}^{2} (Ω_{B})$ for any $β > 1 - \frac{2 (k + 1)}{3}$ , such that $\begin{matrix} (3.15) & u_{0, 0} = \sum_{m = 1}^{k} c_{m}^{\pm} {(r^{\pm})}^{\frac{2 m}{3}} w_{m, 0, \pm} (θ^{\pm}) + u_{0, 0, k}, \end{matrix}$ where $c_{m}^{\pm}$ are real constants continuously depending on $‖ f ‖_{V_{2, β}^{0} (Ω)}$ . Here again, the expansion (3.15) could also be obtained using the method of separation of variables.

3.2.2. A singular problem defining $u_{0, 1}^{δ}$

To illustrate the fact that the macroscopic terms of higher orders cannot always be variational (i.e. belonging to $H^{1} (Ω_{T}) \cap H^{1} (Ω_{B})$ ), let us consider the problem satisfied by $u_{0, 1}^{δ}$ , investigating the regularity of $g_{0, 1}^{δ}$ and $h_{0, 1}^{δ}$ defined in (3.1b) and (3.1c) (we deliberately omit the term $u_{1, 0}^{δ}$ for a while). In view of the asymptotic expansion (3.15) of $u_{0, 0}$ , $\begin{matrix} g_{0, 1}^{δ} \sim c_{0, 1, \pm} {(r^{\pm})}^{- 1 / 3} and h_{0, 1}^{δ} \sim d_{0, 1, \pm} {(r^{\pm})}^{- 4 / 3} \end{matrix}$ as $r^{\pm}$ tends to zeros. The constants $c_{0, 1, \pm}$ and $d_{0, 1, \pm}$ can be explicitly determined (but, there is not need to write their complete expression). As a consequence, $g_{0, 1}^{δ}$ does not belong to $H_{0, 0}^{1 / 2} (Γ)$ and $h_{0, 1}^{δ}$ is not in $L^{2} (Γ)$ . It follows that we are not able to construct $u_{0, 1}^{δ} \in H^{1} (Ω_{T} \cup Ω_{B})$ . However, we shall see that it is possible to build a function $u_{0, 1}^{δ}$ that blows up as ${(r^{\pm})}^{- 1 / 3}$ as $r^{\pm}$ tends to 0. Since this function is not in $H^{1} (Ω_{T} \cup Ω_{B})$ , we say that this function is singular. To distinguish from singular functions, we denote functions in $H^{1} (Ω_{T} \cup Ω_{B})$ as regular (so not meaning $C^{\infty}$ -regular functions).

Remark 3.3.
The previous analysis explains why, contrary to the case of an infinite thin periodic layer (see [14,34]), it is not possible to construct an asymptotic expansion of the form $\begin{matrix} u^{δ} (x_{1}, x_{2}) = \sum_{n \in N} δ^{n} (u_{n} (x) + Π_{n} (x_{1}, X)), \end{matrix}$ where $u_{n} \in H_{0, Γ_{D}}^{1} (Ω_{T}) \cap H_{Γ_{D}}^{1} (Ω_{B})$ and $Π_{n}$ are periodic functions with respect to $X_{1}$ exponentially decaying as $X_{2}$ tends to $\pm \infty$ .
Remark 3.4.
Since it is not possible to construct regular macroscopic terms, we shall construct singular ones. Nevertheless, the exact solution $u^{δ}$ is not singular. As a consequence the far field expansion (1.6), which contains singular terms, can not be valid in the immediate surrounding of the two corners. Here, a near field expansion (1.11) has to be introduced, which replace the singular solution behavior towards the corners in their immediate neighborhood.

3.3. Two families of macroscopic singularities $s_{- m, q}^{\pm}$

In this section, we introduce two families of functions, that are $s_{- m, q}^{+}$ and $s_{- m, q}^{-}$ for the right and left corner, that will facilitate the definition of the macroscopic terms. The functions are defined recursively in q for each $m \in N ∖ {0}$ . The following subsection is dedicated to the definition of $s_{- m, 0}^{\pm}$ , where the functions $s_{- m, q}^{\pm}$ , $q \in N ∖ {0}$ are defined by induction afterwards.

3.3.1. Harmonic singularities $s_{- m, 0}^{\pm}$ ( $m \in N ∖ {0}$ )

For any positive integer n, the terms $u_{n, 0}^{δ}$ are harmonic in Ω. It does not imply that they vanish because we allow for singular behaviors in the vicinity of the two corners. The present subsection is dedicated to the definition of a set of harmonic functions that admit singularities in the vicinity of the two corners $x_{0}^{\pm}$ . The forthcoming analysis is done for the right corner $x_{O}^{+}$ but a strictly similar approach may be carried out for the left corner. To start with, we exhibit a sequence of harmonic functions $s_{- m, 0}^{+}$ that behave like $r^{- 2 m / 3}$ in the vicinity of $x_{0}^{+}$ , and which are regular in the vicinity of $x_{0}^{-}$ .

Proposition 3.5.
Let $m \in N ∖ {0}$ . There exists a unique harmonic function $s_{- m, 0}^{+}$ vanishing on $Γ_{D}$ of the form $\begin{matrix} (3.16) & s_{- m, 0}^{+} = {(r^{+})}^{- \frac{2 m}{3}} w_{- m, 0, +} (θ^{+}) χ_{L}^{+} + {\tilde{s}}_{- m, 0}^{+}, \end{matrix}$ where ${\tilde{s}}_{- m, 0}^{+}$ belongs to $H_{0}^{1} (Ω)$ .
Proof.
Remarking that $Δ {\tilde{s}}_{- m, 0}^{\pm}$ belongs to $L^{2} (Ω)$ , the proof of Proposition 3.5 directly follows from the Lax–Milgram lemma. □

It is worth noting that $s_{- m, 0}^{+}$ does not depend on the cut off function $χ_{L}^{+}$ . Besides, it is easily verified that $s_{- m, 0}^{+}$ belongs to $V_{2, β}^{2} (Ω_{T}) \cap V_{2, β}^{2} (Ω_{B})$ for any $β > 1 + \frac{2 m}{3}$ . For instance, it belongs to $V_{2, \frac{2 m}{3} + \frac{7}{6}}^{2} (Ω_{T}) \cap V_{2, \frac{2 m}{3} + \frac{7}{6}}^{2} (Ω_{B})$ . Naturally, for $m \in N ∖ {0}$ , we can also prove the existence of a set of functions $s_{- m, 0}^{-}$ of the form $\begin{matrix} (3.17) & s_{- m, 0}^{-} = {(r^{-})}^{- \frac{2 m}{3}} w_{- m, 0, -} (θ^{-}) χ_{L}^{-} + {\tilde{s}}_{- m, 0}^{-}, {\tilde{s}}_{- m, 0}^{-} \in H_{0}^{1} (Ω) . \end{matrix}$

As for $u_{0, 0}$ , we shall write an explicit asymptotic expansion of $s_{- m, 0}^{+}$ in the vicinity of the two corners. Applying Proposition 3.2 to the function ${\tilde{s}}_{- m, 0}^{+}$ (noting that $Δ {\tilde{s}}_{- m, 0}^{+}$ vanishes for $r^{\pm} < L / 2$ ), we can prove the following proposition:
Proposition 3.6.
Let $m \in N ∖ {0}$ and $k \in N$ . Then, there exist a function $r_{- m, k, +}^{+}$ belonging to $V_{2, β}^{2} (Ω_{T}) \cap V_{2, β}^{2} (Ω_{B})$ for any $β > 1 - \frac{2 (k + 1)}{3}$ , and k real coefficients $ℓ_{q}^{+} (s_{- m, 0}^{+})$ , $1 ⩽ q ⩽ k$ , such that $\begin{matrix} (3.18) & s_{- m, 0}^{+} = {(r^{+})}^{- \frac{2 m}{3}} w_{- m, 0, +} (θ^{+}) + \sum_{q = 1}^{k} ℓ_{q}^{+} (s_{- m, 0}^{+}) {(r^{+})}^{λ_{q}} w_{q, 0, +} (θ^{+}) + r_{- m, k, +}^{+} . \end{matrix}$ Analogously, there exist a function $r_{- m, k, -}^{+}$ belonging to $V_{2, β}^{2} (Ω_{T}) \cap V_{2, β}^{2} (Ω_{B})$ for any $β > 1 - \frac{2 (k + 1)}{3}$ and k real coefficients $ℓ_{q}^{-} (s_{- m, 0}^{+})$ , $1 ⩽ q ⩽ k$ , such that $\begin{matrix} (3.19) & s_{- m, 0}^{+} = \sum_{q = 1}^{k} ℓ_{q}^{-} (s_{- m, 0}^{+}) {(r^{-})}^{λ_{q}} w_{q, 0, -} (θ^{-}) + r_{- m, k, -}^{+} . \end{matrix}$ Moreover, for any $β > 1 - \frac{2 (k + 1)}{3}$ , there exists a constant C such that $\begin{array}{rcl} \sum_{q = 1}^{k} (| ℓ_{q}^{+} (s_{- m, 0}^{+}) | + | ℓ_{q}^{-} (s_{- m, 0}^{+}) |) + {‖ r_{- m, k, +}^{+} ‖}_{V_{2, β}^{2} (Ω_{T})} + {‖ r_{- m, k, +}^{+} ‖}_{V_{2, β}^{2} (Ω_{B})} \\ + {‖ r_{- m, k, -}^{+} ‖}_{V_{2, β}^{2} (Ω_{T})} + {‖ r_{- m, k, -}^{+} ‖}_{V_{2, β}^{2} (Ω_{B})} \\ (3.20) & ⩽ C ({‖ s_{- m, 0}^{+} ‖}_{V_{2, \frac{2 m}{3} + \frac{7}{6}}^{2} (Ω_{T})} + {‖ s_{- m, 0}^{+} ‖}_{V_{2, \frac{2 m}{3} + \frac{7}{6}}^{2} (Ω_{B})}) . \end{array}$

The formulas (3.18), (3.19) provide asymptotic expansions of $s_{m, 0}^{+}$ in the neighborhood of $x_{O}^{\pm}$ . Again, despite their apparent complexity, they are essentially modal expansions of ${\tilde{s}}_{- m, 0}^{+}$ that can be also obtained using the separation of variables. Note that the remainder $r_{- m, k, +}^{+}$ is orthogonal to the functions $w_{q, 0, +}$ , for $q ⩽ k$ : $\begin{matrix} \int_{I^{+}} r_{- m, k, +}^{+} (r^{+}, θ^{+}) w_{q, 0, +} (θ^{+}) d θ^{+} = 0 \forall q ⩽ k, \end{matrix}$ if $r^{\pm}$ is small enough (i.e., where $χ_{L}^{\pm} = 1$ ). In this case, the coefficients $ℓ_{q}^{\pm} (s_{- m, 0}^{+})$ can be computed as $\begin{matrix} (3.21) & ℓ_{q}^{\pm} (s_{- m, 0}^{+}) = {(r^{\pm})}^{- \frac{2 q}{3}} \int_{I^{\pm}} {\tilde{s}}_{- m, 0}^{+} (r^{\pm}, θ^{\pm}) w_{q, 0, \pm} (θ^{\pm}) d θ^{\pm} . \end{matrix}$ Remark 3.7.
It is known [26, Chapter 6] that any function $v \in V_{2, β}^{2} (Ω_{T}) \cap V_{2, β}^{2} (Ω_{B})$ for $β > 1 + \frac{2 m}{3}$ satisfying $Δ v = 0$ in Ω is a linear combination of the functions $s_{- k, 0}^{\pm}$ , $1 ⩽ k ⩽ m$ .

3.3.2. The families $s_{m, q}^{\pm}$ , $m \in N ∖ {0}$ , $q \in N ∖ {0}$

In order to construct the macroscopic terms, it is useful to introduce the family of functions $s_{- m, q}^{+}$ , $(m, q) \in {(N^{*})}^{2}$ (remember that $N^{*} = N ∖ {0}$ ), corresponding to the ‘propagation’ of $s_{- m, 0}^{+}$ (recursively) through the transmissions conditions (3.1b), (3.1c):

Proposition 3.8.
For any $(m, q) \in {(N^{})}^{2}$ there exists a unique function* $s_{- m, q}^{+} \in V_{2, β}^{2} (Ω_{T}) \cap V_{2, β}^{2} (Ω_{B})$ for $β > 1 + \frac{2 m}{3} + q$ satisfying $\begin{matrix} (3.22) & \{\begin{matrix} - Δ s_{- m, q}^{+} = 0 in Ω_{T} \cap Ω_{B}, \\ s_{- m, q}^{+} = 0 on Γ_{D}, \\ {[s_{- m, q}^{+} (x_{1}, 0)]}_{Γ} = \sum_{p = 1}^{q} D_{p}^{t} \partial_{x_{1}}^{p} {⟨ s_{- m, q - p}^{+} (x_{1}, 0) ⟩}_{Γ} \\ + \sum_{p = 1}^{q} D_{p}^{n} \partial_{x_{1}}^{p - 1} {⟨ \partial_{x_{2}} s_{- m, q - p}^{+} (x_{1}, 0) ⟩}_{Γ}, \\ {[\partial_{x_{2}} s_{- m, q}^{+} (x_{1}, 0)]}_{Γ} = \sum_{p = 1}^{q} N_{p + 1}^{t} \partial_{x_{1}}^{p + 1} {⟨ s_{- m, q - p}^{+} (x_{1}, 0) ⟩}_{Γ} \\ + \sum_{p = 1}^{q} N_{p + 1}^{n} \partial_{x_{1}}^{p} {⟨ \partial_{x_{2}} s_{- m, q - p}^{+} (x_{1}, 0) ⟩}_{Γ}, \end{matrix} \end{matrix}$ which admits the following decompositions
For any $k \in N$ , there exists a function $r_{- m, q, k, +}^{+}$ belonging to $V_{2, β^{'}}^{2} (Ω_{B}) \cap V_{2, β^{'}}^{2} (Ω_{T})$ for any $β^{'} > 1 - \frac{2 (k + 1)}{3}$ and real constants $ℓ_{n}^{+} (s_{- m, q - p}^{+})$ , $0 ⩽ p ⩽ q$ , $1 ⩽ n < (k + 1) + \frac{3}{2} p$ such that $\begin{array}{rcl} s_{- m, q}^{+} & = & {(r^{+})}^{- \frac{2 m}{3} - q} w_{- m, q, +} (θ^{+}, ln r^{+}) \\ (3.23) & + \sum_{p = 0}^{q} \sum_{1 ⩽ n < (k + 1) + \frac{3}{2} p} ℓ_{n}^{+} (s_{- m, q - p}^{+}) {(r^{+})}^{λ_{n} - p} w_{n, p, +} (θ^{+}, ln r^{+}) + r_{- m, q, k, +}^{+}, \end{array}$ where $w_{n, 0, +} (θ^{+}, ln r^{+}) = w_{n, 0, +} (θ^{+})$ are given in Proposition 3.2 and, for $p ⩾ 1$ , $w_{n, p, +} (θ^{+}, ln r^{+})$ are polynomials in $ln r^{+}$ whose coefficients (functions of $θ^{+}$ ) belong to $C^{\infty} ([0, π]) \cap C^{\infty} ([π, \frac{3 π}{2}])$ (here $[a, b]$ denotes the closure of the interval $(a, b)$ ) .

For any $k \in N$ , there exists a function $r_{- m, q, k, -}^{+}$ belonging to $V_{2, β^{'}}^{2} (Ω_{T}) \cap V_{2, β^{'}}^{2} (Ω_{B})$ for any $β^{'} > 1 - \frac{2 (k + 1)}{3}$ and real constants $ℓ_{n}^{-} (s_{- m, q - p}^{+})$ , $0 ⩽ p ⩽ q$ , $1 ⩽ n < (k + 1) + \frac{3}{2} p$ such that $\begin{matrix} (3.24) & s_{- m, q}^{+} = \sum_{p = 0}^{q} \sum_{1 ⩽ n < (k + 1) + \frac{3}{2} p} ℓ_{n}^{-} (s_{- m, q - p}^{+}) {(r^{-})}^{λ_{n} - p} w_{n, p, -} (θ^{-}, ln r^{-}) + r_{- m, q, k, -}^{+}, \end{matrix}$ where $w_{n, 0, -} (θ^{-}, ln r^{-}) = w_{n, 0, -} (θ^{-})$ (see Proposition 3.2 ) and $w_{n, p, -} (θ^{-}, ln r^{-})$ are polynomials in $ln r^{-}$ whose coefficients (functions of $θ^{-}$ ) belong to $C^{\infty} ([0, \frac{π}{2}]) \cap C^{\infty} ([\frac{π}{2}, \frac{3 π}{2}])$ .

A complete proof of Proposition 3.8 can be found in [18] (Appendix C). It is strongly based on the explicit resolution of the Laplace equation in so-called infinite conical domains for particular right-hand sides of the form $r^{λ} {(ln r)}^{n}$ , $λ \in R$ , $n \in N$ (see Section 6.4.2 in [26] for similar results). As for $s_{- m, 0}^{+}$ , the proof consists of constructing an explicit lift of the singular part of the jump values (3.23) in order to reduce the problem to a variational one (as already done for $s_{- m, 0}^{+}$ ). Remark 3.9.
In the same way, for each $m \in N^{}$ we can define by induction a sequence of functions ${(s_{- m, q}^{-})}_{q \in N^{}}$ as follows: $s_{- m, q}^{-}$ is the unique function belonging to $V_{2, β}^{2} (Ω_{T}) \cap V_{2, β}^{2} (Ω_{B})$ for any $β > 1 + \frac{2 m}{3} + q$ that satisfies the transmission problem obtained from (3.22) by substituting $s_{- m, q - p}^{+}$ for $s_{- m, q - p}^{-}$ in the jump conditions, and the asymptotic expansions obtained interchanging (3.23) and (3.24), replacing the superscripts plus by superscripts minus.

3.3.3. Annotations to the singular functions

Let us comment the results of the previous proposition and of Proposition 3.6:

For $m > 0$ fixed, the family ${(s_{- m, q}^{+})}_{q \in N}$ provides particular singular solutions to (3.1).

The exponents λ of $r^{-}$ and $r^{+}$ appearing in the asymptotic expansions (3.23), (3.24) are singular exponents $λ \in Λ$ as well as ‘shifted’ singular exponents of the form $λ = λ_{n} - p$ , $λ_{n} \in Λ$ , where the integer p is between 1 and q. The most singular part of $s_{- m, q}^{+}$ in the vicinity of $x_{O}^{+}$ is ${(r^{+})}^{- \frac{2}{3} m - q} w_{- m, q, +} (θ^{+}, ln r^{+})$ , while the most singular part of $s_{- m, q}^{+}$ in the vicinity of $x_{O}^{-}$ is $ℓ_{1}^{-} (s_{- m, 0}^{+}) {(r^{-})}^{\frac{2}{3} - q} w_{1, q, -} (θ^{-}, ln r^{-})$ . Consequently $s_{- m, q}^{+}$ is ‘less singular’ in the vicinity of the left corner than in the vicinity of the right one.

The function $s_{- m, q}^{+}$ depends only on the functions $s_{- m, p}^{+}$ for $p ⩽ q$ . In others words, providing that $n \neq m$ , the definition of the families ${s_{- m, q}^{+}, q \in N}$ , ${s_{- n, q}^{+}, q \in N}$ are independently defined.

The functions $w_{n, q, \pm}$ are defined in (A.7). However, for the forthcoming derivation of the asymptotic expansion and its analysis their explicit expression is not important. Even so these functions will appear again in the definition of the near field singularities (see Lemma 4.3).

Problem (3.22) alone does not uniquely determine the function $s_{- m, q}^{+}$ . Indeed, in view of Remark 3.7 the solution of (3.22) is defined only up to a linear combination of ${s_{- n, 0}^{\pm}, n ⩽ m + \frac{3}{2} q}$ . However, imposing additionally the singular behavior close to the corners given by (3.23) and (3.24) (using the fact that the functions $w_{n, q, \pm}$ are uniquely determined) restores the uniqueness (cf. Remark 3.7).

For a given $k \in N$ , the constants $ℓ_{n}^{\pm} (s_{- m, q}^{+})$ , $1 ⩽ q ⩽ k$ , and the remainders $r_{- m, q, k, \pm}^{+}$ satisfy an estimate of the form (3.20) that has been omitted for the sake of concision.

The constants $ℓ_{n}^{\pm} (s_{- m, q}^{+})$ are intrinsic to the singularity functions and can be obtained by similar kind of formulas as (3.21). Each constant $ℓ_{n}^{\pm} (s_{- m, q}^{+})$ appears in the decomposition of several singularity functions in (3.23) and (3.24).

3.4. An explicit expression for the macroscopic terms

These part is dedicated to the derivation of a quasi-explicit formula of the macroscopic terms $u_{n, q}^{δ}$ by introducing particular solutions to problem (3.1). As mentioned before, we shall allow $u_{n, q}^{δ}$ to be singular. In view of the previous construction, we shall impose that $\begin{array}{rcl} u_{n, q}^{δ} \in V_{2, β}^{2} (Ω_{T}) \cap V_{2, β}^{2} (Ω_{B}) for any β > 1 + \frac{2 n}{3} + q, n > 0, \\ u_{0, q}^{δ} \in V_{2, β}^{2} (Ω_{T}) \cap V_{2, β}^{2} (Ω_{B}) for any β > \frac{1}{3} + q . \end{array}$

3.4.1. The macroscopic terms $u_{0, q}^{δ}$ , $q \in N$

We remind that the limit macroscopic field $u_{0, 0}$ (remember that $u_{0, 0}^{δ} = u_{0, 0} \in H_{0, Γ_{D}}^{1} (Ω)$ ) satisfies problem (3.2) with $f = f \in L^{2} (Ω_{T}) \cap L^{2} (Ω_{B})$ , $g = 0$ and $h = 0$ (see Section 3.2.1). In this subsection we define the functions $u_{0, q}^{δ} = u_{0, q}$ in the large class of possible singular solutions of (3.1) by imitating the iterative procedure of the previous subsection for the definition of the singular functions $s_{- m, q}^{+}$ (i.e., by ‘propagating’ $u_{0, 0}$ (recursively) through the transmission conditions (3.1b), (3.1c)), by which in turn no additional singular functions are added.

Proposition 3.10.
For any $q \in N^{}$ there exists a unique function* $u_{0, q}^{δ} = u_{0, q} \in V_{2, β}^{2} (Ω_{T}) \cap V_{2, β}^{2} (Ω_{B})$ for $β > \frac{1}{3} + q$ of (3.1) which admits the following decompositions:
For any $k \in N$ , there exists a function $r_{0, q, k, +}^{+}$ belonging to $V_{2, β^{'}}^{2} (Ω_{B}) \cap V_{2, β^{'}}^{2} (Ω_{T})$ for $β^{'} > 1 - \frac{2 (k + 1)}{3}$ and real constants $ℓ_{n}^{+} (u_{0, q - p})$ , $1 ⩽ p ⩽ q$ , $1 ⩽ n < (k + 1) + \frac{3}{2} p$ such that $\begin{matrix} (3.25) & u_{0, q} = \sum_{p = 0}^{q} \sum_{1 ⩽ n < (k + 1) + \frac{3}{2} p} ℓ_{n}^{+} (u_{0, q - p}) {(r^{+})}^{λ_{n} - p} w_{n, p, +} (θ^{+}, ln r^{+}) + r_{0, q, k, +}^{+} (r^{+}, θ^{+}) . \end{matrix}$

Analogously, for any $k \in N$ , there exists a function $r_{0, q, k, -}^{+}$ belonging to $V_{2, β^{'}}^{2} (Ω_{T}) \cap V_{2, β^{'}}^{2} (Ω_{B})$ for any $β^{'} > 1 - \frac{2 (k + 1)}{3}$ and real constants $ℓ_{n}^{-} (u_{0, q - p})$ , $1 ⩽ p ⩽ q$ , $1 ⩽ n < (k + 1) + \frac{3}{2} p$ such that $\begin{matrix} (3.26) & u_{0, q} = \sum_{p = 0}^{q} \sum_{1 ⩽ n ⩽ (k + 1) + \frac{3}{2} p} ℓ_{n}^{-} (u_{0, q - p}) {(r^{-})}^{λ_{n} - p} w_{n, p, -} (θ^{-}, ln r^{-}) + r_{0, q, k, -}^{+} (r^{-}, θ^{-}) . \end{matrix}$

Note that the functions $w_{n, p, \pm}$ in (3.25) and (3.26) were used already in Proposition 3.8 and are defined in (A.7). Similar to the singular functions the constants $ℓ_{n}^{\pm} (u_{0, p})$ are intrinsic and fixed, when $u_{0, 0}$ is fixed. From now on we consider the macroscopic terms $u_{0, q}^{δ} = u_{0, q}$ to be defined by Proposition 3.10.
3.4.2. The macroscopic terms $u_{n, q}^{δ}$ , $n \in N ∖ {0}$ , $q \in N$

We construct $u_{n, q}^{δ}$ as follows:

Proposition 3.11.
Let $n > 0$ . For any $p \in N$ , let $ℓ_{- k}^{\pm} (u_{n, p}^{δ})$ , $1 ⩽ k ⩽ n$ be $2 n$ given real constants. Then, the family of functions $\begin{matrix} (3.27) & u_{n, q}^{δ} = \sum_{\pm} \sum_{p = 0}^{q} \sum_{k = 1}^{n} ℓ_{- k}^{\pm} (u_{n, p}^{δ}) s_{- k, q - p}^{\pm}, q \in N, \end{matrix}$ satisfies the family of macroscopic problems (3.1). Moreover, the term $u_{n, q}^{δ}$ belongs to $V_{2, β}^{2} (Ω_{T}) \cap V_{2, β}^{2} (Ω_{B})$ for any $β > 1 + \frac{2 n}{3} + q$ .

We remind that the functions $s_{- m, q}^{+}$ are defined in Proposition 3.5 for $q = 0$ and Proposition 3.8 for $q > 0$ and $s_{- m, q}^{-}$ are defined in (3.17) ( $q = 0$ ) and Remark 3.9 ( $q > 0$ ).
Proof.
The ‘most singular’ part of $u_{n, q}^{δ}$ , $n > 0$ , defined by (3.27) corresponds to $\sum_{\pm} ℓ_{- n}^{\pm} (u_{n, 0}^{δ}) s_{- n, q}^{\pm}$ , which, in view of Proposition 3.8 belongs to $V_{2, β}^{2} (Ω_{T}) \cap V_{2, β}^{2} (Ω_{B})$ for any $β > 1 + \frac{2 n}{3} + q$ . As a consequence $u_{n, q}^{δ}$ belongs to $V_{2, β}^{2} (Ω_{T}) \cap V_{2, β}^{2} (Ω_{B})$ for any $β > 1 + \frac{2 n}{3} + q$ . Next, let us show by induction on q that the family ${(u_{n, q}^{δ})}_{q \in N}$ is a particular solution to the family of problems (3.1). The base step ( $q = 0$ ) is trivial. For the induction step, it is clear that $u_{n, q}^{δ}$ is harmonic in $Ω_{T}$ and in $Ω_{B}$ and fulfills homogeneous Dirichlet boundary conditions on $Γ_{D}$ . It remains to show the jump conditions across Γ. Substituting the definition (3.27) into (3.1b) we can assert that $\begin{array}{rcl} {[u_{n, q}^{δ}]}_{Γ} & = & \sum_{\pm} \sum_{p = 0}^{q} \sum_{k = 1}^{n} ℓ_{- k}^{\pm} (u_{n, p}^{δ}) \sum_{r = 1}^{q - p} (D_{r}^{t} \partial_{x_{1}}^{r} {⟨ s_{- k, q - p - r} ⟩}_{Γ} + D_{r}^{n} \partial_{x_{1}}^{r - 1} {⟨ \partial_{x_{2}} s_{- k, q - p - r} ⟩}_{Γ}) . \end{array}$ Interchanging the sum over r and p, using the induction hypothesis, we get the expected jump: $\begin{array}{rcl} {[u_{n, q}^{δ}]}_{Γ} & = & \sum_{r = 1}^{q} (D_{r}^{t} \partial_{x_{1}}^{r} + D_{r}^{n} \partial_{x_{1}}^{r - 1}) \sum_{\pm} \sum_{p = 0}^{q - r} \sum_{k = 1}^{n} ℓ_{- k}^{\pm} (u_{n, p}^{δ}) {⟨ s_{- k, q - p - r} ⟩}_{Γ} \\ = & \sum_{r = 1}^{q} (D_{r}^{t} \partial_{x_{1}}^{r} + D_{r}^{n} \partial_{x_{1}}^{r - 1}) {⟨ u_{n, q - r}^{δ} ⟩}_{Γ} . \end{array}$ The condition for the normal jump follows accordingly, and the proof is complete. □

Let $n > 0$ and $q \in N$ be fixed. We emphasize that the function $u_{n, q}^{δ}$ (defined by (3.27)) is determined up to the specification of the $2 n$ constants $ℓ_{- k}^{\pm} (u_{n, q}^{δ})$ , $1 ⩽ k ⩽ n$ (there are $2 n$ degrees of freedom). The matching procedure will provide a way to choose these constants in order to ensure the matching of far and near field expansions in the matching areas.
3.4.3. Expression for the boundary layers correctors

Assume now that $u_{n, q}^{δ}$ is defined by (3.27). Then, inserting this definition into the formula (2.30) defining the boundary layer correctors $Π_{n, q}^{δ}$ , we find them to be given by $\begin{matrix} Π_{0, q}^{δ} = Π_{0, q} = \sum_{p = 0}^{q} W_{p}^{t} \partial_{x_{1}}^{p} {⟨ u_{0, q - p} (x_{1}, 0) ⟩}_{Γ} + \sum_{p = 1}^{q} W_{p}^{n} \partial_{x_{1}}^{p - 1} {⟨ \partial_{x_{2}} u_{0, q - p} (x_{1}, 0) ⟩}_{Γ}, \end{matrix}$ and, for $n > 0$ , $\begin{array}{rcl} Π_{n, q}^{δ} & = & \sum_{\pm} \sum_{k = 1}^{n} (\sum_{p = 0}^{q} W_{p}^{t} \sum_{r = 0}^{q - p} ℓ_{- k}^{\pm} (u_{n, r}^{δ}) \partial_{x_{1}}^{p} {⟨ s_{- k, q - p - r}^{\pm} ⟩}_{Γ} \\ + \sum_{p = 1}^{q} W_{p}^{n} \sum_{r = 0}^{q - p} ℓ_{- k}^{\pm} (u_{n, r}^{δ}) \partial_{x_{1}}^{p - 1} {⟨ \partial_{x_{2}} s_{- k, q - p - r}^{\pm} ⟩}_{Γ}) . \end{array}$

3.4.4. Asymptotic of the far field terms close to the corners

Thanks to the previous formulas, we have a complete asymptotic expansion describing the behavior of both macroscopic and boundary layer correctors terms in the vicinity of the reentrant corners: for any $k \in N$ there exists a function $u_{n, q, k, +}$ belonging to $V_{2, β}^{2} (Ω_{T}) \cap V_{2, β}^{2} (Ω_{B})$ for any $β > 1 - \frac{2 (k + 1)}{3}$ such that $\begin{matrix} (3.28) & u_{n, q}^{δ} = \sum_{r = 0}^{q} \sum_{m = - n}^{k + \frac{3}{2} r} a_{n, q - r, m, +}^{δ} {(r^{+})}^{\frac{2 m}{3} - r} w_{m, r, +} (θ^{+}, ln r^{+}) + u_{n, q, k, +}, \end{matrix}$ where, for any $(n, j, m) \in N^{2} \times Z$ , $\begin{matrix} (3.29) & a_{0, j, m, +}^{δ} = ℓ_{m}^{+} (u_{0, j}), a_{n, j, m, +}^{δ} = \sum_{\pm} \sum_{k = max (1, - m)}^{n} \sum_{p = 0}^{j} ℓ_{- k}^{\pm} (u_{n, p}^{δ}) ℓ_{m}^{+} (s_{- k, j - p}^{\pm}), n > 0 . \end{matrix}$ Here, we have used the convention that $w_{0, r, \pm} = 0$ for any $r \in N$ , that $ℓ_{m}^{\pm} (s_{- k, p}^{+}) = 0$ for any $m < - k$ and $p \in N$ . Moreover, the notation $\sum_{m = - n}^{k + \frac{3}{2} r}$ denotes the sum over the integers $m \in Z$ such that $- n ⩽ m ⩽ k + \frac{3}{2} r$ (the integer index m does not exceed $⌊ k + \frac{3}{2} r ⌋$ , where $⌊ a ⌋$ and $⌈ a ⌉$ denote the largest integer not greater or the smallest integer not less than a, respectively).

For $m < 0$ , the expression of $a_{n, j, m, +}$ can be simplified since $ℓ_{m}^{+} (s_{- k, q - p - r}^{-})$ vanishes, and $ℓ_{m}^{+} (s_{- k, q - p - r}^{+}) = 0$ unless $k = - m$ and $q - p - r = 0$ (in the latter case, $ℓ_{m}^{+} (s_{m, 0}^{+}) = 1$ ): $\begin{matrix} (3.30) & a_{n, j, m, +}^{δ} = ℓ_{m}^{+} (u_{n, j}^{δ}) . \end{matrix}$ One can also give an asymptotic expansion for $Π_{n, q}^{δ}$ for ${(x_{O}^{+})}_{1} - x_{1}$ sufficiently small (we remind that $x_{O}^{\pm} = ({(x_{O}^{\pm})}_{1}, {(x_{O}^{\pm})}_{2})$ denotes the coordinates of the vertice $x_{O}^{\pm}$ ): $\begin{matrix} (3.31) & Π_{n, q}^{δ} = \sum_{r = 0}^{q} \sum_{m = - n}^{k + \frac{3}{2} r} {({(x_{O}^{+})}_{1} - x_{1})}^{\frac{2 m}{3} - r} a_{n, q - r, m, +}^{δ} p_{m, r, +} (ln ({(x_{O}^{+})}_{1} - x_{1}), \frac{x_{1}}{δ}, \frac{x_{2}}{δ}) + Π_{n, q, k, +}, \end{matrix}$ where, $\begin{array}{rcl} p_{m, r, +} (ln t, \frac{x_{1}}{δ}, \frac{x_{2}}{δ}) \\ (3.32) & = \sum_{p = 0}^{r} g_{m, r - p, p, +}^{t} (ln t) W_{p}^{t} (\frac{x_{1}}{δ}, \frac{x_{2}}{δ}) + \sum_{p = 1}^{r} g_{m, r - p, p, +}^{n} (ln t) W_{p}^{n} (\frac{x_{1}}{δ}, \frac{x_{2}}{δ}) . \end{array}$ The functions $g_{m, r, q, +}^{n}$ and $g_{m, r, q, +}^{t}$ are polynomials in $ln t$ . Their definitions are given in (A.13), (A.14). The remainder $Π_{n, q, k, +}$ can be written as $\begin{matrix} (3.33) & Π_{n, q, k, +} (x_{1}, X) = \sum_{p = 0}^{q} ⟨ w_{n, q, p}^{t} (x_{1}, 0) ⟩ W_{p}^{t} (X) + \sum_{p = 0}^{q} ⟨ w_{n, q, p}^{n} (x_{1}, 0) ⟩ W_{p}^{n} (X), \end{matrix}$ where one can verify (using a weighted elliptic regularity argument, see [26, Corollary 6.3.3]) that the functions $w_{n, q, p}^{t}$ and $w_{n, q, p}^{n}$ belong to $V_{2, β}^{2} (Ω_{T}) \cap V_{2, β}^{2} (Ω_{B})$ for any $β > 1 - \frac{2 (k + 1)}{3}$ .

Naturally, similar asymptotic expansions occur in the vicinity of the left corner.

4. Analysis of the near field equations and near field singularities

The near field terms $U_{n, q, \pm}^{δ}$ satisfy Laplace problems (see (1.18)) posed in the unbounded domain ${\hat{Ω}}^{\pm}$ (defined in (1.12)) of the form $\begin{matrix} (4.1) & \{\begin{matrix} - Δ u = f & in {\hat{Ω}}^{\pm}, \\ u = 0 & on \partial K^{\pm}, \\ \partial_{n} u = g & on {\hat{Γ}}_{hole}^{\pm} = \partial {\hat{Ω}}^{\pm} ∖ \partial K^{\pm} . \end{matrix} \end{matrix}$ In this section, we first present a functional framework to solve the model problem (4.1) (Section 4.1). We pay particular attention to the asymptotic behavior of the solutions at infinity (Proposition 4.5). Based on this result, we construct two families $S_{q}^{\pm}$ , $q \in N^{*}$ , of ‘near field’ singularities, i.e., solutions to (4.1) with $f = 0$ but growing at infinity as ${(R^{\pm})}^{\frac{2 q}{3}}$ (Section 4.2). Finally, we use these singularities to write a quasi-explicit formula (see (4.21)) for the near fields terms $U_{n, q, \pm}^{δ}$ (Section 4.3). Here again, most of the results are explained for the problems posed in ${\hat{Ω}}^{+}$ but similar results hold for ${\hat{Ω}}^{-}$ .

4.1. General results of existence and asymptotics of the solution at infinity

4.1.1. Variational framework

As fully described in Section 3.3 in [10], the standard space to solve problem (4.1) is $\begin{matrix} (4.2) & V ({\hat{Ω}}^{+}) = {v \in H_{loc}^{1} ({\hat{Ω}}^{+}), \nabla v \in L^{2} ({\hat{Ω}}^{+}), \frac{v}{\sqrt{1 + {(R^{+})}^{2}}} \in L^{2} ({\hat{Ω}}^{+}), v = 0 on \partial K^{+}}, \end{matrix}$ which, equipped with the norm $\begin{matrix} (4.3) & ‖ v ‖_{V ({\hat{Ω}}^{+})} = {({‖ \frac{v}{\sqrt{1 + {(R^{+})}^{2}}} ‖}_{L^{2} ({\hat{Ω}}^{+})}^{2} + ‖ \nabla v ‖_{L^{2} ({\hat{Ω}}^{+})}^{2})}^{1 / 2}, \end{matrix}$ is a Hilbert space. The variational problem associated with problem (4.1) is the following: $\begin{matrix} find u \in V ({\hat{Ω}}^{+}) such that a (u, v) = \int_{{\hat{Ω}}^{+}} f (X) v (X) d X + \int_{{\hat{Γ}}_{hole}^{+}} g (X) v (X) d σ, \forall v \in V ({\hat{Ω}}^{+}), \end{matrix}$ where $a (u, v) = \int_{{\hat{Ω}}^{+}} \nabla u (X) \cdot \nabla v (X) d X$ . It is proved in [10, Proposition 3.6] (cf. also [39, Lemma 2.2]), that $\begin{matrix} \int_{{\hat{Γ}}_{hole}^{+}} g (X) v (X) d σ ⩽ C {‖ {(1 + {(R^{+})}^{2})}^{1 / 4} g ‖}_{L^{2} ({\hat{Γ}}_{hole}^{+})} ‖ v ‖_{V ({\hat{Ω}}^{+})}, \end{matrix}$ and that the bilinear form $a$ is coercive on $V ({\hat{Ω}}^{+})$ (the seminorm of the gradient $‖ \nabla v ‖_{L^{2} ({\hat{Ω}}^{+})}$ is a norm on $V ({\hat{Ω}}^{+})$ ). As a consequence, the following well-posedness result holds:

Proposition 4.1.
Assume that $\sqrt{1 + {(R^{+})}^{2}} f \in L^{2} ({\hat{Ω}}^{+})$ and ${(1 + {(R^{+})}^{2})}^{1 / 4} g \in L^{2} ({\hat{Γ}}_{hole}^{+})$ . Then, problem (4.1) has a unique solution $u \in V ({\hat{Ω}}^{+})$ .

4.1.2. Asymptotic expansion at infinity

As usual when dealing with matched asymptotic expansions, it is important (for the matching procedure) to be able to write an asymptotic expansion of the near fields as $R^{\pm}$ tends to infinity. In the present case, because of the presence of the thin layer of periodic holes this is far from being trivial: there is no separation of variables. However, Theorem 4.1 in [32] helps to answer this difficult question.

For the statement of the next results, we need to consider a new family of weighted Sobolev spaces. For $ℓ \in N$ (in the sequel, we shall only consider $ℓ \in {0, 1, 2}$ ), we introduce the space $V_{β, γ}^{ℓ} ({\hat{Ω}}^{+})$ defined as the completion of $C_{c}^{\infty} (\overline{{\hat{Ω}}^{+}})$ with respect to the norm $\begin{matrix} (4.4) & ‖ v ‖_{V_{β, γ}^{ℓ} ({\hat{Ω}}^{+})} = \sum_{p = 0}^{ℓ} {‖ {(1 + R^{+})}^{β - γ - δ_{p, 0}} ρ^{γ - ℓ + p + δ_{p, 0}} \nabla^{p} v ‖}_{L^{2} ({\hat{Ω}}^{+})} ρ = 1 + (1 + R^{+}) | θ^{+} - π | . \end{matrix}$ The norm $‖ \cdot ‖_{V_{β, γ}^{2} ({\hat{Ω}}^{+})}$ is a non-uniform weighted norm. The weight varies with the angle $θ^{+}$ . Away from the periodic layer, i.e., for $| θ^{+} - π | ⩾ ε$ for some $ε > 0$ and $R^{+}$ sufficiently large, we recover the classical weighted Sobolev norm $V_{β}^{2} (K^{+})$ (cf. (3.5)): $\begin{matrix} (4.5) & ‖ v ‖_{V_{β}^{ℓ} (K^{+})} = {(\sum_{p = 0}^{ℓ} {‖ {(R^{+})}^{β - ℓ + p} \nabla^{p} v ‖}_{L^{2} (K^{+})}^{2})}^{1 / 2} . \end{matrix}$ Indeed, in this part $ρ \sim 1 + R^{+}$ for $R^{+} \to \infty$ . In contrast, close to the layer, i.e., for $θ^{+} \to π$ for $R^{+}$ fixed, we have $ρ \to 1$ , and the global weight in (4.4) becomes ${(1 + R^{+})}^{β - γ - δ_{p, 0}}$ .

In the classical weighted Sobolev norm (4.5), the weight ${(R^{+})}^{β - ℓ + p}$ depends on the derivative ( $p = 0$ or $p = 1$ ) under consideration. It increases by one at each derivative. This is linked to the fact that the gradient of a function of the form ${(R^{+})}^{λ} g (θ)$ , which is given by ${(R^{+})}^{λ - 1} (λ g (θ) e_{r} + g^{'} (θ) e_{θ})$ , decays more rapidly than the function itself as $R^{+}$ tends to $+ \infty$ (comparing ${(R^{+})}^{λ - 1}$ and ${(R^{+})}^{λ}$ ). This property does not hold anymore for a function of the form ${(X_{1}^{+})}^{λ} g (X_{1}^{+}, X_{2}^{+})$ where $X^{+} = (X_{1}^{+}, X_{2}^{+}) = R^{+} (cos θ^{+}, sin θ^{+})$ and $g \in V^{+} (B)$ (g is periodic with respect to $X_{1}^{+}$ and exponentially decaying with respect to $X_{2}^{+}$ ). Indeed, in this case $\begin{matrix} \nabla ({(X_{1}^{+})}^{λ} g) = (λ {(X_{1}^{+})}^{λ - 1} g + {(X_{1}^{+})}^{λ} \partial_{X_{1}^{+}} g) e_{1} + {(X_{1}^{+})}^{λ} \partial_{X_{2}^{+}} g e_{2}, \end{matrix}$ which does not decrease as ${(X_{1}^{+})}^{λ - 1}$ . This remark gives a first intuition of the necessity to introduce a weighted space with a weight adjusted in the vicinity of the periodic layer, i.e., for $θ^{+} \to π$ . Note that in the case of Dirichlet boundary conditions on the holes, the appropriate weighted space to consider is slightly different (see [30]).

To be used later we mention the following properties related to these new function spaces (see [18], Section D.1, for the proof):

Proposition 4.2.
Let $γ \in (\frac{1}{2}, 1)$ , $p \in {1, 2}$ , $λ \in R$ , $q \in N$ . Let χ and $χ_{\pm}$ be the cut-off functions defined in (1.8) and (1.17) .
The function $v_{1} = χ^{(p)} (X_{2}^{+}) χ_{-} (X_{1}^{+}) {(R^{+})}^{λ} {(ln R^{+})}^{q}$ belongs to $V_{β, γ}^{0} ({\hat{Ω}}^{+})$ providing that $\begin{matrix} β < γ - λ + 1 / 2 . \end{matrix}$

The function $v_{2} = χ (R^{+}) {(R^{+})}^{λ} {(ln R^{+})}^{q}$ belongs to $V_{β, γ}^{2} ({\hat{Ω}}^{+})$ providing that $\begin{matrix} β < 1 - λ . \end{matrix}$

Let $w = w (X_{1}^{+}, X_{2}^{+})$ be a 1-periodic function with respect to $X_{1}^{+}$ such that $‖ w e^{| X_{2}^{+} | / 2} ‖_{L^{2} (B)} < + \infty$ . Then, the function $v_{3} = χ_{-} (X_{1}^{+}) | X_{1}^{+} |^{λ - 1} {(ln | X_{1}^{+} |)}^{q} w (X_{1}^{+}, X_{2}^{+})$ belongs to $V_{β, γ}^{0} ({\hat{Ω}}^{+})$ providing that $\begin{matrix} β < γ - λ + 3 / 2 . \end{matrix}$

Let $w \in V^{+} (B) \cap H_{loc}^{2} (B)$ such that $\begin{matrix} \int_{B} ({| \partial_{X_{1}^{+}}^{2} w |}^{2} + {| \partial_{X_{2}^{+}}^{2} w |}^{2} + | \partial_{X_{1}^{+}} \partial_{X_{2}^{+}} w |^{2}) e^{| X_{2}^{+} |} d X^{+} < + \infty . \end{matrix}$

Then, the function $v_{4} = χ_{-} (X_{1}^{+}) | X_{1}^{+} |^{λ - 1} {(ln R^{+})}^{q} w (X_{1}^{+}, X_{2}^{+})$ belongs to $V_{β, γ}^{2} ({\hat{Ω}}^{+})$ providing that $\begin{matrix} β < γ - λ + 1 / 2 . \end{matrix}$

In absence of the periodic layer the solutions of the near field equations might be written as linear combination of harmonic functions ${(R^{+})}^{λ_{m}} w_{m, 0, +} (ln R^{+}, θ^{+})$ , $m \in N^{}$ for $R^{+} \to \infty$ where $w_{m, r, +}$ have been defined (A.7). With the periodic layer the behavior far above the layer remains the same, but has to be corrected by $| X_{1}^{+} |^{λ_{m}} p_{m, 0, +} (ln | X_{1}^{+} |, X_{1}^{+}, X_{2}^{+})$ (with $p_{m, r, +}$ defined in (3.32)) to fulfill the homogeneous boundary conditions on ${\hat{Γ}}_{hole}$ . This correction is not harmonic and has a particular decay rate for $R^{+} \to \infty$ . It can be (macroscopically) corrected by ${(R^{+})}^{λ_{m} - 1} w_{m, 1, +}$ and in the neighbourhood of the layer by $| X_{1}^{+} |^{λ_{m} - 1} p_{m, 1, +} (ln | X_{1}^{+} |, X_{1}^{+}, X_{2}^{+})$ . Then, through a consecutive correction in the form (the cut-off function χ has been defined in (1.8)) $\begin{matrix} (4.6) & \sum_{r = 0}^{p} ({(R^{+})}^{λ_{m} - r} w_{m, r, +} (ln R^{+}, θ^{+}) χ (X_{2}^{+}) + χ (X_{1}^{+}) {| X_{1}^{+} |}^{λ_{m} - r} p_{m, r, +} (ln | X_{1}^{+} |, X_{1}^{+}, X_{2}^{+})) \end{matrix}$ the Laplacian becomes more and more decaying for $R^{+} \to \infty$ , where any decay rate can be achieved, which becomes, at least formally, zero for $p \to \infty$ . The previous observation will be justified in a more rigorous form in the following lemma which turn out to be very useful in the sequel.

For this let us introduce a smooth cut-off function $χ_{macro, +}$ (see Fig. 4) that satisfies $\begin{matrix} (4.7) & χ_{macro, +} (X_{1}^{+}, X_{2}^{+}) = \{\begin{matrix} χ (X_{2}^{+}) & for X_{1}^{+} < - 1, \\ 1 & for X_{1}^{+} > - \frac{1}{4}, \\ 1 & for X_{1}^{+} > - 1, | X_{2}^{+} | > 3, \end{matrix} \end{matrix}$ and for $m \in Z ∖ {0}$ the asymptotic block (we adopt this notion from [32]) $\begin{array}{rcl} U_{m, p, +} (X_{1}^{+}, X_{2}^{+}) & = & χ (R^{+}) \sum_{r = 0}^{p} (χ_{macro, +} (X_{1}^{+}, X_{2}^{+}) {(R^{+})}^{λ_{m} - r} w_{m, r, +} (R^{+}, θ^{+}) \\ (4.8) & + χ_{-} (X_{1}^{+}) {| X_{1}^{+} |}^{λ_{m} - r} p_{m, r, +} (ln | X_{1}^{+} |, X_{1}^{+}, X_{2}^{+})), \end{array}$ where the cut-off function $χ_{-}$ has been defined in (1.17).
Lemma 4.3.
Under the condition* $γ \in (1 / 2, 1)$ the Laplacian $Δ U_{m, p, +}$ of the asymptotic block $U_{m, p, +}$ belongs to $V_{β, γ}^{0} ({\hat{Ω}}^{+})$ for any β that satisfies $\begin{matrix} (4.9) & β < 2 - λ_{m} + p . \end{matrix}$

Fig. 4.
Schematic representation of the cut-off function $χ_{macro, +}$ defined in (4.7).

The functions $p_{m, q, +}$ are polynomials in $ln | X_{1}^{+} |$ . They are periodic with respect to $X_{1}^{+}$ and exponentially decaying as $X_{2}^{+}$ tends to $\pm \infty$ . The functions $w_{m, q, +}$ are polynomials in $ln R^{+}$ . Note that, for $R^{+}$ large enough, if $X_{1}^{+} < - 1$ , $\begin{matrix} (4.10) & U_{m, p, +} = \sum_{r = 0}^{p} ({(R^{+})}^{λ_{m} - r} w_{m, r, +} (ln R^{+}, θ^{+}) χ (X_{2}^{+}) + {| X_{1}^{+} |}^{λ_{m} - r} p_{m, r, +} (ln | X_{1}^{+} |, X_{1}^{+}, X_{2}^{+})) \end{matrix}$ and, if $X_{1}^{+} > - 1$ , $\begin{matrix} (4.11) & U_{m, p, +} = \sum_{r = 0}^{p} {(R^{+})}^{λ_{m} - r} w_{m, r, +} . \end{matrix}$ The usage of the cut-off functions $χ_{macro, +}$ , $χ_{-} (X_{1}^{+})$ , and $χ (R^{+})$ in (4.8) is simply a technical way to construct function defined on the whole domain ${\hat{Ω}}^{+}$ (and more precisely belonging to $H_{loc}^{2} ({\hat{Ω}}^{+})$ ) that coincides with $U_{m, p, +}$ for large $R^{+}$ as given in (4.10) and (4.11). The proof of Lemma 4.3 is postponed in Section A.3.
Remark 4.4.
In view of Proposition 4.2, reminding that $p_{m, 0} (ln | X_{1}^{+} |, X_{1}^{+}, X_{2}^{+})$ is proportional to $(1 - χ (X_{2}^{+}))$ , for any $γ \in (1 / 2, 1)$ , the asymptotic block $U_{m, p, +}$ belongs to $V_{β, γ}^{2} ({\hat{Ω}}^{+})$ for any $β < 1 - λ_{m}$ .

Defining $χ_{macro, -} (X_{1}^{-}, X_{2}^{-}) = χ_{macro, +} (- X_{1}^{-}, X_{2}^{-})$ , we can also define the asymptotic blocks associated with ${\hat{Ω}}^{-}$ as follows: $\begin{array}{rcl} U_{m, p, -} & = & χ (R^{-}) \sum_{r = 0}^{p} (χ_{macro, -} (X_{1}^{-}, X_{2}^{-}) {(R^{-})}^{λ_{m} - r} w_{m, r, -} (ln R^{-}, θ^{-}) \\ (4.12) & + χ_{+} (X_{1}^{-}) {| X_{1}^{-} |}^{λ_{m} - r} p_{m, r, -} (ln | X_{1}^{-} |, X_{1}^{-}, X_{2}^{-})) . \end{array}$

We are now in a position to write the main result of this subsection, which proves that for $R^{+}$ large and for sufficiently decaying right-hand sides, the solutions of problem (4.1) can be decomposed into a sum of radial contributions corrected by periodic exponentially decaying correctors in the vicinity of the layer of equispaced holes. In the following, a real number β is said to be admissible if $β - 1 \notin Λ$ .
Proposition 4.5.
Let $k \in N$ and $γ \in (1 / 2, 1)$ . Assume that $f \in V_{β, γ}^{0} ({\hat{Ω}}^{+})$ for some admissible $β > max (3, 1 + \frac{2 (k + 1)}{3})$ (and so $\sqrt{1 + {(R^{+})}^{2}} f \in L^{2} ({\hat{Ω}}^{+})$ ) and that $g \in H^{1 / 2} ({\hat{Γ}}_{hole})$ is compactly supported. Then, the unique solution u of problem (4.1) belongs to $V ({\hat{Ω}}^{+}) \cap V_{β^{'}, γ}^{2} ({\hat{Ω}}^{+})$ for any $β^{'} < \frac{1}{2}$ . Moreover, u admits, for $γ - 1$ sufficiently small, the decomposition $\begin{matrix} (4.13) & u = \sum_{n = 1}^{k} L_{- n} (u) U_{- n, p (n), +} + \tilde{u}, p (n) = max (1, ⌈ \frac{2}{3} (k - n) ⌉), \end{matrix}$ where the asymptotic blocks $U_{- n, p (n), +}$ are defined in (4.8) , $L_{- n} (u)$ , $1 ⩽ n ⩽ k$ , denote k constants, and, the remainder $\tilde{u} \in V_{β^{0}, γ}^{2} ({\hat{Ω}}^{+})$ for any $β^{0}$ such that $β^{0} < β$ . In addition, there exists a constant $C > 0$ such that $\begin{matrix} (4.14) & ‖ \tilde{u} ‖_{V_{β^{0}, γ}^{2} ({\hat{Ω}}^{+})} + \sum_{n = 1}^{k} | L_{- n} (u) | ⩽ C (‖ f ‖_{V_{β, γ}^{0} ({\hat{Ω}}^{+})} + ‖ g ‖_{H^{1 / 2} ({\hat{Γ}}_{hole})}) . \end{matrix}$

We remind that $⌈ a ⌉$ (used in (4.13)) stands for the smallest integer not less than a.

The proof of Proposition 4.5 (see Section B.3 in [18]), deeply relies on successive applications of the following lemma, which is a direct adaptation of Theorem 4.1 in [32]. To long to be presented in this paper, its proof requires the use of involved tools of complex analysis that are fully described in [32].
Lemma 4.6.
Let $γ \in (\frac{1}{2}, 1)$ . Let $β^{1}$ and $β^{2}$ be two admissible exponents such that $β^{1} < β^{2}$ and $β^{2} - β^{1} < 1$ . Assume that $u \in V_{β^{1}, γ}^{2} ({\hat{Ω}}^{+})$ satisfies problem (4.1) with $f \in V_{β^{2}, γ}^{0} ({\hat{Ω}}^{+})$ and a compactly supported $g \in H^{1 / 2} ({\hat{Γ}}_{hole})$ . Then, for $γ - 1 < 0$ sufficiently small u admits the decomposition $\begin{matrix} (4.15) & u = \sum_{\frac{3}{2} (1 - β^{2}) < k < \frac{3}{2} (1 - β^{1})} c_{k} U_{k, 3, +} + \tilde{u}, \end{matrix}$ where the asymptotic blocks $U_{k, 3, +}$ are defined in (4.8) and for any admissible $β^{0} \in (β^{1}, β^{2})$ the remainder $\tilde{u} \in V_{β^{0}, γ}^{2} ({\hat{Ω}}^{+})$ . In this case, there exists a positive constant $C > 0$ such that $\begin{matrix} (4.16) & ‖ \tilde{u} ‖_{V_{β^{0}, γ}^{2} ({\hat{Ω}}^{+})} + \sum_{\frac{3}{2} (1 - β^{2}) < k < \frac{3}{2} (1 - β^{1})} | c_{k} | ⩽ C (‖ u ‖_{V_{β^{1}, γ}^{2} ({\hat{Ω}}^{+})} + ‖ f ‖_{V_{β^{2}, γ}^{0} ({\hat{Ω}}^{+})} + ‖ g ‖_{H^{1 / 2} ({\hat{Γ}}_{hole})}) . \end{matrix}$

We emphasize that the powers of $R^{+}$ (or $X_{1}^{+}$ ) appearing in (4.13) (see (4.8)) are of the form $- \frac{2}{3} n - q$ , $n \in N^{*}$ , $q \in N$ . Thus, they coincide with the ones obtained for the far field part (see for instance Proposition 3.8). Moreover, as can be expected, the ‘leading’ singular exponents $\frac{2 n}{3}$ correspond to those of the problem without periodic layer (here again, as for the macroscopic terms).
Remark 4.7.
The assumption on g in Proposition 4.5 and Lemma 4.6 could be weakened by using the trace spaces associated with the weighted Sobolev spaces $V_{β, γ}^{ℓ} ({\hat{Ω}}^{+})$ (see [32]).
4.2. Two families of near field singularities

This subsection is dedicated to the construction of two families of functions $S_{m}^{\pm}$ , $m \in N^{*}$ , hereinafter referred to as the near field singularities for the right and left corner, satisfying the homogeneous Poisson problems $\begin{matrix} (4.17) & \{\begin{matrix} - Δ S_{m}^{\pm} = 0 & in {\hat{Ω}}^{\pm}, \\ S_{m}^{\pm} = 0 & on \partial K^{\pm}, \\ \partial_{n} S_{m}^{\pm} = 0 & on \partial {\hat{Ω}}^{\pm} ∖ \partial K^{\pm}, \end{matrix} \end{matrix}$ and behaving like ${(R^{\pm})}^{λ_{m}} w_{m, 0, \pm} (θ^{^{\pm}})$ for $R^{\pm}$ large, where $w_{m, 0, +} = sin (\frac{2 n}{3} θ^{+})$ and $w_{m, 0, -} = sin (\frac{2 n}{3} (θ^{-} - \frac{π}{2}))$ (defined in Proposition 3.2).

Proposition 4.8.
There exists a unique function $S_{m}^{+} \in V_{β, γ}^{2} ({\hat{Ω}}^{+})$ for any $β < 1 - λ_{m}$ and $γ \in (1 / 2, 1)$ , satisfying the homogeneous equation (4.17) such that the function $\begin{matrix} (4.18) & {\tilde{S}}_{m}^{+} = S_{m}^{+} - U_{m, ⌈ 1 + λ_{m} ⌉, +} \end{matrix}$ belongs to $V ({\hat{Ω}}^{+})$ . Moreover, for any $k \in N^{}$ , choosing* $1 - γ$ sufficiently small, there exists a function $R_{m, k} \in V_{β^{0}, γ}^{2} ({\hat{Ω}}^{+})$ for any admissible $β^{0} < 1 + \frac{2 (k + 1)}{3}$ and k constants $L_{- n} (S_{m}^{+})$ ( $1 ⩽ k ⩽ n$ ) such that $S_{m}^{+}$ admits the decomposition $\begin{matrix} (4.19) & S_{m}^{+} = U_{m, ⌈ \frac{2 (k + m)}{3} ⌉, +} + \sum_{n = 1}^{k} L_{- n} (S_{m}^{+}) U_{- n, ⌈ \frac{2 (k - n)}{3} ⌉, +} + R_{m, k} . \end{matrix}$ In addition, for any $β < 1 - λ_{m}$ , there is a constant $C > 0$ such that $\begin{matrix} (4.20) & \sum_{n = 1}^{k} | L_{- n} (S_{m}^{+}) | + ‖ R_{m, k} ‖_{V_{2, β^{0}}^{2} ({\hat{Ω}}^{+})} ⩽ C {‖ S_{m}^{+} ‖}_{V_{2, β}^{2} ({\hat{Ω}}^{+})} . \end{matrix}$

In the same way as the near field singularities for the right corner $S_{m}^{+}$ , the near field singularities for the left corner $S_{m}^{-}$ can be defined. Note that for $m ⩾ 2$ there are several functions satisfying the homogeneous equations (4.17) and behaving like ${(R^{+})}^{λ_{m}} w_{m, 0, +}$ (leading term) for large $R^{+}$ . Indeed, admitting the existence of the functions $S_{m}^{+}$ , any function of the form $S_{m}^{+} + \sum_{k = 1}^{m - 1} a_{k} S_{k}^{+}$ would also fulfill these requirements. Nevertheless, the (4.19) restores the uniqueness by fixing (arbitrary) $a_{k}$ , $k = 1, \dots, m - 1$ to 0. Proof.
The proof is classical and is very similar to the proof of Propositions 3.5, 3.6 and 3.8. We first prove the existence of $S_{m}^{+}$ . The function ${\tilde{S}}_{m}^{+}$ satisfies $\begin{matrix} \{\begin{matrix} - Δ {\tilde{S}}_{m}^{+} = {\tilde{f}}_{m} & in {\hat{Ω}}^{+}, \\ {\tilde{S}}_{m}^{+} = 0 & on \partial K^{+}, \\ \partial_{n} {\tilde{S}}_{m}^{+} = 0 & on \partial {\hat{Ω}}^{+} ∖ \partial K^{+}, \end{matrix} {\tilde{f}}_{m} = - Δ U_{m, ⌈ 1 + λ_{m} ⌉, +} . \end{matrix}$ In view of Lemma 4.3, ${\tilde{f}}_{m}$ belongs to $V_{2, β}^{0} ({\hat{Ω}}^{+})$ for $β < 2 - λ_{m} + ⌈ 1 + λ_{m} ⌉$ . Noting that $2 - λ_{m} + ⌈ 1 + λ_{m} ⌉ ⩾ 3$ , Proposition 4.1 ensures the existence and uniqueness of ${\tilde{S}}_{m}^{+} \in V ({\hat{Ω}}^{+})$ , and, hence, the existence of $S_{m}^{+}$ . Uniqueness of $S_{m}^{+}$ follows directly from the fact that difference of two possible solutions is in the variational space $V ({\hat{Ω}}^{+})$ and satisfies (4.17). Finally, the asymptotic behavior for large $R^{+}$ results from a direct application of Proposition 4.5 to the function ${\tilde{S}}_{m}^{+} - U_{m, p, +}$ choosing p sufficiently large so that $Δ ({\tilde{S}}_{m}^{+} - U_{m, p, +})$ belongs to $V_{β, γ}^{2} ({\hat{Ω}}^{+})$ for a real number $β > max (3, 1 + \frac{2 (k + 1)}{3})$ (which is, thanks to Lemma 4.3, always possible). □

4.3. An explicit expression for the near field terms

As done for the macroscopic terms in Section 3.4, we can write a quasi-explicit formula for the near field terms $U_{n, q, \pm}^{δ}$ . We shall impose that the functions $U_{n, q, \pm}^{δ}$ do not blow up faster than ${(R^{+})}^{λ_{n}}$ for $R^{+} \to \infty$ . Since $U_{n, q, \pm}^{δ}$ satisfies the near field equations (1.18), it is natural to construct $U_{n, q, \pm}^{δ}$ as a linear combination of the near field singularities $S_{k}^{\pm}$ , $1 ⩽ k ⩽ n$ , namely $\begin{matrix} (4.21) & U_{n, q, \pm}^{δ} = \sum_{k = 1}^{n} L_{k} (U_{n, q, \pm}^{δ}) S_{k}^{\pm}, \end{matrix}$ where $L_{k} (U_{n, q, \pm}^{δ})$ are constants that will be determined by the matching procedure and that might depend on δ. Naturally, the functions $U_{n, q, \pm}$ (defined by (4.21)) satisfy the near field equations (1.18) and belong to $V_{β, γ}^{2} ({\hat{Ω}}^{\pm})$ for any $β < 1 - λ_{n}$ and $γ \in (1 / 2, 1)$ . It is worth noting that the definition (4.21) implies that $\begin{matrix} (4.22) & U_{0, q, \pm}^{δ} = 0 \forall q \in N . \end{matrix}$

Asymptotic behavior for large $R^{+}$ . To conclude this section, we slightly anticipate the upcoming matching procedure by writing the behavior of $U_{n, q, +}^{δ}$ at infinity. Thanks to the asymptotic behavior of $S_{m}^{+}$ for $R^{+} \to \infty$ (Proposition 4.8), we see that, for any $K \in N^{*}$ , $\begin{matrix} (4.23) & U_{n, q, +}^{δ} = \sum_{k = 1}^{n} \sum_{l = - k}^{K} L_{k} (U_{n, q, +}^{δ}) L_{- l} (S_{k}^{+}) U_{- l, ⌈ \frac{2 (K - l)}{3} ⌉, +} + R_{n, q, K, +}, \end{matrix}$ where $R_{n, q, K, +} \in V_{2, β^{0}}^{2} ({\hat{Ω}}^{+})$ for any $β^{0} < 1 + \frac{2 (K + 1)}{3}$ . Here, for the sake of concision, we have posed $\begin{matrix} L_{r} (S_{q}^{+}) = 0 for any r \in N such that 0 ⩽ r ⩽ q - 1, \end{matrix}$ and $U_{0, j, +} = 0$ for any $j \in N$ . Then, substituting the asymptotic blocks $U_{- l, k + 2 - l, +}$ for their explicit expression (4.8), we obtain the decomposition $\begin{array}{rcl} U_{n, q, +}^{δ} & = & χ_{macro, +} (X) \sum_{l = - n}^{K} \sum_{r = 0}^{⌈ \frac{2 (K - l)}{3} ⌉} A_{n, q, - l, +}^{δ} {(R^{+})}^{- \frac{2 l}{3} - r} w_{- l, r, +} (R^{+}, θ^{+}) \\ (4.24) & + χ_{-} (X_{1}^{+}) \sum_{l = - n}^{K} \sum_{r = 0}^{⌈ \frac{2 (K - l)}{3} ⌉} A_{n, q, - l, +}^{δ} {| X_{1}^{+} |}^{- \frac{2 l}{3} - r} p_{- l, r, +} (ln | X_{1}^{+} |, X_{1}^{+}, X_{2}^{+}) + R_{n, q, K, +}, \end{array}$ where we have used the convention $w_{0, r, +} = 0$ and $p_{0, r, +} = 0$ for any $r \in Z$ . Here, $\begin{matrix} (4.25) & \forall (n, q, l) \in N^{2} \times Z, A_{n, q, l, +}^{δ} = \sum_{k = max (1, l)}^{n} L_{k} (U_{n, q, +}^{δ}) L_{l} (S_{k}^{+}) . \end{matrix}$ Note that for $n = 0$ , $A_{n, q, l, +}^{δ} = 0$ . Moreover, for $l ⩾ 0$ , $L_{l} (S_{k}) = δ_{l, k}$ , and consequently $\begin{matrix} (4.26) & A_{n, q, l, +}^{δ} = L_{l} (U_{n, q, +}^{δ}) . \end{matrix}$ Finally, with the change of index $- l \to m$ and summing up over n and q, we can formally obtain an asymptotic series of the near field: For $X_{1}^{+} < - 1$ , $\begin{array}{rcl} \sum_{(n, q) \in N^{2}} δ^{\frac{2 n}{3} + q} U_{n, q, +}^{δ} & = & \sum_{(n, q) \in N^{2}} δ^{\frac{2 n}{3} + q} \sum_{m = - \infty}^{n} A_{n, q, m, +}^{δ} \sum_{r \in N} ({(R^{+})}^{\frac{2 m}{3} - r} w_{m, r, +} (θ^{+}, ln R^{+}) χ (X_{2}^{+}) \\ (4.27) & + {| X_{1}^{+} |}^{\frac{2 m}{3} - r} p_{m, r, +} (ln | X_{1}^{+} |, X^{+})) \end{array}$ and, for $X_{1}^{+} > - 1$ , $\begin{matrix} (4.28) & \sum_{(n, q) \in N^{2}} δ^{\frac{2 n}{3} + q} U_{n, q, +}^{δ} = \sum_{(n, q) \in N^{2}} δ^{\frac{2 n}{3} + q} \sum_{m = - \infty}^{n} A_{n, q, m, +}^{δ} \sum_{r \in N} {(R^{+})}^{\frac{2 m}{3} - r} w_{m, r, +} (θ^{+}, ln R^{+}) . \end{matrix}$

The near field terms $U_{n, q, -}^{δ}$ can be decomposed in strictly similar way by substituting formally the superscript plus into a superscript minus in (4.24) and (4.25). Here, it should be noted that the way to compute $L_{l} (U_{n, q, -}^{δ})$ is different how $L_{l} (U_{n, q, +}^{δ})$ are computed.

5. Matching procedure and construction of the far and near field terms

We are now in the position to write the matching conditions that account for the asymptotic coincidence of the far field expansion with the near field expansion in the matching areas. Based on the matching conditions, we provide an iterative algorithm to define all the terms of the far and near field expansion (to any order), which have not been fixed yet.

5.1. Far field expansion expressed in the microscopic variables

We start with writing the formal expansion of the far field $\sum_{(n, q) \in N^{2}} δ^{\frac{2 n}{3} + q} (u_{n, q}^{δ} (x) χ (x_{2} / δ) + Π_{n, q}^{δ} (x_{1}, \frac{x}{δ}))$ (cf. (1.7)) in the matching area located in the vicinity of the right corner (i.e. for small $r^{+}$ ). Collecting (3.28) and (3.31), summing over the pair of indices $(n, q) \in N^{2}$ and applying the change of scale $x^{+} / δ = (x - x_{O}^{+}) / δ \to X^{+}$ and so $r^{+} / δ \to R^{+}$ we formally obtain for $X_{1} < - 1$ , $\begin{array}{rcl} \sum_{(n, q) \in N} δ^{\frac{2 n}{3} + q} (u_{n, q}^{δ} (x) χ (\frac{x_{2}}{δ}) + Π_{n, q}^{δ} (x_{1}, \frac{x}{δ})) \\ = \sum_{(n, q) \in N^{2}} δ^{\frac{2}{3} n + q} \sum_{m = - \infty}^{n} a_{n - m, q, m, +}^{δ} \sum_{r \in N} ({(R^{+})}^{\frac{2 m}{3} - r} w_{m, r, +} (θ^{+}, ln R^{+} + ln δ) χ (X_{2}^{+}) \\ (5.1) & + {| X_{1}^{+} |}^{\frac{2 m}{3} - r} p_{m, r, +} (ln | X_{1}^{+} | + ln δ, X_{1}^{+}, X_{2}^{+})), \end{array}$ and for $X_{1} > - 1$ $\begin{matrix} (5.2) & \sum_{(n, q) \in N} δ^{\frac{2 n}{3} + q} u_{n, q}^{δ} (x) = \sum_{(n, q) \in N^{2}} δ^{\frac{2}{3} n + q} \sum_{m = - \infty}^{n} a_{n - m, q, m, +}^{δ} \sum_{r \in N} {(R^{+})}^{\frac{2 m}{3} - r} w_{m, r, +} (θ^{+}, ln R^{+} + ln δ) . \end{matrix}$ Note, that the coefficients $a_{n, q, m, +}^{δ}$ , defined in (3.29), depend for $n > 0$ on δ only through the constants $ℓ_{- k}^{\pm} (u_{n, p}^{δ})$ , which we are going to fix in the matching process. In the equations (5.1) and (5.2) the terms $w_{m, r, +}$ and $p_{m, r, +}$ appear with a second shifted argument, i.e., $ln R^{+} + ln δ$ instead of $ln R^{+}$ and $ln | X_{1}^{+} | + ln δ$ instead of $ln | X_{1}^{+} |$ . The following lemma is a reformulation of these terms as linear combinations of non-shifted ones and will prove very useful in the matching procedure. It is based essentially on the fact that the terms $w_{m, r, +}$ are polynomials in the second argument and $p_{m, r, +}$ in the first. The proof of the lemma finds itself in Section A.4.

Lemma 5.1.
The equalities $\begin{matrix} (5.3) & w_{m, r, \pm} (θ^{\pm}, ln R^{\pm} + ln δ) = \sum_{k = 0}^{⌊ r / 2 ⌋} w_{m - 3 k, r - 2 k, \pm} (θ^{\pm}, ln R^{\pm}) \sum_{i = 0}^{k} C_{m, 2 k, i, \pm} {(ln δ)}^{i}, \end{matrix}$ and $\begin{matrix} (5.4) & p_{m, r, \pm} (ln | X_{1}^{\pm} | + ln δ, X^{\pm}) = \sum_{k = 0}^{⌊ r / 2 ⌋} p_{m - 3 k, r - 2 k, \pm} (ln | X_{1}^{\pm} |, X^{\pm}) \sum_{i = 0}^{k} C_{m, 2 k, i, \pm} {(ln δ)}^{i} \end{matrix}$ hold true, where for any integer i, k such that $0 ⩽ i ⩽ k$ and using the notation $w_{m, 2 k, \pm} (θ^{\pm}, ln R^{\pm}) = \sum_{i = 0}^{k} {(ln R^{\pm})}^{i} w_{m, 2 k, i, \pm} (θ^{+})$ , the constants $C_{m, 2 k, i \pm}$ are given by $\begin{matrix} (5.5) & \begin{matrix} C_{m, 2 k, i, \pm} = \frac{4}{3 π} \int_{I^{\pm}} w_{m, 2 k, i, \pm} (θ^{\pm}) w_{m - 3 k, 0, \pm} (θ^{\pm}) d θ^{\pm}, \\ I^{+} = (0, \frac{3 π}{2}), I^{-} = (- \frac{π}{2}, π) . \end{matrix} \end{matrix}$
Remark 5.2.
With the convention that $w_{0, 0, \pm} = 0$ it follows that $C_{m, 2 k, i, \pm} = 0$ for $k = m / 3$ . Moreover, in view of the orthogonality conditions (A.16) $C_{m, 2 k, 0, \pm}$ always vanishes if $k \neq 0$ and $C_{m, 0, 0, \pm} = 1$ .

Inserting (5.3) into (5.2) and noting that $\frac{2}{3} (m - 3 k) - (r - 2 k) = \frac{2}{3} m - r$ , we obtain $\begin{array}{rcl} \sum_{(n, q) \in N^{2}} δ^{\frac{2 n}{3} + q} u_{n, q}^{δ} & = & \sum_{(n, q) \in N^{2}} δ^{\frac{2}{3} n + q} \sum_{m = - \infty}^{n} a_{n - m, q, m, +}^{δ} \\ \times \sum_{r \in N} \sum_{k = 0}^{⌊ r / 2 ⌋} {(R^{+})}^{\frac{2 (m - 3 k)}{3} - (r - 2 k)} w_{m - 3 k, r - 2 k, +} (θ^{+}, ln R^{+}) \sum_{i = 0}^{k} C_{m, 2 k, i, +} {(ln δ)}^{i} . \end{array}$ Then, the changes of indices $r - 2 k \to r$ and $m - 3 k \to m$ give $\begin{array}{rcl} \sum_{(n, q) \in N^{2}} δ^{\frac{2 n}{3} + q} u_{n, q}^{δ} & = & \sum_{(n, q) \in N^{2}} δ^{\frac{2}{3} n + q} \sum_{m = - \infty}^{n} {\tilde{a}}_{n - m, q, m, +}^{δ} \\ (5.6) & \times \sum_{r \in N} {(R^{+})}^{\frac{2 m}{3} - r} w_{m, r} (θ^{+}, ln R^{+}), X_{1}^{+} > 0, \end{array}$ where $\begin{matrix} (5.7) & \forall (n, q, l) \in N^{2} \times Z, {\tilde{a}}_{n, q, m, +}^{δ} = \sum_{k = 0}^{⌊ n / 3 ⌋} a_{n - 3 k, q, m + 3 k, +}^{δ} \sum_{i = 0}^{k} C_{m + 3 k, 2 k, i, +} {(ln δ)}^{i} . \end{matrix}$ In particular, for $m < 0$ , thanks to (3.30) (and using Remark 5.2), we have $\begin{matrix} (5.8) & {\tilde{a}}_{n, q, m, +}^{δ} = ℓ_{m}^{+} (u_{n, q}^{δ}) + \sum_{k = 1}^{⌊ n / 3 ⌋} a_{n - 3 k, q, m + 3 k, +}^{δ} \sum_{i = 0}^{k} C_{m + 3 k, 2 k, i, +} {(ln δ)}^{i} . \end{matrix}$ Analogously, for $X_{1}^{+} < - 1$ , we obtain, $\begin{array}{rcl} \sum_{(n, q) \in N} δ^{\frac{2 n}{3} + q} (u_{n, q}^{δ} (x) χ (\frac{x_{2}}{δ}) + Π_{n, q}^{δ} (x_{1}, \frac{x}{δ})) \\ = \sum_{(n, q) \in N^{2}} δ^{\frac{2}{3} n + q} \sum_{m = - \infty}^{n} {\tilde{a}}_{n - m, q, m, +}^{δ} \\ (5.9) & \times \sum_{r \in N} ({(R^{+})}^{\frac{2 m}{3} - r} w_{m, r, +} (θ^{+}, ln R^{+}) χ (X_{2}) + {| X_{1}^{+} |}^{\frac{2 m}{3} - r} p_{m, r, +} (ln | X_{1}^{+} |, X^{+})) . \end{array}$ The previous two expressions have to be compared with formula (4.27) and (4.28), in which the coefficients $A_{n, q, m, +}^{δ}$ are still not determined, since the constants $L_{m} (U_{n, q, +}^{δ})$ , $m = 1, \dots, n$ are not fixed yet. We aim to match the expansions in the matching zone and, hence, define these constants uniquely.
5.2. Derivation of the matching conditions

Arrived at this point, the derivation of the matching conditions is straight-forward. It suffices to identify formally all terms of the expansions (5.9) and (5.6) of the far field with all terms of the expansions (4.27) and (4.28) for the near field. We end up with the following set of conditions: $\begin{matrix} (5.10) & A_{n, q, m, +}^{δ} = {\tilde{a}}_{n - m, q, m, +}^{δ}, \forall (n, q) \in N^{2}, and m \in Z, m ⩽ n, \end{matrix}$ where $A_{n, q, m, +}^{δ}$ and ${\tilde{a}}_{n - m, q, m, +}^{δ}$ were defined in (4.25) and (5.7). As the coefficients $A_{n, q, m, +}^{δ}$ are linear combinations of the constants $L_{m} (U_{n, q, +}^{δ})$ , $m = 1, \dots, n$ and the coefficients ${\tilde{a}}_{n, q, m, +}^{δ}$ are linear combinations of the constants $ℓ_{- m}^{+} (u_{n, p}^{δ})$ , $m = 1, \dots, n$ for some $p \in N$ we aim now to obtain conditions between those constants. Here, we will proceed separately for the cases $m > 0$ , $m = 0$ and $m < 0$ .

For $0 < m ⩽ n$ , using the equality (4.26), we have for any $(n, q) \in N^{*} \times N$ , $\begin{matrix} (5.11) & L_{m} (U_{n, q, +}^{δ}) = {\tilde{a}}_{n - m, q, m, +}^{δ} . \end{matrix}$

For $m = 0$ , for any $(n, q) \in N$ , there is nothing to be matched. Indeed, both left and right-hand sides of (5.10) vanish ( ${\tilde{a}}_{n, q, 0, +} = 0$ because $C_{3 k, 2 k, i, +} = 0$ , see Remark 5.2).

For $m < 0$ , in view of (5.8) and substituting $A_{n, q, m, +}$ for its definition (4.25), we have for any $(n, q) \in N^{2}$ , $\begin{array}{rcl} ℓ_{m}^{+} (u_{n - m, q}^{δ}) & = & - \sum_{k = 1}^{⌊ (n - m) / 3 ⌋} a_{n - m - 3 k, q, m + 3 k, +}^{δ} \sum_{i = 0}^{k} C_{m + 3 k, 2 k, i, +} {(ln δ)}^{i} \\ (5.12) & + \sum_{k = 1}^{n} L_{k} (U_{n, q, +}^{δ}) L_{m} (S_{k}^{+}), \end{array}$ which may also be red as follows: for any $(n, q, m) \in N^{3}$ such that, $n ⩾ 1$ , and $1 ⩽ m ⩽ n$ , $\begin{matrix} (5.13) & ℓ_{- m}^{+} (u_{n, q}^{δ}) = - \sum_{k = 1}^{⌊ n / 3 ⌋} a_{n - 3 k, q, - m + 3 k, +}^{δ} \sum_{i = 0}^{k} C_{- m + 3 k, 2 k, i, +} {(ln δ)}^{i} + \sum_{k = 1}^{n - m} L_{k} (U_{n - m, q, +}^{δ}) L_{- m} (S_{k}^{+}) . \end{matrix}$

Here again, we can write similar matching conditions for the matching area located close to the left corner. These conditions link the macroscopic terms $u_{n, q}^{δ}$ to the near field terms $U_{n, q, -}^{δ}$ : for $1 ⩽ m ⩽ n$ and for any $(n, q) \in N^{*} \times N$ , $\begin{matrix} (5.14) & L_{m} (U_{n, q, -}^{δ}) = {\tilde{a}}_{n - m, q, m, -}^{δ}, \end{matrix}$ and, for any $(n, q, m) \in N^{3}$ such that, $n ⩾ 1$ , and $1 ⩽ m ⩽ n$ , $\begin{matrix} (5.15) & ℓ_{- m}^{-} (u_{n, q}^{δ}) = - \sum_{k = 1}^{⌊ n / 3 ⌋} a_{n - 3 k, q, - m + 3 k, -}^{δ} \sum_{i = 0}^{k} C_{- m + 3 k, 2 k, i, -} {(ln δ)}^{i} + \sum_{k = 1}^{n - m} L_{k} (U_{n - m, q, -}^{δ}) L_{- m} (S_{k}^{-}) . \end{matrix}$ Here, ${\tilde{a}}_{n, q, m}$ are defined by $\begin{matrix} \forall (n, q, l) \in N^{2} \times Z, {\tilde{a}}_{n, q, m, -}^{δ} = \sum_{k = 0}^{⌊ n / 3 ⌋} a_{n - 3 k, q, m + 3 k, -}^{δ} \sum_{i = 0}^{k} C_{m + 3 k, 2 k, i, -} {(ln δ)}^{i}, \end{matrix}$ with $\begin{array}{rcl} \forall (j, m) \in N \times Z, \\ a_{0, j, m, -}^{δ} = ℓ_{m}^{-} (s_{0, j}), and a_{n, j, m, -}^{δ} = \sum_{\pm} \sum_{k = max (1, - m)}^{n} \sum_{p = 0}^{j} ℓ_{- k}^{\pm} (u_{n, p}^{δ}) ℓ_{m}^{-} (s_{- k, j - p}^{\pm}) if n > 0 . \end{array}$

5.3. Construction of the terms of the asymptotic expansions

The matching conditions then allow us to construct the far field terms $u_{n, q}^{δ}$ and $Π_{n, q}^{δ}$ , and the near field terms $U_{n, q, \pm}^{δ}$ by induction on n. The base case is obvious since we have seen that the macroscopic terms $u_{0, q}^{δ}$ are entirely determined by Proposition 3.10, the boundary layer correctors $Π_{0, q}$ are defined by (2.30), and the near field terms $U_{0, q, \pm}^{δ} = 0$ , $q \in N$ by (4.22).

Then, assuming that $u_{m, q}^{δ}$ and $U_{m, q, \pm}^{δ}$ are constructed for any $m ⩽ n - 1$ , we will see that (5.11), (5.13) and (5.14), (5.15) permit to define both $u_{n, q}^{δ}$ (and consequently $Π_{n, q}^{δ}$ ) and $U_{n, q, \pm}^{δ}$ for any $q \in N$ .

Far field terms. We remind that, for a given $q \in N$ , the complete definition of the macroscopic terms $u_{n, q}^{δ}$ only requires the knowledge of the $ℓ_{- m}^{\pm} (u_{n, q}^{δ})$ for any integer m between 1 and n. In fact, the conditions (5.13) define exactly $ℓ_{- m}^{+} (u_{n, q}^{δ})$ : in the right-hand side of (5.13), the quantities $L_{- m} (S_{k}^{+})$ are known ( $S_{k}^{+}$ is uniquely defined) and $L_{k} (U_{n - m, q, +}^{δ})$ are known since $U_{n - m, q, +}^{δ}$ are already defined (induction hypothesis). In addition, since $\begin{matrix} a_{n - 3 k, q, - m + 3 k, +}^{δ} = \{\begin{matrix} \sum_{\pm} \sum_{r = max (1, m - 3 k)}^{n - 3 k} \sum_{p = 0}^{q} ℓ_{- r}^{\pm} (u_{n - 3 k, p}^{δ}) ℓ_{- m + 3 k}^{+} (s_{- r, q - p}^{\pm}) & if n - 3 k \neq 0, \\ ℓ_{- m + 3 k}^{+} (s_{0, q}) & if n - 3 k = 0, \end{matrix} \end{matrix}$ the coefficient $a_{n - 3 k, q, - m + 3 k, +}^{δ}$ is well defined for any k such that $1 ⩽ k ⩽ ⌊ n / 3 ⌋$ ( $ℓ_{- j}^{+} (u_{n - 3 k, q})$ is known by the induction hypothesis). Naturally the conditions (5.15) define $ℓ_{- m}^{-} (u_{n, q}^{δ})$ is the same way. Finally, the definition of the boundary layer terms $Π_{n, q}^{δ}$ follows from (2.30).

Near field terms. Similarly, the definition of the near field terms $U_{n, q, \pm}^{δ}$ requires the specification of the quantities $L_{m} (U_{n, q, \pm}^{δ})$ for any integer m between 1 and n. The condition (5.11) exactly provides this missing information for $U_{n, q, +}^{δ}$ . Indeed, in the right-hand side of (5.11), the computation of ${\tilde{a}}_{n - m, q, m, +}^{δ}$ requires the knowledge of $\begin{matrix} a_{n - m - 3 k, q, m + 3 k, +}^{δ} = \{\begin{matrix} \sum_{\pm} \sum_{r = 0}^{n - m - 3 k} \sum_{p = 0}^{q} ℓ_{- r}^{\pm} (u_{n - m - 3 k, p}^{δ}) ℓ_{m + 3 k}^{+} (s_{- r, q - p}^{\pm}) & if n - m - 3 k \neq 0, \\ ℓ_{m + 3 k}^{+} (s_{0, q}) & if n - m - 3 k = 0 . \end{matrix} \end{matrix}$ for k between 0 and $⌊ (n - m) / 3 ⌋$ . But, since $m > 0$ , $ℓ_{- r}^{+} (u_{n - m - 3 k, p}^{δ})$ are well defined thanks to the induction hypothesis. Then, $U_{n, q, +}^{δ}$ is entirely determined. In the same way, the condition (5.14) allows us to define $U_{n, q, -}^{δ}$ as well, replacing all occurrences of $ℓ_{m}^{+} (s_{- r, q - p}^{\pm})$ by $ℓ_{m}^{-} (s_{- r, q - p}^{\pm})$ , $m \in Z$ and all occurrences $ℓ_{m}^{+} (s_{0, q})$ by $ℓ_{m}^{-} (s_{0, q})$ , $m \in Z$ in the previous formulas.

Remark 5.3.
We point out that the variables n and q play a very different roles in the recursive construction of the terms of the asymptotic expansion. Indeed, the construction is by induction only in n. At the step n, we construct $u_{n, q}^{δ}$ , $Π_{n, q}^{δ}$ and $U_{n, q, \pm}^{δ}$ for any $q \in N$ .

6. Justification of the asymptotic expansion

To finish this paper, we shall prove Theorem 1.6, which shows the convergence of the truncated macroscopic series toward the exact solution in a fixed domains that excludes a neighbourhood of the corners $x_{O}^{\pm}$ and the interface Γ.

As usual for this kind of work (see e.g. [24, Section 3], [21, Section 5.1], [17, Section 4]), the proof is based on the construction of an approximation $u_{N_{0}}^{δ}$ of the solution $u^{δ}$ on the whole domain $Ω^{δ}$ obtained from the four truncated series (at order $N_{0}$ ) of the macroscopic terms, the boundary layer terms and the near field terms:

The truncated macroscopic series $u_{macro, N_{0}}^{δ}$ : we introduce the macroscopic cut-off function $\begin{array}{rcl} χ_{macro, total}^{δ} (x) & = & χ_{+} (\frac{x_{1} - L}{δ}) χ_{-} (\frac{x_{1} + L}{δ}) χ (\frac{x_{2}}{δ}) \\ (6.1) & + \sum_{\pm} χ_{macro, \pm} (\frac{x_{1} \mp L}{δ}, \frac{x_{2}}{δ}) (1 - χ_{\mp} (\frac{x_{1} \mp L}{δ})), \end{array}$ which is equal to 1 for $| x_{1} | > L$ , and which coincides with $χ (\frac{x_{2}}{δ})$ in the region $| x_{1} | < L - δ$ (see Fig. 5). The cut-off functions χ, $χ_{\pm}$ and $χ_{macro, \pm}$ are defined in (1.8)–(1.17) and (4.7). We then define the macroscopic approximation as follows: $\begin{matrix} (6.2) & u_{macro, N_{0}}^{δ} = χ_{macro, total}^{δ} (x) \sum_{(n, q) \in D_{0}} δ^{\frac{2}{3} n + q} u_{n, q}^{δ} (x) . \end{matrix}$

The truncated periodic corrector series $Π_{N_{0}}^{δ}$ : it is given by $\begin{matrix} (6.3) & Π_{N_{0}}^{δ} (x) = (1 - χ (x_{2})) χ_{+} (\frac{x_{1} - L}{δ}) χ_{-} (\frac{x_{1} + L}{δ}) \sum_{(n, q) \in D_{0}} δ^{\frac{2}{3} n + q} Π_{n, q}^{δ} (x_{1}, \frac{x}{δ}) . \end{matrix}$ The function $χ_{+} (\frac{x_{1} - L}{δ}) χ_{-} (\frac{x_{1} + L}{δ})$ permits us to localize the function $Π_{N_{0}}^{δ} (x)$ in the domain $| x_{1} | < L$ while the introduction of the function $(1 - χ (x_{2}))$ ensures that $Π_{N_{0}}^{δ} (x)$ vanishes on $Γ_{D}$ .

The truncated near field series $U_{N_{0}, \pm}^{δ} (x)$ : $\begin{matrix} (6.4) & U_{N_{0}, \pm}^{δ} (x) = \sum_{(n, q) \in D_{0}} δ^{\frac{2}{3} n + q} U_{n, q}^{δ} (\frac{x}{δ}) . \end{matrix}$

Fig. 5.

Schematic representation of the cut-off function $χ_{macro, total}^{δ}$ .

We shall construct a global approximation $u_{N_{0}}^{δ}$ that coincides with $U_{N_{0}, \pm}^{δ}$ in the vicinity of the two corners, with $Π_{N_{0}}^{δ}$ in the vicinity of the periodic layer and with $u_{macro, N_{0}}^{δ}$ away from the corners and the periodic layer. To do so, we introduce the cut off functions $\begin{matrix} (6.5) & χ_{+}^{δ} (x) = χ (\frac{r^{+}}{η (δ)}) and χ_{-}^{δ} (x) = χ (\frac{r^{-}}{η (δ)}), \end{matrix}$ where $η (δ) : R^{+} \to R^{+}$ is a smooth function such that $\begin{matrix} (6.6) & lim_{δ \to 0} η (δ) = 0 and lim_{δ \to 0} \frac{η (δ)}{δ} = + \infty . \end{matrix}$ For instance for $s \in (0, 1)$ , $η (δ) = δ^{s}$ satisfies these conditions. Finally the global approximation $u_{N_{0}}^{δ}$ of $u^{δ}$ is defined by $\begin{matrix} (6.7) & u_{N_{0}}^{δ} (x) = χ_{+}^{δ} (x) U_{N_{0}, +} + χ_{-}^{δ} (x) U_{N_{0}, -} + (1 - χ_{+}^{δ} (x) - χ_{-}^{δ} (x)) (u_{macro, N_{0}}^{δ} (x) + Π_{N_{0}}^{δ} (x)) . \end{matrix}$ Note that $u_{N_{0}}^{δ}$ belongs to $H_{Γ_{D}}^{1} (Ω^{δ})$ but does not satisfy homogeneous Neumann boundary conditions on $Γ^{δ}$ .

The aim of this part is to estimate the $H^{1}$ -norm of the error $e_{N_{0}}^{δ} = u^{δ} - u_{N_{0}}^{δ}$ in $Ω^{δ}$ (we remind that $u^{δ} \in H_{Γ_{D}}^{1} (Ω^{δ})$ is the ‘exact’ solution, i.e. the solution of problem (1.3)). It is in fact sufficient to estimate the residue $Δ e_{N_{0}}^{δ}$ and the Neumann trace $\partial_{n} e_{N_{0}}^{δ}$ . Then, the estimation of $‖ e_{N_{0}}^{δ} ‖_{H^{1} (Ω^{δ})}$ directly results from a straightforward modification of the uniform stability estimate (1.5) (Proposition 1.1): there exists a constant $C > 0$ such that, for δ small enough, $\begin{matrix} (6.8) & {‖ e_{N_{0}}^{δ} ‖}_{H^{1} (Ω^{δ})} ⩽ C ({‖ Δ e_{N_{0}}^{δ} ‖}_{L^{2} (Ω^{δ})} + {‖ \partial_{n} e_{N_{0}}^{δ} ‖}_{L^{2} (Γ^{δ})}) . \end{matrix}$ The main work of this part consists in proving the following proposition:

Proposition 6.1.

There exist a constant $C > 0$ and a constant $δ_{0} > 0$ such that, for any $ε > 0$ , for any $δ \in (0, δ_{0})$ , $\begin{matrix} (6.9) & {‖ Δ e_{N_{0}}^{δ} ‖}_{L^{2} (Ω^{δ})} + {‖ \partial_{n} e_{N_{0}}^{δ} ‖}_{L^{2} (Γ^{δ})} ⩽ C (δ^{- 2 - ε} {(\frac{δ}{η (δ)})}^{N_{0}} + δ^{- 1} η {(δ)}^{N_{0} - \frac{1}{6} - ε}) . \end{matrix}$

As a direct corollary, choosing $η (δ) = \sqrt{δ}$ , $ε = \frac{1}{2}$ , we obtain the following global error estimate: there exist a constant $C > 0$ and a constant $δ_{0} > 0$ such that, for any $δ \in (0, δ_{0})$ , $\begin{matrix} (6.10) & {‖ e_{N_{0}}^{δ} ‖}_{H^{1} (Ω^{δ})} ⩽ C δ^{\frac{N_{0}}{2} - \frac{5}{2}} . \end{matrix}$ Since $e_{N_{0}}^{δ}$ coincides with $u^{δ} - \sum_{(n, q) \in D_{N_{0}}} δ^{\frac{2}{3} n + q} u_{n, q}^{δ}$ in $Ω_{α}$ for δ small enough, Theorem 1.6 follows from (6.10) and the triangular inequality.

Remark 6.2.

We emphasize that $u_{N_{0}}^{δ}$ is certainly not the best choice to minimize the global error. As shown in [15], a global estimate based on the truncated far and near field terms obtained by the compound method might provide a better global error. Nevertheless, for the sake of simplicity and since we are mainly interested in the macroscopic error estimate (that can always be made optimal thanks to the triangular inequality), we prefer using here $u_{N_{0}}^{δ}$ .

Sketch of the proof of Proposition 6.1.

In this paper, we only explain the main ingredients of the proof but a detailed proof may be found in [18] (Section 6). Remarking that the supports of the derivatives of $χ_{+}^{δ}$ and $χ_{-}^{δ}$ are disjoint (for δ small enough), using additionally that $Δ U_{N_{0}, \pm}^{\pm} = 0$ , we can decompose the residue as $\begin{matrix} (6.11) & - Δ e_{N_{0}}^{δ} = E_{\mod} + E_{match}, \end{matrix}$ where, $\begin{matrix} (6.12) & E_{match} = - \sum_{\pm} [Δ, χ_{\pm}^{δ} (x)] (U_{N_{0}, \pm} - u_{macro, N_{0}}^{δ} (x) - Π_{N_{0}}^{δ} (x)), \end{matrix}$ and $\begin{matrix} (6.13) & E_{\mod} = f - (1 - χ_{+}^{δ} (x) - χ_{-}^{δ} (x)) Δ (u_{macro, N_{0}}^{δ} (x) + Π_{N_{0}}^{δ} (x)) . \end{matrix}$ $E_{match}$ represents the matching error. Its support, which coincides with the union of the supports of $\nabla χ_{+}^{δ}$ and $\nabla χ_{-}^{δ}$ , is included in the union of the rings $η (δ) < | r^{\pm} | < 2 η (δ)$ . It measures the mismatch between the far and near field truncated expansions in the matching zones. Because of the matching principle, this term is expected to be small. $E_{\mod}$ , representing the modeling error (or consistency error), measures how the expansion fails to satisfies the original Laplace problem.

As for the residue on the boundary, is it easily seen that $\begin{matrix} (6.14) & \partial_{n} e_{N_{0}}^{δ} = \sum_{\pm} \partial_{n} χ_{\pm}^{δ} (x) (U_{N_{0}, \pm} - Π_{N_{0}}^{δ} (x)), \end{matrix}$ so that the error on the boundary data is supported in the matching areas, and therefore can be treated as the matching error. The remainder of the proof consists in proving the following error estimates for the matching and modeling errors: There exist a positive constant $C > 0$ and a positive number $δ_{0} > 0$ , such that, for any $δ \in (0, δ_{0})$ , $\begin{matrix} (6.15) & ‖ E_{\mod} ‖_{L^{2} (Ω^{δ})} + ‖ E_{match} ‖_{L^{2} (Ω^{δ})} + {‖ \partial_{n} e_{N_{0}}^{δ} ‖}_{L^{2} (Γ^{δ})} ⩽ C (δ^{- 2 - ε} {(\frac{δ}{η (δ)})}^{N_{0}} + δ^{- 1} η {(δ)}^{N_{0} - \frac{1}{6} - ε}) . \end{matrix}$ Although long and technical, the proof of the previous estimate is rather standard. □

Footnotes

Acknowledgements

The authors gratefully acknowledge the financial support from the Einstein Foundation Berlin, and the second author thanks the research center MATHEON for its support. Part of this work was carried out where B. Delourme was on research leave at Laboratoire POEMS, INRIA-Saclay, ENSTA, UMR CNRS 2706, France.

Proofs of technical results

References

[1]

Abboud and

Ammari , Diffraction at a curved grating: TM and TE cases, homogenization, J. Math. Anal. Appl. 202(3) (1996), 995–1026.

[2]

Achdou , Etude de la réflexion d’une onde électromagnétique par un métal recouvert d’un revêtement métallisé, Technical report, INRIA, 1989.

[3]

Achdou , Effect of a thin metallized coating on the reflection of an electromagnetic wave, C. R. Acad. Sci. Paris, Ser. I 314(3) (1992), 217–222.

[4]

Achdou ,

Pironneau and

Valentin , Effective boundary conditions for laminar flows over periodic rough boundaries, J. Comput. Phys. 147(1) (1998), 187–218.

[5]

Artola and

Cessenat , Diffraction d’une onde électromagnétique par une couche composite mince accolée à un corps conducteur épais. I. Cas des inclusions fortement conductrices, C. R. Acad. Sci. Paris, Ser. I 313(5) (1991), 231–236.

[6]

Artola and

Cessenat , Scattering of an electromagnetic wave by a slender composite slab in contact with a thick perfect conductor. II. Inclusions (or coated material) with high conductivity and high permeability, C. R. Acad. Sci. Paris, Ser. I 313(6) (1991), 381–385.

[7]

Bendali ,

Makhlouf and

Tordeux , Field behavior near the edge of a microstrip antenna by the method of matched asymptotic expansions, Quart. Appl. Math. 69(4) (2011), 691–721.

[8]

Bonnaillie-Noël ,

Dambrine ,

Tordeux and

Vial , Interactions between moderately close inclusions for the Laplace equation, Math. Models Meth. Appl. Sci. 19(10) (2009), 1853–1882.

[9]

Bresch and

Milisic , High order multi-scale wall-laws, Part I: The periodic case, Quart. Appl. Math. 68(2) (2010), 229–253.

10.

[10]

Caloz ,

Costabel ,

Dauge and

Vial , Asymptotic expansion of the solution of an interface problem in a polygonal domain with thin layer, Asymptot. Anal. 50(1–2) (2006), 121–173.

11.

[11]

Chapman ,

Hewett and

Trefethen , The mathematics of the Faraday cage, SIAM Review (2015), to appear.

12.

[12]

I.S.

Ciuperca ,

Jai and

Poignard , Approximate transmission conditions through a rough thin layer: The case of periodic roughness, European J. Appl. Math. 21(1) (2010), 51–75.

13.

[13]

Claeys , On the theoretical justification of Pocklington’s equation, Math. Models Meth. Appl. Sci. 19(8) (2009), 1325–1355.

14.

[14]

Claeys and

Delourme , High order asymptotics for wave propagation across thin periodic interfaces, Asymptot. Anal. 83(1–2) (2013), 35–82.

15.

[15]

Dauge ,

Tordeux and

Vial , Selfsimilar perturbation near a corner: Matching versus multiscale expansions for a model problem, in: Around the Research of Vladimir Maz’ya. II, Int. Math. Ser. (N. Y.), Vol. 12, Springer-Verlag, New York, 2010, pp. 95–134.

16.

[16]

Delourme , Modèles et asymptotiques des interfaces fines et périodiques en électromagnétisme, PhD thesis, Université Pierre et Marie Curie, Paris, 2010.

17.

[17]

Delourme ,

Haddar and

Joly , Approximate models for wave propagation across thin periodic interfaces, J. Math. Pures Appl. (9) 98(1) (2012), 28–71.

18.

[18]

Delourme ,

Schmidt and

Semin , When a thin periodic layer meets corners: Asymptotic analysis of a singular Poisson problem, 2015, available at: arXiv:1506.06964v1.

19.

[19]

Grisvard , Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, Vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985.

20.

[20]

Grisvard , Singularities in Boundary Value Problems, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], Vol. 22, Masson, Paris; Springer-Verlag, Berlin, 1992.

21.

[21]

Haddar ,

Joly and

H.M.

Nguyen , Generalized impedance boundary conditions for scattering by strongly absorbing obstacles: The scalar case, Math. Models Meth. Appl. Sci. 15(8) (2005), 1273–1300.

22.

[22]

A.M.

Il’in , Matching of Asymptotic Expansions of Solutions of Boundary Value Problems, Translations of Mathematical Monographs, Vol. 102, American Mathematical Society, Providence, RI, 1992. Translated from the Russian by V. Minachin.

23.

[23]

Joly and

Tordeux , Matching of asymptotic expansions for wave propagation in media with thin slots. I. The asymptotic expansion, Multiscale Model. Simul. 5(1) (2006), 304–336 (electronic).

24.

[24]

Joly and

Tordeux , Matching of asymptotic expansions for waves propagation in media with thin slots. II. The error estimates, ESAIM: Math. Model. Numer. Anal. 42(2) (2008), 193–221.

25.

[25]

V.A.

Kondrat’ev , Boundary-value problems for elliptic equations in conical regions, Dokl. Akad. Nauk SSSR 153 (1963), 27–29.

26.

[26]

V.A.

Kozlov ,

V.G.

Mazya and

Rossmann , Elliptic Boundary Value Problems in Domains with Point Singularities, Mathematical Surveys and Monographs, Vol. 52, American Mathematical Society, Providence, RI, 1997.

27.

[27]

A.L.

Madureira and

Valentin , Asymptotics of the Poisson problem in domains with curved rough boundaries, SIAM J. Math. Anal. 38(5) (2007), 1450–1473 (electronic).

28.

[28]

Makhlouf , Justification et amélioration de modèles d’antenne patch par la méthode des développements asymptotiques raccordés, PhD thesis, Institut National des Sciences Appliquées de Toulouse, 2008.

29.

[29]

Maz’ya ,

Nazarov and

Plamenevskij , Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains, Vol. I, Operator Theory: Advances and Applications, Vol. 111, Birkhäuser Verlag, Basel, 2000, Translated from the German by Georg Heinig and Christian Posthoff.

30.

[30]

S.A.

Nazarov , Asymptotic behavior of the solution of the Dirichlet problem in an angular domain with a periodically changing boundary, Mat. Zametki 49(5) (1991), 86–96, 159.

31.

[31]

S.A.

Nazarov , Asymptotic behavior of the solution and the modeling of the Dirichlet problem in an angular domain with a rapidly oscillating boundary, Algebra i Analiz 19(2) (2007), 183–225.

32.

[32]

S.A.

Nazarov , The Neumann problem in angular domains with periodic and parabolic perturbations of the boundary, Tr. Mosk. Mat. Obs. 69 (2008), 182–241.

33.

[33]

Nicaise and

A.-M.

Sändig , General interface problems. I, II, Math. Meth. Appl. Sci. 17(6) (1994), 395–429, 431–450.

34.

[34]

J.-R.

Poirier ,

Bendali and

Borderies , Impedance boundary conditions for the scattering of time-harmonic waves by rapidly varying surfaces, IEEE Trans. Antennas and Propagation 54(3) (2006), 995–1005.

35.

[35]

J.-R.

Poirier ,

Bendali ,

Borderies and

Tournier , High order asymptotic expansion for the scattering of fast oscillating periodic surfaces, in: Proc. 9th Int. Conf. on Mathematical and Numerical Aspects of Waves Propagation (Waves 2009), Pau, France, 2009.

36.

[36]

Sánchez-Palencia , Nonhomogeneous Media and Vibration Theory, Lecture Notes in Physics, Vol. 127, Springer-Verlag, Berlin, 1980.

37.

[37]

Sánchez-Palencia , Un problème d’écoulement lent d’un fluide incompressible au travers d’une paroi finement perforée, in: Homogenization Methods: Theory and Applications in Physics (Bréau-sans-Nappe, 1983), Collect. Dir. Études Rech. Élec. France, Vol. 57, Eyrolles, Paris, 1985, pp. 371–400.

38.

[38]

Van Dyke , Perturbation Methods in Fluid Mechanics, Applied Mathematics and Mechanics, Vol. 8, Academic Press, New York, 1964.

39.

[39]

Vial , Analyse multi-échelle et conditions aux limites approchées pour un probleme avec couche mince dans un domaine à coin, PhD thesis, École Normale Supérieure de Cachan, Paris, 2003.

On the homogenization of thin perforated walls of finite length

Abstract

Keywords

Introduction

1. Description of the problem and main results

1.1. Description of the problem

1.2.3. Matching principle

1.3. Far and near field equations

1.3.1. Far field equations: Macroscopic and boundary layer correctors equations

1.3.2. Near field equations

1.4. Outlook of the paper and main result

2.2.1. Step 0: [ u 0 ] Γ and Π 0

2.2.2. Step 1: [ ∂ x 2 u 0 ] Γ , [ u 1 ] Γ , and Π 1

2.2.3. Step 2: [ ∂ x 2 u 1 ] Γ ( [ u 2 ] Γ and Π 2 )

2.3. Transmission conditions up to any order

3.1. General results of existence for transmission problem

3.1.1. Variational framework

3.2.1. The limit macroscopic term and its behavior in the vicinity of the corners

3.2.2. A singular problem defining u 0 , 1 δ

3.3.1. Harmonic singularities s − m , 0 ± ( m ∈ N ∖ { 0 } )

3.4. An explicit expression for the macroscopic terms

3.4.1. The macroscopic terms u 0 , q δ , q ∈ N

3.4.4. Asymptotic of the far field terms close to the corners

4. Analysis of the near field equations and near field singularities

4.1. General results of existence and asymptotics of the solution at infinity

4.1.1. Variational framework

Proposition 4.1. Assume that 1 + ( R + ) 2 f ∈ L 2 ( Ω ˆ + ) and ( 1 + ( R + ) 2 ) 1 / 4 g ∈ L 2 ( Γ ˆ hole + ) . Then, problem (4.1) has a unique solution u ∈ V ( Ω ˆ + ) . 4.1.2. Asymptotic expansion at infinity

5. Matching procedure and construction of the far and near field terms

5.1. Far field expansion expressed in the microscopic variables

5.3. Construction of the terms of the asymptotic expansions

Footnotes

Acknowledgements

Proofs of technical results

References

2.2.1. Step 0: ${[u_{0}]}_{Γ}$ and $Π_{0}$

2.2.2. Step 1: ${[\partial_{x_{2}} u_{0}]}_{Γ}$ , ${[u_{1}]}_{Γ}$ , and $Π_{1}$

2.2.3. Step 2: ${[\partial_{x_{2}} u_{1}]}_{Γ}$ ( ${[u_{2}]}_{Γ}$ and $Π_{2}$ )

3.2.2. A singular problem defining $u_{0, 1}^{δ}$

3.3.1. Harmonic singularities $s_{- m, 0}^{\pm}$ ( $m \in N ∖ {0}$ )

3.4.1. The macroscopic terms $u_{0, q}^{δ}$ , $q \in N$

Proposition 4.1.
Assume that $\sqrt{1 + {(R^{+})}^{2}} f \in L^{2} ({\hat{Ω}}^{+})$ and ${(1 + {(R^{+})}^{2})}^{1 / 4} g \in L^{2} ({\hat{Γ}}_{hole}^{+})$ . Then, problem (4.1) has a unique solution $u \in V ({\hat{Ω}}^{+})$ .

4.1.2. Asymptotic expansion at infinity