# Asymptotic analysis at any order of Helmholtz's problem in a corner with a thin layer: an algebraic approach

Cédric Baudet

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Cédric Baudet. Asymptotic analysis at any order of Helmholtz's problem in a corner with a thin layer: an algebraic approach. 2024. hal-04582276

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# Asymptotic analysis at any order of Helmholtz's problem in a corner with a thin layer: an algebraic approach

Cédric Baudet

## ► To cite this version:

Cédric Baudet. Asymptotic analysis at any order of Helmholtz's problem in a corner with a thin layer: an algebraic approach. Asymptotic Analysis, inPress. hal-04582276v2

HAL Id: hal-04582276

https://hal.science/hal-04582276v2

Submitted on 4 Sep 2025

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archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

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émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

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Distributed under a Creative Commons Attribution 4.0 International License
# Asymptotic analysis at any order of Helmholtz's problem in a corner with a thin layer: an algebraic approach

Cédric Baudet*

May 14, 2024

**Abstract:** We consider the Helmholtz equation in an angular sector partially covered by a homogeneous layer of small thickness, denoted $\varepsilon$. We propose in this work an asymptotic expansion of the solution with respect to $\varepsilon$ at any order. This is done using matched asymptotic expansion, which consists here in introducing different asymptotic expansions of the solution in three subdomains: the vicinity of the corner, the layer and the rest of the domain. These expansions are linked through matching conditions. The presence of the corner makes these matching conditions delicate to derive because the fields have singular behaviors. Our approach is to reformulate these matching conditions purely algebraically by writing all asymptotic expansions as formal series. By using algebraic calculus we reduce the matching conditions to scalar relations linking the singular behaviors of the fields. These relations have a convolutive structure and involve some coefficients that can be computed analytically. Our asymptotic expansion is justified rigorously with error estimates.

**Keywords:** asymptotic analysis, Helmholtz's equation, matched asymptotic expansions, corner singularities, algebraic formal series.

* POEMS, CNRS, Inria, ENSTA Paris, Institut Polytechnique de Paris, 91120 Palaiseau, France.
Email: cedric.baudet@ensta-paris.fr

## Introduction

Problems that involve thin layers appear in many areas, from composite materials engineering [35] to biology [12], including elasticity [8, 20], fluid mechanics [31, 1, 24] and electrochemistry [37]. Applications are especially numerous in electromagnetism, let us mention the studies of thin dielectric layers [22, 26, 34], ferromagnetic films [5, 21] and the skin effect [15]. All these situations are numerically challenging because they require finely meshing the thin structure, which is very costly when its thickness is very small compared to the wavelength and the size of the objects. In this work we propose to overcome this difficulty by using an asymptotic expansion of the solution, such that each term of the expansion is cheaper to compute than the solution itself.

Infinite planar layers and smooth curved layers were studied during the 90s in [9, 19]. Their method is to stretch the layer in its transverse direction into a standard layer of thickness 1, and look for a Taylor-type asymptotic expansion as a sum of integer powers of the original thickness, denoted $\varepsilon$. The terms of this expansion can be computed by induction. Those results were later extended to heterogeneous and periodic layers in [3, 2, 4] and more recently in [16, 10].

Here we want to handle more realistic situations where the coating has angles or covers only partially the obstacle. We consider a two-dimensional model where the domain is the union of an infinite angular sector and the coating, potentially with a perturbation at the corner of size proportional to $\varepsilon$. This was studied for Poisson's problems in [11, 6, 7], providing an asymptotic expansion at any order and approximate models. These works show the presence of non-integer powers of $\varepsilon$ and integer powers of $\ln \varepsilon$ in the asymptotic expansion, that are linked to the corner singularities of the solution. That asymptotic expansion at any order was generalized to periodic layer in [17], still for Poisson's problems. In comparison, Helmholtz's problems not only present the same difficulties, but they also lead to more complex singularities, which prompted us to introduce new and more efficient algebraic calculus tools in order to
# Asymptotic analysis at any order of Helmholtz's problem in a corner with a thin layer: an algebraic approach

Cédric Baudet*

August 24, 2025

**Abstract:** We consider the Helmholtz equation in an angular sector partially covered by a homogeneous layer of small thickness, denoted $\varepsilon$. We propose in this work an asymptotic expansion of the solution with respect to $\varepsilon$ at any order. This is done using matched asymptotic expansion, which consists here in introducing different asymptotic expansions of the solution in three subdomains: the vicinity of the corner, the layer and the rest of the domain. These expansions are linked through matching conditions. The presence of the corner makes these matching conditions delicate to derive because the fields have singular behaviors. Our approach is to reformulate these matching conditions purely algebraically by writing all asymptotic expansions as formal series. By using algebraic calculus we reduce the matching conditions to scalar relations linking the singular behaviors of the fields. These relations have a convolutive structure and involve some coefficients that can be computed analytically. Our asymptotic expansion is justified rigorously with error estimates.

**Keywords:** asymptotic analysis, Helmholtz's equation, matched asymptotic expansions, corner singularities, algebraic formal series.

* POEMS, CNRS, Inria, ENSTA, Institut Polytechnique de Paris, 91120 Palaiseau, France.

Email: cedric.baudet@ensta.fr

## Introduction

Problems that involve thin layers appear in many areas, from composite materials engineering [42] to biology [13], including elasticity [8, 22], fluid mechanics [38, 1, 28] and electrochemistry [44]. Applications are especially numerous in electromagnetism, let us mention the studies of thin dielectric layers [26, 32, 41], ferromagnetic films [5, 25] and the skin effect [16]. All these situations are numerically challenging because they require finely meshing the thin structure, which is very costly when its thickness is very small compared to the wavelength and the size of the objects. In this work we propose to overcome this difficulty by using an asymptotic expansion of the solution, such that each term of the expansion is cheaper to compute than the solution itself.

Infinite planar layers and smooth curved layers were studied during the 90s in [9, 20]. Their method is to stretch the layer in its transverse direction into a standard layer of thickness 1, and look for a Taylor-type asymptotic expansion as a sum of integer powers of the original thickness, denoted $\varepsilon$. The terms of this expansion can be computed by induction. Those results were later extended to heterogeneous and periodic layers in [3, 2, 4] and more recently in [17, 11].

Here we want to handle more realistic situations where the coating has angles or covers only partially the obstacle. We consider a two-dimensional model where the domain is the union of an infinite angular sector and the coating, potentially with a perturbation at the corner of size proportional to $\varepsilon$. This was studied for Poisson's problems in [12, 6, 7], providing an asymptotic expansion at any order and approximate models. These works show the presence of non-integer powers of $\varepsilon$ and integer powers of $\ln \varepsilon$ in the asymptotic expansion, that are linked to the corner singularities of the solution. See similar studies for eigenvalue problems in [34, 24, 23]. That asymptotic expansion at any order was generalized to periodic layer in [18], still for Poisson's problems. As for the Helmholtz equation, [29] proved a similar asymptotic expansion at any order in the related case of a half plane with a thin slot. In comparison, Helmholtz's
obtain an expansion at any order. Let us mention also that [33] proposes an asymptotic expansion of Helmholtz's problem up to order 2 in presence of periodic layers.

We can identify in these works two methods of analysis: multiscale asymptotic expansions and matched asymptotic expansions (see [38, 28, 29] and [36, 23, 18] respectively for a general presentation). They both involve two types of fields: “far fields” depending on the macroscopic scale described by $(x, y)$ and “near fields” depending on the microscopic scale described by $(\frac{x}{\epsilon}, \frac{y}{\epsilon})$. In multiscale expansions, far and near fields are defined in the whole domain and the near fields tend to 0 towards infinity so that they describe a boundary layer effect in the “near zone” (the vicinity of the corner or the layer, depending on the situation). In contrast, matched asymptotic expansions involve near fields only in the near zone and far fields only in the “far zone” (the rest of the domain), and the near and far fields have to coincide in an intermediate zone.

In this paper, we chose the method of matched expansions. In addition, we propose a new algebraic approach to derive the matching conditions, that are especially intricate in our problem. It avoids specific cumbersome calculations, replacing them with abstract generalizable ones. We believe that this approach gives a better understanding of the structure of the asymptotic expansion at any order. It reveals a convolutive structure and it provides explicit expressions to compute exactly and very cheaply the constants that appear in the obtained matching formulas.

We consider the Helmholtz equation with absorption because it brings obvious well-posedness and stability of the problem uniformly in $\epsilon$, which allows us to focus on asymptotic expansion techniques. The case without absorption requires to design a specific radiation condition, that will be the object of a future paper. Moreover, we apply a Dirichlet condition on the boundary. The extension to Neumann is not obvious and will be presented in a forthcoming article.

This paper is organized as follows. In Section 1, we define the problem, state the main result and introduce the method based on matched asymptotic expansion. The matching condition around the corner are derived using an algebraic approach in Section 2. It is the most original part of the article. In Section 3, we introduce appropriate frameworks which allow to define uniquely the terms of the asymptotic expansion. Error estimates are performed in Section 4, proving the main result of the paper.

**Acknowledgment:** I would like to thank Sonia Fliss and Patrick Joly for the helpful discussions we had about the writing of this paper.

# Contents

1 **Setting of the problem and the method** 3

1.1 Definition of the problem and main result 3

1.2 The matched asymptotic expansion method 4

2 **Matching conditions** 5

2.1 Algebraic preliminaries 7

2.2 Definition of the spaces $\mathcal{A}$ 9

2.3 Tools for solving the Poisson and Helmholtz equations in the spaces $\mathcal{A}$ 11

2.4 Asymptotic behaviors w.r.t. $r$ of solutions of model problems 14

2.5 Specifying of the matching conditions 17

3 **Construction of the asymptotic expansion** 23

3.1 Existence and uniqueness for far fields-like problems 25

3.2 Existence and uniqueness for corner fields-like problems 26

3.3 Construction of the fields 29

3.4 Practical way to build the far fields 30

4 **Error estimates** 33

A **Appendix: proof of Lemma 2.8** 37
problems involving finite layers not only present the same difficulties, but they also lead to much more complex singularities, which prompted us to introduce new and more efficient algebraic calculus tools in order to obtain an expansion at any order. Let us mention also that [10, 32], resp. [40], propose asymptotic expansions of Helmholtz's problems up to order 2 in presence of homogeneous, resp. periodic, layers.

We can identify in these works two methods of analysis: multiscale asymptotic expansions and matched asymptotic expansions (see [45, 35, 36] and [43, 27, 19] respectively for a general presentation). They both involve two types of fields: “far fields” depending on the macroscopic scale described by $(x, y)$ and “near fields” depending on the microscopic scale described by $(\frac{x}{\epsilon}, \frac{y}{\epsilon})$. In multiscale expansions, far and near fields are defined in the whole domain and the near fields tend to 0 towards infinity so that they describe a boundary layer effect in the “near zone” (the vicinity of the corner or the layer, depending on the situation). In contrast, matched asymptotic expansions involve near fields only in the near zone and far fields only in the “far zone” (the rest of the domain), and the near and far fields have to coincide in an intermediate zone.

In this paper, we chose the method of matched expansions. In addition, we propose a new algebraic approach to derive the matching conditions, as they seem too intricate in our problem to be reasonably obtained at any order by classical means, especially if one wants to treat any corner angle. It lies on three main ideas: a general explicit expression of the singularities, operators to generate and manipulate efficiently families of singularities, and formal series to perform rigorous calculations on all orders at once. It avoids specific cumbersome calculations, replacing them with abstract generalizable ones. We believe that this approach gives a better understanding of the structure of the asymptotic expansion at any order. It reveals a convolutive structure and it provides explicit expressions to compute exactly and very cheaply the constants that appear in the obtained matching formulas.

We consider the Helmholtz equation with absorption because it brings obvious well-posedness and stability of the problem uniformly in $\epsilon$, which allows us to focus on asymptotic expansion techniques. The case without absorption requires to design a specific radiation condition, that will be the object of a future paper. We do not put any restriction on the corner angle. Moreover, we apply a Dirichlet condition on the boundary. The extension to Neumann is not obvious and will be presented in a forthcoming article.

This paper is organized as follows. In Section 1, we define the problem, state the main result and introduce the method based on matched asymptotic expansion. The matching condition around the corner are derived using an algebraic approach in Section 2. It is the most original part of the article. In Section 3, we introduce appropriate frameworks which allow to define uniquely the terms of the asymptotic expansion. Error estimates are performed in Section 4, proving the main result of the paper.

**Acknowledgment:** I would like to thank Sonia Fliss and Patrick Joly for the helpful discussions we had about the writing of this paper.

# Contents

1 **Setting of the problem and the method** 3

1.1 Definition of the problem and main result 3

1.2 The matched asymptotic expansion method 4

2 **Matching conditions** 6

2.1 Algebraic preliminaries 7

2.2 Definition of the spaces $\mathcal{A}$ 9

2.3 Tools for solving the Poisson and Helmholtz equations in the spaces $\mathcal{A}$ 11

2.4 Asymptotic behaviors w.r.t. $r$ of solutions of model problems 15

2.5 Specifying of the matching conditions 18

3 **Construction of the asymptotic expansion** 23

3.1 Existence and uniqueness for far fields-like problems 24

3.2 Existence and uniqueness for corner fields-like problems 26

3.3 Construction of the fields 28
<table>
  <tbody>
    <tr>
      <td>3.4</td>
      <td>Practical way to build the far fields</td>
      <td>30</td>
    </tr>
    <tr>
      <td>4</td>
      <td>Error estimates</td>
      <td>32</td>
    </tr>
    <tr>
      <td>A</td>
      <td>Appendix: proof of Lemma 2.8</td>
      <td>36</td>
    </tr>
    <tr>
      <td>B</td>
      <td>Appendix: asymptotic behaviors w.r.t. r</td>
      <td>37</td>
    </tr>
    <tr>
      <td>B.1</td>
      <td>Proof of Theorem 2.23: asymptotic behavior for corner fields-like problems</td>
      <td>37</td>
    </tr>
    <tr>
      <td>B.2</td>
      <td>Proof of Theorem 2.22: asymptotic behavior for far-and-layer fields-like problems</td>
      <td>45</td>
    </tr>
    <tr>
      <td>References</td>
      <td></td>
      <td>47</td>
    </tr>
  </tbody>
</table>

# 1 Setting of the problem and the method

## 1.1 Definition of the problem and main result

To describe the domain, let us introduce $\Theta \in (0, 2\pi)$, $\Omega := \{(r \cos \theta, r \sin \theta) \mid r \in \mathbb{R}_+^*, \theta \in (0, \Theta)\}$, $\Lambda := \mathbb{R}_+^* \times (-1, 0)$, $\Gamma = \mathbb{R}_+^* \times \{0\}$, $\Sigma_\Omega = \{(r \cos \Theta, r \sin \Theta) \mid r \in \mathbb{R}_+^*\}$ and $\Sigma_\Lambda := \mathbb{R}_+^* \times \{-1\}$. All these sets are shown in Figure 1. Then let $\Omega_1 \subset \mathbb{R}^2$ be an open set that coincides with $\Omega \cup \Gamma \cup \Lambda$ outside of the disc $B(0, R_c)$ for some $R_c \in \mathbb{R}_+^*$. In addition, let $\mu, \rho \in L^\infty(\Omega_1)$ be two functions greater than a positive constant (ellipticity assumption), and equal to $\mu_0$ and $\rho_0$ in $\Omega \setminus B(0, R_c)$ and to $\mu_1$ and $\rho_1$ in $\Lambda \setminus B(0, R_c)$. See Figures 2 and 3 for different configurations.

<figure><img src="image_2.png" /><figcaption>Figure 1: The domains $\Omega$ (on the left) and $\Lambda$ (on the right)</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Figure 2: The domain $\Omega_1$ with a configuration example of $\Omega_1 \cap B(0, R_c)$</figcaption></figure>

Let $\varepsilon > 0$. The physical domain is given by $\Omega_\varepsilon := \{(x, y) \in \mathbb{R}^2 \mid (x_\varepsilon, y_\varepsilon) \in \Omega_1\}$. We introduce the scaled coefficients $\mu_\varepsilon : (x, y) \in \Omega_\varepsilon \mapsto \mu(x_\varepsilon, y_\varepsilon)$ and $\rho_\varepsilon : (x, y) \in \Omega_\varepsilon \mapsto \rho(x_\varepsilon, y_\varepsilon)$, and the scaled variables $X := \frac{x}{x_\varepsilon}$ and $Y := \frac{y}{y_\varepsilon}$. Let $\omega \in C \setminus \mathbb{R}$ and $f_s \in H^{-1}(\Omega)$ a source term s.t. $\text{dist}(\text{supp}(f_s), \Gamma) > 0$. We denote $u_\varepsilon$ the unique solution in $H_0^1(\Omega_\varepsilon)$ of

$$ \text{div}(\mu_\varepsilon \nabla u_\varepsilon) + \omega^2 \rho_\varepsilon u_\varepsilon = f_s \quad \text{in } \Omega_\varepsilon \qquad (1.1) $$
Let $\varepsilon > 0$. The physical domain is given by $\Omega_\varepsilon := \{(x, y) \in \mathbb{R}^2 \mid (\frac{x}{\varepsilon}, \frac{y}{\varepsilon}) \in \Omega_1\}$. We introduce the scaled coefficients $\mu_\varepsilon : (x, y) \in \Omega_\varepsilon \mapsto \mu(\frac{x}{\varepsilon}, \frac{y}{\varepsilon})$ and $\rho_\varepsilon : (x, y) \in \Omega_\varepsilon \mapsto \rho(\frac{x}{\varepsilon}, \frac{y}{\varepsilon})$, and the scaled variables $X := \frac{x}{\varepsilon}$ and $Y := \frac{y}{\varepsilon}$. Let $\omega \in \mathbb{C} \setminus \mathbb{R}$ and $f_s \in H^{-1}(\Omega)$ a source term s.t. $\text{dist}(\text{supp}(f_s), \Gamma) > 0$. We denote $u_\varepsilon$ the unique solution in $H_0^1(\Omega_\varepsilon)$ of

$$ \operatorname{div}(\mu_{\varepsilon} \nabla u_{\varepsilon}) + \omega^2 \rho_{\varepsilon} u_{\varepsilon} = f_s \quad \text{in } \Omega_{\varepsilon} \qquad (1.1) $$

$\mathfrak{I}(\omega) \neq 0$ is a technical assumption that makes this problem well-posed (it suffices to use the Lax-Milgram theorem) with a stability constant independent of $\varepsilon$:

$$ \exists C > 0, \forall f_s \in H^{-1}(\Omega), \forall \varepsilon > 0, \quad \|u_\varepsilon\|_{H^1(\Omega_\varepsilon)} \le C \|f_s\|_{H^{-1}(\Omega)} \qquad (1.2) $$

The case $\mathfrak{I}(\omega) = 0$ is an open question and will be the object of a future work.

The main result of this paper is given in the following theorem, proven in Section 4, page 36.

**Theorem 1.1: asymptotic expansion of $u_\varepsilon$**

Let $\mathbb{P} := \mathbb{N} + \frac{\pi}{\Theta}\mathbb{N}$. There exist $(n_p) \in \mathbb{N}^\mathbb{P}$ and a family $(u_{p,\ell})_{p \in \mathbb{P}, \ell \in [0, n_p]}$ of elements of $H_{\text{loc}}^1(\Omega)$ that can be build recursively w.r.t. $p$ (see Theorem 3.17 for the construction) such that

$$ \forall P \in \mathbb{R}_+, \forall \delta > 0, \quad \left\| u_{\varepsilon} - \sum_{p \in \mathbb{P} \cap [0, P]} \sum_{\ell=0}^{n_p} \varepsilon^p \ln^{\ell} \varepsilon u_{p, \ell} \right\|_{H^1(\Omega \setminus B(0, \delta))} = o(\varepsilon^P) \quad \text{when } \varepsilon \to 0. $$

The presence of integer powers of $\varepsilon$ is entirely classical in asymptotic analysis. Integer powers of $\varepsilon^{\pi/\Theta}$ and $\ln \varepsilon$ can be found in other asymptotic expansions involving corners, see [11, 6, 7]. Theorem 1.1 can be extended to the case where $\Sigma_\Omega$ is covered by another layer (see Remark 2.7 for a useful point).

**Notations:** We denote $(x, y)$ the cartesian coordinates, $(r, \theta)$ the polar coordinates with $\theta \in [0, 2\pi)$, $B(0, r)$ the disc of $\mathbb{R}^2$ of radius $r$ centered at $(0, 0)$, $k_i := \omega\sqrt{\rho_i/\mu_i}$ for any $i \in \{0, 1\}$ and $\alpha := e^{-i\Theta}$.

## 1.2 The matched asymptotic expansion method

To take into account the different behaviors of the solution in the layer, near the corner and far from the corner and the layer, we divide $\Omega_\varepsilon$ in three zones, illustrated in Figure 4. In each zone we postulate an asymptotic expansion in powers of $\varepsilon$ and $\ln \varepsilon$, called “ansatz”.

<figure><img src="image_2.png" /><figcaption>Figure 4: Zones of the matched asymptotic expansion</figcaption></figure>

Let us assume that for any $(p, \ell) \in \mathbb{P} \times \mathbb{N}$ there exist three functions independent of $\varepsilon$ – namely $u_{p,\ell}$ defined on $\Omega$ called “far field”, $U_{p,\ell}$ defined on $\Lambda$ called “layer field” and $S_{p,\ell}$ defined on $\Omega_1$ called “corner field” – such that $u_\varepsilon$ is formally written as:

$$ u_{\varepsilon}(x, y) = \sum_{p \in \mathbb{P}} \sum_{\ell \in \mathbb{N}} \varepsilon^p \ln^{\ell} \varepsilon u_{p, \ell}(x, y) \quad \text{when } r := \|(x, y)\| \ge \sqrt{\varepsilon} \text{ and } (x, y) \notin \mathbb{R}_+ \times (-\varepsilon, 0) \text{ (far zone),} $$
<figure><img src="image_4.png" /><figcaption>Figure 3: Other configuration examples of $\Omega_1 \cap B(0, R_c)$ for different values of $\Theta$</figcaption></figure>

$\mathfrak{I}(\omega) \neq 0$ is a technical assumption that makes this problem well-posed (it suffices to use the Lax-Milgram theorem) with a stability constant independent of $\varepsilon$:

$$ \exists C > 0, \forall f_s \in H^{-1}(\Omega), \forall \varepsilon > 0, \quad \|u_\varepsilon\|_{H^1(\Omega_\varepsilon)} \le C \|f_s\|_{H^{-1}(\Omega)} \quad (1.2) $$

The case $\mathfrak{I}(\omega) = 0$ is an open question and will be the object of a future work.

The main result of this paper is given in the following theorem, proven in Section 4, page 36.

**Theorem 1.1: asymptotic expansion of $u_\varepsilon$**

Let $\mathbb{P} := \mathbb{N} + \frac{\pi}{\Theta}\mathbb{N}$. There exist $(n_p) \in \mathbb{N}^\mathbb{P}$ and a family $(u_{p,\ell})_{p \in \mathbb{P}, \ell \in [0, n_p]}$ of elements of $H_{\text{loc}}^1(\Omega)$ that can be build recursively w.r.t. $p$ (see Theorem 3.17 for the construction) such that

$$ \forall P \in \mathbb{R}_+, \forall \delta > 0, \quad \left\| u_\varepsilon - \sum_{p \in \mathbb{P} \cap [0, P]} \sum_{\ell=0}^{n_p} \varepsilon^p \ln^\ell \varepsilon u_{p,\ell} \right\|_{H^1(\Omega \setminus B(0, \delta))} = o(\varepsilon^P) \quad \text{when } \varepsilon \to 0. $$

The presence of integer powers of $\varepsilon$ is entirely classical in asymptotic analysis. Integer powers of $\varepsilon^{\pi/\Theta}$ and $\ln \varepsilon$ can be found in other asymptotic expansions involving corners, see [12, 6, 7]. Theorem 1.1 can be extended to the case where $\Sigma_\Omega$ is covered by another layer (see Remark 2.7 for a useful point).

**Notations:** We denote $(x, y)$ the cartesian coordinates, $(r, \theta)$ the polar coordinates with $\theta \in [0, 2\pi)$, $B(0, r)$ the disc of $\mathbb{R}^2$ of radius $r$ centered at $(0, 0)$, $k_i := \omega\sqrt{\rho_i/\mu_i}$ for any $i \in \{0, 1\}$ and $\alpha := e^{-i\Theta}$.

## 1.2 The matched asymptotic expansion method

To take into account the different behaviors of the solution in the layer, near the corner and far from the corner and the layer, we divide $\Omega_\varepsilon$ in three zones, illustrated in Figure 4. In each zone we postulate an asymptotic expansion in powers of $\varepsilon$ and $\ln \varepsilon$, called “ansatz”.

Let us assume that for any $(p, \ell) \in \mathbb{P} \times \mathbb{N}$ there exist three functions independent of $\varepsilon$ – namely $u_{p,\ell}$ defined on $\Omega$ called “far field”, $U_{p,\ell}$ defined on $\Lambda$ called “layer field” and $S_{p,\ell}$ defined on $\Omega_1$ called “corner field” – such that $u_\varepsilon$ is formally written as:

* $u_\varepsilon(x, y) = \sum_{p \in \mathbb{P}} \sum_{\ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon u_{p,\ell}(x, y)$ when $r := \|(x, y)\| \ge \sqrt{\varepsilon}$ and $(x, y) \notin \mathbb{R}_+ \times (-\varepsilon, 0)$ (far zone),

* $u_\varepsilon(x, y) = \sum_{p \in \mathbb{P}} \sum_{\ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon U_{p,\ell}\left(x, \frac{y}{\varepsilon}\right)$ when $x \ge \sqrt{\varepsilon}$ and $y \in (-\varepsilon, 0)$ (layer zone),

* $u_\varepsilon(x, y) = \sum_{p \in \mathbb{P}} \sum_{\ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon S_{p,\ell}\left(\frac{x}{\varepsilon}, \frac{y}{\varepsilon}\right)$ when $r \le 2\sqrt{\varepsilon}$ (corner zone).

**Remark:** We will see in Proposition 3.14 that: $\forall p \in \mathbb{P}, \exists n_p \in \mathbb{N}, \forall \ell > n_p, u_{p,\ell} = 0$ and $U_{p,\ell} = 0$ and $S_{p,\ell} = 0$.
• $u_{\varepsilon}(x, y) = \sum_{p \in \mathbb{P}} \sum_{\ell \in \mathbb{N}} \varepsilon^p \ln^{\ell} \varepsilon U_{p, \ell} \left(x, \frac{y}{\varepsilon}\right)$ when $x \ge \sqrt{\varepsilon}$ and $y \in (-\varepsilon, 0)$ (layer zone),

• $u_{\varepsilon}(x, y) = \sum_{p \in \mathbb{P}} \sum_{\ell \in \mathbb{N}} \varepsilon^p \ln^{\ell} \varepsilon S_{p, \ell} \left(\frac{x}{\varepsilon}, \frac{y}{\varepsilon}\right)$ when $r \le 2\sqrt{\varepsilon}$ (corner zone).

**Remark:** We will see in Proposition 3.14 that: $\forall p \in \mathbb{P}, \exists n_p \in \mathbb{N}, \forall \ell > n_p, u_{p,\ell} = 0$ and $U_{p,\ell} = 0$ and $S_{p,\ell} = 0$.

Injecting the above sums in the Helmholtz equation, using that $\partial_x^2[\varphi(\frac{x}{\varepsilon})] = \varepsilon^{-2}[\partial_X^2\varphi](X = \frac{x}{\varepsilon})$ and $\partial_y^2[\varphi(\frac{y}{\varepsilon})] = \varepsilon^{-2}[\partial_Y^2\varphi](Y = \frac{y}{\varepsilon})$ for any function $\varphi$, and formally identifying the powers of $\varepsilon$ and $\ln \varepsilon$, one can easily derive the following volume equations and edge conditions for the various fields.

$$ \left\{ \begin{array}{ll} \mu_0 \Delta u_{p,\ell} + \omega^2 \rho_0 u_{p,\ell} = f & \text{in } \Omega \\ u_{p,\ell} = U_{p,\ell} & \text{on } \Gamma \\ u_{p,\ell} = 0 & \text{on } \Sigma_{\Omega} \end{array} \right. \quad (1.3) \quad \left\{ \begin{array}{ll} \mu_1 \partial_Y^2 U_{p,\ell} = -(\mu_1 \partial_x^2 + \omega^2 \rho_1) U_{p-2,\ell} & \text{in } \Lambda \\ \mu_1 \partial_Y U_{p,\ell} = \mu_0 \partial_y u_{p-1,\ell} & \text{on } \Gamma \\ U_{p,\ell} = 0 & \text{on } \Sigma_{\Lambda} \end{array} \right. $$

$$ \left\{ \begin{array}{ll} \operatorname{div}(\mu \nabla S_{p,\ell}) = -\omega^2 \rho S_{p-2,\ell} & \text{in } \Omega_1 \\ S_{p,\ell} = 0 & \text{on } \partial\Omega_1 \end{array} \right. \qquad (1.5) $$

where we denote by convention $u_{p,\ell} = 0$, $U_{p,\ell} = 0$ and $S_{p,\ell} = 0$ for any $(p, \ell) \in \mathbb{R} \setminus \mathbb{P} \times \mathbb{N}$, and $\delta_{i,j} := 1$ if $i = j$ and 0 if not.

**Remarks:**

• The condition $\mu_1 \partial_Y U_{p,\ell} = \mu_0 \partial_y u_{p-1,\ell}$ is included in the problem satisfied by $U_{p,\ell}$ whereas $u_{p,\ell} = U_{p,\ell}$ is included in the problem satisfied by $u_{p,\ell}$ so that the construction is inductive: $u_{p-1,\ell}$ allows to build $U_{p,\ell}$, which allows to build $u_{p,\ell}$.

• The problem satisfied by $U_{p,\ell}$ depends only on $Y$, the variable $x$ playing the role of a parameter.

(1.3)-(1.5) would be sufficient to uniquely define the fields, if they were in their natural variational spaces (e.g. $H^1(\Omega)$ for $u_{p,\ell}$). But we need to take into account a matching condition: the far and corner fields must coincide in the intersection of the far and corner zones, and similarly for the layer and corner fields. These intersections form the matching zone (see Figure 4). Given that $\varepsilon \to 0$ and $\frac{\sqrt{\varepsilon}}{\varepsilon} \to \infty$, this zone tends to (0, 0) w.r.t. the far and layer fields, but it tends to infinity w.r.t. the corner fields. Thus, the matching condition links the asymptotic behavior of far and layer fields at the corner to the one of corner fields at infinity:

$$ \left\{ \begin{array}{ll} \sum \varepsilon^p \ln^\ell \varepsilon u_{p,\ell}(x, y) \approx \sum \varepsilon^p \ln^\ell \varepsilon S_{p,\ell}(\frac{x}{\varepsilon}, \frac{y}{\varepsilon}) & \text{in } \Omega \text{ when } r \to 0 \text{ and } \frac{r}{\varepsilon} \to \infty \\ \sum \varepsilon^p \ln^\ell \varepsilon U_{p,\ell}(x, \frac{y}{\varepsilon}) \approx \sum \varepsilon^p \ln^\ell \varepsilon S_{p,\ell}(\frac{x}{\varepsilon}, \frac{y}{\varepsilon}) & \text{in } \Lambda \text{ when } x \to 0 \text{ and } \frac{x}{\varepsilon} \to \infty \end{array} \right. \qquad (1.6) $$

We will see that the far fields $u_{p,\ell}$ have an asymptotic expansion at the corner which is roughly a sum of powers of $r$, some of which are positive (like in a Taylor expansion e.g.). The matching conditions imply that these positive powers of $r$ have to appear in the asymptotic expansions of the corner fields at infinity. We call them singularities for the corner fields. Conversely, the asymptotic expansion of the corner fields at infinity contain negative powers of $r$ corresponding to the decay of the variational part, and these powers must be found in the far fields, which corresponds to singularities at the corner. Thus the fields cannot be searched in their natural variational spaces.

## 2 Matching conditions

This section establishes the matching condition linking corner fields to far and layer fields. This is by far the most difficult relation to derive, while all the others have been easily stated in (1.3)-(1.5). In this section we assume that the various fields exist and that they satisfy (1.3)-(1.5) and we give a necessary
<figure><img src="image_5.png" /><figcaption>Figure 4: Zones of the matched asymptotic expansion</figcaption></figure>

Injecting the above sums in the Helmholtz equation, using that $\partial_x^2[\varphi(\frac{x}{\epsilon})] = \epsilon^{-2}[\partial_X^2\varphi](X = \frac{x}{\epsilon})$ and $\partial_y^2[\varphi(\frac{y}{\epsilon})] = \epsilon^{-2}[\partial_Y^2\varphi](Y = \frac{y}{\epsilon})$ for any function $\varphi$, and formally identifying the powers of $\epsilon$ and $\ln\epsilon$, one can easily derive the following volume equations and edge conditions for the various fields.

$$ \left\{ \begin{array}{ll} \mu_0 \Delta u_{p,\ell} + \omega^2 \rho_0 u_{p,\ell} = f \delta_{p,0} \delta_{\ell,0} & \text{in } \Omega \\ u_{p,\ell} = U_{p,\ell} & \text{on } \Gamma \\ u_{p,\ell} = 0 & \text{on } \Sigma_{\Omega} \end{array} \right. \quad (1.3) \quad \left\{ \begin{array}{ll} \mu_1 \partial_Y^2 U_{p,\ell} = -(\mu_1 \partial_x^2 + \omega^2 \rho_1) U_{p-2,\ell} & \text{in } \Lambda \\ \mu_1 \partial_Y U_{p,\ell} = \mu_0 \partial_y u_{p-1,\ell} & \text{on } \Gamma \\ U_{p,\ell} = 0 & \text{on } \Sigma_{\Lambda} \end{array} \right. $$

$$ \left\{ \begin{array}{ll} \text{div}(\mu \nabla S_{p,\ell}) = -\omega^2 \rho S_{p-2,\ell} & \text{in } \Omega_1 \\ S_{p,\ell} = 0 & \text{on } \partial\Omega_1 \end{array} \right. \qquad (1.5) $$

where we denote by convention $u_{p,\ell} = 0$, $U_{p,\ell} = 0$ and $S_{p,\ell} = 0$ for any $(p, \ell) \in \mathbb{R} \setminus \mathbb{P} \times \mathbb{N}$, and $\delta_{i,j} := 1$ if $i = j$ and $0$ if not.

### Remarks:

* The condition $\mu_1 \partial_Y U_{p,\ell} = \mu_0 \partial_y u_{p-1,\ell}$ is included in the problem satisfied by $U_{p,\ell}$ whereas $u_{p,\ell} = U_{p,\ell}$ is included in the problem satisfied by $u_{p,\ell}$ so that the construction is inductive: $u_{p-1,\ell}$ allows to build $U_{p,\ell}$, which allows to build $u_{p,\ell}$.

* The problem satisfied by $U_{p,\ell}$ depends only on $Y$, the variable $x$ playing the role of a parameter.

(1.3)–(1.5) would be sufficient to uniquely define the fields, if they were in their natural variational spaces (e.g. $H^1(\Omega)$ for $u_{p,\ell}$). But we need to take into account a matching condition: the far and corner fields must coincide in the intersection of the far and corner zones, and similarly for the layer and corner fields. These intersections form the matching zone (see Figure 4). Given that $\epsilon \to 0$ and $\frac{\sqrt{\epsilon}}{\epsilon} \to \infty$, this zone tends to (0,0) w.r.t. the far and layer fields, but it tends to infinity w.r.t. the corner fields. Thus, the matching condition links the asymptotic behavior of far and layer fields at the corner to the one of corner fields at infinity:

$$ \left\{ \begin{array}{ll} \sum \epsilon^p \ln^\ell \epsilon u_{p,\ell}(x, y) \approx \sum \epsilon^p \ln^\ell \epsilon S_{p,\ell}(\frac{x}{\epsilon}, \frac{y}{\epsilon}) & \text{in } \Omega \text{ when } r \to 0 \text{ and } \frac{r}{\epsilon} \to \infty \\ \sum \epsilon^p \ln^\ell \epsilon U_{p,\ell}(x, \frac{y}{\epsilon}) \approx \sum \epsilon^p \ln^\ell \epsilon S_{p,\ell}(\frac{x}{\epsilon}, \frac{y}{\epsilon}) & \text{in } \Lambda \text{ when } x \to 0 \text{ and } \frac{x}{\epsilon} \to \infty \end{array} \right. \qquad (1.6) $$

We will see that the far fields $u_{p,\ell}$ have an asymptotic expansion at the corner which is roughly a sum of powers of $r$, some of which are positive (like in a Taylor expansion e.g.). The matching conditions imply that these positive powers of $r$ have to appear in the asymptotic expansions of the corner fields at infinity. We call them singularities for the corner fields. Conversely, the asymptotic expansion of the corner fields at infinity contain negative powers of $r$ corresponding to the decay of the variational part, and these powers must be found in the far fields, which corresponds to singularities at the corner. Thus the fields cannot be searched in their natural variational spaces. To derive problems that define uniquely the fields, we must specify their singular part (see Theorems 3.6 and 3.11). These parts are fixed by the matching conditions as explained in the next section.
and sufficient condition for the matching assumptions (1.6) to be satisfied. Our approach is based on an algebraic formulation of the problem, that reveals the structure of the matching relations by a rigorous algebraic calculus.

To perform the matching of the corner fields with the far and the layer fields at the same time, we merge the latter two into a single field denoted $\mathbf{u}_{p,\ell}$ and called “far-and-layer field”. It is defined on a new domain $\Pi$, defined as follows:

• If $\Theta \le \frac{3\pi}{2}$, then $\Pi := \Omega \sqcup \Gamma \sqcup \Lambda$ (disjoint union) and it is an open of $\mathbb{R}^2$.

• If $\Theta > \frac{3\pi}{2}$, then $\Omega$ and $\Lambda$ intersect as subsets of $\mathbb{R}^2$, so the previous definition is not valid anymore (see Figure 5). Thus, we define $\Pi$ as the disjoint gluing of $\Omega$ and $\Lambda$ on $\Gamma$ (which is a flat Riemannian manifold).

<figure><img src="image_3.png" /><figcaption>Figure 5: The domain $\Pi$ is equal to $\Omega \sqcup \Gamma \sqcup \Lambda$ when $\Theta \le \frac{3\pi}{2}$ (open subset of $\mathbb{R}^2$) and it is a flat Riemannian manifold when $\Theta > \frac{3\pi}{2}$ ($\Omega \cap \Lambda \ne \emptyset$).</figcaption></figure>

For all $(p, \ell) \in \mathbb{P} \times \mathbb{N}$, we define $\mathbf{u}_{p,\ell} := \begin{cases} u_{p,\ell} & \text{in } \Omega \\ U_{p,\ell} & \text{in } \Lambda \end{cases}$ and the generalized radial variable $\mathbf{r} := \begin{cases} r & \text{in } \Omega \\ x & \text{in } \Lambda \end{cases}$

A straightforward reformulation of (1.3) and (1.4) gives that for any $(p, \ell) \in \mathbb{P} \times \mathbb{N}$:

$$ \left\{ \begin{array}{ll} \mu_0 \Delta \mathbf{u}_{p,\ell} + \omega^2 \rho_0 \mathbf{u}_{p,\ell} = f \delta_{p,0} \delta_{\ell,0} & \text{in } \Omega \\ \mu_1 \partial_Y^2 \mathbf{u}_{p,\ell} = -(\mu_1 \partial_x^2 + \omega^2 \rho_1) \mathbf{u}_{p-2,\ell} & \text{in } \Lambda \\ \mu_1 \partial_Y \mathbf{u}_{p,\ell|Y=0^{-}} = \mu_0 \partial_y \mathbf{u}_{p-1,\ell|y=0^{+}} & \text{on } \Gamma \\ \mathbf{u}_{p,\ell|y=0^{+}} - \mathbf{u}_{p,\ell|Y=0^{-}} = 0 & \text{on } \Gamma \\ \mathbf{u}_{p,\ell} = 0 & \text{on } \Sigma_{\Omega} \cup \Sigma_{\Lambda} \end{array} \right. \quad (2.1) $$

where by convention $\mathbf{u}_{p,\ell} = 0$ when $(p, \ell) \in \mathbb{R} \setminus \mathbb{P} \times \mathbb{N}$.

Let us give some starting point ideas to dive into this section. The matching assumption under study links the asymptotic behaviors of $\sum_{p,\ell} \varepsilon^p \ln^\ell \varepsilon \mathbf{u}_{p,\ell}$ when $\mathbf{r} \to 0$ and $\sum_{p,\ell} \varepsilon^p \ln^\ell \varepsilon S_{p,\ell}$ when $\mathbf{r} \to \infty$. So we can begin with a look at the asymptotic of $\mathbf{u}_{p,\ell}$ when $\mathbf{r} \to 0$, especially on $\Omega$ because it is the most interesting part. First, by (2.1), $u_{0,0}$ satisfies $\mu_0 \Delta u_{0,0} + \omega^2 \rho_0 u_{0,0} = 0$ in the vicinity of the corner in $\Omega$, with homogeneous Dirichlet condition on $\partial\Omega$. So using separation of variables, it is easy to show that:

$$ u_{0,0}(r, \theta) = \sum_{r \le 1} \sigma_d(\mathbf{u}_{0,0}) J_d(k_0 r) \sin(d\theta) = \sum_{d \in \mathbb{N}^*} \sigma_d(\mathbf{u}_{0,0}) \left( \sum_{n=0}^{\infty} a_{d,n} r^{d+2n} \right) \sin(d\theta) \quad (2.2) $$

with $\sigma_d(\mathbf{u}_{0,0})$ and $a_{d,n}$ some coefficients in $\mathbb{C}$, $J_d$ Bessel functions of the first kind and $k_0 := \omega \sqrt{\rho_0/\mu_0}$. Then, one can show that (2.1) implies that $u_{1,0}$ satisfies the Helmholtz equation but with condition $\mu_1 u_{1,0} = \mu_0 \partial_y u_{0,0|y=0^+}$ on $\Gamma$. Using (2.2), one can show that there exist some functions $f_{q,i}$ and coefficients $\sigma_d(\mathbf{u}_{1,0})$ s.t.:

$$ u_{1,0} = \sum_{r \le 1} r^{d-1} (f_{d,0}(\theta) + \ln(r) f_{d,1}(\theta)) + \sum_{d \in \mathbb{N}^*} \sigma_d(\mathbf{u}_{1,0}) J_d(k_0 r) \sin(d\theta) $$

where the first sum is a particular solution of Helmholtz's equation that has trace $\frac{\mu_0}{\mu_1}$, $\partial_y u_{0|y=0^+}$ on $\Gamma$, and the second one is a homogeneous solution. More generally, $u_{p,\ell}$ has a similar decomposition with
## 2 Matching conditions

This section establishes the matching condition linking corner fields to far and layer fields. This is by far the most difficult relation to derive, while all the others have been easily stated in (1.3)-(1.5). In this section we assume that the various fields exist and that they satisfy (1.3)-(1.5) and we give a necessary and sufficient condition for the matching assumptions (1.6) to be satisfied. Our approach is based on an algebraic formulation of the problem, that reveals the structure of the matching relations by a rigorous algebraic calculus.

To perform the matching of the corner fields with the far and the layer fields at the same time, we merge the latter two into a single field denoted $\mathbf{u}_{p,\ell}$ and called “far-and-layer field”. It is defined on a new domain $\Pi$, defined as follows:

* If $\Theta \le \frac{3\pi}{2}$, then $\Pi := \Omega \sqcup \Gamma \sqcup \Lambda$ (disjoint union) and it is an open of $\mathbb{R}^2$.

* If $\Theta > \frac{3\pi}{2}$, then $\Omega$ and $\Lambda$ intersect as subsets of $\mathbb{R}^2$, so the previous definition is not valid anymore (see Figure 5). Thus, we define $\Pi$ as the disjoint gluing of $\Omega$ and $\Lambda$ on $\Gamma$ (which is a flat Riemannian manifold).

<figure><img src="image_6.png" /><figcaption>Figure 5: The domain $\Pi$ is equal to $\Omega \sqcup \Gamma \sqcup \Lambda$ when $\Theta \le \frac{3\pi}{2}$ (open subset of $\mathbb{R}^2$) and it is a flat Riemannian manifold when $\Theta > \frac{3\pi}{2}$ ($\Omega \cap \Lambda \ne \emptyset$).</figcaption></figure>

For all $(p, \ell) \in \mathbb{P} \times \mathbb{N}$, we define $\mathbf{u}_{p,\ell} := \begin{cases} u_{p,\ell} & \text{in } \Omega \\ U_{p,\ell} & \text{in } \Lambda \end{cases}$ and the generalized radial variable $\mathbf{r} := \begin{cases} r & \text{in } \Omega \\ x & \text{in } \Lambda \end{cases}$

A straightforward reformulation of (1.3) and (1.4) gives that for any $(p, \ell) \in \mathbb{P} \times \mathbb{N}$:

$$ \left\{ \begin{array}{ll} \mu_0 \Delta \mathbf{u}_{p,\ell} + \omega^2 \rho_0 \mathbf{u}_{p,\ell} = f \delta_{p,0} \delta_{\ell,0} & \text{in } \Omega \\ \mu_1 \partial_Y^2 \mathbf{u}_{p,\ell} = -(\mu_1 \partial_x^2 + \omega^2 \rho_1) \mathbf{u}_{p-2,\ell} & \text{in } \Lambda \\ \mu_1 \partial_Y \mathbf{u}_{p,\ell|Y=0^-} = \mu_0 \partial_y \mathbf{u}_{p-1,\ell|y=0^+} & \text{on } \Gamma \\ \mathbf{u}_{p,\ell|y=0^+} - \mathbf{u}_{p,\ell|Y=0^-} = 0 & \text{on } \Gamma \\ \mathbf{u}_{p,\ell} = 0 & \text{on } \Sigma_{\Omega} \cup \Sigma_{\Lambda} \end{array} \right. \quad (2.1) $$

where by convention $\mathbf{u}_{p,\ell} = 0$ when $(p, \ell) \in \mathbb{R} \setminus \mathbb{P} \times \mathbb{N}$.

Let us give some starting point ideas to dive into this section. The matching assumption under study links the asymptotic behaviors of $\sum_{p,\ell} \epsilon^p \ln^\ell \epsilon \mathbf{u}_{p,\ell}$ when $\mathbf{r} \to 0$ and $\sum_{p,\ell} \epsilon^p \ln^\ell \epsilon S_{p,\ell}$ when $\mathbf{r} \to \infty$. So we can begin with a look at the asymptotic of $\mathbf{u}_{p,\ell}$ when $\mathbf{r} \to 0$, especially on $\Omega$ because it is the most interesting part. First, by (2.1), $u_{0,0}$ satisfies $\mu_0 \Delta u_{0,0} + \omega^2 \rho_0 u_{0,0} = 0$ in the vicinity of the corner in $\Omega$, with homogeneous Dirichlet condition on $\partial\Omega$. So using separation of variables, it is easy to show that:

$$ u_{0,0}(r, \theta) = \sum_{r \le 1} \sigma_d(\mathbf{u}_{0,0}) J_d(k_0 r) \sin(d\theta) = \sum_{d \in \mathbb{N}^*} \sigma_d(\mathbf{u}_{0,0}) \left( \sum_{n=0}^{\infty} a_{d,n} r^{d+2n} \right) \sin(d\theta) \quad (2.2) $$

with $\sigma_d(\mathbf{u}_{0,0})$ and $a_{d,n}$ some coefficients in $\mathbb{C}$, $J_d$ Bessel functions of the first kind and $k_0 := \omega \sqrt{\rho_0/\mu_0}$. Then, one can show that (2.1) implies that $u_{1,0}$ satisfies the Helmholtz equation with condition $\mu_1 u_{1,0} = \mu_0 \partial_y u_{0,0|y=0^+}$ on $\Gamma$. Using (2.2), one can show that there exist some functions $f_{q,i}$ and coefficients
potentially lower powers of $r$ (the smallest power being $-p$) and higher powers of $\ln r$.

$u_{1,0}(\Omega)$ is not in $H^1$ because it behaves as $\frac{1}{r}$ when $r \to 0$. More generally $u_{p,\ell}$ is not in $H^1(\Omega)$. It can be decomposed as the sum of a $H^1$ function and a function that is singular at $(0,0)$. Proposition 3.6 will show how to find solutions in such spaces and it reveals that they depend on some coefficients $\sigma_d(\mathbf{u}_{p,\ell})$ for $d < 0$ (corresponding to the singular part) while the coefficients for $d > 0$ (corresponding to the variational part) are uniquely fixed. So the $\sigma_d(\mathbf{u}_{p,\ell})$ for $d < 0$ are degrees of freedom in the construction of far-and-layer fields, and they will be fixed by the matching conditions. Likewise, the corner fields have similar decompositions when $\mathbf{r} \to \infty$, and singular coefficients $\sigma_d(S_{p,\ell})$ for $d > 0$ (but here singular when $\mathbf{r} \to \infty$) that will be fixed by the matching conditions.

In Section 2.1 we introduce algebraic formal series later used to perform efficient calculations on the asymptotic expansions. In Section 2.2 we define spaces of explicit functions called $\mathcal{A}(\dots)$ that contain all possible singularities at 0 (for the far-and-layer fields) or at $\infty$ (for the corner fields). In the previous paragraph, it corresponds to the functions $r^q \ln^i r f_{q,i}(\theta)$. In Section 2.3 we give tools to compute these singularities. In Section 2.4, we write the asymptotic expansions w.r.t. $\mathbf{r}$ of the various fields. Finally, in Section 2.5, we re-express the matching conditions (1.6) with equations that can be used to build the fields.

## 2.1 Algebraic preliminaries

To handle infinite series that may not converge, e.g. "$\sum_{p,\ell} \epsilon^p \ln^\ell \epsilon u_{p,\ell}$", we use the algebraic notion of formal series introduced in this section. Let $E$ be a vector space and $(E_i)_{i \in I}$ be a family of vector subspaces of $E$. To begin, let us remind that $\sum_{i \in I} E_i$ designates the vector subspace of $E$ made of finite sums of elements of the $E_i$. If this sum is direct, we denote it $\bigoplus_{i \in I} E_i$. From now on, we assume that the sum is direct. In order to deal with infinite sums we introduce the following definition.

**Notation 2.1:** Let us denote

$$ \forall (\varphi_i) \in \prod_{i \in I} E_i, \quad \sum_{i \in I} \varphi_i := (\varphi_i) \quad \text{and} \quad \sum_{i \in I} E_i := \prod_{i \in I} E_i. $$

Note the boldness of the symbol $\sum$. $\sum_{i \in I} \varphi_i$ is not a real sum that can be computed, but just a notation called “formal series”. Its support is defined as $\{i \in I | \varphi_i \neq 0\}$. In additional for any $J \subset I$ that contains this support, we also denote $\sum_{j \in J} \varphi_j := (\varphi_i)_{i \in I}$.

There is a canonical injection $\bigoplus E_i \to \sum E_i$, that maps any sum $\sum_{i \in I} \varphi_i$ with finite support (and $\forall i, \varphi_i \in E_i$) to the formal series $\sum_{i \in I} \varphi_i$. So we can consider in practice that $\bigoplus E_i$ is included in $\sum E_i$.

We will use Notation 2.1 with $I = \mathbb{R}$ and $E_d = \mathcal{A}_d(\dots)$ a space of functions that behave like $\mathbf{r}^d$ defined in Section 2.2. In Section 2.3 we build some operators in the spaces $\mathcal{A}$ that have a translation action on the index $d$. We say that they have a “degree” (cf. Definition 2.2 and Figure 6). That allows us to naturally extend them to the formal series of the spaces $\mathcal{A}$ via the construction below.

**Definition 2.2: operators with a degree**

Let $F$ be another vector space and $(F_d)_{d \in \mathbb{R}}$ be a family of subspaces of $F$ s.t. the sum $\sum F_d$ is direct.

Let $f: \bigoplus E_d \to \bigoplus F_d$ be a linear map and let $d_0 \in \mathbb{R}$.

We say that $f$ has degree $d_0$ iff: $\forall d \in \mathbb{R}, \forall \varphi \in E_d, f(\varphi) \in F_{d+d_0}$. In this case we denote $\deg f := d_0$ and we extend $f$ from $\sum E_d$ to $\sum F_d$ by setting $f(\sum_{d \in \mathbb{R}} \varphi_d) := \sum_{d \in \mathbb{R}} f(\varphi_d)$ for any $(\varphi_d)$.
$\sigma_d(\mathbf{u}_{1,0})$ s.t.:

$$u_{1,0} = \sum_{r \ll 1} r^{d-1} (f_{d,0}(\theta) + \ln(r) f_{d,1}(\theta)) + \sum_{d \in \frac{\pi}{\Theta} \mathbb{N}^*} \sigma_d(\mathbf{u}_{1,0}) J_d(k_0 r) \sin(d\theta)$$

where the first sum is a particular solution of Helmholtz's equation that has trace $\frac{\mu_0}{\mu_1} \partial_y u_{0|y=0+}$ on $\Gamma$, and the second one is a homogeneous solution. More generally, the behavior of $\mathbf{u}_{p,\ell}$ when $\mathbf{r} \to 0$ (resp. $S_{p,\ell}$ when $\mathbf{r} \to \infty$) has the following form:

$$\sum_{n=0}^{\infty} r^{d_n} \ln^{k_n} r \cdot g_{n,1}(\theta) \quad \text{in } \Omega \quad \text{and} \quad \sum_{n=0}^{\infty} r^{d_n} \ln^{k_n} r \cdot g_{n,2}(Y) \quad \text{in } \Lambda$$

where $d_n$ is an increasing (resp. decreasing) sequence in $\mathbb{Z} + \frac{\pi}{\Theta}\mathbb{Z}$ and $k_n \in \mathbb{N}$. In order to know whether the behaviors of the far-and-layer fields and the corner fields match, we need to describe the functions $g_{n,i}$. In the literature, they are usually built inductively, by solving 1D problems [12, 18, 40]. However the several inductive source terms of (2.1) imply that the $g_{n,i}$ do not depend on each other only through linear sequences, but rather through a tree of dependencies that grows exponentially as one builds more terms of the asymptotic expansion (see Remark 2.27).

To overcome the difficulty of matching such complex singularities, the first original idea of this section is to decompose them on a basis of functions. Thus, only the coefficients of the decompositions need to be matched. This somehow mimics the simplicity of infinite periodic layers, for which a basis consists of the integer powers of the distance to the layer, since the fields behaviors in the matching zone are polynomial. We provide explicit expressions of those basis functions. Therefore all singularities have analytical expressions, which allow to compute them both exactly and very quickly. This last point is an original contribution compared to the existing literature.

The second idea of this section is to define operators in order to describe how singularities are linked to each other. These operators are defined from the solution of Poisson-like problems in $\Pi$, given by (2.9). They can be composed, added and factorized in all possible ways, which, in contrast to linear sequences of functions, offers sufficient flexibility to describe all singularities. An important result (Theorem 2.26) is that all singularities are generated, through these operators, by the fundamental singularities $\phi_d := r^d \sin(d\theta)$, $d \in \frac{\pi}{\Theta}\mathbb{Z}^*$ (which are part of the basis). It follows that it is sufficient to match only the coefficients of the fields behaviors on the $\phi_d$ (denoted $\sigma_d(\mathbf{u}_{p,\ell})$ for the far fields and $\sigma_d(S_{p,\ell})$ for the near fields). More precisely only the coefficients involved in the non variational behaviors of the fields need to be imposed, i.e. the $\sigma_d(\mathbf{u}_{p,\ell})$ with $d < 0$ and the $\sigma_d(S_{p,\ell})$ with $d > 0$.

The last main idea is to use formal series to treat all orders at once and to focus on the calculative nature of the matching, without having to deal with sum truncations and asymptotic remainders like $o(r^d)$. Formal series are usually used in asymptotic analysis in an intuitive way like we did in Section 1. Here we provided rigorous foundations to formal series in order to use them as a proof tool and to justify non trivial operations on them, such as applying operators, and even formal series of operators. Let us mention that [21] also used an approach with formal series to study a problem with a thin layer, but with more elementary algebraic tools.

All these ideas form a system of powerful calculus tools that are suitable to face the sheer complexity of the matching.

In Section 2.1 we introduce the algebraic formal series. In Section 2.2 we define the spaces $\mathcal{A}(\dots)$ of explicit functions that contain the singularities. In Section 2.3 we build the mentioned operators on singularities. In Section 2.4, we write the asymptotic expansions w.r.t. $\mathbf{r}$ of the various fields. Finally, in Section 2.5, we re-express the matching conditions (1.6) with equations that can be used to build the fields (Theorem 2.30).

## 2.1 Algebraic preliminaries

To handle infinite series that may not converge, e.g. "$\sum_{p,\ell} \varepsilon^p \ln^\ell \varepsilon u_{p,\ell}$", we use the algebraic notion of formal series introduced in this section. Let $E$ be a vector space and $(E_i)_{i \in I}$ be a family of vector
<figure><img src="image_4.png" /><figcaption>Figure 6: Schematic illustration of an operator that has degree $d_0$ (here $d_0 > 0$)</figcaption></figure>

**Definition 2.3:** Let $\sum_{d \in \mathbb{R}} \varphi_d$ be an element of $\sum E_d$ with a support bounded from below and $d_{\inf} := \inf \text{supp}(\varphi_d)$. For any linear map $f: \bigoplus E_d \to \bigoplus E_d$ that has a positive degree, we define

$$ \sum_{n=0}^{\infty} f^n \left( \sum_{d \in \mathbb{R}} \varphi_d \right) := \sum_{d \in \mathbb{R}} \sum_{n \in \mathbb{N}} f^n(\varphi_{d-n \deg f}) \quad (2.3) $$

(where $f^n$ is the n-th iterated composition of $f$). See Figure 7. More generally, for any finite set $\mathcal{F}$ of linear maps $\bigoplus E_d \to \bigoplus E_d$ that have positive degrees, we set

$$ \sum_{n=0}^{\infty} \sum_{(f_1, \dots, f_n) \in \mathcal{F}^n} f_1 \circ \dots \circ f_n \left( \sum_{d \in \mathbb{R}} \varphi_d \right) := \sum_{d \in \mathbb{R}} \sum_{n \in \mathbb{N}} \sum_{(f_1, \dots, f_n) \in \mathcal{F}^n} f_1 \circ \dots \circ f_n (\varphi_{d-\sum_{1 \le i \le n} \deg f_i}). \quad (2.4) $$

We also denote it as $\langle \mathcal{F} \rangle (\sum \varphi_d)$, or $\langle \tilde{f}_1, \dots, \tilde{f}_k \rangle (\sum \varphi_d)$ if $\mathcal{F} = \{\tilde{f}_1, \dots, \tilde{f}_k\}$. This is well-defined because, for any $d$, the sums over $n$ in the right-hand sides of (2.3)–(2.4) have a finite number of non-zero terms and belong to $E_d$.

If $\text{supp}(\varphi_d)$ is bounded from above and the elements of $\mathcal{F}$ have a negative degree, we can do the same definition.

<figure><img src="image_5.png" /><figcaption>Figure 7: Schematic illustration of $\sum_{n=0}^{\infty} f^n$ for $f$ a linear map that has a positive degree</figcaption></figure>

In Section 2.5, we introduce formal series with powers of $\varepsilon$ in order to express the matching condition. They are defined similarly to Notation 2.1: for any set $P \subset \mathbb{R}$ and any family $(\tilde{E}_p)_{p \in P}$ of vector spaces, we denote

$$ \forall (\varphi_p) \in \prod_{p \in P} \tilde{E}_p, \quad \sum_{p \in P} \varepsilon^p \varphi_p := (\varphi_p) \quad \text{and} \quad \sum_{p \in P} \varepsilon^p \tilde{E} := \prod_{p \in P} \tilde{E}_p. \quad (2.5) $$

Again this is only a notation and here $\varepsilon$ is not a real number but an algebraic indeterminate. This is similar to Notation 2.1 and we will later choose to use either notation depending on the physical meaning of the formal series.

For instance, the case $\tilde{E} = \mathbb{C}$ and $P = \mathbb{N}$ gives the classical set of formal power series, usually denoted $\mathbb{C}[[\varepsilon]]$ (see [30, 32]). The Taylor approximations at 0 of any smooth function $f$ can be represented by $\sum_{p=0}^{\infty} \varepsilon^p \frac{f^{(p)}(0)}{p!} \in \mathbb{C}[[\varepsilon]]$. Truncations of this series give approximations at a given order. We will use similar representations for the asymptotic expansion of $u_\varepsilon$.

Let us take $P := \mathbb{R}$, $\tilde{E} := \sum_{d \in \mathbb{R}} E_d$ and $\tilde{E}_p := \tilde{E}$ for any $p$. Let $p_0 \in \mathbb{R}$ and $f: \bigoplus E_d \to \bigoplus E_d$ a linear
subspaces of E. To begin, let us remind that $\sum_{i \in I} E_i$ designates the vector subspace of E made of finite sums of elements of the $E_i$. If this sum is direct, we denote it $\bigoplus_{i \in I} E_i$. From now on, we assume that the sum is direct. In order to deal with infinite sums we introduce the following definition.

**Notation 2.1:** Let us denote

$$ \forall (\varphi_i) \in \prod_{i \in I} E_i, \quad \sum_{i \in I} \varphi_i := (\varphi_i) \quad \text{and} \quad \sum_{i \in I} E_i := \prod_{i \in I} E_i. $$

Note the boldness of the symbol $\sum$. $\sum_{i \in I} \varphi_i$ is not a real sum that can be computed, but just a notation called “formal series”. Its support is defined as $\{i \in I | \varphi_i \neq 0\}$. In additional for any $J \subset I$ that contains this support, we also denote $\sum_{j \in J} \varphi_j := (\varphi_i)_{i \in I}$.

There is a canonical injection $\bigoplus E_i \to \sum E_i$, that maps any sum $\sum_{i \in I} \varphi_i$ with finite support (and $\forall i, \varphi_i \in E_i$) to the formal series $\sum_{i \in I} \varphi_i$. So we can consider in practice that $\bigoplus E_i$ is included in $\sum E_i$.

We will use Notation 2.1 with $I = \mathbb{R}$ and $E_d = \mathcal{A}_d(\dots)$ a space of functions that behave like $\mathbf{r}^d$ defined in Section 2.2. In Section 2.3 we build some operators in the spaces $\mathcal{A}$ that have a translation action on the index $d$. We say that they have a “degree” (cf. Definition 2.2 and Figure 6). That allows us to naturally extend them to the formal series of the spaces $\mathcal{A}$ via the construction below.

**Definition 2.2:** operators with a degree

Let $F$ be another vector space and $(F_d)_{d \in \mathbb{R}}$ be a family of subspaces of $F$ s.t. the sum $\sum F_d$ is direct.

Let $f: \bigoplus E_d \to \bigoplus F_d$ be a linear map and let $d_0 \in \mathbb{R}$.

We say that $f$ has degree $d_0$ iff: $\forall d \in \mathbb{R}, \forall \varphi \in E_d, f(\varphi) \in F_{d+d_0}$. In this case we denote $\deg f := d_0$ and we extend $f$ from $\sum E_d$ to $\sum F_d$ by setting $f(\sum_{d \in \mathbb{R}} \varphi_d) := \sum_{d \in \mathbb{R}} f(\varphi_d)$ for any $(\varphi_d)$.

<figure><img src="image_7.png" /><figcaption>Figure 6: Schematic illustration of an operator that has degree $d_0$ (here $d_0 > 0$)</figcaption></figure>

**Definition 2.3:** Let $\sum_{d \in \mathbb{R}} \varphi_d$ be an element of $\sum E_d$ with a support bounded from below and $d_{\inf} := \inf \text{supp}(\varphi_d)$. For any linear map $f: \bigoplus E_d \to \bigoplus E_d$ that has a positive degree, we define

$$ \sum_{n=0}^{\infty} f^n \left( \sum_{d \in \mathbb{R}} \varphi_d \right) := \sum_{d \in \mathbb{R}} \sum_{n \in \mathbb{N}} f^n \left( \varphi_{d-n \deg f} \right) \quad (2.3) $$

(where $f^n$ is the $n$-th iterated composition of $f$). See Figure 7. More generally, for any finite set $\mathcal{F}$ of linear maps $\bigoplus E_d \to \bigoplus E_d$ that have positive degrees, we set

$$ \sum_{n=0}^{\infty} \sum_{(f_1, \dots, f_n) \in \mathcal{F}^n} f_1 \circ \dots \circ f_n \left( \sum_{d \in \mathbb{R}} \varphi_d \right) := \sum_{d \in \mathbb{R}} \sum_{n \in \mathbb{N}} \sum_{(f_1, \dots, f_n) \in \mathcal{F}^n} f_1 \circ \dots \circ f_n \left( \varphi_{d - \sum_{1 \le i \le n} \deg f_i} \right). \quad (2.4) $$

We also denote it as $\langle \mathcal{F} \rangle (\sum \varphi_d)$, or $\langle \tilde{f}_1, \dots, \tilde{f}_k \rangle (\sum \varphi_d)$ if $\mathcal{F} = \{\tilde{f}_1, \dots, \tilde{f}_k\}$. This is well-defined because, for any $d$, the sums over $n$ in the right-hand sides of (2.3)–(2.4) have a finite number of non-zero terms and belong to $E_d$.

If $\text{supp}(\varphi_d)$ is bounded from above and the elements of $\mathcal{F}$ have a negative degree, we can do the same definition.
map that has a degree $d_0$. For any $(\varphi_p) \in \tilde{E}^{\mathbb{R}}$, we define:

$$ (\varepsilon^{p_0} f)\left(\sum_{p \in \mathbb{R}} \varepsilon^p \varphi_p\right) := \sum_{p \in \mathbb{R}} \varepsilon^p f(\varphi_{p-p_0}) \quad (2.6) $$

These kind of linear maps $\sum_p \varepsilon^p \tilde{E} \to \sum_p \varepsilon^p \tilde{E}$ are the one said to have a degree. We denote $\deg(\varepsilon^{p_0} f) := (p_0, d_0)$.

**Definition 2.4:** Let $\sum_{(p,d) \in \mathbb{R}^2} \varepsilon^p \varphi_{p,d} \in \sum_{p \in \mathbb{R}} \varepsilon^p \tilde{E}$. Let $\mathcal{G}$ be a finite set of linear maps $\sum \varepsilon^p \tilde{E} \to \sum \varepsilon^p \tilde{E}$ that have degrees. There is a finite set $\mathcal{F}$ of linear maps $\bigoplus E_d \to \bigoplus E_d$ and $(p_f)_f \in \mathbb{R}^\mathcal{F}$ s.t. $\mathcal{G} = \{\varepsilon^{p_f} f \mid f \in \mathcal{F}\}$. We assume that there is $v \in \mathbb{R}^2$ s.t. $\{(\langle p, d \rangle, v) \mid (p, d) \in \mathbb{R}^2, \varphi_{p,d} \neq 0\}$ is bounded from below and $\forall g \in \mathcal{G}, \langle \deg g, v \rangle > 0$. We denote:

$$ \sum_{n=0}^{\infty} \sum_{(g_1, \dots, g_n) \in \mathcal{G}^n} g_1 \circ \dots \circ g_n \left( \sum_{(p,d) \in \mathbb{R}^2} \varepsilon^p \varphi_{p,d} \right) := \sum_{(p,d) \in \mathbb{R}^2} \varepsilon^p \sum_{n \in \mathbb{N}} \sum_{(f_1, \dots, f_n) \in \mathcal{F}^n} f_1 \circ \dots \circ f_n (\varphi_{p-\sum_i p_{f_i}, d-\sum_i \deg f_i}). $$

which is well-defined in $\sum \varepsilon^p \tilde{E}$. We also denote it as $\langle \mathcal{G} \rangle (\sum_{p,d} \varepsilon^p \varphi_{p,d})$.

**Definition 2.5:** Let “$\ln \varepsilon$” be here an algebraic indeterminate independent from the indeterminate $\varepsilon$. We denote $E[\ln \varepsilon]$ the set of polynomials with coefficients in $E$. More precisely it is the set of elements of $E^\mathbb{N}$ with finite support and, for any $(\varphi_\ell) \in E[\ln \varepsilon]$, we denote $\sum_{\ell \in \mathbb{N}} \ln^\ell \varepsilon \varphi_\ell := (\varphi_\ell)$.

## 2.2 Definition of the spaces $\mathcal{A}$

In [14, p.10], Costabel and Dauge build a similar asymptotic expansion for the Poisson equation in the half plane with mixed boundary condition: Neumann in a part of the boundary and Robin $u + \varepsilon \partial_n u = 0$ in another. They quickly mention that their singularities can be written as $\mathbb{R}[(-z)^q P(\ln(-z))]$ with $z = x + iy, q \in \mathbb{R}$ and $P$ a real polynomial. To define the spaces $\mathcal{A}$, we adapted this idea to take into account the layer, the angle $\Theta$ and the Helmholtz equation. These simple expressions give both powerful algebraic tools for the theory and fast precise algorithms for the numerical resolution (see Section 2.3).

**Definition 2.6: the spaces $\mathcal{A}$**

Let $\alpha := e^{-i\Theta}$. We define in $\Omega$ the complex variable $z := x + iy = re^{i\theta}$. For all $q \in \mathbb{R}$, we take the following conventions: $(\alpha z)^q := r^q e^{qi(\theta-\Theta)}$, $\overline{\alpha z}^q := r^q e^{-qi(\theta-\Theta)}$ and $\log(\alpha z) := \ln r + i(\theta - \Theta)$. Let $d \in \mathbb{R}$. We denote:

* $\mathcal{A}_d(\Omega)$ the vector subspace of $\mathcal{C}^0(\Omega, \mathbb{C})$ generated by the functions $z \mapsto \Im[(\alpha z)^q \overline{\alpha z}^k P(\log(\alpha z))]$ with $q \in \mathbb{R}$, $k \in \mathbb{N}$, $q+k=d$ and $P \in \mathbb{R}[T]$,

* $\mathcal{A}_d(\Lambda) := \{(x, Y) \mapsto x^d Q(\ln x, Y) \mid Q \in \mathbb{C}[T, Y] \text{ and } Q(T, -1) = 0\}$,

* $\mathcal{A}_d(\Gamma) := \{x \mapsto x^d Q(\ln x) \mid Q \in \mathbb{C}[T]\}$,

* $\mathcal{A}_d(\Pi) := \{\varphi \in \mathcal{C}^0(\Pi, \mathbb{C}) \mid \varphi|_{\Omega} \in \mathcal{A}_d(\Omega) \text{ and } \varphi|_{\Lambda} \in \mathcal{A}_d(\Lambda)\}$,

* and for any $D \in \{\Pi, \Omega, \Lambda, \Gamma\}$, $\mathcal{A}(D) := \sum_{d \in \mathbb{R}} \mathcal{A}_d(D)$ (cf. the introduction of Section 2.1).

Note that elements of $\mathcal{A}(\Omega)$ and $\mathcal{A}(\Pi)$ vanish on $\Sigma_\Omega$, and elements of $\mathcal{A}(\Lambda)$ and $\mathcal{A}(\Pi)$ vanish on $\Sigma_\Lambda$. In addition, elements of $\mathcal{A}(\Omega)$ are naturally functions depending on the polar coordinates. For instance:

* $\Im[(\alpha z)^q \overline{\alpha z}^k] = r^{q+k} \sin((q-k)(\theta - \Theta))$

* $\Im[(\alpha z)^q \overline{\alpha z}^k \log(\alpha z)] = r^{q+k} [\ln(r) \cos((q-k)(\theta - \Theta)) - (\theta - \Theta) \sin((q-k)(\theta - \Theta))]$

Note also that in this definition we used the variables $x, y$, which are relevant for far fields, but all the tools developed in this section can also be used for corner fields, replacing $(x, y)$ by $(X, Y)$.
<figure><img src="image_8.png" /><figcaption>Figure 7: Schematic illustration of $\sum_{n=0}^{\infty} f^n$ for $f$ a linear map that has a positive degree</figcaption></figure>

In Section 2.5, we introduce formal series with powers of $\epsilon$ in order to express the matching condition. They are defined similarly to Notation 2.1: for any set $P \subset \mathbb{R}$ and any family $(\tilde{E}_p)_{p \in P}$ of vector spaces, we denote

$$ \forall (\varphi_p) \in \prod_{p \in P} \tilde{E}_p, \quad \sum_{p \in P} \varepsilon^p \varphi_p := (\varphi_p) \quad \text{and} \quad \sum_{p \in P} \varepsilon^p \tilde{E}_p := \prod_{p \in P} \tilde{E}_p. \qquad (2.5) $$

Again this is only a notation and here $\epsilon$ is not a real number but an algebraic indeterminate. This is similar to Notation 2.1 and we will later choose to use either notation depending on the physical meaning of the formal series.

For instance, the case $\tilde{E} = \mathbb{C}$ and $P = \mathbb{N}$ gives the classical set of formal power series, usually denoted $\mathbb{C}[[\epsilon]]$ (see [37, 39]). The Taylor approximations at 0 of any smooth function $f$ can be represented by $\sum_{p=0}^{\infty} \varepsilon^p \frac{f^{(p)}(0)}{p!} \in \mathbb{C}[[\epsilon]]$. Truncations of this series give approximations at a given order. We will use similar representations for the asymptotic expansion of $u_\epsilon$.

Let us take $P := \mathbb{R}$, $\tilde{E} := \sum_{d \in \mathbb{R}} E_d$ and $\tilde{E}_p := \tilde{E}$ for any $p$. Let $p_0 \in \mathbb{R}$ and $f : \bigoplus E_d \to \bigoplus E_d$ a linear map that has a degree $d_0$. For any $(\varphi_p) \in \tilde{E}^{\mathbb{R}}$, we define:

$$ (\varepsilon^{p_0} f) \left( \sum_{p \in \mathbb{R}} \varepsilon^p \varphi_p \right) := \sum_{p \in \mathbb{R}} \varepsilon^p f(\varphi_{p-p_0}) \qquad (2.6) $$

These kind of linear maps $\sum_p \varepsilon^p \tilde{E} \to \sum_p \varepsilon^p \tilde{E}$ are the one said to have a degree. We denote $\deg(\varepsilon^{p_0} f) := (p_0, d_0)$.

**Definition 2.4:** Let $\sum_{(p,d) \in \mathbb{R}^2} \varepsilon^p \varphi_{p,d} \in \sum_{p \in \mathbb{R}} \varepsilon^p \tilde{E}$. Let $\mathcal{G}$ be a finite set of linear maps $\sum \varepsilon^p \tilde{E} \to \sum \varepsilon^p \tilde{E}$ that have degrees. There is a finite set $\mathcal{F}$ of linear maps $\bigoplus E_d \to \bigoplus E_d$ and $(p_f)_f \in \mathbb{R}^\mathcal{F}$ s.t. $\mathcal{G} = \{\varepsilon^{p_f} f \mid f \in \mathcal{F}\}$. We assume that there is $v \in \mathbb{R}^2$ s.t. $\{(\langle p, d \rangle, v) \mid (p, d) \in \mathbb{R}^2, \varphi_{p,d} \neq 0\}$ is bounded from below and $\forall g \in \mathcal{G}, \langle \deg g, v \rangle > 0$. We denote:

$$ \sum_{n=0}^{\infty} \sum_{(g_1, \dots, g_n) \in \mathcal{G}^n} g_1 \circ \dots \circ g_n \left( \sum_{(p,d) \in \mathbb{R}^2} \varepsilon^p \varphi_{p,d} \right) := \sum_{(p,d) \in \mathbb{R}^2} \varepsilon^p \sum_{n \in \mathbb{N}} \sum_{(f_1, \dots, f_n) \in \mathcal{F}^n} f_1 \circ \dots \circ f_n (\varphi_{p-\sum_i p_{f_i}}, d-\sum_i \deg f_i). $$

which is well-defined in $\sum \varepsilon^p \tilde{E}$. We also denote it as $\langle \mathcal{G} \rangle (\sum_{p,d} \varepsilon^p \varphi_{p,d})$.

**Definition 2.5:** Let “$\ln \epsilon$” be here an algebraic indeterminate independent from the indeterminate $\epsilon$. We denote $E[\ln \epsilon]$ the set of polynomials with coefficients in $E$. More precisely it is the set of elements of $E^\mathbb{N}$ with finite support and, for any $(\varphi_\ell) \in E[\ln \epsilon]$, we denote $\sum_{\ell \in \mathbb{N}} \ln^\ell \varepsilon \varphi_\ell := (\varphi_\ell)$.

## 2.2 Definition of the spaces $\mathcal{A}$

In [15, p.10], Costabel and Dauge build a similar asymptotic expansion for the Poisson equation in the half plane with mixed boundary condition: Neumann in a part of the boundary and Robin $u + \varepsilon \partial_n u = 0$
**Remark 2.7:** Definition 2.6 can be extended to the case where $\Sigma_{\Omega}$ is covered by another layer, by defining $\mathcal{A}_d(\Omega)$ as the vector space generated by the functions $z \mapsto \mathfrak{S}[(\alpha z)^q \bar{\alpha} \bar{z}^k P(\log(\alpha z))]$ and $z \mapsto \mathfrak{S}[z^q \bar{z}^k P(\log z)]$ with $q \in \mathbb{R}$, $k \in \mathbb{N}$, $q + k = d$ and $P \in \mathbb{R}[T]$.

In order to build particular solutions of PDEs in $\mathcal{A}$, we will need the three following lemmas. The proof of the first one can be found in Appendix A.

**Lemma 2.8:** For any $D \in \{\Pi, \Omega, \Gamma, \Lambda\}$, we have the following decomposition: $\mathcal{A}(D) = \bigoplus_{d \in \mathbb{R}} \mathcal{A}_d(D)$. Furthermore, for any $d \in \mathbb{R}$, $\mathcal{A}_d(\Omega)$ can itself be decomposed as follows:

$$ \mathcal{A}_d(\Omega) = \bigoplus_{(q,k) \in \mathbb{R} \times \mathbb{N}} \{z \mapsto \lambda \mathfrak{S}[(\alpha z)^q \bar{\alpha} \bar{z}^k P(\log(\alpha z))] \mid \lambda \in \mathbb{C}, P \in \mathbb{R}[T] \text{ and } \mathcal{P}(q, k, P)\} \quad (2.7) $$

where $\mathcal{P}$ is the property defined by $\mathcal{P}(q, k, P) := (q \notin \mathbb{N} \text{ or } q > k \text{ or } P(0) = 0)$.

**Remark:** The condition $\mathcal{P}$ is a way to exclude the functions $z \mapsto \mathfrak{S}[(\alpha z)^q \bar{\alpha} \bar{z}^k]$ with $q \in \mathbb{N}$ and $q < k$, which are already present in the direct sum as they are equal to $z \mapsto -\mathfrak{S}[(\alpha z)^k \bar{\alpha} \bar{z}^q]$.

Let $\varphi$ be a function of $\mathcal{A}(\Omega)$ of the form $\mathfrak{S}[(\alpha z)^q \bar{\alpha} \bar{z}^k P(\log(\alpha z))]$. Note that on $\Gamma$, $\varphi$ is equal to $x^{q+k} \mathfrak{S}[\alpha^{q-k} P(\ln x - i\Theta)]$. Let us define $\mathfrak{S}[\alpha^{q-k} P(T - i\Theta)] := \sum_{i=1}^{\deg P} \mathfrak{S}(a_i) T^i$ in $\mathbb{C}[T]$, where $\sum_{i=1}^{\deg P} a_i T^i := \alpha^{q-k} P(T - i\Theta)$. Then $\varphi|_{\Gamma}(x) = x^{q+k} Q(\ln x)$ for some $Q \in \mathbb{R}[X]$, which implies $\varphi|_{\Gamma} \in \mathcal{A}(\Gamma)$. Conversely, for future constructions, it will be important to solve the equation:

$$ \text{given } Q \in \mathbb{R}[T], \text{ find } P \in \mathbb{R}[T] \text{ s.t. } \mathfrak{S}[\alpha^d P(T - i\Theta)] = Q(T). \quad (2.8) $$

**Lemma 2.9:** Let $\alpha := e^{-i\Theta}$, $d \in \mathbb{R}$ and $Q \in \mathbb{R}[T]$.

1. If $d \in \mathbb{R} \setminus \frac{\pi}{\Theta}\mathbb{Z}$, then there is a unique solution $P \in \mathbb{R}[T]$ of (2.8). Moreover $\deg P = \deg Q$. We denote the solution $L_d(Q)$.

2. If $d \in \frac{\pi}{\Theta}\mathbb{Z}$, then the set of solutions of (2.8) is of the form $\{P_0 + c \mid c \in \mathbb{R}\}$ with $P_0 \in \mathbb{R}[T]$ and $\deg(P_0) = \deg(Q) + 1$. We denote $L_d(Q)$ the unique solution that vanishes at 0.

In both cases, $L_d$ is a linear map from $\mathbb{R}[T]$ into itself.

**Proof:** There are two cases whether the coefficient of $\deg m$ of $\mathfrak{S}[\alpha^d(T - i\Theta)^m]$ vanishes or not.

1. If $d \in \mathbb{R} \setminus \frac{\pi}{\Theta}\mathbb{Z}$, then $\alpha^d \in \mathbb{C} \setminus \mathbb{R}$, so: $\forall m \in \mathbb{N}$, $\deg \mathfrak{S}[\alpha^d(T - i\Theta)^m] = m$. Therefore, $(\mathfrak{S}[\alpha^d(T - i\Theta)^m])_{m \in \mathbb{N}}$ is a basis of $\mathbb{R}[T]$. So writing $Q$ in this basis gives a unique solution of (2.8).

2. If $d \in \frac{\pi}{\Theta}\mathbb{Z}$, since $\alpha^d \in \mathbb{R}$, we have: $\forall m \in \mathbb{N}$, $\deg \mathfrak{S}[\alpha^d(T - i\Theta)^m] = m - 1$. So in this case $(\mathfrak{S}[\alpha^d(T - i\Theta)^m])_{m \in \mathbb{N}^*}$ is a basis of $\mathbb{R}[T]$. Thus, $F: P \mapsto \mathfrak{S}[\alpha^d P(T - i\Theta)]$ is surjective and its kernel is the set of constant polynomials. Its restriction to $E := \{P \in \mathbb{R}[T] \mid P(0) = 0\}$ is therefore an isomorphism and we set $L_d := (F|_E)^{-1}$. Finally, for any $Q \in \mathbb{R}[T]$ we have $F^{-1}(\{Q\}) = \{L_d(Q)\} + \text{Ker } F$. $\square$

For any $q \in \mathbb{R}$, the maps $Q \in \mathbb{C}[T] \mapsto \mathfrak{S}[x^q Q(\ln x)] \in \mathcal{A}(\Gamma)$ and $Q \in \mathbb{C}[T, Y] \mapsto \mathfrak{S}[x^q Q(\ln x, Y)] \in \mathcal{A}(\Lambda)$ are clearly injective. In the following lemma we investigate the injectivity of $P \in \mathbb{R}[T] \mapsto \mathfrak{S}[(\alpha z)^q \bar{\alpha} \bar{z}^k P(\log(\alpha z))] \in \mathcal{A}(\Omega)$.

**Lemma 2.10:** Let $(q, k) \in \mathbb{R} \times \mathbb{N}$. The map $P \mapsto \mathfrak{S}[(\alpha z)^q \bar{\alpha} \bar{z}^k P(\log(\alpha z))]$ is injective from the set of real polynomials $P$ for which $\mathcal{P}(q, k, P)$ is true into $\mathcal{A}(\Omega)$.

**Proof:** For any $\theta \in (0, \Theta)$ and $r \in \mathbb{R}^*$, we have $\varphi(re^{i\theta}) = r^{q+k} \mathfrak{S}[e^{i(q-k)(\theta-\Theta)} P(\ln r + i(\theta - \Theta))] = r^{q+k} \mathfrak{S}[\alpha'^{q-k} P(\ln r - i\Theta')]$ with $\Theta' := \Theta - \theta$ and $\alpha' := e^{-i\Theta'}$.
in another. They quickly mention that their singularities can be written as $\Re[(z)^q P(\ln(-z))]$ with $z = x + iy, q \in \mathbb{R}$ and $P$ a real polynomial. To define the spaces $\mathcal{A}$, we adapted this idea to take into account the layer, the angle $\Theta$ and the Helmholtz equation. These simple expressions give both powerful algebraic tools for the theory and fast precise algorithms for the numerical resolution (see Section 2.3).

**Definition 2.6: the spaces $\mathcal{A}$**

Let $\alpha := e^{-i\Theta}$. We define in $\Omega$ the complex variable $z := x + iy = re^{i\theta}$. For all $q \in \mathbb{R}$, we take the following conventions: $(\alpha z)^q := r^q e^{qi(\theta-\Theta)}$, $\overline{\alpha z}^q := r^q e^{-qi(\theta-\Theta)}$ and $\log(\alpha z) := \ln r + i(\theta - \Theta)$. Let $d \in \mathbb{R}$. We denote:

* $\mathcal{A}_d(\Omega)$ the vector subspace of $C^0(\Omega, \mathbb{C})$ generated by the functions $z \mapsto \Im[(\alpha z)^q \overline{\alpha z}^k P(\log(\alpha z))]$ with $q \in \mathbb{R}, k \in \mathbb{N}, q+k=d$ and $P \in \mathbb{R}[T]$,

* $\mathcal{A}_d(\Lambda) := \{(x, Y) \mapsto x^d Q(\ln x, Y) \mid Q \in \mathbb{C}[T, Y] \text{ and } Q(T, -1) = 0\}$,

* $\mathcal{A}_d(\Gamma) := \{x \mapsto x^d Q(\ln x) \mid Q \in \mathbb{C}[T]\}$,

* $\mathcal{A}_d(\Pi) := \{\varphi \in C^0(\Pi, \mathbb{C}) \mid \varphi|_{\Omega} \in \mathcal{A}_d(\Omega) \text{ and } \varphi|_{\Lambda} \in \mathcal{A}_d(\Lambda)\}$,

* and for any $D \in \{\Pi, \Omega, \Lambda, \Gamma\}$, $\mathcal{A}(D) := \sum_{d \in \mathbb{R}} \mathcal{A}_d(D)$ (cf. the introduction of Section 2.1).

Note that elements of $\mathcal{A}(\Omega)$ and $\mathcal{A}(\Pi)$ vanish on $\Sigma_{\Omega}$, and elements of $\mathcal{A}(\Lambda)$ and $\mathcal{A}(\Pi)$ vanish on $\Sigma_{\Lambda}$. In addition, elements of $\mathcal{A}(\Omega)$ are naturally functions depending on the polar coordinates. For instance:

$$ \Im[(\alpha z)^q \overline{\alpha z}^k] = r^{q+k} \sin((q-k)(\theta - \Theta)) $$

$$ \Im[(\alpha z)^q \overline{\alpha z}^k \log(\alpha z)] = r^{q+k} [\ln(r) \cos((q-k)(\theta - \Theta)) - (\theta - \Theta) \sin((q-k)(\theta - \Theta))] $$

Note also that in this definition we used the variables $x, y$, which are relevant for far fields, but all the tools developed in this section can also be used for corner fields, replacing $(x, y)$ by $(X, Y)$.

**Remark 2.7:** Definition 2.6 can be extended to the case where $\Sigma_{\Omega}$ is covered by another layer, by defining $\mathcal{A}_d(\Omega)$ as the vector space generated by the functions $z \mapsto \Im[(\alpha z)^q \overline{\alpha z}^k P(\log(\alpha z))]$ and $z \mapsto \Im[z^q \overline{z}^k P(\log z)]$ with $q \in \mathbb{R}, k \in \mathbb{N}, q+k=d$ and $P \in \mathbb{R}[T]$.

In order to build particular solutions of PDEs in $\mathcal{A}$, we will need the three following lemmas. The proof of the first one can be found in Appendix A.

**Lemma 2.8:** For any $D \in \{\Pi, \Omega, \Gamma, \Lambda\}$, we have the following decomposition: $\mathcal{A}(D) = \bigoplus_{d \in \mathbb{R}} \mathcal{A}_d(D)$. Furthermore, for any $d \in \mathbb{R}$, $\mathcal{A}_d(\Omega)$ can itself be decomposed as follows:

$$ \mathcal{A}_d(\Omega) = \bigoplus_{\substack{(q,k) \in \mathbb{R} \times \mathbb{N} \\ q+k=d}} \operatorname{Span}_{\mathbb{C}} \{z \mapsto \Im[(\alpha z)^q \overline{\alpha z}^k P(\log(\alpha z))] \mid P \in \mathbb{R}[T] \text{ and } \mathcal{P}(q, k, P)\} \quad (2.7) $$

where $\mathcal{P}$ is the property defined by $\mathcal{P}(q, k, P) := (q \notin \mathbb{N} \text{ or } q > k \text{ or } P(0) = 0)$.

**Remark:** The condition $\mathcal{P}$ is a way to exclude the functions $z \mapsto \Im[(\alpha z)^q \overline{\alpha z}^k]$ with $q \in \mathbb{N}$ and $q < k$, which are already present in the direct sum as they are equal to $z \mapsto -\Im[(\alpha z)^k \overline{\alpha z}^q]$.

Let $\varphi$ be a function of $\mathcal{A}(\Omega)$ of the form $\Im[(\alpha z)^q \overline{\alpha z}^k P(\log(\alpha z))]$. Note that on $\Gamma$, $\varphi$ is equal to $x^{q+k} \Im[\alpha^{q-k} P(\ln x - i\Theta)]$. Let us define $\Im[\alpha^{q-k} P(T - i\Theta)] := \sum_{i=1}^{\deg P} \Im(a_i) T^i$ in $\mathbb{C}[T]$, where $\sum_{i=1}^{\deg P} a_i T^i := \alpha^{q-k} P(T - i\Theta)$. Then $\varphi|_{\Gamma}(x) = x^{q+k} Q(\ln x)$ for some $Q \in \mathbb{R}[X]$, which implies $\varphi|_{\Gamma} \in \mathcal{A}(\Gamma)$. Conversely, for future constructions, it will be important to solve the equation:

$$ \text{given } Q \in \mathbb{R}[T], \text{ find } P \in \mathbb{R}[T] \text{ s.t. } \Im[\alpha^d P(T - i\Theta)] = Q(T). \quad (2.8) $$
* If $q \neq k$, we can choose $\theta$ so that $q - k \in \mathbb{R} \setminus \frac{\pi}{\Theta'}\mathbb{Z}$. So Lemma 2.9 applied to $(\Theta', \alpha')$ instead of $(\Theta, \alpha)$ implies that $P$ is unique.

* Otherwise, $q - k = 0 \in \frac{\pi}{\Theta'}\mathbb{Z}$ for any $\theta$. So according to 2.9, $P$ is unique up to a constant a priori. But the property $\mathcal{P}$ implies that $P(0) = 0$, so this constant is fixed. $\square$

## 2.3 Tools for solving the Poisson and Helmholtz equations in the spaces $\mathcal{A}$

In this section, we show how to solve canonical problems set in $\Pi$ in the spaces $\mathcal{A}$. More precisely, let $(\psi_{\Omega}, \psi_{\Lambda}, \psi_{\Gamma}) \in \mathcal{A}(\Omega) \times \mathcal{A}(\Lambda) \times \mathcal{A}(\Gamma)$, we look for the solutions $\varphi \in \mathcal{A}(\Pi)$ of: problems of the form

$$ 
\left\{ 
\begin{array}{ll}
\Delta\varphi = \psi_{\Omega} & \text{in } \Omega \\
\partial_{Y}^{2}\varphi = \psi_{\Lambda} & \text{in } \Lambda \\
\partial_{Y}\varphi|_{Y=0^{-}} = \psi_{\Gamma} & \text{on } \Gamma
\end{array}
\right. 
\qquad (2.9) 
$$

Note that by definition of $\mathcal{A}(\Pi)$, $\varphi$ also satisfies $\varphi|_{Y=0^{+}} - \varphi|_{Y=0^{-}}$ on $\Gamma$ and $\varphi|_{\Sigma_{\Omega} \cup \Sigma_{\Lambda}} = 0$. Solving this system will enable us to build in Section 2.4 the asymptotic expansion of $u_{p,l}$ and $S_{p,l}$. Indeed, note for instance that this system is identical to (2.1) except for the first line. We first describe the homogeneous solutions of (2.9), then build explicitly some particular solutions. Since functions of $\mathcal{A}$ are uniquely determined by some polynomials (see Lemma 2.10), we are able to code an exact, very fast and memory-thrifty solver of (2.9). This is one of the key advantages of the $\mathcal{A}$ framework.

**Definition 2.11:** For any $d \in \frac{\pi}{\Theta'}\mathbb{Z}^*$, we define on $\Omega$ the function $\phi_d^\Omega := (-1)^{d\Theta/\pi}r^d \sin(d\theta) = r^d \sin(d(\theta - \Theta)) = \mathfrak{S}[(\alpha z)^d] \in \mathcal{A}_d(\Omega)$ and $\phi_d \in \mathcal{A}_d(\Pi)$ its extension by 0 in $\Lambda$.

These functions play an important role in the sequel because they solve the homogeneous Laplace equation in $\Omega$, resp. $\Pi$.

**Proposition 2.12: the Laplace problem in $\mathcal{A}$**

1. Span($\{\phi_d^\Omega | d \in \frac{\pi}{\Theta'}\mathbb{Z}^*\}$) is the set of solutions in $\mathcal{A}(\Omega)$ of

$$ 
\left\{ 
\begin{array}{ll}
\Delta\varphi = 0 & \text{in } \Omega \\
\varphi = 0 & \text{on } \Gamma \cup \Sigma_{\Omega}
\end{array}
\right. 
\qquad (2.10) 
$$

2. Span($\{\phi_d | d \in \frac{\pi}{\Theta'}\mathbb{Z}^*\}$) is the set of solutions in $\mathcal{A}(\Pi)$ of

$$ 
\left\{ 
\begin{array}{ll}
\Delta\varphi = 0 & \text{in } \Omega \\
\partial_{Y}^{2}\varphi = 0 & \text{in } \Lambda \\
\partial_{Y}\varphi|_{Y=0^{-}} = 0 & \text{on } \Gamma
\end{array}
\right. 
\qquad (2.11) 
$$

**Proof:**

1. $\phi_d^\Omega$ is clearly solution of (2.10) for any $d \in \frac{\pi}{\Theta'}\mathbb{Z}^*$. Conversely, let $\varphi$ be a solution. Let us denote

$$ \forall d \in \frac{\pi}{\Theta'}\mathbb{N}^*, \forall r \in \mathbb{R}_+^*, \quad c_d(r) := \frac{2}{\Theta} \int_0^\Theta \varphi(r, \theta) \sin(d\theta) d\theta. $$

Using separation of variables and $\varphi|_{\partial\Omega} = 0$, it is easy to show that $\varphi = \sum_{d \in \frac{\pi}{\Theta'}\mathbb{N}^*} c_d(r) \sin(d\theta)$ with convergence in $H^2(\Omega \cap \{r_1 < r < r_2\})$ for any $0 < r_1 < r_2 < \infty$. Since $\Delta\varphi = 0$, we have $(r \frac{d}{dr})^2 c_d = d^2 c_d$ for any $d$. Hence we get: $\forall d \in \frac{\pi}{\Theta'}\mathbb{N}^*, \exists a_d, a_{-d} \in \mathbb{C}, \forall r \in \mathbb{R}_+^*, c_d(r) = a_d r^d + a_{-d} r^{-d}$. Moreover, by definition of $\mathcal{A}(\Omega)$, there is $q \in \mathbb{R}_+^*$ s.t. $\varphi \in \sum_{d \in [-q, q]} \mathcal{A}_d(\Omega)$. So

$$ \forall d \in \frac{\pi}{\Theta'}\mathbb{Z}^* \cap (q, \infty), \quad a_d = \lim_{r \to \infty} r^{-d} c_d(r) = \lim_{r \to \infty} \frac{2}{\Theta} \int_0^\Theta r^{-d} \varphi(r, \theta) \sin(d\theta) d\theta = 0. $$

Similarly, looking at $r \to 0$ one gets: $\forall d \in \frac{\pi}{\Theta'}\mathbb{Z}^* \cap (-\infty, -q)$, $a_d = 0$. So $\varphi = \sum_{d \in \frac{\pi}{\Theta'}\mathbb{Z}^* \cap [-q, q]} a_d \operatorname{sgn}(d) \phi_d^\Omega$ (where $\operatorname{sgn}(d) := d/|d|$), which is a finite sum. Therefore, $\varphi$ is in the desired span.
**Lemma 2.9:** Let $\alpha := e^{-i\Theta}$, $d \in \mathbb{R}$ and $Q \in \mathbb{R}[T]$.

1. If $d \in \mathbb{R} \setminus \frac{\pi}{\Theta}\mathbb{Z}$, then there is a unique solution $P \in \mathbb{R}[T]$ of (2.8). Moreover $\deg P = \deg Q$. We denote the solution $L_d(Q)$.

2. If $d \in \frac{\pi}{\Theta}\mathbb{Z}$, then the set of solutions of (2.8) is of the form $\{P_0 + c \mid c \in \mathbb{R}\}$ with $P_0 \in \mathbb{R}[T]$ and $\deg(P_0) = \deg(Q) + 1$. We denote $L_d(Q)$ the unique solution that vanishes at 0.

In both cases, $L_d$ is a linear map from $\mathbb{R}[T]$ into itself.

*Proof:* There are two cases whether the coefficient of degree $m$ of $\mathfrak{S}[\alpha^d(T - i\Theta)^m]$ vanishes or not.

1. If $d \in \mathbb{R} \setminus \frac{\pi}{\Theta}\mathbb{Z}$, then $\alpha^d \in \mathbb{C} \setminus \mathbb{R}$, so: $\forall m \in \mathbb{N}$, $\deg \mathfrak{S}[\alpha^d(T - i\Theta)^m] = m$. Therefore, $(\mathfrak{S}[\alpha^d(T - i\Theta)^m])_{m \in \mathbb{N}}$ is a basis of $\mathbb{R}[T]$. So writing $Q$ in this basis gives a unique solution of (2.8).

2. If $d \in \frac{\pi}{\Theta}\mathbb{Z}$, since $\alpha^d \in \mathbb{R}$, we have: $\forall m \in \mathbb{N}$, $\deg \mathfrak{S}[\alpha^d(T - i\Theta)^m] = m - 1$. So in this case $(\mathfrak{S}[\alpha^d(T - i\Theta)^m])_{m \in \mathbb{N}^*}$ is a basis of $\mathbb{R}[T]$. Thus, $F: P \mapsto \mathfrak{S}[\alpha^d P(T - i\Theta)]$ is surjective and its kernel is the set of constant polynomials. Its restriction to $E := \{P \in \mathbb{R}[T] \mid P(0) = 0\}$ is therefore an isomorphism and we set $L_d := (F|_E)^{-1}$. Finally, for any $Q \in \mathbb{R}[T]$ we have $F^{-1}(\{Q\}) = \{L_d(Q)\} + \text{Ker } F$. $\square$

For any $q \in \mathbb{R}$, the maps $Q \in \mathbb{C}[T] \mapsto \mathfrak{S}[x^q Q(\ln x)] \in \mathcal{A}(\Gamma)$ and $Q \in \mathbb{C}[T, Y] \mapsto \mathfrak{S}[x^q Q(\ln x, Y)] \in \mathcal{A}(\Lambda)$ are clearly injective. In the following lemma we investigate the injectivity of $P \in \mathbb{R}[T] \mapsto \mathfrak{S}[(\alpha z)^q \overline{\alpha z}^k P(\log(\alpha z))] \in \mathcal{A}(\Omega)$.

**Lemma 2.10:** Let $(q, k) \in \mathbb{R} \times \mathbb{N}$. The map $P \mapsto \mathfrak{S}[(\alpha z)^q \overline{\alpha z}^k P(\log(\alpha z))]$ is injective from the set of real polynomials $P$ for which $P(q, k, P)$ is true into $\mathcal{A}(\Omega)$.

*Proof:* For any $\theta \in (0, \Theta)$ and $r \in \mathbb{R}_+^*$, we have $\varphi(re^{i\theta}) = r^{q+k} \mathfrak{S}[e^{i(q-k)(\theta-\Theta)}P(\ln r + i(\theta - \Theta))] = r^{q+k} \mathfrak{S}[\alpha'^{q-k}P(\ln r - i\Theta')]$ with $\Theta' := \Theta - \theta$ and $\alpha' := e^{-i\Theta'}$.

* If $q \neq k$, we can choose $\theta$ so that $q - k \in \mathbb{R} \setminus \frac{\pi}{\Theta}\mathbb{Z}$. So Lemma 2.9 applied to $(\Theta', \alpha')$ instead of $(\Theta, \alpha)$ implies that $P$ is unique.

* Otherwise, $q - k = 0 \in \frac{\pi}{\Theta}\mathbb{Z}$ for any $\theta$. So according to 2.9, $P$ is unique up to a constant a priori. But the property $P$ implies that $P(0) = 0$, so this constant is fixed. $\square$

## 2.3 Tools for solving the Poisson and Helmholtz equations in the spaces $\mathcal{A}$

In this section, we show how to solve canonical problems set in $\Pi$ in the spaces $\mathcal{A}$. More precisely, let $(\psi_\Omega, \psi_\Lambda, \psi_\Gamma) \in \mathcal{A}(\Omega) \times \mathcal{A}(\Lambda) \times \mathcal{A}(\Gamma)$, we look for the solutions $\varphi \in \mathcal{A}(\Pi)$ of: problems of the form

$$ \left\{ \begin{array}{ll} \Delta\varphi = \psi_{\Omega} & \text{in } \Omega \\ \partial_{Y}^{2}\varphi = \psi_{\Lambda} & \text{in } \Lambda \\ \partial_{Y}\varphi|_{Y=0^{-}} = \psi_{\Gamma} & \text{on } \Gamma \end{array} \right. \qquad (2.9) $$

Note that by definition of $\mathcal{A}(\Pi)$, $\varphi$ also satisfies $\varphi|_{Y=0^+} - \varphi|_{Y=0^-}$ on $\Gamma$ and $\varphi|_{\Sigma_\Omega \cup \Sigma_\Lambda} = 0$. Solving this system will enable us to build in Section 2.4 the asymptotic expansion of $u_{p,l}$ and $S_{p,l}$. Indeed, note for instance that this system is identical to (2.1) except for the first line. We first describe the homogeneous solutions of (2.9), then build explicitly some particular solutions. Since functions of $\mathcal{A}$ are uniquely determined by some polynomials (see Lemma 2.10), we are able to code an exact, very fast and memory-thrifty solver of (2.9). This is one of the key advantages of the $\mathcal{A}$ framework.

**Definition 2.11:** For any $d \in \frac{\pi}{\Theta}\mathbb{Z}^*$, we define on $\Omega$ the function $\phi_d^\Omega := (-1)^{d\Theta/\pi}r^d \sin(d\theta) = r^d \sin(d(\theta - \Theta)) = \mathfrak{S}[(\alpha z)^d] \in \mathcal{A}_d(\Omega)$ and $\phi_d \in \mathcal{A}_d(\Pi)$ its extension by 0 in $\Lambda$.
2. Any solution of the system vanishes in $\Lambda$, so point 2 easily follows from point 1. $\square$

Let us now define the following linear forms $\sigma_d$ which satisfy $\sigma_d(\phi_q) = \delta_{d,q}$ for any $d, q \in \pi_{\Theta}Z^*$ and which enable us to “project” any element of $\mathcal{A}(\Pi)$ on $\text{Span}(\{\phi_d \mid d \in \pi_{\Theta}Z^*\})$. These linear forms appear later as key singularity coefficients in the matching condition.

**Definition 2.13: linear forms $\sigma_d$**

Let $d \in \pi_{\Theta}Z^*$. For any $(q, k, P) \in \mathbb{R} \times \mathbb{N} \times \mathbb{R}[T]$ s.t. $\mathcal{P}(q, k, P)$ is true and $\varphi : z \mapsto \mathfrak{S}[(\alpha z)^q \bar{\alpha} z^k P(\log(\alpha z))],$ let us define:

$$ egin{cases} \sigma_d(\varphi) = 0 & \text{if } q \neq d \text{ or } k \neq 0 \\ \sigma_d(\mathfrak{S}[(\alpha z)^d P(\log(\alpha z))]) = P(0) & \text{otherwise} \end{cases} $$

It is well-defined by Lemma 2.10. By Lemma 2.8, $\sigma_d$ can be extended into a linear form $\mathcal{A}(\Omega) \to \mathbb{C}$.
Finally, for any $\varphi \in \mathcal{A}(\Pi)$, $\sigma_d(\varphi) := \sigma_d(\varphi|_{\Omega})$.

Let us now build particular solutions of (2.9). By linearity, it suffices to build particular solutions of three sub-problems. According to Definition 2.2, for any $D_1, D_2 \in \{\Pi, \Omega, \Gamma, \Lambda\}$, we say that a linear map $F: \mathcal{A}(D_1) \to \mathcal{A}(D_2)$ has degree $d \in \mathbb{R}$ iff: $\forall q \in \mathbb{R}, \forall \varphi \in \mathcal{A}_q(D_1), F(\varphi) \in \mathcal{A}_{q+d}(D_2)$.

**Proposition 2.14: particular solutions of (2.9)**

Let us denote $\mathcal{A}_{\perp}(\Pi) := \{\varphi \in \mathcal{A}(\Pi) \mid \forall d \in \pi_{\Theta}Z^*, \sigma_d(\varphi) = 0\}$, which is a supplementary of $\text{Span}(\{\varphi_d \mid d \in \pi_{\Theta}Z^*\})$ in $\mathcal{A}(\Pi)$.

1. For any $\psi_{\Omega} \in \mathcal{A}(\Omega)$ there exists a unique solution $\varphi_{\Delta} \in \mathcal{A}_{\perp}(\Pi)$ of

$$ egin{cases} \Delta\varphi = \psi_{\Omega} & \text{in } \Omega \\ \partial_{Y}^{2}\varphi = 0 & \text{in } \Lambda \\ \partial_{Y}\varphi|_{Y=0^{-}} = 0 & \text{on } \Gamma \end{cases} \qquad (2.12) $$

The associated map $R_{\Delta}: \psi_{\Omega} \in \mathcal{A}(\Omega) \mapsto \varphi_{\Delta} \in \mathcal{A}_{\perp}(\Pi)$ is linear and has degree 2.

2. For any $\psi_{\Lambda} \in \mathcal{A}(\Lambda)$ there exists a unique solution $\varphi_{\partial_{Y}^{2}} \in \mathcal{A}_{\perp}(\Pi)$ of

$$ egin{cases} \Delta\varphi = 0 & \text{in } \Omega \\ \partial_{Y}^{2}\varphi = \psi_{\Lambda} & \text{in } \Lambda \\ \partial_{Y}\varphi|_{Y=0^{-}} = 0 & \text{on } \Gamma \end{cases} \qquad (2.13) $$

The associated map $R_{\partial_{Y}^{2}}: \psi_{\Lambda} \in \mathcal{A}(\Lambda) \mapsto \varphi_{\partial_{Y}^{2}} \in \mathcal{A}_{\perp}(\Pi)$ is linear and has degree 0.

3. For any $\psi_{\Gamma} \in \mathcal{A}(\Gamma)$ there exists a unique solution $\varphi_{\Gamma} \in \mathcal{A}_{\perp}(\Pi)$ of

$$ egin{cases} \Delta\varphi = 0 & \text{in } \Omega \\ \partial_{Y}^{2}\varphi = 0 & \text{in } \Lambda \\ \partial_{Y}\varphi|_{Y=0^{-}} = \psi_{\Gamma} & \text{on } \Gamma \end{cases} \qquad (2.14) $$

The associated map $R_{\Gamma}: \psi_{\Gamma} \in \mathcal{A}(\Gamma) \mapsto \varphi_{\Gamma} \in \mathcal{A}_{\perp}(\Pi)$ is linear and has degree 0.

Using Propositions 2.12 and 2.14, it is then easy to see that the set of solutions of (2.9) is $R_{\Delta}(\varphi_{\Omega}) + R_{\partial_{Y}^{2}}(\varphi_{\Lambda}) + R_{\Gamma}(\varphi_{\Gamma}) + \text{Span}(\{\phi_{d} \mid d \in \pi_{\Theta}Z^{*}\})$. Moreover, the functions $\varphi_{\Delta}, \varphi_{\partial_{Y}^{2}}$ and $\varphi_{\Gamma}$ in Proposition 2.14 have explicit expressions (see the proof below), which allows to compute them easily in practice.

**Proof:** Proposition 2.12 gives the uniqueness of the solutions $\varphi_{\Delta}, \varphi_{\partial_{Y}^{2}}$ and $\varphi_{\Gamma}$, so only their existence remains to prove. This is done by a construction. For any $D \in \{\Pi, \Omega, \Lambda, \Gamma\}$, let $\mathcal{A}(D, \mathbb{R}) := \mathcal{A}(D) \cap C^{0}(D, \mathbb{R})$. Since $\mathcal{A}(D) = \mathcal{A}(D, \mathbb{R}) \oplus i\mathcal{A}(D, \mathbb{R})$, it suffices to build the solutions when $(\psi_{\Omega}, \psi_{\Lambda}, \psi_{\Gamma}) \in \mathcal{A}(\Omega, \mathbb{R}) \times \mathcal{A}(\Lambda, \mathbb{R}) \times \mathcal{A}(D, \Gamma)$, and then extend it to any source term by complexification.
These functions play an important role in the sequel because they solve the homogeneous Laplace equation in $\Omega$, resp. $\Pi$.

**Proposition 2.12: the Laplace problem in $\mathcal{A}$**

1. Span($\{\phi_d^\Omega | d \in \bar{\Theta}\mathbb{Z}^*\}$) is the set of solutions in $\mathcal{A}(\Omega)$ of

$$ \left\{ \begin{array}{ll} \Delta\varphi = 0 & \text{in } \Omega \\ \varphi = 0 & \text{on } \Gamma \cup \Sigma_{\Omega} \end{array} \right. \qquad (2.10) $$

2. Span($\{\phi_d | d \in \bar{\Theta}\mathbb{Z}^*\}$) is the set of solutions in $\mathcal{A}(\Pi)$ of

$$ \left\{ \begin{array}{ll} \Delta\varphi = 0 & \text{in } \Omega \\ \partial_Y^2\varphi = 0 & \text{in } \Lambda \\ \partial_Y\varphi|_{Y=0^-} = 0 & \text{on } \Gamma \end{array} \right. \qquad (2.11) $$

**Proof:**

1. $\phi_d^\Omega$ is clearly solution of (2.10) for any $d \in \bar{\Theta}\mathbb{Z}^*$. Conversely, let $\varphi$ be a solution. Let us denote

$$ \forall d \in \bar{\Theta}\mathbb{N}^*, \forall r \in \mathbb{R}_+^*, \quad c_d(r) := \frac{2}{\Theta} \int_0^\Theta \varphi(r, \theta) \sin(d\theta) \, d\theta. $$

Using separation of variables and $\varphi|_{\partial\Omega} = 0$, it is easy to show that $\varphi = \sum_{d \in \bar{\Theta}\mathbb{N}^*} c_d(r) \sin(d\theta)$ with convergence in $H^2(\Omega \cap \{r_1 < r < r_2\})$ for any $0 < r_1 < r_2 < \infty$. Since $\Delta\varphi = 0$, we have $(r \frac{d}{dr})^2 c_d = d^2 c_d$ for any $d$. Hence we get: $\forall d \in \bar{\Theta}\mathbb{N}^*, \exists a_d, a_{-d} \in \mathbb{C}, \forall r \in \mathbb{R}_+^*, c_d(r) = a_d r^d + a_{-d} r^{-d}$. Moreover, by definition of $\mathcal{A}(\Omega)$, there is $q \in \mathbb{R}_+^*$ s.t. $\varphi \in \sum_{d \in [-q, q]} \mathcal{A}_d(\Omega)$. So

$$ \forall d \in \bar{\Theta}\mathbb{Z}^* \cap (q, \infty), \quad a_d = \lim_{r \to \infty} r^{-d} c_d(r) = \lim_{r \to \infty} \frac{2}{\Theta} \int_0^\Theta \underbrace{r^{-d} \varphi(r, \theta)}_{\to 0} \sin(d\theta) \, d\theta = 0. $$

Similarly, looking at $r \to 0$ one gets: $\forall d \in \bar{\Theta}\mathbb{Z}^* \cap (-\infty, -q)$, $a_d = 0$. So $\varphi = \sum_{d \in \bar{\Theta}\mathbb{Z}^* \cap [-q, q]} a_d \operatorname{sgn}(d) \phi_d^\Omega$ (where $\operatorname{sgn}(d) := d/|d|$), which is a finite sum. Therefore, $\varphi$ is in the desired span.

2. Any solution of the system vanishes in $\Lambda$, so point 2 easily follows from point 1. $\square$

Let us now define the following linear forms $\sigma_d$ which satisfy $\sigma_d(\phi_q) = \delta_{d,q}$ for any $d, q \in \bar{\Theta}\mathbb{Z}^*$ and which enable us to “project” any element of $\mathcal{A}(\Pi)$ on $\operatorname{Span}(\{\phi_d | d \in \bar{\Theta}\mathbb{Z}^*\})$. These linear forms appear later as key singularity coefficients in the matching condition.

**Definition 2.13: linear forms $\sigma_d$**

Let $d \in \bar{\Theta}\mathbb{Z}^*$. For any $(q, k, P) \in \mathbb{R} \times \mathbb{N} \times \mathbb{R}[T]$ s.t. $P(q, k, P)$ is true and $\varphi : z \mapsto \mathfrak{S}[(\alpha z)^q \overline{\alpha z}^k P(\log(\alpha z))]$, let us define:

$$ \left\{ \begin{array}{ll} \sigma_d(\varphi) = 0 & \text{if } q \neq d \text{ or } k \neq 0 \\ \sigma_d(\mathfrak{S}[(\alpha z)^d P(\log(\alpha z))]) = P(0) & \text{otherwise} \end{array} \right. $$

It is well-defined by Lemma 2.10. By Lemma 2.8, $\sigma_d$ can be extended into a linear form $\mathcal{A}(\Omega) \to \mathbb{C}$. Finally, for any $\varphi \in \mathcal{A}(\Pi)$, $\sigma_d(\varphi) := \sigma_d(\varphi|_{\Omega})$.

Let us now build particular solutions of (2.9). By linearity, it suffices to build particular solutions of three sub-problems. According to Definition 2.2, for any $D_1, D_2 \in \{\Pi, \Omega, \Gamma, \Lambda\}$, we say that a linear map $F : \mathcal{A}(D_1) \to \mathcal{A}(D_2)$ has degree $d \in \mathbb{R}$ iff: $\forall q \in \mathbb{R}, \forall \varphi \in \mathcal{A}_q(D_1), F(\varphi) \in \mathcal{A}_{q+d}(D_2)$.
1. According to Lemma 2.8, it suffices to build $\varphi_{\Delta}$ when $\psi_{\Omega} = \mathfrak{F}[(\alpha z)^q \bar{\alpha} z^k P_{\psi}(\log(\alpha z))]$ with $(q, k, P_{\psi}) \in \mathbb{R} \times \mathbb{N} \times \mathbb{R}[T]$ s.t. $\mathcal{P}(q, k, P_{\psi})$ is true. First, $\varphi_{\Delta|\Lambda} = 0$ because $\varphi_{\Delta}$ satisfies:

$$ \left\{ \begin{array}{ll} \partial_Y^2 \varphi_{\Delta} = 0 & \text{in } \Lambda \\ \partial_Y \varphi_{\Delta|_{Y=0^-}} = 0 & \text{on } \Gamma \\ \varphi_{\Delta} = 0 & \text{on } \Sigma_{\Lambda} \end{array} \right. $$

Given that $\Delta = 4\partial_z\partial_{\bar{z}}$, we have for any $\varphi_1 : z \mapsto \mathfrak{F}[(\alpha z)^{q_1} \bar{\alpha} z^{k_1} P_1(\log(\alpha z))]$ that

$$ \Delta\varphi_1 = 4 \mathfrak{F}[(\alpha z)^{q_1-1} k_1 \bar{\alpha} z^{k_1-1} (q_1 P_1 + P'_1)(\log(\alpha z))] $$

So taking $q_1 := q + 1$, $k_1 := k + 1$ and $P_1 \in \mathbb{R}[T]$ a solution of $4k_1(q_1P_1 + P'_1) = P_{\psi}$, we have $\Delta\varphi_1 = \psi_{\Omega}$ in $\Omega$.

* If $q_1 = 0$, $P_1$ is unique up to a constant, Moreover we can write

$$ \varphi_1 = \mathfrak{F}[\bar{\alpha} z^{k_1} (P_1(\log(\alpha z)) - P_1(0))] - P_1(0) \mathfrak{F}[(\alpha z)^{k_1}] $$

where each term satisfy the property $\mathcal{P}$. So for any $d \in \frac{\pi}{\Theta}\mathbb{Z}^*$ different from $k_1$, we have $\sigma_d(\varphi_1) = 0$, while $\sigma_{k_1}(\varphi_1) = -P_1(0)$ if $k_1 \in \mathbb{N} \cap \frac{\pi}{\Theta}\mathbb{Z}^*$. Taking $P_1(0) := 0$ thus gives: $\forall d \in \frac{\pi}{\Theta}\mathbb{Z}^*, \sigma_d(\varphi_1) = 0$.

* If $q_1 \neq 0$, there is a unique solution $P_1$. Given that $q_1 \neq 0$ and $k_1 \neq 0$, Definition 2.13 implies that: $\forall d \in \frac{\pi}{\Theta}\mathbb{Z}^*, \sigma_d(\varphi_1) = 0$.

However, we cannot set $\varphi_{\Delta|\Omega} = \varphi_1$, because $\varphi_1$ does not vanish on $\Gamma$. Let us then introduce $\varphi_2 : z \mapsto \mathfrak{F}[(\alpha z)^{q+k+2} P_2(\log(\alpha z))]$ with $P_2(T) := L_{q+k+2}(\mathfrak{F}[\alpha^{q-k}P_1(T - i\Theta)])$ that satisfies by Lemma 2.9:

$$ \left\{ \begin{array}{ll} \Delta\varphi_2 = 0 & \text{in } \Omega \\ \varphi_2 = x^{q+k+2}\mathfrak{F}[\alpha^{q+k+2}P_2(\ln x - i\Theta)] = x^{q+k+2}\mathfrak{F}[\alpha^{q-k}P_1(\ln x - i\Theta)] = \varphi_1 & \text{on } \Gamma \end{array} \right. $$

In addition Lemma 2.9 implies that $q + k + 2 \in \frac{\pi}{\Theta}\mathbb{Z}^* \Rightarrow P_2(0) = 0$, so: $\forall d \in \frac{\pi}{\Theta}\mathbb{Z}^*, \sigma_d(\varphi_2) = 0$. Finally we set $\varphi_{\Delta|\Omega} := \varphi_1 - \varphi_2$, which is in $\mathcal{A}_{\perp}(\Pi)$.

2. Similarly, it suffices to build $\varphi_{\partial_Y^2}$ for $\psi_{\Lambda} = x^q Q_{\psi}(\ln x, Y)$. We look for $\varphi_{\partial_Y^2}$ of the form $x^q Q_{\varphi}(\ln x, Y)$ in $\Lambda$ with $Q_{\varphi} \in \mathbb{R}[T, Y]$. Then necessarily we have

$$ \left\{ \begin{array}{l} \partial_Y^2 Q_{\varphi} = Q_{\psi} \\ \partial_Y Q_{\varphi}(\cdot, 0) = 0 \\ Q_{\varphi}(\cdot, -1) = 0 \end{array} \right. $$

This uniquely defines $Q_{\varphi}$. Taking $P_{\varphi} := L_q(Q_{\varphi}(\cdot, 0))$ and $\varphi_{\partial_Y^2|\Omega} := \mathfrak{F}[(\alpha z)^q P_{\varphi}(\log(\alpha z))]$ then implies that $\Delta\varphi_{\partial_Y^2} = 0$ in $\Omega$, $\varphi$ continuous and $\varphi_{\partial_Y^2} \in \mathcal{A}_{\perp}(\Pi)$.

3. Again it suffices to consider $\psi_{\Gamma} = x^q P_{\psi}(\ln x)$ with $P_{\psi} \in \mathbb{R}[T]$. We take $\varphi_{\Gamma}$ of the same form as in point 2. Then $\partial_Y\varphi_{\Gamma|_{Y=0^-}} = x^q \partial_Y Q_{\varphi}(\ln x, 0)$ so it suffices to take the polynomial solutions of:

$$ \left\{ \begin{array}{l} \partial_Y^2 Q_{\varphi} = 0 \\ \partial_Y Q_{\varphi}(\cdot, 0) = P_{\psi} \\ Q_{\varphi}(\cdot, -1) = 0 \\ P_{\varphi} = L_q(Q_{\varphi}(\cdot, 0)) \end{array} \right. $$

Finally the linearity of Problems 2.12-2.14 and the uniqueness of $\varphi_{\Delta}$, $\varphi_{\partial_Y^2}$ and $\varphi_{\Gamma}$ imply that $R_{\Delta}$, $R_{\partial_Y^2}$ and $R_{\Gamma}$ are linear maps. $\square$

Moreover, we will need analogous operators in $\mathcal{A}(\Omega)$ to build the far fields in Theorem 3.6. The proof is entirely similar to the one of Proposition 2.14, so we omit it. Again, these operators have explicit expressions.
**Proposition 2.14: particular solutions of (2.9)**

Let us denote $\mathcal{A}_{\perp}(\Pi) := \{\varphi \in \mathcal{A}(\Pi) \mid \forall d \in \frac{\pi}{\Theta}\mathbb{Z}^*, \sigma_d(\varphi) = 0\}$, which is a supplementary of $\text{Span}(\{\varphi_d \mid d \in \frac{\pi}{\Theta}\mathbb{Z}^*\})$ in $\mathcal{A}(\Pi)$.

1. For any $\psi_{\Omega} \in \mathcal{A}(\Omega)$ there exists a unique solution $\varphi_{\Delta} \in \mathcal{A}_{\perp}(\Pi)$ of

$$ \left\{ \begin{array}{ll} \Delta\varphi = \psi_{\Omega} & \text{in } \Omega \\ \partial_{Y}^{2}\varphi = 0 & \text{in } \Lambda \\ \partial_{Y}\varphi|_{Y=0^{-}} = 0 & \text{on } \Gamma \end{array} \right. \qquad (2.12) $$

The associated map $R_{\Delta}: \psi_{\Omega} \in \mathcal{A}(\Omega) \mapsto \varphi_{\Delta} \in \mathcal{A}_{\perp}(\Pi)$ is linear and has degree 2.

2. For any $\psi_{\Lambda} \in \mathcal{A}(\Lambda)$ there exists a unique solution $\varphi_{\partial_{Y}^{2}} \in \mathcal{A}_{\perp}(\Pi)$ of

$$ \left\{ \begin{array}{ll} \Delta\varphi = 0 & \text{in } \Omega \\ \partial_{Y}^{2}\varphi = \psi_{\Lambda} & \text{in } \Lambda \\ \partial_{Y}\varphi|_{Y=0^{-}} = 0 & \text{on } \Gamma \end{array} \right. \qquad (2.13) $$

The associated map $R_{\partial_{Y}^{2}}: \psi_{\Lambda} \in \mathcal{A}(\Lambda) \mapsto \varphi_{\partial_{Y}^{2}} \in \mathcal{A}_{\perp}(\Pi)$ is linear and has degree 0.

3. For any $\psi_{\Gamma} \in \mathcal{A}(\Gamma)$ there exists a unique solution $\varphi_{\Gamma} \in \mathcal{A}_{\perp}(\Pi)$ of

$$ \left\{ \begin{array}{ll} \Delta\varphi = 0 & \text{in } \Omega \\ \partial_{Y}^{2}\varphi = 0 & \text{in } \Lambda \\ \partial_{Y}\varphi|_{Y=0^{-}} = \psi_{\Gamma} & \text{on } \Gamma \end{array} \right. \qquad (2.14) $$

The associated map $R_{\Gamma}: \psi_{\Gamma} \in \mathcal{A}(\Gamma) \mapsto \varphi_{\Gamma} \in \mathcal{A}_{\perp}(\Pi)$ is linear and has degree 0.

Using Propositions 2.12 and 2.14, it is then easy to see that the set of solutions of (2.9) is $R_{\Delta}(\varphi_{\Omega}) + R_{\partial_{Y}^{2}}(\varphi_{\Lambda}) + R_{\Gamma}(\varphi_{\Gamma}) + \text{Span}(\{\phi_{d} \mid d \in \frac{\pi}{\Theta}\mathbb{Z}^{*}\})$. Moreover, the functions $\varphi_{\Delta}$, $\varphi_{\partial_{Y}^{2}}$ and $\varphi_{\Gamma}$ in Proposition 2.14 have explicit expressions (see the proof below), which allows to compute them easily in practice.

**Proof:** Proposition 2.12 gives the uniqueness of the solutions $\varphi_{\Delta}$, $\varphi_{\partial_{Y}^{2}}$ and $\varphi_{\Gamma}$, so only their existence remains to prove. This is done by a construction. For any $D \in \{\Pi, \Omega, \Lambda, \Gamma\}$, let $\mathcal{A}(D, \mathbb{R}) := \mathcal{A}(D) \cap C^{0}(D, \mathbb{R})$. Since $\mathcal{A}(D) = \mathcal{A}(D, \mathbb{R}) \oplus i\mathcal{A}(D, \mathbb{R})$, it suffices to build the solutions when $(\psi_{\Omega}, \psi_{\Lambda}, \psi_{\Gamma}) \in \mathcal{A}(\Omega, \mathbb{R}) \times \mathcal{A}(\Lambda, \mathbb{R}) \times \mathcal{A}(D, \Gamma)$, and then extend it to any source term by complexification.

1. According to Lemma 2.8, it suffices to build $\varphi_{\Delta}$ when $\psi_{\Omega} = \mathfrak{F}[(\alpha z)^q \overline{\alpha z}^{k_1} P_{\psi}(\log(\alpha z))]$ with $(q, k, P_{\psi}) \in \mathbb{R} \times \mathbb{N} \times \mathbb{R}[T]$ s.t. $\mathcal{P}(q, k, P_{\psi})$ is true. First, $\varphi_{\Delta|\Lambda} = 0$ because $\varphi_{\Delta}$ satisfies:

$$ \left\{ \begin{array}{ll} \partial_{Y}^{2}\varphi_{\Delta} = 0 & \text{in } \Lambda \\ \partial_{Y}\varphi_{\Delta|Y=0^{-}} = 0 & \text{on } \Gamma \\ \varphi_{\Delta} = 0 & \text{on } \Sigma_{\Lambda} \end{array} \right. $$

Given that $\Delta = 4\partial_{Z}\partial_{\bar{Z}}$, we have for any $\varphi_{1}: z \mapsto \mathfrak{F}[(\alpha z)^{q_1} \overline{\alpha z}^{k_1} P_{1}(\log(\alpha z))]$ that

$$ \Delta\varphi_{1} = 4\mathfrak{F}[(\alpha z)^{q_1-1} k_1 \overline{\alpha z}^{k_1-1} (q_1 P_1 + P'_1)(\log(\alpha z))] $$

So taking $q_1 := q + 1$, $k_1 := k + 1$ and $P_1 \in \mathbb{R}[T]$ a solution of $4k_1(q_1P_1 + P'_1) = P_{\psi}$, we have $\Delta\varphi_1 = \psi_{\Omega}$ in $\Omega$.

* If $q_1 = 0$, $P_1$ is unique up to a constant, Moreover we can write

$$ \varphi_1 = \mathfrak{F}[\overline{\alpha z}^{k_1} (P_1(\log(\alpha z)) - P_1(0))] - P_1(0) \mathfrak{F}[(\alpha z)^{k_1}] $$

where each term satisfy the property $\mathcal{P}$. So for any $d \in \frac{\pi}{\Theta}\mathbb{Z}^*$ different from $k_1$, we have $\sigma_d(\varphi_1) = 0$, while $\sigma_{k_1}(\varphi_1) = -P_1(0)$ if $k_1 \in \mathbb{N} \cap \frac{\pi}{\Theta}\mathbb{Z}^*$. Taking $P_1(0) := 0$ thus gives: $\forall d \in \frac{\pi}{\Theta}\mathbb{Z}^*$, $\sigma_d(\varphi_1) = 0$.

* If $q_1 \neq 0$, there is a unique solution $P_1$. Given that $q_1 \neq 0$ and $k_1 \neq 0$, Definition 2.13 implies that: $\forall d \in \frac{\pi}{\Theta}\mathbb{Z}^*$, $\sigma_d(\varphi_1) = 0$.
**Proposition 2.15:** Let us denote $\mathcal{A}_{\perp}(\Omega) = \{\varphi \in \mathcal{A}(\Omega) \mid \forall d \in \frac{\pi}{\Theta}\mathbb{Z}^*, \sigma_d(\varphi) = 0\}$, which is a supplementary of $\text{Span}(\{\phi_d^\Omega \mid d \in \frac{\pi}{\Theta}\mathbb{Z}^*\})$ in $\mathcal{A}(\Omega)$.

1. For any $\psi_{\Omega} \in \mathcal{A}(\Gamma)$ there exists a unique solution $\varphi_{\Delta} \in \mathcal{A}_{\perp}(\Omega)$ of

$$ \left\{ \begin{array}{ll} \Delta\varphi = \psi_{\Omega} & \text{in } \Omega \\ \varphi = 0 & \text{on } \Gamma \end{array} \right. \qquad (2.15) $$

The associated map $R_{\Delta}^{\Omega}: \psi_{\Gamma} \in \mathcal{A}(\Omega) \mapsto \varphi_{\Delta} \in \mathcal{A}_{\perp}(\Omega)$ is linear and has degree 2.

2. For any $\psi_{\Gamma} \in \mathcal{A}(\Gamma)$ there exists a unique solution $\varphi_{\Delta} \in \mathcal{A}_{\perp}(\Omega)$ of

$$ \left\{ \begin{array}{ll} \Delta\varphi = 0 & \text{in } \Omega \\ \varphi = \psi & \text{on } \Gamma \end{array} \right. \qquad (2.16) $$

The associated map $R_{\Delta}^{\Omega}: \psi_{\Gamma} \in \mathcal{A}(\Gamma) \mapsto \varphi_{\Delta} \in \mathcal{A}_{\perp}(\Omega)$ is linear and has degree 0.

To end this section, let us show that the spaces $\mathcal{A}$ are stable under some differential operators.

**Lemma 2.16:**

1. $\partial_{x|\Lambda}^2 : \varphi \mapsto \partial_{x}^2 \varphi|_{\Lambda}$ maps $\mathcal{A}(\Pi)$ to $\mathcal{A}(\Lambda)$ and has degree $-2$ (see Definition 2.2).

2. $\partial_{y|\Gamma, y=0^+} : \varphi \mapsto \partial_{y} \varphi|_{\Gamma, y=0^+}$ maps $\mathcal{A}(\Pi)$ to $\mathcal{A}(\Gamma)$ and has degree $-1$.

3. $\Delta : \varphi \mapsto \Delta\varphi|_{\Omega}$ maps $\mathcal{A}(\Omega)$ (and $\mathcal{A}(\Pi)$) to $\mathcal{A}(\Omega)$ and has degree $-2$.

Note that, when used on corner fields (which depend on $(X, Y)$), the first two operators will rather be denoted $\partial_{X|\Lambda}^2$ and $\partial_{Y|\Gamma, Y=0^+}$.

*Proof:* It suffices to verify it when $\varphi$ has the form $\Im[(\alpha z)^q \overline{\alpha z}^k P(\log(\alpha z))]$ in $\Omega$ and $x^{q+k} Q(\ln x, Y)$ in $\Lambda$, with $q \in \mathbb{R}$, $k \in \mathbb{N}$, $P[T]$ and $Q \in \mathbb{R}[T, Y]$.

1. We have $\partial_{x}^2 \varphi(x, Y) = x^{q+k-2}((q+k)(q+k-1)+2(q+k)\partial_{T} + \partial_{T}^2)Q(\ln x, Y)$ which is in $\mathcal{A}(\Lambda)$.

2. Let $\psi: z \mapsto (\alpha z)^q \overline{\alpha z}^k P(\log(\alpha z))$. From $\partial_z \psi = \frac{1}{2}(\partial_x \psi - i\partial_y \psi)$ and $\partial_{\bar{z}} \psi = \frac{1}{2}(\partial_x \psi + i\partial_y \psi)$ it follows that $\partial_y \varphi = \Im[\partial_y \psi] = \Re[\partial_z \psi - \partial_{\bar{z}} \psi]$. Therefore,

$$ \begin{aligned} \partial_y \varphi|_{\Gamma, y=0^+} &= \Re[\alpha(\alpha x)^{q-1} \overline{\alpha x}^k (qP + P')(\log(\alpha x)) - (\alpha x)^q \overline{\alpha} k \overline{\alpha x}^{k-1} P(\log(\alpha x))] \\ &= x^{q+k-1} \Re[\alpha^{q-k}((q-k)P + P')(\ln x - i\Theta)] \in \mathcal{A}(\Gamma) \end{aligned} $$

3. Since $\Delta = 4\partial_z \partial_{\bar{z}}$, we have $\Delta\varphi = 4\Im[(\alpha z)^{q-1} k \overline{\alpha z}^{k-1}(qP_1 + P'_1)(\log(\alpha z))]$ which is in $\mathcal{A}(\Omega)$. $\square$

## 2.4 Asymptotic behaviors w.r.t. r of solutions of model problems

In this section we give tools that will be used to compute the asymptotic behaviors of $u_{p,l}$ and $S_{p,l}$ resp. when $r \to 0$ and $r \to \infty$ using the spaces $\mathcal{A}$. To do so, we will use series of elements of $\mathcal{A}$, which is made rigorous by the following definition.

**Definition 2.17: the spaces $\mathcal{A}^{\pm}$**

Let $D \in \{\Pi, \Omega, \Gamma, \Lambda\}$. We denote $\mathcal{A}^{+}(D)$, resp. $\mathcal{A}^{-}(D)$, the set of elements of $\sum_{d \in \mathbb{R}} \mathcal{A}_d(D)$ whose support is included in the image of a sequence that tends to $\infty$, resp. $-\infty$. We write their elements as formal series according to Definition 2.1.

**Remarks:**

* The asymptotic of $u_{p,l}$ when $r \to 0$ involve increasing powers of $r$ so it will be expressed in $\mathcal{A}^{+}(D)$. Similarly, $S_{p,l}$ when $r \to \infty$ involves decreasing powers of $r$, so it will be expressed in $\mathcal{A}^{-}(D)$.
However, we cannot set $\varphi_{\Delta|\Omega} = \varphi_1$, because $\varphi_1$ does not vanish on $\Gamma$. Let us then introduce $\varphi_2 : z \mapsto \mathfrak{F}[(\alpha z)^{q+k+2}P_2(\log(\alpha z))]$ with $P_2(T) := L_{q+k+2}(\mathfrak{F}[\alpha^{q-k}P_1(T - i\Theta)])$ that satisfies by Lemma 2.9:

$$ egin{cases} 
\Delta\varphi_2 = 0 & \text{in } \Omega \\
\varphi_2 = x^{q+k+2}\mathfrak{F}[\alpha^{q+k+2}P_2(\ln x - i\Theta)] = x^{q+k+2}\mathfrak{F}[\alpha^{q-k}P_1(\ln x - i\Theta)] = \varphi_1 & \text{on } \Gamma
\end{cases} $$

In addition Lemma 2.9 implies that $q + k + 2 \in \frac{\pi}{\Theta}\mathbb{Z}^* \Rightarrow P_2(0) = 0$, so: $\forall d \in \frac{\pi}{\Theta}\mathbb{Z}^*, \sigma_d(\varphi_2) = 0$. Finally we set $\varphi_{\Delta|\Omega} := \varphi_1 - \varphi_2$, which is in $A_{\perp}(\Pi)$.

2. Similarly, it suffices to build $\varphi_{\partial_Y^2}$ for $\psi_{\Lambda} = x^q Q_{\psi}(\ln x, Y)$. We look for $\varphi_{\partial_Y^2}$ of the form $x^q Q_{\varphi}(\ln x, Y)$ in $\Lambda$ with $Q_{\varphi} \in \mathbb{R}[T, Y]$. Then necessarily we have

$$ egin{cases} 
\partial_Y^2 Q_{\varphi} = Q_{\psi} \\
\partial_Y Q_{\varphi}(\cdot, 0) = 0 \\
Q_{\varphi}(\cdot, -1) = 0
\end{cases} $$

This uniquely defines $Q_{\varphi}$. Taking $P_{\varphi} := L_q(Q_{\varphi}(\cdot, 0))$ and $\varphi_{\partial_Y^2|\Omega} := \mathfrak{F}[(\alpha z)^q P_{\varphi}(\log(\alpha z))]$ then implies that $\Delta\varphi_{\partial_Y^2} = 0$ in $\Omega$, $\varphi$ continuous and $\varphi_{\partial_Y^2} \in A_{\perp}(\Pi)$.

3. Again it suffices to consider $\psi_{\Gamma} = x^q P_{\psi}(\ln x)$ with $P_{\psi} \in \mathbb{R}[T]$. We take $\varphi_{\Gamma}$ of the same form as in point 2. Then $\partial_Y \varphi_{N|Y=0-} = x^q \partial_Y Q_{\varphi}(\ln x, 0)$ so it suffices to take the polynomial solutions of:

$$ egin{cases} 
\partial_Y^2 Q_{\varphi} = 0 \\
\partial_Y Q_{\varphi}(\cdot, 0) = P_{\psi} \\
Q_{\varphi}(\cdot, -1) = 0 \\
P_{\varphi} = L_q(Q_{\varphi}(\cdot, 0))
\end{cases} $$

Finally the linearity of Problems 2.12-2.14 and the uniqueness of $\varphi_{\Delta}$, $\varphi_{\partial_Y^2}$ and $\varphi_{\Gamma}$ imply that $R_{\Delta}$, $R_{\partial_Y^2}$ and $R_{\Gamma}$ are linear maps. $\square$

Moreover, we will need analogous operators in $A(\Omega)$ to build the far fields in Theorem 3.6. The proof is entirely similar to the one of Proposition 2.14, so we omit it. Again, these operators have explicit expressions.

**Proposition 2.15:** Let us denote $A_{\perp}(\Omega) = \{\varphi \in A(\Omega) \mid \forall d \in \frac{\pi}{\Theta}\mathbb{Z}^*, \sigma_d(\varphi) = 0\}$, which is a supplementary of $\text{Span}(\{\phi_d^{\Omega} \mid d \in \frac{\pi}{\Theta}\mathbb{Z}^*\})$ in $A(\Omega)$.

1. For any $\psi_{\Omega} \in A(\Gamma)$ there exists a unique solution $\varphi_{\Delta} \in A_{\perp}(\Omega)$ of

$$ egin{cases} 
\Delta\varphi = \psi_{\Omega} & \text{in } \Omega \\
\varphi = 0 & \text{on } \Gamma
\end{cases} \qquad (2.15) $$

The associated map $R_{\Delta}^{\Omega} : \psi_{\Gamma} \in A(\Omega) \mapsto \varphi_{\Delta} \in A_{\perp}(\Omega)$ is linear and has degree 2.

2. For any $\psi_{\Gamma} \in A(\Gamma)$ there exists a unique solution $\varphi_{\Gamma} \in A_{\perp}(\Omega)$ of

$$ egin{cases} 
\Delta\varphi = 0 & \text{in } \Omega \\
\varphi = \psi & \text{on } \Gamma
\end{cases} \qquad (2.16) $$

The associated map $R_{\Gamma}^{\Omega} : \psi_{\Gamma} \in A(\Gamma) \mapsto \varphi_{\Gamma} \in A_{\perp}(\Omega)$ is linear and has degree 0.

To end this section, let us show that the spaces $A$ are stable under some differential operators.

**Lemma 2.16:**

1. $\partial_{x|\Lambda}^2 : \varphi \mapsto \partial_{x}^2 \varphi|_{\Lambda}$ maps $A(\Pi)$ to $A(\Lambda)$ and has degree $-2$ (see Definition 2.2).

2. $\partial_{y|\Gamma, y=0+} : \varphi \mapsto \partial_{y} \varphi|_{\Gamma, y=0+}$ maps $A(\Pi)$ to $A(\Gamma)$ and has degree $-1$.

3. $\Delta : \varphi \mapsto \Delta\varphi|_{\Omega}$ maps $A(\Omega)$ (and $A(\Pi)$) to $A(\Omega)$ and has degree $-2$.
• As seen in Section 2.1, $\mathcal{A}(D)$ is included in $\mathcal{A}^+(D)$ and $\mathcal{A}^-(D)$. But elements of $\mathcal{A}^+(D)$ are not in general $D \to \mathbb{R}$ functions, as the formal series may diverge pointwise.

Using Definitions 2.2, we can extend $\partial_{x|\Lambda}^2$, $\partial_{y|\Gamma, y=0+}$, $R_{\partial_Y^2}$, $R_N$ and $R_\Delta$ to the spaces $\mathcal{A}^\pm$. We also use the notation $\langle \cdot, \cdot \rangle$ introduced in Definition 2.3. E.g. $\langle -k_0^2 R_\Delta \rangle = \sum_{n=0}^\infty (-k_0^2 R_\Delta)^n$. Moreover, we extend $\sigma_d$ to $\mathcal{A}^\pm(\Pi)$ for any $d \in \frac{\pi}{\Theta}\mathbb{Z}^*$, by setting $\sigma_d(\sum_{d'} \varphi_{d'}) := \sigma_d(\varphi_d)$ for any $\sum_{d'} \varphi_{d'} \in \mathcal{A}^\pm(\Pi)$.

**Lemma 2.18:** Let $g^0 \in \mathcal{A}^+(\Lambda)$ and $h^0 \in \mathcal{A}^+(\Gamma)$. The solutions in $\mathcal{A}^+(\Pi)$ of

$$ 
\left\{ 
\begin{array}{ll}
\mu_0 \Delta \mathbf{u}^0 + \omega^2 \rho_0 \mathbf{u}^0 = 0 & \text{in } \Omega \\
\mu_1 \partial_Y^2 \mathbf{u}^0 = g & \text{in } \Lambda \\
\mu_1 \partial_{Y|Y=0-} \mathbf{u}^0 = h & \text{on } \Gamma
\end{array}
\right. 
\qquad (2.17) 
$$

are the formal series of the following form (where $\sigma_d(\mathbf{u}^0)$ vanishes when $d$ is small enough)

$$ \mathbf{u}^0 = \langle -k_0^2 R_\Delta \rangle \left( \frac{1}{\mu_1} R_{\partial_Y^2}(g^0) + \frac{1}{\mu_1} R_N(h^0) + \sum_{d \in \frac{\pi}{\Theta} \mathbb{Z}^*} \sigma_d(\mathbf{u}^0) \phi_d \right) \qquad (2.18) $$

**Proof:** Let $\mathbf{v}^0 := (\text{id} + k_0^2 R_\Delta)\mathbf{u}^0$. Since $\langle -k_0^2 R_\Delta \rangle = \sum_{n=0}^\infty (-k_0^2 R_\Delta)^n$, $\text{id} + k_0^2 R_\Delta$ is the inverse of $\langle -k_0^2 R_\Delta \rangle$, so $\mathbf{u}^0 = \langle -k_0^2 R_\Delta \rangle \mathbf{v}^0$. Moreover, Proposition 2.14 states that $\Delta \circ R_\Delta = \text{id}$, $\partial_{Y|\Lambda}^2 \circ R_\Delta = 0$ and $\partial_{Y|\Gamma, Y=0-} \circ R_\Delta = 0$, which imply resp. $(\mu_0 \Delta + \omega^2 \rho_0) \circ \langle -k_0^2 R_\Delta \rangle = \mu_0 \Delta \circ (\text{id} + k_0^2 R_\Delta) \circ \langle -k_0^2 R_\Delta \rangle = \mu_0 \Delta$, $\partial_{Y|\Lambda}^2 \circ \langle -k_0^2 R_\Delta \rangle = \partial_{Y|\Lambda}^2$ and $\partial_{Y|\Gamma, Y=0-} \circ \langle -k_0^2 R_\Delta \rangle = \partial_{Y|\Gamma, Y=0-}^2$. Therefore, (2.17) is equivalent to

$$ 
\left\{ 
\begin{array}{ll}
\mu_0 \Delta \mathbf{v}^0 = 0 & \text{in } \Omega \\
\mu_1 \partial_Y^2 \mathbf{v}^0 = g^0 & \text{in } \Lambda \\
\mu_1 \partial_Y \mathbf{v}_{|Y=0-}^0 = h^0 & \text{on } \Gamma
\end{array}
\right. 
\qquad (2.19) 
$$

Then Propositions 2.12 and 2.14 imply that the solutions of (2.19) in $\mathcal{A}^+(\Pi)$ are the $\frac{1}{\mu_1} R_{\partial_Y^2}(g^0) + \frac{1}{\mu_1} R_N(h^0) + \sum_{d \in \frac{\pi}{\Theta} \mathbb{Z}^*} c_d \phi_d$ where $c_d$ vanishes for small enough $d$. Finally we have $c_d = \sigma_d(\mathbf{v}^0) = \sigma_d(\mathbf{u}^0)$ for any $d \in \frac{\pi}{\Theta} \mathbb{Z}^*$ because $\sigma_d \circ R_\Delta$, $\sigma_d \circ R_{\partial_Y^2}$ and $\sigma_d \circ R_N$ vanish by Proposition 2.14. $\square$

**Lemma 2.19:** Let $F_\Omega^\infty \in \mathcal{A}^-(\Omega)$ and $F_\Lambda^\infty \in \mathcal{A}^-(\Lambda)$. The solutions in $\mathcal{A}^-(\Pi)$ of

$$ 
\left\{ 
\begin{array}{ll}
\mu_0 \Delta S^\infty = F_\Omega^\infty & \text{in } \Omega \\
\mu_1 \Delta S^\infty = F_\Lambda^\infty & \text{in } \Lambda \\
\mu_0 \partial_{Y|Y=0+} S^\infty - \mu_1 \partial_{Y|Y=0-} S^\infty = 0 & \text{on } \Gamma
\end{array}
\right. 
\qquad (2.20) 
$$

are the formal series of the following form (where $\sigma_d(S^\infty)$ vanishes when $d$ is big enough)

$$ S^\infty = \left\langle -R_{\partial_Y^2} \circ \partial_{X|\Lambda}^2, \frac{\mu_0}{\mu_1} R_N \circ \partial_{Y|\Gamma, Y=0+} \right\rangle \left( \frac{1}{\mu_0} R_\Delta(F_\Omega^\infty) + \frac{1}{\mu_1} R_{\partial_Y^2}(F_\Lambda^\infty) + \sum_{d \in \frac{\pi}{\Theta} \mathbb{Z}^*} \sigma_d(S^\infty) \phi_d \right) \quad (2.21) $$

**Proof:** This is similar to Lemma 2.18. Let $R_1 := -R_{\partial_Y^2} \circ \partial_{X|\Lambda}^2$, $R_2 := \frac{\mu_0}{\mu_1} R_N \circ \partial_{Y|\Gamma, Y=0+}$ and $\mathbf{v}^\infty := (\text{id} - R_1 - R_2)S^\infty$. By Definition 2.3 we have

$$ \langle R_1, R_2 \rangle = \sum_{n=0}^{\infty} \sum_{(i_1, \dots, i_n) \in \{1, 2\}^n} R_{i_1} \circ \dots \circ R_{i_n} = \sum_{n=0}^{\infty} (R_1 + R_2)^n = (\text{id} - R_1 - R_2)^{-1}. $$

So $S^\infty = \langle R_1, R_2 \rangle \mathbf{v}^\infty$. Moreover, Proposition 2.14 implies that $\Delta \circ R_1 = \Delta \circ R_2 = 0$, $\partial_{Y|\Lambda}^2 \circ R_1 = -\partial_{X|\Lambda}^2$, $\partial_{Y|\Lambda}^2 \circ R_2 = 0$, $\partial_{Y|\Gamma, Y=0-} \circ R_1 = 0$ and $\partial_{Y|\Gamma, Y=0-} \circ R_2 = \frac{\mu_0}{\mu_1} \partial_{Y|\Gamma, Y=0+}$. We deduce that $\Delta \circ \langle R_1, R_2 \rangle = \Delta$, $\langle \partial_{Y|\Lambda}^2 + \partial_{X|\Lambda}^2 \rangle \circ \langle R_1, R_2 \rangle = \partial_{Y|\Lambda}^2 \circ (\text{id} - R_1) \circ \langle R_1, R_2 \rangle = \partial_{Y|\Lambda}^2 \circ (\text{id} - R_1 - R_2) \circ \langle R_1, R_2 \rangle = \partial_{Y|\Lambda}^2$
Note that, when used on corner fields (which depend on $(X, Y)$), the first two operators will rather be denoted $\partial_{X|\Lambda}^2$ and $\partial_{Y|\Gamma, Y=0+}$.

**Proof:** It suffices to verify it when $\varphi$ has the form $\mathfrak{F}[(\alpha z)^q \overline{\alpha z}^k P(\log(\alpha z))]$ in $\Omega$ and $x^{q+k} Q(\ln x, Y)$ in $\Lambda$, with $q \in \mathbb{R}$, $k \in \mathbb{N}$, $P[T]$ and $Q \in \mathbb{R}[T, Y]$.

1. We have $\partial_x^2 \varphi(x, Y) = x^{q+k-2}((q+k)(q+k-1)+2(q+k)\partial_T + \partial_T^2)Q(\ln x, Y)$ which is in $\mathcal{A}(\Lambda)$.

2. Let $\psi: z \mapsto (\alpha z)^q \overline{\alpha z}^k P(\log(\alpha z))$. From $\partial_z \psi = \frac{1}{2}(\partial_x \psi - i\partial_y \psi)$ and $\partial_{\bar{z}} \psi = \frac{1}{2}(\partial_x \psi + i\partial_y \psi)$ it follows that $\partial_y \varphi = \mathfrak{F}[\partial_y \psi] = \mathfrak{F}[\partial_z \psi - \partial_{\bar{z}} \psi]$. Therefore,

$$ \begin{aligned} \partial_y \varphi|_{\Gamma, y=0+} &= \mathfrak{F}[\alpha(\alpha x)^{q-1} \overline{\alpha \bar{x}}^k (qP + P')(\log(\alpha x)) - (\alpha x)^q \overline{\alpha k} \overline{\alpha \bar{x}}^{k-1} P(\log(\alpha x))] \\ &= x^{q+k-1} \mathfrak{F}[\alpha^{q-k}((q-k)P + P')(\ln x - i\Theta)] \in \mathcal{A}(\Gamma) \end{aligned} $$

3. Since $\Delta = 4\partial_z \partial_{\bar{z}}$, we have $\Delta\varphi = 4\mathfrak{F}[(\alpha z)^{q-1} k \overline{\alpha z}^{k-1}(qP_1 + P'_1)(\log(\alpha z))]$ which is in $\mathcal{A}(\Omega)$. $\square$

## 2.4 Asymptotic behaviors w.r.t. r of solutions of model problems

In this section we give tools that will be used to compute the asymptotic behaviors of $u_{p,l}$ and $S_{p,l}$ resp. when $r \to 0$ and $r \to \infty$ using the spaces $\mathcal{A}$. To do so, we will use series of elements of $\mathcal{A}$, which is made rigorous by the following definition.

**Definition 2.17: the spaces $\mathcal{A}^\pm$**

Let $D \in \{\Pi, \Omega, \Gamma, \Lambda\}$. We denote $\mathcal{A}^+(D)$, resp. $\mathcal{A}^-(D)$, the set of elements of $\sum_{d \in \mathbb{R}} \mathcal{A}_d(D)$ whose support is included in the image of a sequence that tends to $\infty$, resp. $-\infty$. We write their elements as formal series according to Definition 2.1.

**Remarks:**

* The asymptotic of $u_{p,l}$ when $r \to 0$ involve increasing powers of $r$ so it will be expressed in $\mathcal{A}^+(D)$. Similarly, $S_{p,l}$ when $r \to \infty$ involves decreasing powers of $r$, so it will be expressed in $\mathcal{A}^-(D)$.

* As seen in Section 2.1, $\mathcal{A}(D)$ is included in $\mathcal{A}^+(D)$ and $\mathcal{A}^-(D)$. But elements of $\mathcal{A}^\pm(D)$ are not in general $D \to \mathbb{R}$ functions, as the formal series may diverge pointwise.

Using Definitions 2.2, we can extend $\partial_{x|\Lambda}^2$, $\partial_{y|\Gamma, y=0+}$, $R_{\partial_y^2}$, $R_N$ and $R_\Delta$ to the spaces $\mathcal{A}^\pm$. We also use the notation $\langle \cdot \rangle$ introduced in Definition 2.3. E.g. $\langle -k_0^2 R_\Delta \rangle = \sum_{n=0}^\infty (-k_0^2 R_\Delta)^n$. Moreover, we extend $\sigma_d$ to $\mathcal{A}^\pm(\Pi)$ for any $d \in \frac{\pi}{\Theta}\mathbb{Z}^*$, by setting $\sigma_d(\sum_{d'} \varphi_{d'}) := \sigma_d(\varphi_d)$ for any $\sum_{d'} \varphi_{d'} \in \mathcal{A}^\pm(\Pi)$.

**Lemma 2.18:** Let $g^0 \in \mathcal{A}^+(\Lambda)$ and $h^0 \in \mathcal{A}^+(\Gamma)$. The solutions in $\mathcal{A}^+(\Pi)$ of

$$ \left\{ \begin{array}{ll} \mu_0 \Delta \mathbf{u}^0 + \omega^2 \rho_0 \mathbf{u}^0 = 0 & \text{in } \Omega \\ \mu_1 \partial_y^2 \mathbf{u}^0 = g^0 & \text{in } \Lambda \\ \mu_1 \partial_{Y|Y=0-} \mathbf{u}^0 = h^0 & \text{on } \Gamma \end{array} \right. \qquad (2.17) $$

are the formal series of the following form (where $\sigma_d(\mathbf{u}^0)$ vanishes when $d$ is small enough)

$$ \mathbf{u}^0 = \langle -k_0^2 R_\Delta \rangle \left( \frac{1}{\mu_1} R_{\partial_y^2}(g^0) + \frac{1}{\mu_1} R_N(h^0) + \sum_{d \in \frac{\pi}{\Theta} \mathbb{Z}^*} \sigma_d(\mathbf{u}^0) \phi_d \right) \qquad (2.18) $$

**Proof:** Let $\mathbf{v}^0 := (\mathrm{id} + k_0^2 R_\Delta)\mathbf{u}^0$. Since $\langle -k_0^2 R_\Delta \rangle = \sum_{n=0}^\infty (-k_0^2 R_\Delta)^n$, $\mathrm{id} + k_0^2 R_\Delta$ is the inverse of $\langle -k_0^2 R_\Delta \rangle$, so $\mathbf{u}^0 = \langle -k_0^2 R_\Delta \rangle \mathbf{v}^0$. Moreover, Proposition 2.14 states that $\Delta \circ R_\Delta = \mathrm{id}$, $\partial_{Y|\Lambda}^2 \circ R_\Delta = 0$ and $\partial_{Y|\Gamma, Y=0-} \circ$
and similarly $(\mu_0 \partial_{Y|\Gamma, Y=0^+} - \mu_1 \partial_{Y|\Gamma, Y=0^-}) \circ \langle R_1, R_2 \rangle = \mu_1 \partial_{Y|\Gamma, Y=0^-}$. Therefore, (2.20) is equivalent to

$$ \left\{ \begin{array}{ll} \mu_0 \Delta \mathbf{v}^{\infty} = F_{\Omega}^{\infty} & \text{in } \Omega \\ \mu_1 \partial_{Y}^{2} \mathbf{v}^{\infty} = F_{\Lambda}^{\infty} & \text{in } \Lambda \\ \mu_1 \partial_{Y} \mathbf{v}_{|Y=0^{-}}^{\infty} = 0 & \text{on } \Gamma \end{array} \right. \qquad (2.22) $$

Then Propositions 2.12 and 2.14 imply that the solutions of (2.22) in $\mathcal{A}^+(\Pi)$ are the $\frac{1}{\mu_0} R_{\Delta}(F_{\Omega}^{\infty}) + \frac{1}{\mu_1} R_{\partial_Y^2}(F_{\Lambda}^{\infty}) + \sum_{d \in \frac{\pi}{\hat{\Omega}} Z^*} c_d \phi_d$ where $c_d = 0$ for big enough $d$. Finally, $c_d = \sigma_d(S^{\infty})$ as for Lemma 2.18. $\square$

**Definition 2.20:** Let $d \in \mathbb{R}$ and $a \in \{0, \infty\}$. We define $o_{\partial}$, a kind of differentiable small $o$, as follows.

* For any $\varphi : \Omega \to \mathbb{R}$, we say that $\varphi = o_{\partial}(r^d)$ when $r \to a$ if $\varphi$ is $C^{\infty}$ in a vicinity of $r = a$ and $\forall (j, k) \in \mathbb{N}^2$, $\partial_r^j \partial_{\theta}^k \varphi = o(r^{d-j})$ uniformly w.r.t. $\theta$ when $r \to a$.

* For any $\varphi : \Lambda \to \mathbb{R}$, we say that $\varphi = o_{\partial}(x^d)$ when $x \to a$ if $\varphi$ is $C^{\infty}$ in a vicinity of $x = a$ and $\forall (j, k) \in \mathbb{N}^2$, $\partial_r^j \partial_{Y}^k \varphi = o(x^{d-j})$ uniformly w.r.t. $Y$ when $x \to a$.

* For any $\varphi : \Gamma \to \mathbb{R}$, we say that $\varphi = o_{\partial}(x^d)$ when $x \to a$ if $\varphi$ is $C^{\infty}$ in a vicinity of $x = a$ and $\forall j \in \mathbb{N}$, $\partial_r^j \varphi = o(x^{d-j})$ when $x \to a$.

* For any $\varphi : \Pi \to \mathbb{R}$, we say that $\varphi = o_{\partial}(\mathbf{r}^d)$ when $\mathbf{r} \to a$ if $\varphi|_{\Omega} = o_{\partial}(r^d)$ and $\varphi|_{\Lambda} = o_{\partial}(x^d)$.

**Definition 2.21:** Let $d \in \mathbb{R}$, $D \in \{\Pi, \Omega, \Gamma, \Lambda\}$ and $\varphi = \sum_{q \in \mathbb{R}} \varphi_q$ in $\mathcal{A}^+(D)$ or $\mathcal{A}^-(D)$ with $\varphi_q \in \mathcal{A}_q(D)$ for all $q \in \mathbb{R}$. We denote $T_{\le d}(\varphi) := \sum_{q \in \mathbb{R}, q \le d} \varphi_q$ and $T_{\ge d}(\varphi) := \sum_{q \in \mathbb{R}, q \ge d} \varphi_q$ the truncations of $\varphi$ below and above $d$.

Using Lemmas 2.18-2.19 and Kondratiev's theory (involving weighted Sobolev spaces, Laplace's transform and the residue theorem), we proved the following theorems, giving asymptotic behaviors for solutions of model problems of the type of far-and-layer fields and corner fields. The proofs can be found in Appendix B.

**Theorem 2.22:** Let $\mathbf{u} \in H_{\text{loc}}^1(\Pi)$, $f \in \mathcal{D}'(\Omega)$, $g \in L_{\text{loc}}^2(\Lambda)$ and $h \in L_{\text{loc}}^2(\Gamma)$ be such that

$$ \left\{ \begin{array}{ll} \mu_0 \Delta \mathbf{u} + \omega^2 \rho_0 \mathbf{u} = f & \text{in } \Omega \\ \mu_1 \partial_{Y}^2 \mathbf{u} = g & \text{in } \Lambda \\ \mu_1 \partial_{Y|Y=0^{-}} \mathbf{u} = h & \text{on } \Gamma \\ \mathbf{u}_{|y=0^+} - \mathbf{u}_{|Y=0^{-}} = 0 & \text{on } \Gamma \\ \mathbf{u} = 0 & \text{on } \Sigma_{\Omega} \cup \Sigma_{\Lambda} \end{array} \right. \qquad (2.23) $$

We assume that:

* $f$ vanishes in the vicinity of the corner $(0, 0)$,

* there exists $g^0 \in \mathcal{A}^+(\Lambda)$ s.t.: $\forall d \in \mathbb{R}$, $g(x, Y) = T_{\le d}(g^0)(x, Y) + o_{\partial}(x^d)$ when $x \to 0$,

* there exists $h^0 \in \mathcal{A}^+(\Gamma)$ s.t.: $\forall d \in \mathbb{R}$, $h(x) = T_{\le d}(h^0)(x) + o_{\partial}(x^d)$ when $x \to 0$,

* there is $\eta > 0$, $u^v \in H^1(\Omega \cap B(0, \eta))$ and $\varphi \in \mathcal{A}(\Omega)$ s.t. $\mathbf{u}|_{\Omega \cap B(0, \eta)} = u^v + \varphi$.

Then there is $\mathbf{u}^0 \in \mathcal{A}^+(\Pi)$ that has the form (2.18) s.t.: $\forall d \in \mathbb{R}$, $\mathbf{u} = T_{\le d}(\mathbf{u}^0) + o_{\partial}(\mathbf{r}^d)$ when $\mathbf{r} \to 0$.

**Theorem 2.23:** Let $S \in H_{\text{loc}}^1(\Omega_1)$ and $F \in L_{\text{loc}}^2(\bar{\Omega}_1)$ (i.e. $L^2$ on any bounded subset of $\Omega_1$) such that

$$ \left\{ \begin{array}{ll} \text{div}(\mu \nabla S) = F & \text{in } \Omega_1 \\ S = 0 & \text{on } \partial \Omega_1 \end{array} \right. \qquad (2.24) $$

We assume that:

* there exists $F_{\Omega}^{\infty} \in \mathcal{A}^-(\Omega)$ such that $\forall d \in \mathbb{R}$, $F_{|\Omega} = T_{\ge d}(F_{\Omega}^{\infty}) + o_{\partial}(r^d)$ when $r \to \infty$,

* there exists $F_{\Lambda}^{\infty} \in \mathcal{A}^-(\Lambda)$ such that $\forall d \in \mathbb{R}$, $F_{|\Lambda} = T_{\ge d}(F_{\Lambda}^{\infty}) + o_{\partial}(x^d)$ when $x \to \infty$,

* $S$ belongs to $V + \chi_{\infty}\mathcal{A}(\Pi)$ (the space in which the corner fields will be built in Section 3.2).
$R_Δ = 0$, which imply resp. $(\mu_0\Delta + \omega^2\rho_0) \circ \langle -k_0^2 R_Δ \rangle = \mu_0\Delta \circ (\mathrm{id} + k_0^2 R_Δ) \circ \langle -k_0^2 R_Δ \rangle = \mu_0\Delta$, $\partial_{Y|\Lambda}^2 \circ \langle -k_0^2 R_Δ \rangle = \partial_{Y|\Lambda}^2$ and $\partial_{Y|\Gamma, Y=0^-} \circ \langle -k_0^2 R_Δ \rangle = \partial_{Y|\Gamma, Y=0^-}^2$. Therefore, (2.17) is equivalent to

$$ \left\{ \begin{array}{ll} \mu_0 \Delta \mathbf{v}^0 = 0 & \text{in } \Omega \\ \mu_1 \partial_Y^2 \mathbf{v}^0 = g^0 & \text{in } \Lambda \\ \mu_1 \partial_Y \mathbf{v}_{|Y=0^-}^0 = h^0 & \text{on } \Gamma \end{array} \right. \qquad (2.19) $$

Then Propositions 2.12 and 2.14 imply that the solutions of (2.19) in $\mathcal{A}^+(\Pi)$ are the $\frac{1}{\mu_1} R_{\partial_Y^2}(g^0) + \frac{1}{\mu_1} R_N(h^0) + \sum_{d \in \frac{\pi}{\Theta} Z^*} c_d \phi_d$ where $c_d$ vanishes for small enough $d$. Finally we have $c_d = \sigma_d(\mathbf{v}^0) = \sigma_d(\mathbf{u}^0)$ for any $d \in \frac{\pi}{\Theta} Z^*$ because $\sigma_d \circ R_Δ$, $\sigma_d \circ R_{\partial_Y^2}$ and $\sigma_d \circ R_N$ vanish by Proposition 2.14. $\square$

**Lemma 2.19:** Let $F_{\Omega}^{\infty} \in \mathcal{A}^{-}(\Omega)$ and $F_{\Lambda}^{\infty} \in \mathcal{A}^{-}(\Lambda)$. The solutions in $\mathcal{A}^{-}(\Pi)$ of

$$ \left\{ \begin{array}{ll} \mu_0 \Delta S^\infty = F_\Omega^\infty & \text{in } \Omega \\ \mu_1 \Delta S^\infty = F_\Lambda^\infty & \text{in } \Lambda \\ \mu_0 \partial_{Y|Y=0^+} S^\infty - \mu_1 \partial_{Y|Y=0^-} S^\infty = 0 & \text{on } \Gamma \end{array} \right. \qquad (2.20) $$

are the formal series of the following form (where $\sigma_d(S^\infty)$ vanishes when $d$ is big enough)

$$ S^{\infty} = \left\langle -R_{\partial_Y^2} \circ \partial_{X|\Lambda}^2, \frac{\mu_0}{\mu_1} R_N \circ \partial_{Y|\Gamma, Y=0^+} \right\rangle \left( \frac{1}{\mu_0} R_\Delta(F_\Omega^\infty) + \frac{1}{\mu_1} R_{\partial_Y^2}(F_\Lambda^\infty) + \sum_{d \in \frac{\pi}{\Theta} Z^*} \sigma_d(S^\infty) \phi_d \right) \quad (2.21) $$

**Proof:** This is similar to Lemma 2.18. Let $R_1 := -R_{\partial_Y^2} \circ \partial_{X|\Lambda}^2$, $R_2 := \frac{\mu_0}{\mu_1} R_N \circ \partial_{Y|\Gamma, Y=0^+}$ and $\mathbf{v}^\infty := (\mathrm{id} - R_1 - R_2)S^\infty$. By Definition 2.3 we have

$$ \langle R_1, R_2 \rangle = \sum_{n=0}^{\infty} \sum_{(i_1, \dots, i_n) \in \{1, 2\}^n} R_{i_1} \circ \dots \circ R_{i_n} = \sum_{n=0}^{\infty} (R_1 + R_2)^n = (\mathrm{id} - R_1 - R_2)^{-1}. $$

So $S^\infty = \langle R_1, R_2 \rangle \mathbf{v}^\infty$. Moreover, Proposition 2.14 implies that $\Delta \circ R_1 = \Delta \circ R_2 = 0$, $\partial_{Y|\Lambda}^2 \circ R_1 = -\partial_{X|\Lambda}^2$, $\partial_{Y|\Lambda}^2 \circ R_2 = 0$, $\partial_{Y|\Gamma, Y=0^-} \circ R_1 = 0$ and $\partial_{Y|\Gamma, Y=0^-} \circ R_2 = \frac{\mu_0}{\mu_1} \partial_{Y|\Gamma, Y=0^+}$. We deduce that $\Delta \circ \langle R_1, R_2 \rangle = \Delta$,

$$ (\partial_{Y|\Lambda}^2 + \partial_{X|\Lambda}^2) \circ \langle R_1, R_2 \rangle = \partial_{Y|\Lambda}^2 \circ (\mathrm{id} - R_1) \circ \langle R_1, R_2 \rangle = \partial_{Y|\Lambda}^2 \circ (\mathrm{id} - R_1 - R_2) \circ \langle R_1, R_2 \rangle = \partial_{Y|\Lambda}^2 $$

and similarly $(\mu_0 \partial_{Y|\Gamma, Y=0^+} - \mu_1 \partial_{Y|\Gamma, Y=0^-}) \circ \langle R_1, R_2 \rangle = \mu_1 \partial_{Y|\Gamma, Y=0^-}$. Therefore, (2.20) is equivalent to

$$ \left\{ \begin{array}{ll} \mu_0 \Delta \mathbf{v}^\infty = F_\Omega^\infty & \text{in } \Omega \\ \mu_1 \partial_Y^2 \mathbf{v}^\infty = F_\Lambda^\infty & \text{in } \Lambda \\ \mu_1 \partial_Y \mathbf{v}_{|Y=0^-}^\infty = 0 & \text{on } \Gamma \end{array} \right. \qquad (2.22) $$

Then Propositions 2.12 and 2.14 imply that the solutions of (2.22) in $\mathcal{A}^+(\Pi)$ are the $\frac{1}{\mu_1} R_\Delta(F_\Omega^\infty) + \frac{1}{\mu_1} R_{\partial_Y^2}(F_\Lambda^\infty) + \sum_{d \in \frac{\pi}{\Theta} Z^*} c_d \phi_d$ where $c_d = 0$ for big enough $d$. Finally, $c_d = \sigma_d(S^\infty)$ as for Lemma 2.18. $\square$

**Definition 2.20:** Let $d \in \mathbb{R}$ and $a \in \{0, \infty\}$. We define $o_\partial$, a kind of differentiable small $o$, as follows.

* For any $\varphi : \Omega \to \mathbb{R}$, we say that $\varphi = o_\partial(r^d)$ when $r \to a$ if $\varphi$ is $C^\infty$ in a vicinity of $r = a$ and $\forall (j, k) \in \mathbb{N}^2$, $\partial_r^j \partial_\theta^k \varphi = o(r^{d-j})$ uniformly w.r.t. $\theta$ when $r \to a$.

* For any $\varphi : \Lambda \to \mathbb{R}$, we say that $\varphi = o_\partial(x^d)$ when $x \to a$ if $\varphi$ is $C^\infty$ in a vicinity of $x = a$ and $\forall (j, k) \in \mathbb{N}^2$, $\partial_r^j \partial_\theta^k \varphi = o(x^{d-j})$ uniformly w.r.t. $Y$ when $x \to a$.

* For any $\varphi : \Gamma \to \mathbb{R}$, we say that $\varphi = o_\partial(x^d)$ when $x \to a$ if $\varphi$ is $C^\infty$ in a vicinity of $x = a$ and $\forall j \in \mathbb{N}$, $\partial_r^j \varphi = o(x^{d-j})$ when $x \to a$.

* For any $\varphi : \Pi \to \mathbb{R}$, we say that $\varphi = o_\partial(\mathbf{r}^d)$ when $\mathbf{r} \to a$ if $\varphi|_{\Omega} = o_\partial(r^d)$ and $\varphi|_{\Lambda} = o_\partial(x^d)$.
Then there is $S^\infty \in \mathcal{A}^-(\Pi)$ that has the form (2.21) s.t.: $\forall d \in \mathbb{R}$, $S = T_{\ge d}(S^\infty) + o_\partial(\mathbf{r}^d)$ when $\mathbf{r} \to \infty$.

Note that, since $g^0 \in \mathcal{A}^+(\Lambda)$, we have $T_{\le d}(g^0) \in \mathcal{A}(\Lambda)$, so the formula $g = T_{\le d}(g^0) + o_\partial(x^d)$ makes sense. The same applies to the truncations of $h^0$, $\mathbf{u}^0$, $F_\Omega^\infty$ and $F_\Lambda^\infty$ and $S^\infty$.

A consequence of Theorems 2.22–2.23 is Proposition 3.13 that states that for any $(p, \ell)$ there is $\mathbf{u}_{p,\ell}^0 \in \mathcal{A}^+(\Pi)$ and $S_{p,\ell}^\infty \in \mathcal{A}^-(\Pi)$ s.t.:

$$ \forall d \in \mathbb{R}, \quad \left\{ \begin{array}{ll} \mathbf{u}_{p,\ell} = T_{\le d}(\mathbf{u}_{p,\ell}^0) + o_\partial(\mathbf{r}^d) & \text{when } \mathbf{r} \to 0 \\ S_{p,\ell} = T_{\ge d}(S_{p,\ell}^\infty) + o_\partial(\mathbf{r}^d) & \text{when } \mathbf{r} \to \infty \end{array} \right. $$

In the rest of this section, we will assume that such formal series exist. In addition, given the equations satisfied by $\mathbf{u}_{p,\ell}$ and $S_{p,\ell}$ (see (2.1) and (1.5)), (2.18) and (2.21) rewrite here as

$$ \mathbf{u}_{p,\ell}^0 = \langle -k_0^2 R_\Delta \rangle \left( -R_{\partial_Y^2} \circ (\partial_{x|\Lambda}^2 + k_1^2)(\mathbf{u}_{p-2,\ell}^0) + \frac{\mu_0}{\mu_1} R_N \circ \partial_{Y|\Gamma, y=0^+} (\mathbf{u}_{p-1,\ell}^0) + \sum_{d \in \frac{\pi}{\mathbb{Z}}^*} \sigma_d(\mathbf{u}_{p,\ell}^0) \phi_d \right) \quad (2.25) $$

$$ S_{p,\ell}^{\infty} = \langle -R_{\partial_Y^2} \circ \partial_{X|\Lambda}^2, \frac{\mu_0}{\mu_1} R_N \circ \partial_{Y|\Gamma, Y=0^+} \rangle \left( -k_0^2 R_\Delta (S_{p-2,\ell|\Omega}^\infty) - k_1^2 R_{\partial_Y^2} (S_{p-2,\ell|\Lambda}^\infty) + \sum_{d \in \frac{\pi}{\mathbb{Z}}^*} \sigma_d (S_{p,\ell}^\infty) \phi_d \right) $$

where by convention $\mathbf{u}_{p,\ell}^0$ and $S_{p,\ell}^\infty$ vanish when $p \in \mathbb{R} \setminus \mathbb{P}$. Therefore $(\mathbf{u}_{p,\ell})_{p,\ell}$ and $(S_{p,\ell}^\infty)_{p,\ell}$ are uniquely defined by $(\sigma_d(\mathbf{u}_{p,\ell}^0))_{d,p,\ell}$ and $(\sigma_d(S_{p,\ell}^\infty))_{d,p,\ell}$. When $d < 0$ (resp. $d > 0$), $\phi_d$ is non-variational for the far-and-layer fields (resp. corner fields), and Theorems 3.6 and 3.11 show that $\sigma_d(\mathbf{u}_{p,\ell}^0)$ (resp. $\sigma_d(S_{p,\ell}^\infty)$) can be fixed arbitrarily. The rest of this section is devoted to finding how to fix them in order to satisfy the matching conditions (1.6). On the contrary, when $d > 0$ (resp. $d < 0$), $\phi_d$ is variational for the far-and-layer fields (resp. corner fields) and the values of $\sigma_d(\mathbf{u}_{p,\ell}^0)$ (resp. $\sigma_d(S_{p,\ell}^\infty)$) are uniquely defined once $(\sigma_d'(\mathbf{u}_{p,\ell}^0))_{d'<0}$ (resp. $(\sigma_d'(S_{p,\ell}^\infty))_{d'>0}$) has been fixed. Ways to compute these values numerically will be investigated in a future work.

## 2.5 Specifying of the matching conditions

In this section we express $\sum_{p,\ell} \varepsilon^p \ln^\ell \varepsilon \cdot \mathbf{u}_{p,\ell}^0$ and $\sum_{p,\ell} \varepsilon^p \ln^\ell \varepsilon \cdot S_{p,\ell}^\infty$ in function of the $\sigma_d(\mathbf{u}_{p,\ell}^0)$ and $\sigma_d(S_{p,\ell}^\infty)$, we then rewrite rigorously the matching conditions (1.6), and we finally show that they are equivalent to a set of equations on the coefficients $\sigma_d(\cdot)$. Here $\sum_{p,\ell} \varepsilon^p \ln^\ell \varepsilon \cdot \mathbf{u}_{p,\ell}^0$ and $\sum_{p,\ell} \varepsilon^p \ln^\ell \varepsilon \cdot S_{p,\ell}^\infty$ are formal series that belong resp. to the spaces $\mathcal{A}_\varepsilon^+(\Pi)$ and $\mathcal{A}_\varepsilon^-(\Pi)$ defined below.

In this section, “$\varepsilon$” and “$\ln \varepsilon$” denote two algebraic indeterminates independent of each other (so they are not numbers).

**Definition 2.24:** We denote

$$ \mathcal{A}_{\varepsilon}^{+}(\Pi) := \sum_{p \in \mathbb{P}} \varepsilon^p \sum_{d \in \mathbb{P}-p} \mathcal{A}_d(\Pi)[\ln \varepsilon] \quad \text{and} \quad \mathcal{A}_{\varepsilon}^{-}(\Pi) := \sum_{p \in \mathbb{P}} \varepsilon^p \sum_{d \in p-\mathbb{P}} \mathcal{A}_d(\Pi)[\ln \varepsilon]. $$

According to Section 2.1, we write their elements as formal series like

$$ \sum_{p \in \mathbb{P}} \sum_{d \in \mathbb{P}-p} \sum_{\ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon \cdot \varphi_{p,d,\ell}, \quad \text{resp.} \quad \sum_{p \in \mathbb{P}} \sum_{d \in p-\mathbb{P}} \sum_{\ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon \cdot \varphi_{p,d,\ell}. $$

**Ansatz 2.25:** We assume that

$$ \sum_{p \in \mathbb{P}, \ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon \cdot \mathbf{u}_{p,\ell}^0 \in \mathcal{A}_\varepsilon^+(\Pi) \quad \text{and} \quad \sum_{p \in \mathbb{P}, \ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon \cdot S_{p,\ell}^\infty \in \mathcal{A}_\varepsilon^-(\Pi). $$

Compared to the ansatz of Section 1.2, it adds that, for any $(p, \ell) \in \mathbb{P} \times \mathbb{N}$, $\mathbf{u}_{p,\ell}$ has components only in the $\mathcal{A}_d(\Pi)$ s.t. $d \in \mathbb{P}-p$ and $S_{p,\ell}$ has only in the $\mathcal{A}_d(\Pi)$ s.t. $d \in p-\mathbb{P}$. This is necessary for the matching
**Definition 2.21:** Let $d \in \mathbb{R}$, $D \in \{\Pi, \Omega, \Gamma, \Lambda\}$ and $\varphi = \sum_{q \in \mathbb{R}} \varphi_q$ in $\mathcal{A}^+(D)$ or $\mathcal{A}^-(D)$ with $\varphi_q \in \mathcal{A}_q(D)$ for all $q \in \mathbb{R}$. We denote $T_{\le d}(\varphi) := \sum_{q \in \mathbb{R}, q \le d} \varphi_q$ and $T_{\ge d}(\varphi) := \sum_{q \in \mathbb{R}, q \ge d} \varphi_q$ the truncations of $\varphi$ below and above $d$.

Using Lemmas 2.18–2.19 and Kondrat’ev’s theory (involving weighted Sobolev spaces, Laplace’s transform and the residue theorem), we proved the following theorems, giving asymptotic behaviors for solutions of model problems of the type of far-and-layer fields and corner fields. The proofs can be found in Appendix B.

**Theorem 2.22:** Let $\mathbf{u} \in H_{\text{loc}}^1(\Pi)$, $f \in \mathcal{D}'(\Omega)$, $g \in L_{\text{loc}}^2(\Lambda)$ and $h \in L_{\text{loc}}^2(\Gamma)$ be such that

$$ 
\left\{ 
\begin{array}{ll}
\mu_0 \Delta \mathbf{u} + \omega^2 \rho_0 \mathbf{u} = f & \text{in } \Omega \\
\mu_1 \partial_Y^2 \mathbf{u} = g & \text{in } \Lambda \\
\mu_1 \partial_{Y|Y=0^{-}} \mathbf{u} = h & \text{on } \Gamma \\
\mathbf{u}_{|y=0^+} - \mathbf{u}_{|Y=0^{-}} = 0 & \text{on } \Gamma \\
\mathbf{u} = 0 & \text{on } \Sigma_{\Omega} \cup \Sigma_{\Lambda}
\end{array}
\right.
\qquad (2.23) 
$$

We assume that:

* $f$ vanishes in the vicinity of the corner $(0, 0)$,

* there exists $g^0 \in \mathcal{A}^+(\Lambda)$ s.t.: $\forall d \in \mathbb{R}$, $g(x, Y) = T_{\le d}(g^0)(x, Y) + o_{\partial}(x^d)$ when $x \to 0$,

* there exists $h^0 \in \mathcal{A}^+(\Gamma)$ s.t.: $\forall d \in \mathbb{R}$, $h(x) = T_{\le d}(h^0)(x) + o_{\partial}(x^d)$ when $x \to 0$,

* there is $\eta > 0$, $u^v \in H^1(\Omega \cap B(0, \eta))$ and $\varphi \in \mathcal{A}(\Omega)$ s.t. $\mathbf{u}_{|\Omega \cap B(0, \eta)} = u^v + \varphi$.

Then there is $\mathbf{u}^0 \in \mathcal{A}^+(\Pi)$ that has the form (2.18) s.t.: $\forall d \in \mathbb{R}$, $\mathbf{u} = T_{\le d}(\mathbf{u}^0) + o_{\partial}(\mathbf{r}^d)$ when $\mathbf{r} \to 0$.

**Theorem 2.23:** Let $S \in H_{\text{loc}}^1(\Omega_1)$ and $F \in L_{\text{loc}}^2(\overline{\Omega}_1)$ (i.e. $L^2$ on any bounded subset of $\Omega_1$) such that

$$ 
\left\{ 
\begin{array}{ll}
\text{div}(\mu \nabla S) = F & \text{in } \Omega_1 \\
S = 0 & \text{on } \partial \Omega_1
\end{array}
\right.
\qquad (2.24) 
$$

We assume that:

* there exists $F_{\Omega}^{\infty} \in \mathcal{A}^-(\Omega)$ such that $\forall d \in \mathbb{R}$, $F_{|\Omega} = T_{\ge d}(F_{\Omega}^{\infty}) + o_{\partial}(r^d)$ when $r \to \infty$,

* there exists $F_{\Lambda}^{\infty} \in \mathcal{A}^-(\Lambda)$ such that $\forall d \in \mathbb{R}$, $F_{|\Lambda} = T_{\ge d}(F_{\Lambda}^{\infty}) + o_{\partial}(x^d)$ when $x \to \infty$,

* $S$ belongs to $V + \chi_{\infty}\mathcal{A}(\Pi)$ (the space in which the corner fields will be built in Section 3.2).

Then there is $S^{\infty} \in \mathcal{A}^-(\Pi)$ that has the form (2.21) s.t.: $\forall d \in \mathbb{R}$, $S = T_{\ge d}(S^{\infty}) + o_{\partial}(\mathbf{r}^d)$ when $\mathbf{r} \to \infty$.

Note that, since $g^0 \in \mathcal{A}^+(\Lambda)$, we have $T_{\le d}(g^0) \in \mathcal{A}(\Lambda)$, so the formula $g = T_{\le d}(g^0) + o_{\partial}(x^d)$ makes sense. The same applies to the truncations of $h^0$, $\mathbf{u}^0$, $F_{\Omega}^{\infty}$ and $F_{\Lambda}^{\infty}$ and $S^{\infty}$.

A consequence of Theorems 2.22–2.23 is Proposition 3.13 that states that for any $(p, \ell)$ there is $\mathbf{u}_{p, \ell}^0 \in \mathcal{A}^+(\Pi)$ and $S_{p, \ell}^{\infty} \in \mathcal{A}^-(\Pi)$ s.t.:

$$ 
\forall d \in \mathbb{R}, \quad \left\{ 
\begin{array}{ll}
\mathbf{u}_{p, \ell} = T_{\le d}(\mathbf{u}_{p, \ell}^0) + o_{\partial}(\mathbf{r}^d) & \text{when } \mathbf{r} \to 0 \\
S_{p, \ell} = T_{\ge d}(S_{p, \ell}^{\infty}) + o_{\partial}(\mathbf{r}^d) & \text{when } \mathbf{r} \to \infty
\end{array}
\right. 
$$

In the rest of this section, we will assume that such formal series exist. In addition, given the equations satisfied by $\mathbf{u}_{p, \ell}$ and $S_{p, \ell}$ (see (2.1) and (1.5)), (2.18) and (2.21) rewrite here as

$$ 
\mathbf{u}_{p, \ell}^0 = \langle -k_0^2 R_{\Delta} \rangle \left( -R_{\partial_Y^2} \circ (\partial_{x|\Lambda}^2 + k_1^2)(\mathbf{u}_{p-2, \ell}^0) + \frac{\mu_0}{\mu_1} R_N \circ \partial_{y|\Gamma, y=0^+} (\mathbf{u}_{p-1, \ell}^0) + \sum_{d \in \mathbb{Z}^*} \sigma_d(\mathbf{u}_{p, \ell}^0) \phi_d \right) \quad (2.25) 
$$

$$ 
S_{p, \ell}^{\infty} = \langle -R_{\partial_Y^2} \circ \partial_{X|\Lambda}^2, \frac{\mu_0}{\mu_1} R_N \circ \partial_{Y|\Gamma, Y=0^+} \rangle \left( -k_0^2 R_{\Delta}(S_{p-2, \ell|\Omega}^{\infty}) - k_1^2 R_{\partial_Y^2} (S_{p-2, \ell|\Lambda}^{\infty}) + \sum_{d \in \mathbb{Z}^*} \sigma_d(S_{p, \ell}^{\infty}) \phi_d \right) 
$$
<figure><img src="image_6.png" /><figcaption>Figure 8: Points of $\{(d, p) \mid p \in \mathbb{P} \text{ and } d \in \mathbb{P} - p\}$ for $\Theta = \frac{3\pi}{2}$ (on the left) and $\Theta = 2$ (on the right)</figcaption></figure>

because e.g. we will see that for any $\varphi \in \varepsilon^p A_d(\Pi)[\ln \varepsilon] \subset A_{\bar{\varepsilon}}^{-}(\Pi)$ that is a term of $\sum \varepsilon^p \ln^\ell \varepsilon S_{p,\ell}^{\infty}$, $\varphi(\frac{\cdot}{\bar{\varepsilon}})$ must appear in $\sum \varepsilon^p \ln^\ell \varepsilon \mathbf{u}_{p,\ell}^0$ and $\varphi(\frac{\cdot}{\bar{\varepsilon}}) \in \varepsilon^{p-d} A_d(\Pi)[\ln \varepsilon]$, so $p-d \in \mathbb{P}$.

$$
\begin{align*}
\mathcal{R}_{\varepsilon}^{+} &:= \left\{ -k_0^2 R_{\Delta}, -\varepsilon^2 R_{\partial_Y^2} \circ \partial_{x|\Lambda}^2, -\varepsilon^2 k_1^2 R_{\partial_Y^2}, \varepsilon \frac{\mu_0}{\mu_1} R_N \circ \partial_{y|\Gamma, y=0^+} \right\} \\
\mathcal{R}_{\varepsilon}^{-} &:= \left\{ -\varepsilon^2 k_0^2 R_{\Delta}, -R_{\partial_Y^2} \circ \partial_{X|\Lambda}^2, -\varepsilon^2 k_1^2 R_{\partial_Y^2}, \frac{\mu_0}{\mu_1} R_N \circ \partial_{Y|\Gamma, Y=0^+} \right\}
\end{align*}
$$

where the above operators are defined as in (2.6). For any $R \in \mathcal{R}_{\varepsilon}^{+} \cup \mathcal{R}_{\varepsilon}^{-}$, we denote $(\deg_{\varepsilon} R, \deg_{\Lambda} R) := \deg R$. See Figure 9. Thanks to Definition 2.4, we can consider $\langle \mathcal{R}_{\varepsilon}^{\pm} \rangle$ which is well-defined on $\mathcal{A}_{\varepsilon}^{\pm}(\Pi)$ (it suffices to take $v := (\pm 1, 2)$ in Definition 2.4). Moreover one can check that it maps $\mathcal{A}_{\varepsilon}^{\pm}(\Pi)$ into itself.

<figure><img src="image_7.png" /><figcaption>Figure 9: Degrees of the elements of $\mathcal{R}_{\varepsilon}^{+}$ (on the left) and $\mathcal{R}_{\varepsilon}^{-}$ (on the right)</figcaption></figure>

**Proposition 2.26:** We have the following equalities in $\mathcal{A}_{\varepsilon}^{+}(\Pi)$ and $\mathcal{A}_{\varepsilon}^{-}(\Pi)$ respectively:

$$
\sum_{p \in \mathbb{P}, \ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon \cdot \mathbf{u}_{p, \ell}^0 = \langle \mathcal{R}_\varepsilon^+ \rangle \left( \sum_{p \in \mathbb{P}, \ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon \sum_{d \in \bar{\Theta}^* \cap (\mathbb{P}-p)} \sigma_d(\mathbf{u}_{p, \ell}^0) \phi_d \right) \quad (2.26)
$$

$$
\sum_{p \in \mathbb{P}, \ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon \cdot S_{p, \ell}^\infty = \langle \mathcal{R}_\varepsilon^- \rangle \left( \sum_{p \in \mathbb{P}, \ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon \sum_{d \in \bar{\Theta}^* \cap (p-\mathbb{P})} \sigma_d(S_{p, \ell}^\infty) \phi_d \right) \quad (2.27)
$$

**Proof:** We only prove (2.26), as (2.27) is similar. We could do it by inductively composing (2.25), but we chose instead a proof similar to Lemma 2.19 to avoid heavy calculations. Let $(p, \ell) \in \mathbb{P} \times \mathbb{N}$. Given the equations satisfied by $\mathbf{u}_{p, \ell}$ in (2.1), and the fact that $\mathbf{u}_{p, \ell} = T_{\le d}(\mathbf{u}_{p, \ell}^0) + o_{\partial}(\mathbf{r}^d)$ for any $d \in \mathbb{R}$, we have

$$
\left\{
\begin{array}{ll}
\mu_0 \Delta \mathbf{u}_{p, \ell}^0 + \omega^2 \rho_0 \mathbf{u}_{p, \ell}^0 = 0 & \text{in } \Omega \\
\mu_1 \partial_Y^2 \mathbf{u}_{p, \ell}^0 = -(\mu_1 \partial_x^2 + \omega^2 \rho_1) \mathbf{u}_{p-2, \ell}^0 & \text{in } \Lambda \\
\mu_1 \partial_{Y|Y=0^-} \mathbf{u}_{p, \ell}^0 = \mu_0 \partial_{y|y=0^+} \mathbf{u}_{p-1, \ell}^0 & \text{on } \Gamma
\end{array}
\right.
\qquad (2.28)
$$

Let $\mathbf{u}_{\varepsilon}^{0} := \sum_{p, \ell} \varepsilon^{p} \ln^{\ell} \varepsilon \mathbf{u}_{p, \ell}^{0}$. Summing over $(p, \ell)$ (2.28) times $\varepsilon^{p} \ln^{\ell} \varepsilon$ yields

$$
\left\{
\begin{array}{ll}
\mu_0 \Delta \mathbf{u}_{\varepsilon}^0 + \omega^2 \rho_0 \mathbf{u}_{\varepsilon}^0 = 0 & \text{in } \Omega \\
\mu_1 \partial_Y^2 \mathbf{u}_{\varepsilon}^0 + \varepsilon^2 (\mu_1 \partial_x^2 + \omega^2 \rho_1) \mathbf{u}_{\varepsilon}^0 = 0 & \text{in } \Lambda \\
\mu_1 \partial_{Y|Y=0^-} \mathbf{u}_{\varepsilon}^0 - \varepsilon \mu_0 \partial_{y|y=0^+} \mathbf{u}_{\varepsilon}^0 = 0 & \text{on } \Gamma
\end{array}
\right.
$$
where by convention $\mathbf{u}_{p,\ell}^0$ and $S_{p,\ell}^\infty$ vanish when $p \in \mathbb{R} \setminus \mathbb{P}$. Therefore $(\mathbf{u}_{p,\ell}^0)_{p,\ell}$ and $(S_{p,\ell}^\infty)_{p,\ell}$ are uniquely defined by $(\sigma_d(\mathbf{u}_{p,\ell}^0))_{d,p,\ell}$ and $(\sigma_d(S_{p,\ell}^\infty))_{d,p,\ell}$. When $d < 0$ (resp. $d > 0$), $\phi_d$ is non-variational for the far-and-layer fields (resp. corner fields), and Theorems 3.6 and 3.11 show that $\sigma_d(\mathbf{u}_{p,\ell}^0)$ (resp. $\sigma_d(S_{p,\ell}^\infty)$) can be fixed arbitrarily. The rest of this section is devoted to finding how to fix them in order to satisfy the matching conditions (1.6). On the contrary, when $d > 0$ (resp. $d < 0$), $\phi_d$ is variational for the far-and-layer fields (resp. corner fields) and the values of $\sigma_d(\mathbf{u}_{p,\ell}^0)$ (resp. $\sigma_d(S_{p,\ell}^\infty)$) are uniquely defined once $(\sigma_{d'}(\mathbf{u}_{p,\ell}^0))_{d'<0}$ (resp. $(\sigma_{d'}(S_{p,\ell}^\infty))_{d'>0}$) has been fixed. Ways to compute these values numerically will be investigated in a future work.

## 2.5 Specifying of the matching conditions

In this section we express $\sum_{p,\ell} \varepsilon^p \ln^\ell \varepsilon \cdot \mathbf{u}_{p,\ell}^0$ and $\sum_{p,\ell} \varepsilon^p \ln^\ell \varepsilon \cdot S_{p,\ell}^\infty$ in function of the $\sigma_d(\mathbf{u}_{p,\ell}^0)$ and $\sigma_d(S_{p,\ell}^\infty)$, we then rewrite rigorously the matching conditions (1.6), and we finally show that they are equivalent to a set of equations on the coefficients $\sigma_d(\cdot)$. Here $\sum_{p,\ell} \varepsilon^p \ln^\ell \varepsilon \cdot \mathbf{u}_{p,\ell}^0$ and $\sum_{p,\ell} \varepsilon^p \ln^\ell \varepsilon \cdot S_{p,\ell}^\infty$ are formal series that belong resp. to the spaces $\mathcal{A}_\varepsilon^+(\Pi)$ and $\mathcal{A}_\varepsilon^-(\Pi)$ defined below.

In this section, “$\varepsilon$” and “$\ln \varepsilon$” denote two algebraic indeterminates independent of each other (so they are not numbers).

**Definition 2.24:** We denote

$$ \mathcal{A}_{\varepsilon}^{+}(\Pi) := \sum_{p \in \mathbb{P}} \varepsilon^{p} \sum_{d \in \mathbb{P}-p} \mathcal{A}_{d}(\Pi)[\ln \varepsilon] \quad \text{and} \quad \mathcal{A}_{\varepsilon}^{-}(\Pi) := \sum_{p \in \mathbb{P}} \varepsilon^{p} \sum_{d \in p-\mathbb{P}} \mathcal{A}_{d}(\Pi)[\ln \varepsilon]. $$

According to Section 2.1, we write their elements as formal series like

$$ \sum_{p \in \mathbb{P}} \sum_{d \in \mathbb{P}-p} \sum_{\ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon \cdot \varphi_{p,d,\ell}, \quad \text{resp.} \quad \sum_{p \in \mathbb{P}} \sum_{d \in p-\mathbb{P}} \sum_{\ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon \cdot \varphi_{p,d,\ell}. $$

<figure><img src="image_9.png" /><figcaption>Figure 8: Points of $\{(d, p) \mid p \in \mathbb{P} \text{ and } d \in \mathbb{P}-p\}$ for $\Theta = \frac{2\pi}{3}$ (on the left) and $\Theta = 2$ (on the right)</figcaption></figure>

**Ansatz 2.25:** We assume that

$$ \sum_{p \in \mathbb{P}, \ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon \cdot \mathbf{u}_{p, \ell}^0 \in \mathcal{A}_\varepsilon^+(\Pi) \quad \text{and} \quad \sum_{p \in \mathbb{P}, \ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon \cdot S_{p, \ell}^\infty \in \mathcal{A}_\varepsilon^-(\Pi). $$

Compared to the ansatz of Section 1.2, it adds that, for any $(p, \ell) \in \mathbb{P} \times \mathbb{N}$, $\mathbf{u}_{p,\ell}$ has components only in the $\mathcal{A}_d(\Pi)$ s.t. $d \in \mathbb{P}-p$ and $S_{p,\ell}$ has only in the $\mathcal{A}_d(\Pi)$ s.t. $d \in p-\mathbb{P}$. This is necessary for the matching because e.g. we will see that for any $\varphi \in \varepsilon^p \mathcal{A}_d(\Pi)[\ln \varepsilon] \subset \mathcal{A}_\varepsilon^-(\Pi)$ that is a term of $\sum \varepsilon^p \ln^\ell \varepsilon S_{p,\ell}^\infty$, $\varphi(\frac{\cdot}{\varepsilon})$ must appear in $\sum \varepsilon^p \ln^\ell \varepsilon \mathbf{u}_{p,\ell}^0$ and $\varphi(\frac{\cdot}{\varepsilon}) \in \varepsilon^{p-d} \mathcal{A}_d(\Pi)[\ln \varepsilon]$, so $p-d \in \mathbb{P}$.

$$ \mathcal{R}_{\varepsilon}^{+} := \left\{ -k_{0}^{2} R_{\Delta}, -\varepsilon^{2} R_{\partial_{Y}^{2}} \circ \partial_{x|\Lambda}^{2}, -\varepsilon^{2} k_{1}^{2} R_{\partial_{Y}^{2}}, \varepsilon^{\frac{\mu_{0}}{\mu_{1}}} R_{N} \circ \partial_{y|\Gamma, y=0^{+}} \right\} $$

$$ \mathcal{R}_{\varepsilon}^{-} := \left\{ -\varepsilon^{2} k_{0}^{2} R_{\Delta}, -R_{\partial_{Y}^{2}} \circ \partial_{X|\Lambda}^{2}, -\varepsilon^{2} k_{1}^{2} R_{\partial_{Y}^{2}}, \frac{\mu_{0}}{\mu_{1}} R_{N} \circ \partial_{Y|\Gamma, Y=0^{+}} \right\} $$
Let $\mathbf{v}_{\varepsilon}^{0} := (\mathrm{id} - \sum_{R \in \mathcal{R}_{\varepsilon}^{+}} R)\mathbf{u}_{\varepsilon}^{0}$. It is easy to see that $\mathrm{id} - \sum_{R \in \mathcal{R}_{\varepsilon}^{+}} R$ maps $\mathcal{A}_{\varepsilon}^{+}(\Pi)$ into itself, so $\mathbf{v}_{\varepsilon}^{0} \in \mathcal{A}_{\varepsilon}^{+}(\Pi)$.
By Definition 2.4, we have

$$ \langle \mathcal{R}_{\varepsilon}^{+} \rangle := \sum_{n=0}^{\infty} \sum_{(R_1, \dots, R_n) \in (\mathcal{R}_{\varepsilon}^{+})^n} R_1 \circ \dots \circ R_n = \sum_{n=0}^{\infty} \left( \sum_{R \in \mathcal{R}_{\varepsilon}^{+}} R \right)^n = \left( \mathrm{id} - \sum_{R \in \mathcal{R}_{\varepsilon}^{+}} R \right)^{-1}. $$

So $\mathbf{u}_{\varepsilon}^{0} = \langle \mathcal{R}_{\varepsilon}^{+} \rangle \mathbf{v}_{\varepsilon}^{0}$. Moreover, Proposition 2.14 implies $\forall R \in \mathcal{R}_{\varepsilon}^{+} \setminus \{-k_{0}^{2} R_{\Delta}\}, \Delta \circ R = 0$. Hence

$$ (\mu_0 \Delta + \omega^2 \rho_0) \circ \langle \mathcal{R}_{\varepsilon}^{+} \rangle = \mu_0 \Delta \circ (\mathrm{id} + k_0^2 R_{\Delta}) \circ \langle \mathcal{R}_{\varepsilon}^{+} \rangle = \mu_0 \Delta \circ \left( \mathrm{id} - \sum_{R \in \mathcal{R}_{\varepsilon}^{+}} R \right) \circ \langle \mathcal{R}_{\varepsilon}^{+} \rangle = \mu_0 \Delta $$

Similarly, one can check that $[\mu_1 \partial_{Y|\Lambda}^2 + \varepsilon^2(\mu_1 \partial_{x|\Lambda}^2 + \omega^2 \rho_1)] \circ \langle \mathcal{R}_{\varepsilon}^{+} \rangle = \mu_1 \partial_{Y|\Lambda}^2$ and $(\mu_1 \partial_{Y|\Gamma, Y=0^{-}} - \varepsilon \mu_0 \partial_{Y|\Gamma, Y=0^{+}}) \circ \langle \mathcal{R}_{\varepsilon}^{+} \rangle = \mu_1 \partial_{Y|\Gamma, Y=0^{-}}$. Therefore $\mathbf{v}_{\varepsilon}^{0}$ satisfies

$$ \left\{ \begin{array}{ll} \mu_0 \Delta \mathbf{v}_{\varepsilon}^{0} = 0 & \text{in } \Omega \\ \mu_1 \partial_{Y}^{2} \mathbf{v}_{\varepsilon}^{0} = 0 & \text{in } \Lambda \\ \mu_1 \partial_{Y|Y=0^{-}} \mathbf{v}_{\varepsilon}^{0} = 0 & \text{on } \Gamma \end{array} \right. $$

Then Proposition 2.14 implies that there are numbers $c_{p, \ell, d}$ s.t. $\mathbf{v}_{\varepsilon}^{0} = \sum_{p, \ell, d} \varepsilon^p \ln^{\ell} \varepsilon c_{p, \ell, d} \phi_d$. Finally we have $c_{p, \ell, d} = \sigma_d(\mathbf{u}_{p, \ell}^0)$ for any $(p, \ell, d)$ because Proposition 2.14 gives that $\forall R \in \mathcal{R}_{\varepsilon}^{+}$, $\sigma_d \circ R = 0$. $\square$

Now, we want to define $\mathcal{H}_{\varepsilon} : \mathcal{A}_{\varepsilon}^{+}(\Pi) \to \mathcal{A}_{\varepsilon}^{-}(\Pi)$ as the scaling operator

$$ \forall \varphi \in \mathcal{A}_{\varepsilon}^{+}(\Pi), \quad \mathcal{H}_{\varepsilon}(\varphi) = \left\{ \begin{array}{ll} (X, Y) \mapsto \varphi(\varepsilon X, \varepsilon Y) & \text{in } \Omega \\ (X, Y) \mapsto \varphi(\varepsilon X, Y) & \text{in } \Lambda \end{array} \right. \qquad (2.29) $$

However, $\varepsilon$ is an indeterminate. Let us first define $\mathcal{H}_{\varepsilon}$ for any $\varphi \in \mathcal{A}_{d}(\Pi)$, $d \in \mathbb{R}$. If $\varphi|_{\Omega}$ has the form $\Im[(\alpha z)^q \overline{\alpha z}^k P(\log(\alpha z))]$ with $(q, k, P) \in \mathbb{R} \times \mathbb{N} \times \mathbb{R}[T]$ s.t. $q + k = d$ and $\mathcal{P}(q, k, P)$ is true, we set:

$$ \mathcal{H}_{\varepsilon}(\varphi)|_{\Omega} := \sum_{j=0}^{\deg P} \varepsilon^d \frac{\ln^j \varepsilon}{j!} \Im[(\alpha z)^q \overline{\alpha z}^k P^{(j)}(\log(\alpha z))] \in \varepsilon^d \mathcal{A}_d(\Omega)[\ln \varepsilon]. \qquad (2.30) $$

And if $\varphi|_{\Lambda}$ has the form $x^d Q(\ln(x), Y)$ with $Q \in \mathbb{C}[T]$, we set:

$$ \mathcal{H}_{\varepsilon}(\varphi)|_{\Lambda} := \sum_{j=0}^{\deg Q} \varepsilon^d \frac{\ln^j \varepsilon}{j!} x^d \partial_T^j Q(\ln x, Y) \in \varepsilon^d \mathcal{A}_d(\Lambda)[\ln \varepsilon]. \qquad (2.31) $$

Thanks to Lemmas 2.8 and 2.10, it defines well $\mathcal{H}_{\varepsilon}$ from $\mathcal{A}_d(\Pi)$ to $\varepsilon^d \mathcal{A}_d(\Pi)[\ln \varepsilon]$. Then we extend $\mathcal{H}_{\varepsilon}$ to $\mathcal{A}_{\varepsilon}^{+}(\Pi)$ by setting $\mathcal{H}_{\varepsilon}(\sum_{p,d,\ell} \varepsilon^p \ln^{\ell} \varepsilon \varphi_{p,d,\ell}) := \sum_{p,d} \sum_{\ell} \varepsilon^p \ln^{\ell} \varepsilon \mathcal{H}_{\varepsilon}(\varphi_{p,d,\ell})$ for any $(\varphi_{p,d,\ell})$ s.t. $\varphi_{p,d,\ell} \in \mathcal{A}_d(\Pi)$ for any $(p, d, \ell)$. One can check that is in $\mathcal{A}_{\varepsilon}^{-}(\Pi)$.

### Remarks:

* In practice, we will only use the informal definition of (2.29), but everything we will do can be checked using (2.30) and (2.31).

* $\mathcal{H}_{\varepsilon}$ is invertible and $\mathcal{H}_{\varepsilon}^{-1}$ is roughly the scaling: $\mathcal{H}_{\varepsilon}^{-1}(\varphi) = \left\{ \begin{array}{ll} (x, y) \mapsto \varphi(\frac{x}{\varepsilon}, \frac{y}{\varepsilon}) & \text{in } \Omega \\ (x, Y) \mapsto \varphi(\frac{x}{\varepsilon}, Y) & \text{in } \Lambda \end{array} \right.$

* For the first time, powers of $\ln \varepsilon$ naturally appear because of the power of $\ln r$. This explains why the presence of these powers in the ansatz is necessary from the beginning.

### Definition 2.27:

We rigorously rewrite the matching condition as: $\sum_{p \in \mathbb{P}, \ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon \cdot \mathbf{u}_{p, \ell}^0 = \mathcal{H}_{\varepsilon}^{-1} \left( \sum_{p \in \mathbb{P}, \ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon \cdot S_{p, \ell}^\infty \right)$.

When composing Definition 2.27 and Proposition 2.26, $\mathcal{H}_{\varepsilon}^{-1} \circ \langle \mathcal{R}_{\varepsilon}^{-} \rangle$ appears. Since $\mathcal{R}_{\varepsilon}^{+}$ and $\mathcal{R}_{\varepsilon}^{-}$ only differ on powers of $\varepsilon$ due to the scaling, one could expect that $\mathcal{H}_{\varepsilon}^{-1} \circ \langle \mathcal{R}_{\varepsilon}^{-} \rangle \circ \mathcal{H}_{\varepsilon}$ is equal to $\langle \mathcal{R}_{\varepsilon}^{+} \rangle$. However, $R_{\Delta}$
where the above operators are defined as in (2.6). For any $R \in \mathcal{R}_{\varepsilon}^{+} \cup \mathcal{R}_{\varepsilon}^{-}$, we denote $(\deg_{\varepsilon} R, \deg_{A} R) := \deg R$. See Figure 9. Thanks to Definition 2.4, we can consider $\langle \mathcal{R}_{\varepsilon}^{\pm} \rangle$ which is well-defined on $\mathcal{A}_{\varepsilon}^{\pm}(\Pi)$ (it suffices to take $v := (\pm 1, 2)$ in Definition 2.4). Moreover one can check that it maps $\mathcal{A}_{\varepsilon}^{\pm}(\Pi)$ into itself.

<figure><img src="image_10.png" /><figcaption>Figure 9: Degrees of the elements of $\mathcal{R}_{\varepsilon}^{+}$ (on the left) and $\mathcal{R}_{\varepsilon}^{-}$ (on the right)</figcaption></figure>

**Theorem 2.26:** We have the following equalities in $\mathcal{A}_{\varepsilon}^{+}(\Pi)$ and $\mathcal{A}_{\varepsilon}^{-}(\Pi)$ respectively:

$$ \sum_{p \in \mathbb{P}, \ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon \cdot \mathbf{u}_{p, \ell}^0 = \langle \mathcal{R}_\varepsilon^+ \rangle \left( \sum_{p \in \mathbb{P}, \ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon \sum_{d \in \mathbb{Z}^* \cap (\mathbb{P}-p)} \sigma_d(\mathbf{u}_{p, \ell}^0) \phi_d \right) \quad (2.26) $$

$$ \sum_{p \in \mathbb{P}, \ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon \cdot S_{p, \ell}^\infty = \langle \mathcal{R}_\varepsilon^- \rangle \left( \sum_{p \in \mathbb{P}, \ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon \sum_{d \in \mathbb{Z}^* \cap (p-\mathbb{P})} \sigma_d(S_{p, \ell}^\infty) \phi_d \right) \quad (2.27) $$

*Proof:* We only prove (2.26), as (2.27) is similar. We could do it by inductively by composing (2.25), but we chose instead a proof similar to Lemma 2.19 to avoid heavy calculations. Let $(p, \ell) \in \mathbb{P} \times \mathbb{N}$. Given the equations satisfied by $\mathbf{u}_{p, \ell}$ in (2.1), and the fact that $\mathbf{u}_{p, \ell} = T_{\le d}(\mathbf{u}_{p, \ell}^0) + o_{\partial}(\mathbf{r}^d)$ for any $d \in \mathbb{R}$, we have

$$ \left\{ \begin{array}{ll} \mu_0 \Delta \mathbf{u}_{p, \ell}^0 + \omega^2 \rho_0 \mathbf{u}_{p, \ell}^0 = 0 & \text{in } \Omega \\ \mu_1 \partial_Y^2 \mathbf{u}_{p, \ell}^0 = -(\mu_1 \partial_x^2 + \omega^2 \rho_1) \mathbf{u}_{p-2, \ell}^0 & \text{in } \Lambda \\ \mu_1 \partial_{Y|Y=0^{-}} \mathbf{u}_{p, \ell}^0 = \mu_0 \partial_{y|y=0^{+}} \mathbf{u}_{p-1, \ell}^0 & \text{on } \Gamma \end{array} \right. \quad (2.28) $$

Let $\mathbf{u}_{\varepsilon}^0 := \sum_{p, \ell} \varepsilon^p \ln^\ell \varepsilon \mathbf{u}_{p, \ell}^0$. Summing over $(p, \ell)$ (2.28) times $\varepsilon^p \ln^\ell \varepsilon$ yields

$$ \left\{ \begin{array}{ll} \mu_0 \Delta \mathbf{u}_{\varepsilon}^0 + \omega^2 \rho_0 \mathbf{u}_{\varepsilon}^0 = 0 & \text{in } \Omega \\ \mu_1 \partial_Y^2 \mathbf{u}_{\varepsilon}^0 + \varepsilon^2 (\mu_1 \partial_x^2 + \omega^2 \rho_1) \mathbf{u}_{\varepsilon}^0 = 0 & \text{in } \Lambda \\ \mu_1 \partial_{Y|Y=0^{-}} \mathbf{u}_{\varepsilon}^0 - \varepsilon \mu_0 \partial_{y|y=0^{+}} \mathbf{u}_{\varepsilon}^0 = 0 & \text{on } \Gamma \end{array} \right. \quad (2.29) $$

Let $\mathbf{v}_{\varepsilon}^0 := (\text{id} - \sum_{R \in \mathcal{R}_{\varepsilon}^{+}} R)\mathbf{u}_{\varepsilon}^0$. Let us show that (2.29) implies (2.26). It is easy to see that $\text{id} - \sum_{R \in \mathcal{R}_{\varepsilon}^{+}} R$ maps $\mathcal{A}_{\varepsilon}^{+}(\Pi)$ into itself, so $\mathbf{v}_{\varepsilon}^0 \in \mathcal{A}_{\varepsilon}^{+}(\Pi)$. By Definition 2.4, we have

$$ \langle \mathcal{R}_{\varepsilon}^{+} \rangle := \sum_{n=0}^{\infty} \sum_{(R_1, \dots, R_n) \in (\mathcal{R}_{\varepsilon}^{+})^n} R_1 \circ \dots \circ R_n = \sum_{n=0}^{\infty} \left( \sum_{R \in \mathcal{R}_{\varepsilon}^{+}} R \right)^n = \left( \text{id} - \sum_{R \in \mathcal{R}_{\varepsilon}^{+}} R \right)^{-1}. $$

So $\mathbf{u}_{\varepsilon}^0 = \langle \mathcal{R}_{\varepsilon}^{+} \rangle \mathbf{v}_{\varepsilon}^0$. Moreover, Proposition 2.14 implies $\forall R \in \mathcal{R}_{\varepsilon}^{+} \setminus \{-k_0^2 R_{\Delta}\}, \Delta \circ R = 0$. Hence

$$ (\mu_0 \Delta + \omega^2 \rho_0) \circ \langle \mathcal{R}_{\varepsilon}^{+} \rangle = \mu_0 \Delta \circ (\text{id} + k_0^2 R_{\Delta}) \circ \langle \mathcal{R}_{\varepsilon}^{+} \rangle = \mu_0 \Delta \circ \left( \text{id} - \sum_{R \in \mathcal{R}_{\varepsilon}^{+}} R \right) \circ \langle \mathcal{R}_{\varepsilon}^{+} \rangle = \mu_0 \Delta $$

Similarly, one can check that $[\mu_1 \partial_{Y|\Lambda}^2 + \varepsilon^2(\mu_1 \partial_{x|\Lambda}^2 + \omega^2 \rho_1)] \circ \langle \mathcal{R}_{\varepsilon}^{+} \rangle = \mu_1 \partial_{Y|\Lambda}^2$ and $(\mu_1 \partial_{Y|\Gamma, Y=0^{-}} - \varepsilon \mu_0 \partial_{y|\Gamma, y=0^{+}}) \circ \langle \mathcal{R}_{\varepsilon}^{+} \rangle = \mu_1 \partial_{Y|\Gamma, Y=0^{-}}$. Therefore $\mathbf{v}_{\varepsilon}^0$ satisfies

$$ \left\{ \begin{array}{ll} \mu_0 \Delta \mathbf{v}_{\varepsilon}^0 = 0 & \text{in } \Omega \\ \mu_1 \partial_Y^2 \mathbf{v}_{\varepsilon}^0 = 0 & \text{in } \Lambda \\ \mu_1 \partial_{Y|Y=0^{-}} \mathbf{v}_{\varepsilon}^0 = 0 & \text{on } \Gamma \end{array} \right. $$
(resp. $R_{\partial_2^2}$, $R_N$) picks the particular solution of Equation 2.12 (resp. 2.13, 2.14) whose image by the $\sigma_d$ vanish. This means that they are the solutions in the kernels of the following projectors of $\mathcal{A}_{\varepsilon}^{\pm}(\Pi)$:

$$ \pi_{\sigma}^{\pm} \left( \sum_{p \in \mathbb{P}} \sum_{d \in \pm(\mathbb{P}-p)} \sum_{\ell \in \mathbb{N}} \varepsilon^{p} \ln^{\ell} \varepsilon \cdot \varphi_{p,d,\ell} \right) := \sum_{p \in \mathbb{P}} \sum_{d \in \pm \mathbb{Z}^* \cap \pm(\mathbb{P}-p)} \sum_{\ell \in \mathbb{N}} \varepsilon^{p} \ln^{\ell} \varepsilon \cdot \sigma_d(\varphi_{p,d,\ell}) \phi_d \quad (2.32) $$

for any $(\varphi_{p,d,\ell})$ s.t. $\varphi_{p,d,\ell} \in \mathcal{A}_d(\Pi)$ for any $(p, d, \ell)$. Lemma 2.28 below implies that $\mathcal{H}_{\varepsilon}^{\pm 1}$ does not map Ker $\pi_{\sigma}^{\pm}$ to Ker $\pi_{\sigma}^{\mp}$. Therefore $\mathcal{H}_{\varepsilon}^{-1} \circ R_{\Delta} \circ \mathcal{H}_{\varepsilon}$ selects other solutions than $R_{\Delta}$, and likewise for $R_{\partial_2^2}$ and $R_N$. We will see that it implies that $\mathcal{H}_{\varepsilon}^{-1} \circ \langle \mathcal{R}_{\varepsilon} \rangle \circ \mathcal{H}_{\varepsilon}$ it is equal to $\langle \mathcal{R}_{\varepsilon}^{+} \rangle$ times a correction operator given in Theorem 2.29.

To precise the action of $\mathcal{H}_{\varepsilon}$ on $\pi_{\sigma}^{\pm}$, we introduce $\pi_{\varepsilon}^{\pm}$ and $\pi_{\varepsilon}^{-}$, which map any $\varphi \in \mathcal{A}_d(\Pi)$ of the form $\mathfrak{F}[(\alpha z)^q \bar{\alpha} z^k P(\log(\alpha z))]$ in $\Omega$ (with $(q, k, P) \in \mathbb{R} \times \mathbb{N} \times \mathbb{R}[T]$, $q+k=d$ and $\mathcal{P}(q, k, P)$) to:

$$ \pi_{\varepsilon}^{\pm}(\varphi) := \begin{cases} P(\pm \ln \varepsilon) \phi_d & \text{if } d \in \frac{\pi}{\Theta} \mathbb{Z}^* \text{ and } k=0 \\ 0 & \text{otherwise} \end{cases} \quad (2.33) $$

By Lemmas 2.8 and 2.10, it defines a linear map $\mathcal{A}_d(\Pi) \to \mathcal{A}_d(\Pi)[\ln \varepsilon]$. Then we extend $\pi_{\varepsilon}^{\pm}$ into an endomorphism of $\mathcal{A}_{\varepsilon}^{\pm}(\Pi)$ by setting $\pi_{\varepsilon}^{\pm}(\sum_{p,d,\ell} \varepsilon^p \ln^{\ell} \varepsilon \varphi_{p,d,\ell}) := \sum_{p,d} \sum_{\ell} \varepsilon^p \ln^{\ell} \varepsilon \pi_{\varepsilon}^{\pm}(\varphi_{p,d,\ell})$.

**Lemma 2.28:** We have $\mathcal{H}_{\varepsilon}^{\mp 1} \circ \pi_{\sigma}^{\mp} \circ \mathcal{H}_{\varepsilon}^{\pm 1} = \pi_{\varepsilon}^{\pm}$.

**Proof:** Since the involved operators only depend on the part on $\Omega$, and by Lemma 2.8, it suffices to prove that $\mathcal{H}_{\varepsilon}^{\mp 1} \circ \pi_{\sigma}^{\mp} \circ \mathcal{H}_{\varepsilon}^{\pm 1}(\varphi) = \pi_{\varepsilon}^{\pm}(\varphi)$ for any $\varphi$ of the form $\mathfrak{F}[(\alpha z)^q \bar{\alpha} z^k P(\log(\alpha z))]$ in $\Omega$ (with $(q, k, P) \in \mathbb{R} \times \mathbb{N} \times \mathbb{R}[T]$ and $\mathcal{P}(q, k, P)$). If $k > 0$ or $q \notin \frac{\pi}{\Theta} \mathbb{Z}^*$, then both images vanish, so we can assume that $k=0$ and $q \in \frac{\pi}{\Theta} \mathbb{Z}^*$. Then both images vanish in $\Lambda$, and we have in $\Omega$:

$$ \begin{align*} \mathcal{H}_{\varepsilon}^{\pm 1}(\varphi) & : z \mapsto \varepsilon^{\pm q} \mathfrak{F}[(\alpha z)^q P(\log(\alpha z) \pm \ln \varepsilon)] \quad (\text{with abuse of notation}) \\ \pi_{\sigma}^{\mp} \circ \mathcal{H}_{\varepsilon}^{\pm 1}(\varphi) &= \varepsilon^q P(\ln \varepsilon) \phi_q \\ \mathcal{H}_{\varepsilon}^{\mp 1} \circ \pi_{\sigma}^{\mp} \circ \mathcal{H}_{\varepsilon}^{\pm 1}(\varphi) &= P(\pm \ln \varepsilon) \phi_q = \pi_{\varepsilon}^{\pm}(\varphi) \tag*{\square} \end{align*} $$

The next theorem is the key ingredient to match the far-and-layer fields with the corner fields because it links $\langle \mathcal{R}_{\varepsilon}^{+} \rangle$ to $\langle \mathcal{R}_{\varepsilon}^{-} \rangle$.

**Theorem 2.29:** We have $\mathcal{H}_{\varepsilon}^{\mp 1} \circ \langle \mathcal{R}_{\varepsilon}^{\mp} \rangle \circ \mathcal{H}_{\varepsilon}^{\pm 1} = \langle \mathcal{R}_{\varepsilon}^{\pm} \rangle \circ (\text{id} - \langle \pi_{\varepsilon}^{\pm} | \mathcal{R}_{\varepsilon}^{\pm} \rangle)$ where we denote:

$$ \begin{align*} \langle \pi_{\varepsilon}^{\pm} | \mathcal{R}_{\varepsilon}^{\pm} \rangle &:= \pi_{\varepsilon}^{\pm} \circ \left( \sum_{R \in \mathcal{R}_{\varepsilon}^{\pm}} R \right) \circ \langle \{ (\text{id} - \pi_{\varepsilon}^{\pm}) \circ R \mid R \in \mathcal{R}_{\varepsilon}^{\pm} \} \rangle \\ &= \sum_{n=1}^{\infty} \sum_{(R_1, \dots, R_n) \in (\mathcal{R}_{\varepsilon}^{\pm})^n} \pi_{\varepsilon}^{\pm} \circ R_1 \circ (\text{id} - \pi_{\varepsilon}^{\pm}) \circ R_2 \circ \dots \circ (\text{id} - \pi_{\varepsilon}^{\pm}) \circ R_n \end{align*} $$

**Proof:** We only prove that $\mathcal{H}_{\varepsilon}^{-1} \circ \langle \mathcal{R}_{\varepsilon}^{-} \rangle \circ \mathcal{H}_{\varepsilon} = \langle \mathcal{R}_{\varepsilon}^{+} \rangle \circ (\text{id} - \langle \pi_{\varepsilon}^{+} | \mathcal{R}_{\varepsilon}^{+} \rangle)$, but the other formula works the same. We have $\mathcal{H}_{\varepsilon}^{-1} \circ \langle \mathcal{R}_{\varepsilon}^{-} \rangle \circ \mathcal{H}_{\varepsilon} = \langle \{ \mathcal{H}_{\varepsilon}^{-1} \circ R \circ \mathcal{H}_{\varepsilon} \mid R \in \mathcal{R}_{\varepsilon}^{-} \} \rangle$, so we need to calculate the $\mathcal{H}_{\varepsilon}^{-1} \circ R \circ \mathcal{H}_{\varepsilon}$ for each $R \in \mathcal{R}_{\varepsilon}^{-}$. We claim that:

$$ \begin{align*} \mathcal{H}_{\varepsilon}^{-1} \circ (-\varepsilon^2 k_0^2 R_{\Delta}) &\circ \mathcal{H}_{\varepsilon} = \mathcal{H}_{\varepsilon}^{-1} \circ (\text{id} - \pi_{\sigma}^{-}) \circ \mathcal{H}_{\varepsilon} \circ (-k_0^2 R_{\Delta}) \\ \mathcal{H}_{\varepsilon}^{-1} \circ (-R_{\partial_2^2} \circ \partial_{x|\Lambda}^2) &\circ \mathcal{H}_{\varepsilon} = \mathcal{H}_{\varepsilon}^{-1} \circ (\text{id} - \pi_{\sigma}^{-}) \circ \mathcal{H}_{\varepsilon} \circ (-\varepsilon^2 R_{\partial_2^2} \circ \partial_{x|\Lambda}^2) \\ \mathcal{H}_{\varepsilon}^{-1} \circ (-\varepsilon^2 k_1^2 R_{\partial_2^2}) &\circ \mathcal{H}_{\varepsilon} = \mathcal{H}_{\varepsilon}^{-1} \circ (\text{id} - \pi_{\sigma}^{-}) \circ \mathcal{H}_{\varepsilon} \circ (-\varepsilon^2 k_1^2 R_{\partial_2^2}) \\ \mathcal{H}_{\varepsilon}^{-1} \circ \underbrace{(\frac{\mu_0}{\mu_1} R_N \circ \partial_{y|\Gamma, y=0^+})}_{\in \mathcal{R}_{\varepsilon}^{-}} &\circ \mathcal{H}_{\varepsilon} = \mathcal{H}_{\varepsilon}^{-1} \circ (\text{id} - \pi_{\sigma}^{-}) \circ \mathcal{H}_{\varepsilon} \circ \underbrace{(\varepsilon \frac{\mu_0}{\mu_1} R_N \circ \partial_{y|\Gamma, y=0^+})}_{\in \mathcal{R}_{\varepsilon}^{+}} \end{align*} $$

Let us show the first line (the others are similar).
Then Proposition 2.12 implies that there are numbers $c_{p,\ell,d}$ s.t. $\mathbf{v}_{\varepsilon}^0 = \sum_{p,\ell,d} \varepsilon^p \ln^{\ell} \varepsilon c_{p,\ell,d} \phi_d$. Finally we have $c_{p,\ell,d} = \sigma_d(\mathbf{u}_{p,\ell}^0)$ for any $(p, \ell, d)$ because Proposition 2.14 gives that $\forall R \in \mathcal{R}_{\varepsilon}^+$, $\sigma_d \circ R = 0$. $\square$

**Remark 2.27:** The elements of $\mathcal{A}(\Pi)$ present in (2.26)-(2.27) are the $R_1 \circ \dots \circ R_n(\phi_d)$ with: $\forall i, R_i \in \{-k_0^2 R_{\Delta}, R_{\partial_Y^2} \circ \partial_{x|\Lambda}^2, -k_1^2 R_{\partial_Y^2}, \frac{\mu_0}{\mu_1} R_{\Lambda} \circ \partial_{y|\Gamma, y=0^+}\}$. They cannot be described using only inductive sequences and their number grows exponentially with $n$. That is why we introduced the operators $R_{\Delta}$, $R_{\partial_Y^2}$ and $R_{\Lambda}$. They allow a factorization of (2.26)-(2.27) into a geometric series of operators, that encapsulates all the complexity of the singularities, and the coefficients $\sigma_d(\dots)$, that need to be fixed for the different fields to match. This will allow us to reduce the matching condition to only fixing the coefficients $\sigma_d(\dots)$.

Now, we want to define $\mathcal{H}_{\varepsilon} : \mathcal{A}_{\varepsilon}^{+}(\Pi) \to \mathcal{A}_{\varepsilon}^{-}(\Pi)$ as the scaling operator

$$ \forall \varphi \in \mathcal{A}_{\varepsilon}^{+}(\Pi), \quad \mathcal{H}_{\varepsilon}(\varphi) = \begin{cases} (X, Y) \mapsto \varphi(\varepsilon X, \varepsilon Y) & \text{in } \Omega \\ (X, Y) \mapsto \varphi(\varepsilon X, Y) & \text{in } \Lambda \end{cases} \qquad (2.30) $$

However, $\varepsilon$ is an indeterminate. Let us first define $\mathcal{H}_{\varepsilon}$ for any $\varphi \in \mathcal{A}_d(\Pi)$, $d \in \mathbb{R}$. If $\varphi|_{\Omega}$ has the form $\mathfrak{S}[(\alpha z)^q \bar{\alpha} \bar{z}^k P(\log(\alpha z))]$ with $(q, k, P) \in \mathbb{R} \times \mathbb{N} \times \mathbb{R}[T]$ s.t. $q + k = d$ and $\mathcal{P}(q, k, P)$ is true, we set:

$$ \mathcal{H}_{\varepsilon}(\varphi)|_{\Omega} := \sum_{j=0}^{\deg P} \varepsilon^d \frac{\ln^j \varepsilon}{j!} \mathfrak{S}[(\alpha z)^q \bar{\alpha} \bar{z}^k P^{(j)}(\log(\alpha z))] \in \varepsilon^d \mathcal{A}_d(\Omega)[\ln \varepsilon]. \qquad (2.31) $$

And if $\varphi|_{\Lambda}$ has the form $x^d Q(\ln(x), Y)$ with $Q \in \mathbb{C}[T]$, we set:

$$ \mathcal{H}_{\varepsilon}(\varphi)|_{\Lambda} := \sum_{j=0}^{\deg Q} \varepsilon^d \frac{\ln^j \varepsilon}{j!} x^d \partial_T^j Q(\ln x, Y) \in \varepsilon^d \mathcal{A}_d(\Lambda)[\ln \varepsilon]. \qquad (2.32) $$

Thanks to Lemmas 2.8 and 2.10, it defines well $\mathcal{H}_{\varepsilon}$ from $\mathcal{A}_d(\Pi)$ to $\varepsilon^d \mathcal{A}_d(\Pi)[\ln \varepsilon]$. Then we extend $\mathcal{H}_{\varepsilon}$ to $\mathcal{A}_{\varepsilon}^{+}(\Pi)$ by setting $\mathcal{H}_{\varepsilon}(\sum_{p,d,\ell} \varepsilon^p \ln^{\ell} \varepsilon \varphi_{p,d,\ell}) := \sum_{p,d} \sum_{\ell} \varepsilon^p \ln^{\ell} \varepsilon \mathcal{H}_{\varepsilon}(\varphi_{p,d,\ell})$ for any $(\varphi_{p,d,\ell})$ s.t. $\varphi_{p,d,\ell} \in \mathcal{A}_d(\Pi)$ for any $(p, d, \ell)$. One can check that is in $\mathcal{A}_{\varepsilon}^{-}(\Pi)$.

### Remarks:

* In practice, we will only use the informal definition of (2.30), but everything we will do can be checked using (2.31) and (2.32).

* $\mathcal{H}_{\varepsilon}$ is invertible and $\mathcal{H}_{\varepsilon}^{-1}$ is roughly the scaling: $\mathcal{H}_{\varepsilon}^{-1}(\varphi) = \begin{cases} (x, y) \mapsto \varphi(\frac{x}{\varepsilon}, \frac{y}{\varepsilon}) & \text{in } \Omega \\ (x, Y) \mapsto \varphi(\frac{x}{\varepsilon}, Y) & \text{in } \Lambda \end{cases}$

* For the first time, powers of $\ln \varepsilon$ naturally appear because of the power of $\ln r$. This explains why the presence of these powers in the ansatz is necessary from the beginning.

### Definition 2.28:

We rigorously rewrite the matching condition as: $\sum_{p \in \mathbb{P}, \ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon \cdot \mathbf{u}_{p, \ell}^0 = \mathcal{H}_{\varepsilon}^{-1} \left( \sum_{p \in \mathbb{P}, \ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon \cdot S_{p, \ell}^\infty \right)$.

In order to rewrite the matching condition in terms of the coefficients $\sigma_d(\mathbf{u}_{p,\ell}^0)$ and $\sigma_d(S_{p,\ell}^\infty)$, we define the following projectors of $\mathcal{A}_{\varepsilon}^{\pm}(\Pi)$:

$$ \pi_{\sigma}^{\pm} \left( \sum_{p \in \mathbb{P}} \sum_{d \in \pm(\mathbb{P}-p)} \sum_{\ell \in \mathbb{N}} \varepsilon^p \ln^{\ell} \varepsilon \cdot \varphi_{p,d,\ell} \right) := \sum_{p \in \mathbb{P}} \sum_{d \in \pm \mathbb{Z}^* \cap \pm(\mathbb{P}-p)} \sum_{\ell \in \mathbb{N}} \varepsilon^p \ln^{\ell} \varepsilon \cdot \sigma_d(\varphi_{p,d,\ell}) \phi_d $$

for any $(\varphi_{p,d,\ell})$ s.t. $\varphi_{p,d,\ell} \in \mathcal{A}_d(\Pi)$ for any $(p, d, \ell)$.
Proposition 2.14 implies that for any $\psi \in \mathcal{A}_{\varepsilon}^{-}(\Pi)$, $R_{\Delta}(\psi)$ is a solution of:

$$ egin{cases} 
\Delta\varphi = \psi & \text{in } \Omega \\
\partial_Y^2\varphi = 0 & \text{in } \Lambda \\
\partial_Y\varphi|_{Y=0^{-}} = 0 & \text{on } \Gamma
\end{cases} 
\qquad (2.34) $$

and $\pi_{\sigma}^{-} \circ R_{\Delta} = 0$. Moreover, by Proposition 2.12, the solution of (2.34) is unique modulo $\text{Im } \pi_{\sigma}^{-}$. But one can check that $\varepsilon^{-2}\mathcal{H}_{\varepsilon} \circ R_{\Delta} \circ \mathcal{H}_{\varepsilon}^{-1}(\psi)$ is also solution of (2.34). So the image of $\varepsilon^{-2}\mathcal{H}_{\varepsilon} \circ R_{\Delta} \circ \mathcal{H}_{\varepsilon}^{-1} - R_{\Delta}$ is included in $\text{Im } \pi_{\sigma}^{-}$. Hence $(\text{id} - \pi_{\sigma}^{-}) \circ (\varepsilon^{-2}\mathcal{H}_{\varepsilon} \circ R_{\Delta} \circ \mathcal{H}_{\varepsilon}^{-1} - R_{\Delta}) = 0$. Therefore:

$$ R_{\Delta} = (\text{id} - \pi_{\sigma}^{-}) \circ R_{\Delta} = (\text{id} - \pi_{\sigma}^{-}) \circ \varepsilon^{-2}\mathcal{H}_{\varepsilon} \circ R_{\Delta} \circ \mathcal{H}_{\varepsilon}^{-1} $$

which implies $\mathcal{H}_{\varepsilon}^{-1} \circ R_{\Delta} \circ \mathcal{H}_{\varepsilon} = \mathcal{H}_{\varepsilon}^{-1} \circ (\text{id} - \pi_{\sigma}^{-}) \circ \mathcal{H}_{\varepsilon} \circ \varepsilon^{-2}R_{\Delta}$.

$$ 
\begin{aligned}
& \mathcal{H}_{\varepsilon}^{-1} \circ \langle \mathcal{R}_{\varepsilon}^{-} \rangle \circ \mathcal{H}_{\varepsilon} = \langle \{ \mathcal{H}_{\varepsilon}^{-1} \circ R \circ \mathcal{H}_{\varepsilon} \mid R \in \mathcal{R}_{\varepsilon}^{-} \} \rangle \\
& = \langle \{ \mathcal{H}_{\varepsilon}^{-1} \circ (\text{id} - \pi_{\sigma}^{-}) \circ \mathcal{H}_{\varepsilon} \circ R \mid R \in \mathcal{R}_{\varepsilon}^{+} \} \rangle \\
& = \langle \{(\text{id} - \pi_{\varepsilon}^{+}) \circ R \mid R \in \mathcal{R}_{\varepsilon}^{+} \} \rangle \\
& = \sum_{n=0}^{\infty} \sum_{(R_1, \dots, R_n) \in (\mathcal{R}_{\varepsilon}^{+})^n} (\text{id} - \pi_{\varepsilon}^{+}) \circ R_1 \circ \dots \circ (\text{id} - \pi_{\varepsilon}^{+}) \circ R_n \\
& = \sum_{n=0}^{\infty} \sum_{\substack{(R_1, \dots, R_n) \in (\mathcal{R}_{\varepsilon}^{+})^n \\ (i_1, \dots, i_n) \in \{0, 1\}^n}} (-\pi_{\varepsilon}^{+})^{i_1} \circ R_1 \circ \dots \circ (-\pi_{\varepsilon}^{+})^{i_n} \circ R_n
\end{aligned}
$$

The terms s.t. $(i_1, \dots, i_n) = (0, \dots, 0)$ sum up to $\langle \mathcal{R}_{\varepsilon}^{+} \rangle$, and for the rest we split the sums depending on $\ell := \min\{k \mid i_k = 1\}$:

$$ \mathcal{H}_{\varepsilon}^{-1} \circ \langle \mathcal{R}_{\varepsilon}^{-} \rangle \circ \mathcal{H}_{\varepsilon} = \langle \mathcal{R}_{\varepsilon}^{+} \rangle + \sum_{n=0}^{\infty} \sum_{\ell=1}^{n} \sum_{\substack{(R_1, \dots, R_n) \in (\mathcal{R}_{\varepsilon}^{+})^{n} \\ i_1 = \dots = i_{\ell-1} = 0, i_{\ell+1} = \dots = i_{\ell} = 1 \\ (i_{\ell+1}, \dots, i_n) \in \{0, 1\}^{n-\ell}}} (-\pi_{\varepsilon}^{+})^{i_1} \circ R_1 \circ \dots \circ (-\pi_{\varepsilon}^{+})^{i_n} \circ R_n $$

Denoting $m = n - \ell + 1$, $(\tilde{R}_1, \dots, \tilde{R}_m) := (R_{\ell}, \dots, R_n)$ and $(j_2, \dots, j_m) := (i_{\ell+1}, \dots, i_n)$ yields

$$ 
\begin{aligned}
& \mathcal{H}_{\varepsilon}^{-1} \circ \langle \mathcal{R}_{\varepsilon}^{-} \rangle \circ \mathcal{H}_{\varepsilon} = \langle \mathcal{R}_{\varepsilon}^{+} \rangle + \sum_{\ell,m \in \mathbb{N}^{*}} \sum_{(R_1, \dots, R_{\ell-1}) \in (\mathcal{R}_{\varepsilon}^{+})^{\ell-1}} \sum_{\substack{(\tilde{R}_1, \dots, \tilde{R}_m) \in (\mathcal{R}_{\varepsilon}^{+})^{m} \\ (j_2, \dots, j_m) \in \{0,1\}^{m-1}}} R_1 \circ \dots \circ R_{\ell-1} \circ (-\pi_{\varepsilon}^{+}) \circ \tilde{R}_1 \circ (-\pi_{\varepsilon}^{+})^{j_1} \circ \tilde{R}_2 \circ \dots \circ (-\pi_{\varepsilon}^{+})^{j_m} \circ \tilde{R}_m \\
& = \langle \mathcal{R}_{\varepsilon}^{+} \rangle + \left( \sum_{\ell \in \mathbb{N}^{*}} R_1 \circ \dots \circ R_{\ell-1} \right) \left( \sum_{m \in \mathbb{N}^{*}, (\tilde{R}_k) \in (\mathcal{R}_{\varepsilon}^{+})^{m} \\ (j_k) \in \{0,1\}^{m-1}} (-\pi_{\varepsilon}^{+}) \circ \tilde{R}_1 \circ (-\pi_{\varepsilon}^{+})^{j_2} \circ \tilde{R}_2 \circ \dots \circ (-\pi_{\varepsilon}^{+})^{j_m} \circ \tilde{R}_m \right) \\
& = \langle \mathcal{R}_{\varepsilon}^{+} \rangle - \left( \sum_{\ell' \in \mathbb{N}^{*}, (R_k) \in (\mathcal{R}_{\varepsilon}^{+})^{\ell'}} R_1 \circ \dots \circ R_{\ell'} \right) \left( \sum_{m \in \mathbb{N}^{*}, (\tilde{R}_k) \in (\mathcal{R}_{\varepsilon}^{+})^{m}} \pi_{\varepsilon}^{+} \circ \tilde{R}_1 \circ (\text{id} - \pi_{\varepsilon}^{+}) \circ \tilde{R}_2 \circ \dots \circ (\text{id} - \pi_{\varepsilon}^{+}) \circ \tilde{R}_m \right) \\
& = \langle \mathcal{R}_{\varepsilon}^{+} \rangle - \langle \mathcal{R}_{\varepsilon}^{+} \rangle \circ \langle \pi_{\varepsilon}^{+} \mid \mathcal{R}_{\varepsilon}^{+} \rangle
\end{aligned}
\qquad \square $$

**Corollary 2.30:** The matching condition of Definition 2.27 is equivalent to each of these equalities:

$$ \sum_{p \in \mathbb{P}, \ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon \sum_{d \in \mathbb{Z}^* \cap (\mathbb{P}-p)} \sigma_d(\mathbf{u}_{p,\ell})\phi_d = (\text{id} - \langle \pi_{\varepsilon}^{+} | \mathcal{R}_{\varepsilon}^{+} \rangle) \circ \mathcal{H}_{\varepsilon}^{-1} \left( \sum_{p \in \mathbb{P}, \ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon \sum_{d \in \mathbb{Z}^* \cap (p-\mathbb{P})} \sigma_d(S_{p,\ell}^{\infty})\phi_d \right) $$

$$ \sum_{p \in \mathbb{P}, \ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon \sum_{d \in \mathbb{Z}^* \cap (p-\mathbb{P})} \sigma_d(S_{p,\ell}) \phi_d = (\text{id} - \langle \pi_{\varepsilon}^{-} | \mathcal{R}_{\varepsilon}^{-} \rangle) \circ \mathcal{H}_{\varepsilon} \left( \sum_{p \in \mathbb{P}, \ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon \sum_{d \in \mathbb{Z}^* \cap (\mathbb{P}-p)} \sigma_d(\mathbf{u}_{p,\ell}^0) \phi_d \right) $$

**Proof:** Let us prove only the first condition. Combining Proposition 2.26 with Theorem 2.29 shows that the matching condition is equivalent to:

$$ \langle \mathcal{R}_{\varepsilon}^{+} \rangle \left( \sum_{p, \ell} \varepsilon^p \ln^\ell \varepsilon \sum_d \sigma_d(\mathbf{u}_{p, \ell}^0) \phi_d \right) = \langle \mathcal{R}_{\varepsilon}^{+} \rangle \circ (\text{id} - \langle \pi_{\varepsilon}^{+} | \mathcal{R}_{\varepsilon}^{+} \rangle) \circ \mathcal{H}_{\varepsilon}^{-1} \left( \sum_{p, \ell} \varepsilon^p \ln^\ell \varepsilon \sum_d \sigma_d(S_{p, \ell}^{\infty}) \phi_d \right) \quad (2.35) $$
**Lemma 2.29:** The matching condition of Definition 2.28 is equivalent to each of these equalities:

$$ \sum_{p \in \mathbb{P}, \ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon \sum_{d \in \frac{\pi}{\Theta} \mathbb{Z}^* \cap (\mathbb{P}-p)} \sigma_d(\mathbf{u}_{p,\ell}) \phi_d = \pi_\sigma^+ \circ \mathcal{H}_\varepsilon^{-1} \circ \langle \mathcal{R}_\varepsilon^- \rangle \left( \sum_{p \in \mathbb{P}, \ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon \sum_{d \in \frac{\pi}{\Theta} \mathbb{Z}^* \cap (p-\mathbb{P})} \sigma_d(S_{p,\ell}^\infty) \phi_d \right) \quad (2.33) $$

$$ \sum_{p \in \mathbb{P}, \ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon \sum_{d \in \frac{\pi}{\Theta} \mathbb{Z}^* \cap (p-\mathbb{P})} \sigma_d(S_{p,\ell}) \phi_d = \pi_\sigma^- \circ \mathcal{H}_\varepsilon \circ \langle \mathcal{R}_\varepsilon^+ \rangle \left( \sum_{p \in \mathbb{P}, \ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon \sum_{d \in \frac{\pi}{\Theta} \mathbb{Z}^* \cap (\mathbb{P}-p)} \sigma_d(\mathbf{u}_{p,\ell}^0) \phi_d \right) \quad (2.34) $$

**Proof:** Let us prove only (2.33). By Theorem 2.26, the matching condition of Definition 2.28 is equivalent to:

$$ \langle \mathcal{R}_{\varepsilon}^{+} \rangle \left( \sum_{p, \ell} \varepsilon^p \ln^\ell \varepsilon \sum_d \sigma_d(\mathbf{u}_{p, \ell}^0) \phi_d \right) = \mathcal{H}_{\varepsilon}^{-1} \circ \langle \mathcal{R}_{\varepsilon}^{-} \rangle \circ \left( \sum_{p, \ell} \varepsilon^p \ln^\ell \varepsilon \sum_d \sigma_d(S_{p, \ell}^{\infty}) \phi_d \right) \quad (2.35) $$

Moreover, Proposition 2.14 implies: $\forall R \in \mathcal{R}_\varepsilon^+$, $\pi_\sigma^+ \circ R = 0$. Since $\langle \mathcal{R}_\varepsilon^+ \rangle$ is a sum of the identity and non-trivial products of elements of $\mathcal{R}_\varepsilon^+$, we have $\pi_\sigma^+ \circ \langle \mathcal{R}_\varepsilon^+ \rangle = \pi_\sigma^+ \circ \text{id} = \pi_\sigma^+$. Thus applying $\pi_\sigma^+$ to (2.35) gives (2.33).

It remains to show conversely that (2.33) implies (2.35). Let us denote $A_1$ and $A_2$ the two sides of (2.35). One can check that they are solutions of (2.29). The proof of Theorem 2.26 shows that such solutions have the following form

$$ A_i = \langle \mathcal{R}_\varepsilon^+ \rangle \left( \sum_{p, \ell, d} \varepsilon^p \ln^\ell \varepsilon c_{i, p, \ell, d} \phi_d \right), \quad i \in \{1, 2\} $$

for some numbers $c_{i,p,\ell,d}$. In addition, we have $\pi_\sigma^+(A_i) = \sum_{p,\ell,d} \varepsilon^p \ln^\ell \varepsilon c_{i,p,\ell,d} \phi_d$, so $A_i$ is entirely determined by $\pi_\sigma^+(A_i)$. Since (2.33) states that $\pi_\sigma^+(A_1) = \pi_\sigma^+(A_2)$, we have $A_1 = A_2$, i.e. (2.35). $\square$

**Remark:** Since $\mathcal{R}_\varepsilon^+$ and $\mathcal{R}_\varepsilon^-$ only differ on powers of $\varepsilon$ due to the scaling, one could expect that $\mathcal{H}_\varepsilon^{-1} \circ \langle \mathcal{R}_\varepsilon^- \rangle$ is equal to $\langle \mathcal{R}_\varepsilon^+ \rangle \circ \mathcal{H}_\varepsilon^{-1}$. This would simplify a lot (2.33), which would become simply “$\sigma_d(\mathbf{u}_{p,\ell}^0) = \sigma_d(S_{p+d,\ell}^\infty)$” for any $(d,p,\ell)$. However, $R_\Delta$ (resp. $R_{\partial_Y^2}$, $R_N$) picks the particular solution of Equation 2.12 (resp. 2.13, 2.14) whose image by the $\sigma_d$ vanish. This means that they are the solutions in the kernels of $\pi_\sigma^\pm$. It is easy to see that $\mathcal{H}_\varepsilon^{\pm 1}$ does not map Ker $\pi_\sigma^\pm$ to Ker $\pi_\sigma^\mp$ (e.g. consider an element of $\mathcal{A}_\varepsilon^\pm(\Pi)$ equal to $\mathfrak{I}[(\alpha z)\frac{\pi}{\Theta} \log(\alpha z)]$ on $\Omega$). Therefore $\mathcal{H}_\varepsilon^{-1} \circ R_\Delta$ selects other solutions than $R_\Delta \circ \mathcal{H}_\varepsilon^{-1}$, and likewise for $R_{\partial_Y^2}$ and $R_N$, which yields $\mathcal{H}_\varepsilon^{-1} \circ \langle \mathcal{R}_\varepsilon^- \rangle \neq \langle \mathcal{R}_\varepsilon^+ \rangle \circ \mathcal{H}_\varepsilon^{-1}$. This observation explains the complexity of the matching relations in Theorem 2.30.

For any $(p, \ell)$, let us define $\tau_{p,\ell} : \mathcal{A}_\varepsilon^\pm(\Pi) \to \mathcal{A}^\pm(\Pi)$ by: $\forall \varphi = \sum_{p',\ell'} \varepsilon^{p'} \ln^{\ell'} \varepsilon \varphi_{p',\ell'} \in \mathcal{A}_\varepsilon^\pm(\Pi)$, $\tau_{p,\ell}(\varphi) := \varphi_{p,\ell}$. It allows us to define the matching coefficients, that are for any $(d, d', p, \ell) \in (\frac{\pi}{\Theta} \mathbb{Z}^*)^2 \times \mathbb{P} \times \mathbb{N}$ the following complex numbers:

$$ c_{d, d', p, \ell}^{\mathbf{u} \leftarrow S} := \sigma_d \circ \tau_{p, \ell} \circ \mathcal{H}_{\varepsilon}^{-1} \circ \langle \mathcal{R}_{\varepsilon}^{-} \rangle(\phi_{d'}) \quad \text{and} \quad c_{d, d', p, \ell}^{S \leftarrow \mathbf{u}} := \sigma_d \circ \tau_{p, \ell} \circ \mathcal{H}_{\varepsilon} \circ \langle \mathcal{R}_{\varepsilon}^{+} \rangle(\phi_{d'}) \quad (2.36) $$

In addition, for any $a, b \in \mathbb{R}$, we denote $[a, b]_{\mathbb{P},N} := \{c \in \mathbb{R} \mid c - a \in \mathbb{P} \text{ and } b - c \in \mathbb{N}\}$. It is a finite subset of $[a, b]$.

Theorem 2.30 gives equations to concretely build the fields $\mathbf{u}_{p,\ell}$ and $S_{p,\ell}$ so that they match around the corner. It fixes their non-variational, which are determined by $\sigma_d(\mathbf{u}_{p,\ell}^0)$ when $d < 0$ and $\sigma_d(S_{p,\ell}^\infty)$ when $d > 0$ (see Theorems 3.6 and 3.11). It also provides inductive formulas, depending on the fields with smaller $p$. Moreover these formulas have a convolutive structure w.r.t. $p$ and $\ell$.

**Theorem 2.30:** The matching condition of Definition 2.28 is equivalent to the following set of equations:

$$ \left\{ \begin{array}{l} \forall (d, p, \ell) \in (-\frac{\pi}{\Theta} \mathbb{N}^*) \times \mathbb{P} \times \mathbb{N}, \quad \sigma_d(\mathbf{u}_{p,\ell}^0) = \sum_{p' \in [0, p+d]_{\mathbb{P},N}} \sum_{d' \in \frac{\pi}{\Theta} \mathbb{Z}^*} \sum_{\ell'=0}^{\ell} c_{d, d', p-p', \ell-\ell'}^{\mathbf{u} \leftarrow S} \sigma_d(S_{p', \ell'}^\infty) \quad (2.37a) \\ \\ \forall (d, p, \ell) \in \frac{\pi}{\Theta} \mathbb{N}^* \times \mathbb{P} \times \mathbb{N}, \quad \sigma_d(S_{p,\ell}^\infty) = \sum_{p' \in [0, p-d]_{\mathbb{P},N}} \sum_{d' \in \frac{\pi}{\Theta} \mathbb{Z}^*} \sum_{\ell'=0}^{\ell} c_{d, d', p-p', \ell-\ell'}^{S \leftarrow \mathbf{u}} \sigma_d(\mathbf{u}_{p', \ell'}^0) \quad (2.37b) \end{array} \right. $$

$$ \left\{ \begin{array}{l} \forall (d, p, \ell) \in \left(\frac{\pi}{\Theta} \mathbb{N}^* \times \mathbb{P} \times \mathbb{N}\right), \quad \sigma_d(S_{p,\ell}^\infty) = \sum_{p' \in [0, p-d]_{\mathbb{P},N}} \sum_{d' \in \frac{\pi}{\Theta} \mathbb{Z}^*} \sum_{\ell'=0}^{\ell} c_{d, d', p-p', \ell-\ell'}^{S \leftarrow \mathbf{u}} \sigma_d(\mathbf{u}_{p', \ell'}^0) \quad (2.37b) \\ \\ \forall (d, p, \ell) \in \left(-\frac{\pi}{\Theta} \mathbb{N}^* \times \mathbb{P} \times \mathbb{N}\right), \quad \sigma_d(S_{p,\ell}^\infty) = \sum_{p' \in [0, p+d]_{\mathbb{P},N}} \sum_{d' \in \frac{\pi}{\Theta} \mathbb{Z}^*} \sum_{\ell'=0}^{\ell} c_{d, d', p-p', \ell-\ell'}^{S \leftarrow \mathbf{u}} \sigma_d(\mathbf{u}_{p', \ell'}^0) \quad (2.37b) \end{array} \right. $$
It remains to simplify both $\langle \mathcal{R}_{\varepsilon}^{+} \rangle$. Proposition 2.14 implies: $\forall R \in \mathcal{R}_{\varepsilon}^{+}$, $\pi_{\sigma}^{+} \circ R = 0$. Since $\langle \mathcal{R}_{\varepsilon}^{+} \rangle$ is a sum of the identity and non-trivial products of elements of $\mathcal{R}_{\varepsilon}^{+}$, we have $\pi_{\sigma}^{+} \circ \langle \mathcal{R}_{\varepsilon}^{+} \rangle = \pi_{\sigma}^{+} \circ \text{id} = \pi_{\sigma}^{+}$. So applying $\pi_{\sigma}^{+}$ to (2.35) replaces the factors $\langle \mathcal{R}_{\varepsilon}^{+} \rangle$ by $\pi_{\sigma}^{+}$. Now, Im $\pi_{\sigma}^{+}$ is stable by id $- \langle \pi_{\varepsilon}^{+} | \mathcal{R}_{\varepsilon}^{+} \rangle$ and the projector $\pi_{\sigma}^{+}$ is the identity on it, so we can finally simplify $\pi_{\sigma}^{+}$ on both sides of (2.35). $\square$

For any $(p, \ell)$, let us define $\tau_{p,\ell} : \mathcal{A}_{\varepsilon}^{\pm}(\Pi) \to \mathcal{A}^{\pm}(\Pi)$ by: $\forall \varphi = \sum_{p',\ell'} \varepsilon^{p'} \ln^{\ell'} \varepsilon \varphi_{p',\ell'} \in \mathcal{A}_{\varepsilon}^{\pm}(\Pi)$, $\tau_{p,\ell}(\varphi) := \varphi_{p,\ell}$. It allows us to define the matching coefficients, that are for any $(d, d', p, \ell) \in (\frac{\pi}{\Theta}\mathbb{Z}^{*})^{2} \times \mathbb{P} \times \mathbb{N}$ the following complex numbers:

$$c_{d,d',p,\ell}^{\mathbf{u} \leftarrow S} := \sigma_d \circ \tau_{p+d',\ell} \circ (\text{id} - \langle \pi_{\varepsilon}^{+} | \mathcal{R}_{\varepsilon}^{+} \rangle)(\phi_{d'}) \quad \text{and} \quad c_{d,d',p,\ell}^{S \leftarrow \mathbf{u}} := \sigma_d \circ \tau_{p-d',\ell} \circ (\text{id} - \langle \pi_{\varepsilon}^{-} | \mathcal{R}_{\varepsilon}^{-} \rangle)(\phi_{d'})$$

In addition, for any $a, b \in \mathbb{R}$, we denote $[[a, b]]_{\mathbb{P},\mathbb{N}} := \{c \in \mathbb{R} \mid c - a \in \mathbb{P} \text{ and } b - c \in \mathbb{N}\}$. It is a finite subset of $[a, b]$.

Theorem 2.31 gives equations to concretely build the fields $\mathbf{u}_{p,\ell}$ and $S_{p,\ell}$ so that they match around the corner. It fixes their non-variational, which are determined by $\sigma_d(\mathbf{u}_{p,\ell}^0)$ when $d < 0$ and $\sigma_d(S_{p,\ell}^\infty)$ when $d > 0$ (see Theorems 3.6 and 3.11). It also provides inductive formulas, depending on the fields with smaller $p$. Moreover these formulas have a convolutive structure w.r.t. $p$ and $\ell$.

**Theorem 2.31:** The matching condition of Definition 2.27 is equivalent to the following set of equations:

$$\left\{ \begin{array}{l} \forall (d, p, \ell) \in (-\frac{\pi}{\Theta}\mathbb{N}^*) \times \mathbb{P} \times \mathbb{N}, \quad \sigma_d(\mathbf{u}_{p,\ell}^0) = \sum_{p' \in [[0, p+d]]_{\mathbb{P},\mathbb{N}}} \sum_{d' \in \frac{\pi}{\Theta}\mathbb{Z}^*} \sum_{\ell'=0}^{\ell} c_{d,d',p-p',\ell-\ell'} \cdot \sigma_{d'}(S_{p',\ell'}^{\infty}) \quad (2.36a) \\ \\ \forall (d, p, \ell) \in \frac{\pi}{\Theta}\mathbb{N}^* \times \mathbb{P} \times \mathbb{N}, \quad \sigma_d(S_{p,\ell}^{\infty}) = \sum_{p' \in [[0, p-d]]_{\mathbb{P},\mathbb{N}}} \sum_{d' \in \frac{\pi}{\Theta}\mathbb{Z}^*} \sum_{\ell'=0}^{\ell} c_{d,d',p-p',\ell-\ell'} \cdot \sigma_{d'}(\mathbf{u}_{p',\ell'}^0) \quad (2.36b) \end{array} \right.$$

**Proof:** Let $V^u$ be the set of families of complex numbers $(u_{p,d,\ell})_{p \in \mathbb{P}, d \in \frac{\pi}{\Theta}\mathbb{Z}^* \cap (\mathbb{P}-p), \ell \in \mathbb{N}}$ s.t. for any $(p, d)$ only a finite number of the $u_{p,d,\ell}$ are non-zero. We define $V^S$ similarly by replacing "$d \in \mathbb{P}-p$" with "$d \in p-\mathbb{P}$". Let $\sigma(\mathbf{u}) := (\sigma_d(\mathbf{u}_{p,\ell}^0))_{p,d,\ell} \in V^u$ and $\sigma(S) := (\sigma_d(S_{p,\ell}^\infty))_{p,d,\ell} \in V^S$. We define the following linear maps:

$$\mathbf{P} : V^S \to V^u, \quad (S_{p,d,\ell})_{p,d,\ell} \mapsto \left( \sum_{p' \in [[0, p+d]]_{\mathbb{P},\mathbb{N}}} \sum_{d' \in \frac{\pi}{\Theta}\mathbb{Z}^*} \sum_{\ell'=0}^{\ell} c_{d,d',p-p',\ell-\ell'} \cdot S_{p',d',\ell'} \right)_{p,d,\ell}$$

$$\mathbf{Q} : V^u \to V^S, \quad (u_{p,d,\ell})_{p,d,\ell} \mapsto \left( \sum_{p' \in [[0, p-d]]_{\mathbb{P},\mathbb{N}}} \sum_{d' \in \frac{\pi}{\Theta}\mathbb{Z}^*} \sum_{\ell'=0}^{\ell} c_{d,d',p-p',\ell-\ell'}^{S \leftarrow \mathbf{u}} \cdot u_{p',d',\ell'} \right)_{p,d,\ell}$$

These sums have a finite number of terms, so they are well-defined.

**Step 1:** Let $(d, p, \ell) \in \frac{\pi}{\Theta}\mathbb{Z}^* \times \mathbb{P} \times \mathbb{N}$. Applying $\sigma_d \circ \tau_{p,\ell}$ to the first equation of Corollary 2.30 gives:

$$\begin{align*} \sigma_d(\mathbf{u}_{p,\ell}^0) &= \sum_{p' \in \mathbb{P}, \ell' \in \mathbb{N}} \sum_{d' \in \frac{\pi}{\Theta}\mathbb{Z}^* \cap (\mathbb{P}-p')} \sigma_d \circ \tau_{p-p',\ell-\ell'} \circ (\text{id} - \langle \pi_{\varepsilon}^{+} | \mathcal{R}_{\varepsilon}^{+} \rangle) \circ \mathcal{H}_{\varepsilon}^{-1}(\phi_{d'}) \cdot \sigma_{d'}(S_{p',\ell'}^{\infty}) \\ &= \sum_{p' \in \mathbb{P}, \ell' \in \mathbb{N}} \sum_{d' \in \frac{\pi}{\Theta}\mathbb{Z}^* \cap (\mathbb{P}-p')} \sigma_d \circ \tau_{p-p'+d',\ell-\ell'} \circ (\text{id} - \langle \pi_{\varepsilon}^{+} | \mathcal{R}_{\varepsilon}^{+} \rangle)(\phi_{d'}) \cdot \sigma_{d'}(S_{p',\ell'}^{\infty}) \\ &= \sum_{p' \in \mathbb{P}} \sum_{d' \in \frac{\pi}{\Theta}\mathbb{Z}^*, p'+d' \in \mathbb{P}} \sum_{\ell' \in \mathbb{N}} c_{d,d',p-p',\ell-\ell'}^{\mathbf{u} \leftarrow S} \cdot \sigma_{d'}(S_{p',\ell'}^{\infty}). \end{align*}$$

Moreover, for any $R \in \mathcal{R}_{\varepsilon}^{+}$, we have $\deg_{\varepsilon} R \in \mathbb{N}$ and $\deg_{\varepsilon} R + \deg_{\mathcal{A}} R \in \mathbb{N}$ (see Figure 9). So for $c_{d,d',p-p',\ell-\ell'}^{\mathbf{u} \leftarrow S}$ to be non-zero, we need $p - p' + d' \in \mathbb{N}$ and $(p - p' + d') + d - d' \in \mathbb{N}$. In addition, $\ell - \ell' \in \mathbb{N}$. Therefore, the matching condition in equivalent to $\sigma(\mathbf{u}) = \mathbf{P} \sigma(S)$. Similarly, it is equivalent