**Proof:** Let $V^u$ be the set of families of complex numbers $(u_{p,d,l})_{p \in \mathbb{P}, d \in \frac{\pi}{\Theta}Z^* \cap (\mathbb{P}-p), l \in \mathbb{N}}$ s.t. for any $(p, d)$ only a finite number of the $u_{p,d,l}$ 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:

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

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

**Step 1:** Let $(d, p, \ell) \in \frac{\pi}{\Theta}Z^* \times \mathbb{P} \times \mathbb{N}$. Applying $\sigma_d \circ \tau_{p,\ell}$ to the first equation of Lemma 2.29 gives:

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

because $\sigma_d \circ \tau_{p-p',\ell-\ell'} \circ \pi_\sigma^+ = \sigma_d \circ \tau_{p-p',\ell-\ell'}$. Moreover, for any $R \in \mathcal{R}_\epsilon^-$, we have $\deg_\epsilon R \in \mathbb{N}$ and $\deg_\epsilon R - \deg_\Lambda R \in \mathbb{N}$ (see Figure 9). So for $c_{d,d',p-p',\ell-\ell'}^{u \leftarrow S}$ 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 to $\sigma(S) = \mathbf{Q}\sigma(\mathbf{u})$.

**Step 2:** We define the subspaces $V_\pm^u = \{u \in V^u | \forall (p, d, \ell), u_{p,d,\ell} \neq 0 \Rightarrow \pm d > 0\}$ and likewise $V_\pm^S$. So $\bar{V}^u = \bar{V}_+^u \oplus \bar{V}_-^u$ and similarly for $V^S$. Let $\sigma(\mathbf{u})_\pm$ be the components of $\sigma(\mathbf{u})$ on $V_\pm^u$, and similarly for $\sigma(S)_\pm$. We decompose $\mathbf{P}$ and $\mathbf{Q}$ on those subspaces, which gives in block matrix notation:

$$ \mathbf{P} = \left( \underbrace{\left( \begin{array}{c} \mathbf{P}_+ \\ \mathbf{P}_- \end{array} \right)}_{\mathbf{V}^S} \right) \left\{ \begin{array}{c} \mathbf{V}_+^u \\ \mathbf{V}_-^u \end{array} \right\} \quad \text{and} \quad \mathbf{Q} = \left( \underbrace{\left( \begin{array}{c} \mathbf{Q}_+ \\ \mathbf{Q}_- \end{array} \right)}_{\mathbf{V}_+^u} \right) = \left( \underbrace{\left( \begin{array}{c} \mathbf{Q}_+^+ \\ \mathbf{Q}_-^+ \end{array} \right)}_{\mathbf{V}_-^u} \right) \left\{ \begin{array}{c} \mathbf{V}_+^S \\ \mathbf{V}_-^S \end{array} \right\} $$

The present theorem rewrites as:

$$ \left\{ \begin{aligned} \sigma(\mathbf{u})_- &= \mathbf{P}_- \sigma(S) \\ \sigma(S)_+ &= \mathbf{Q}_+ \sigma(\mathbf{u}) \end{aligned} \right. \qquad (2.38) $$

By Step 1, this is clearly a necessary condition for the matching. It remains to prove that it is sufficient. Let us show that (2.38) implies one of the two conditions of Step 1, e.g. $\sigma(\mathbf{u}) = \mathbf{P}\sigma(S)$.

Note that by Step 1 we have for any $\mathbf{u} \in V^u$ and $\mathbf{S} \in V^S$: $\mathbf{u} = \mathbf{P}\mathbf{S} \Leftrightarrow \mathbf{S} = \mathbf{Q}\mathbf{u}$. So $\mathbf{P}$ and $\mathbf{Q}$ are inverses of each other. Hence $\mathbf{Q}(\sigma(\mathbf{u}) - \mathbf{P}\sigma(S)) = \mathbf{Q}\sigma(\mathbf{u}) - \sigma(S)$. Projecting this onto $V_+^S$ we deduce:

$$ \mathbf{Q}_+^\dagger (\sigma(\mathbf{u})_+ - \mathbf{P}_+ \sigma(S)) + \mathbf{Q}_-^\dagger (\sigma(\mathbf{u})_- - \mathbf{P}_- \sigma(S)) = \underbrace{\mathbf{Q}_+ \sigma(\mathbf{u}) - \sigma(S)_+}_{=0} + \underbrace{\mathbf{Q}_- \sigma(\mathbf{u}) - \sigma(S)_-}_{=0}. $$

Let us show that $\mathbf{Q}_+^\dagger$ is injective.

Let $\mathbf{u} \in V_+^u \setminus \{0\}$ and let us show that $\mathbf{Q}_+^\dagger \mathbf{u} \neq 0$. Let $(p, d, \ell)$ be the smallest triplet for lexicographic order such that $u_{p,d,\ell} \neq 0$. The term of $\mathbf{Q}_+^\dagger \mathbf{u}$ of index $(p+d, d, \ell)$ is:

$$ \sum_{p' \in [0,p]_{\mathbb{P},N}} \sum_{d' \in \frac{\pi}{\Theta}N^*} \sum_{\ell'=0}^{\ell} c_{d,d',p+d-p',\ell-\ell'}^{S \leftarrow u} \cdot \mathbf{u}_{p',d',\ell'} = c_{d,d,d,0}^{S \leftarrow u} \cdot \mathbf{u}_{p,d,\ell}. $$

We claim that it is non-zero because $c_{d,d,d,0}^{S \leftarrow u} = 1$. Indeed, in the sum

$$ \langle \mathcal{R}_\epsilon^- \rangle := \sum_{n=0}^{\infty} \sum_{(R_1, \dots, R_n) \in (\mathcal{R}_\epsilon^-)^n} R_1 \circ R_2 \circ \dots \circ R_n $$

the only term of degree $(0,0)$ w.r.t. $\mathcal{A}$ and $\epsilon$ is the identity, which appears for $n=0$. So the component of $\mathcal{H}_\epsilon \circ \langle \mathcal{R}_\epsilon^- \rangle(\phi_d)$ in $\epsilon^d \mathcal{A}_d(\Pi)[\ln \epsilon]$ is $\epsilon^d \phi_d$. Thus $c_{d,d,d,0}^{S \leftarrow u} = \sigma_d \circ \tau_{d,0} \circ \mathcal{H}_\epsilon \circ \langle \mathcal{R}_\epsilon^- \rangle(\phi_d) = \sigma_d(\phi_d) = 1$. So $\mathbf{Q}_+^\dagger \mathbf{u} \neq 0$.
to $\sigma(S) = Q\sigma(\mathbf{u})$.

Step 2: We define the subspaces $V_{\pm}^{u} = \{\mathbf{u} \in V^{u} | \forall (p, d, \ell), \mathbf{u}_{p,d,\ell} \neq 0 \Rightarrow \pm d > 0\}$ and likewise $V_{\pm}^{s}$. So $V^{u} = V_{+}^{u} \oplus V_{-}^{u}$ and similarly for $V^{s}$. Let $\sigma(\mathbf{u})_{\pm}$ be the components of $\sigma(\mathbf{u})$ on $V_{\pm}^{u}$, and similarly for $\sigma(S)_{\pm}$. We decompose $\mathbb{P}$ and $\mathbb{Q}$ on those subspaces, which gives in block matrix notation:

$$ \mathbf{P} = \left( \begin{array}{c} \mathbf{P}_{+} \\ \mathbf{P}_{-} \\ \hline \mathbf{V}^{s} \end{array} \right) \left\{ \begin{array}{c} \mathbf{V}_{+}^{u} \\ \mathbf{V}_{-}^{u} \\ \hline \mathbf{V}^{s} \end{array} \right\} \quad \text{and} \quad \mathbb{Q} = \left( \begin{array}{c} \mathbb{Q}_{+} \\ \mathbb{Q}_{-} \\ \hline \mathbb{V}_{+}^{u} \\ \mathbb{V}_{-}^{u} \end{array} \right) = \left( \begin{array}{c|c} \mathbb{Q}_{+}^{+} & \mathbb{Q}_{+}^{-} \\ \mathbb{Q}_{-}^{+} & \mathbb{Q}_{-}^{-} \\ \hline \mathbb{V}_{+}^{u} & \mathbb{V}_{-}^{u} \end{array} \right) \left\{ \begin{array}{c} \mathbb{V}_{+}^{s} \\ \mathbb{V}_{-}^{s} \\ \hline \mathbb{V}_{+}^{u} \\ \mathbb{V}_{-}^{u} \end{array} \right\} $$

The present theorem rewrites as:

$$ \left\{ \begin{array}{l} \sigma(\mathbf{u})_{-} = \mathbb{P}_{-} \sigma(S) \\ \sigma(S)_{+} = \mathbb{Q}_{+} \sigma(\mathbf{u}) \end{array} \right. \qquad (2.37) $$

By Step 1, this is clearly a necessary condition for the matching. It remains to prove that it is sufficient. Let us show that (2.37) implies one of the two conditions of Step 1, e.g. $\sigma(\mathbf{u}) = \mathbb{P}\sigma(S)$.

Note that by Step 1 we have for any $\mathbf{u} \in V^{u}$ and $\mathbb{S} \in V^{s}$: $\mathbf{u} = \mathbb{P}\mathbb{S} \iff \mathbb{S} = \mathbb{Q}\mathbf{u}$. So $\mathbb{P}$ and $\mathbb{Q}$ are inverses of each other. Hence $\mathbb{Q}(\sigma(\mathbf{u}) - \mathbb{P}\sigma(S)) = \mathbb{Q}\sigma(\mathbf{u}) - \sigma(S)$. Projecting this onto $V_{+}^{s}$ we deduce:

$$ \mathbb{Q}_{+}^{s}(\sigma(\mathbf{u})_{+} - \mathbb{P}_{+} \sigma(S)) + \mathbb{Q}_{-}^{s}(\sigma(\mathbf{u})_{-} - \mathbb{P}_{-} \sigma(S)) = \mathbb{Q}_{+} \sigma(\mathbf{u}) - \sigma(S)_{+}. $$

Let us show that $\mathbb{Q}_{+}^{s}$ is injective.

Let $\mathbf{u} \in V_{+}^{u} \setminus \{0\}$ and let us show that $\mathbb{Q}_{+}^{s}\mathbf{u} \neq 0$. Let $(p, d, \ell)$ be the smallest triplet for lexicographic order such that $\mathbf{u}_{p,d,\ell} \neq 0$. The term of $\mathbb{Q}_{+}^{s}\mathbf{u}$ of index $(p+d, d, \ell)$ is:

$$ \sum_{p' \in [0, p]_{\mathbb{R}, N}} \sum_{d' \in \frac{\pi}{\Theta} \mathbb{N}^{*}} \sum_{p'+d' \in [0, p+d]_{\mathbb{R}, N}}^{\ell} c_{d,d',p+d-p',\ell-\ell'}^{S \leftarrow \mathbf{u}} \cdot \mathbf{u}_{p',d',\ell'} = c_{d,d,d,0}^{S \leftarrow \mathbf{u}} \cdot \mathbf{u}_{p,d,\ell}. $$

We claim that it is non-zero because $c_{d,d,d,0}^{S \leftarrow \mathbf{u}} = 1$. Indeed, in the sum

$$ \langle \pi_{\varepsilon}^{+} | \mathcal{R}_{\varepsilon}^{+} \rangle := \sum_{n=1}^{\infty} \sum_{(R_1, \dots, R_n) \in (\mathcal{R}_{\varepsilon}^{+})^n} \pi_{\varepsilon}^{+} \circ R_1 \circ (\mathrm{id} - \pi_{\varepsilon}^{+}) \circ R_2 \circ \dots \circ (\mathrm{id} - \pi_{\varepsilon}^{+}) \circ R_n $$

there is no term of degree $(0,0)$ w.r.t. $\mathcal{A}$ and $\varepsilon$. So the component of $\langle \pi_{\varepsilon}^{+} | \mathcal{R}_{\varepsilon}^{+} \rangle(\phi_d)$ in $\varepsilon^0 \mathcal{A}_d(\Pi)[\ln \varepsilon]$ is zero. Thus $c_{d,d,d,0}^{S \leftarrow \mathbf{u}} = \sigma_d \circ \tau_{0,0} \circ (\mathrm{id} - \langle \pi_{\varepsilon}^{+} | \mathcal{R}_{\varepsilon}^{+} \rangle)(\phi_d) = \sigma_d(\phi_d) = 1$. So $\mathbb{Q}_{+}^{s}\mathbf{u} \neq 0$.

We have proven that $\sigma(\mathbf{u})_{+} = \mathbb{P}_{+} \sigma(S)$. Given that $\sigma(\mathbf{u})_{-} = \mathbb{P}_{-} \sigma(S)$, we deduce $\sigma(\mathbf{u}) = \mathbb{P}\sigma(S)$. $\square$

### Remarks 2.32:

* Thanks to the tools of Section 2.3, we can compute exactly and very quickly the coefficients $c_{d,d',p,\ell}^{u \leftarrow s}$ and $c_{d,d',p,\ell}^{s \leftarrow u}$. Moreover, these coefficients depend only on $\Theta$, $\omega$ and $(\mu_0, \mu_1, \rho_0, \rho_1)$, but not on $\Omega_1$ nor precisely on the functions $\mu$ and $\rho$.

* In the sums of Theorem 2.31, the indexes $d$ and $d'$ satisfy $d - d' \in \mathbb{Z} \cap \frac{\pi}{\Theta}\mathbb{Z}$. Indeed, on the one hand $d, d' \in \frac{\pi}{\Theta}\mathbb{Z}^*$. On the other we have in (2.36a) that $p' \in [0, p+d]_{\mathbb{P},N}$ and $p' - d' \in [0, p]_{\mathbb{P},N}$, so $d - d' = (p+d - p') - (p - (p' - d')) \in \mathbb{N} - \mathbb{N} \subset \mathbb{Z}$ (and likewise in (2.36b)). Note that the set $\mathbb{Z} \cap \frac{\pi}{\Theta}\mathbb{Z}$ can be very small. If $\Theta \in \pi\mathbb{Q}$, then $\mathbb{Z} \cap \frac{\pi}{\Theta}\mathbb{Z} = b\mathbb{Z}$ where $\Theta = \pi \frac{a}{b}$ with $(a, b) \in \mathbb{N} \times \mathbb{N}^*$ and $\gcd(a, b) = 1$. Otherwise, $\mathbb{Z} \cap \frac{\pi}{\Theta}\mathbb{Z} = \{0\}$.

## 3 Construction of the asymptotic expansion

Equations 1.3–1.5 and Theorem 2.31 give the equations that the fields $u_{p,\ell}$, $U_{p,\ell}$ and $S_{p,\ell}$ must satisfy. In this section, we will build these fields according to those conditions. First of all, let us express the layer fields with the far fields, so that only two types of fields remain to build. Let $(\mathcal{U}_n) \in \mathbb{R}[Y]^N$ be the unique sequence of polynomials s.t. for any $n \in \mathbb{N}^*$:

$$ \left\{ \begin{array}{l} \mathcal{U}_0'' = 0 \\ \mathcal{U}_0'(0) = \frac{\mu_0}{\mu_1} \\ \mathcal{U}_0(-1) = 0 \end{array} \right. \quad \text{and} \quad \left\{ \begin{array}{l} \mathcal{U}_n'' = -\mathcal{U}_{n-1} \\ \mathcal{U}_n'(0) = 0 \\ \mathcal{U}_n(-1) = 0 \end{array} \right. \qquad (3.1) $$
We have proven that $\sigma(\mathbf{u})_+ = \mathbb{P}_+ \sigma(S)$. Given that $\sigma(\mathbf{u})_- = \mathbb{P}_- \sigma(S)$, we deduce $\sigma(\mathbf{u}) = \mathbb{P} \sigma(S)$. $\square$

**Remarks 2.31:**

* Thanks to the tools of Section 2.3, we can compute exactly and very quickly the coefficients $c_{d,d',p,\ell}^{u \leftarrow S}$ and $c_{d,d',p,\ell}^{S \leftarrow u}$. Moreover, these coefficients depend only on $\Theta$, $\omega$ and $(\mu_0, \mu_1, \rho_0, \rho_1)$, but not on $\Omega_1$ nor precisely on the functions $\mu$ and $\rho$.

* In the sums of Theorem 2.30, the indexes $d$ and $d'$ satisfy $d - d' \in \mathbb{Z} \cap \frac{\pi}{\Theta}\mathbb{Z}$. Indeed, on the one hand $d, d' \in \frac{\pi}{\Theta}\mathbb{Z}^*$. On the other we have in (2.37a) that $p' \in [0, p + d]_{\mathbb{P},\mathbb{N}}$ and $p' - d' \in [0, p]_{\mathbb{P},\mathbb{N}}$, so $d - d' = (p + d - p') - (p - (p' - d')) \in \mathbb{N} - \mathbb{N} \subset \mathbb{Z}$ (and likewise in (2.37b)). Note that the set $\mathbb{Z} \cap \frac{\pi}{\Theta}\mathbb{Z}$ can be very small. If $\Theta \in \pi\mathbb{Q}$, then $\mathbb{Z} \cap \frac{\pi}{\Theta}\mathbb{Z} = b\mathbb{Z}$ where $\Theta = \pi \frac{a}{b}$ with $(a, b) \in \mathbb{N} \times \mathbb{N}^*$ and $\gcd(a, b) = 1$. Otherwise, $\mathbb{Z} \cap \frac{\pi}{\Theta}\mathbb{Z} = \{0\}$.

**3 Construction of the asymptotic expansion**

Equations 1.3–1.5 and Theorem 2.30 give the equations that the fields $u_{p,\ell}$, $U_{p,\ell}$ and $S_{p,\ell}$ must satisfy. In this section, we will build these fields according to those conditions. First of all, let us express the layer fields with the far fields, so that only two types of fields remain to build. Let $(\mathcal{U}_n) \in \mathbb{R}[Y]^\mathbb{N}$ be the unique sequence of polynomials s.t. for any $n \in \mathbb{N}^*$:

$$ \left\{ \begin{array}{l} \mathcal{U}_0'' = 0 \\ \mathcal{U}_0'(0) = \frac{\mu_0}{\mu_1} \\ \mathcal{U}_0(-1) = 0 \end{array} \right. \quad \text{and} \quad \left\{ \begin{array}{l} \mathcal{U}_n'' = -\mathcal{U}_{n-1} \\ \mathcal{U}_n'(0) = 0 \\ \mathcal{U}_n(-1) = 0 \end{array} \right. \qquad (3.1) $$

**Lemma 3.1: expression of the layer fields**

Let us assume that the fields $u_{p,\ell}$ and $U_{p,\ell}$ are regular enough (we will check later that they are). (1.4) implies for any $(p, \ell) \in \mathbb{P} \times \mathbb{N}$ and $(x, Y) \in \Lambda$:

$$ U_{p,\ell}(x, Y) = \sum_{n=0}^{\infty} (\partial_x^2 + k_1^2)^n \partial_y u_{p-1-2n,\ell}(x, 0) \cdot \mathcal{U}_n(Y). \qquad (3.2) $$

where this sum has a finite number of non-zero terms by the convention: $\forall p \in \mathbb{R} \setminus \mathbb{P}, \forall \ell \in \mathbb{N}, u_{p,\ell} := 0$.

**Proof:** There exists an increasing sequence $(p_m)$ s.t. $\mathbb{P} = \{p_m \mid m \in \mathbb{N}\}$. So we can prove the result by induction on $p \in \mathbb{P}$. For $p = 0$, (1.4) states:

$$ \left\{ \begin{array}{ll} \partial_Y^2 U_{0,\ell} = 0 & \text{in } \Lambda \\ \partial_Y U_{0,\ell} = 0 & \text{on } \Gamma \\ U_{0,\ell} = 0 & \text{on } \Sigma_\Lambda \end{array} \right. $$

so $U_{0,\ell} = 0$ for any $\ell$. It is coherent (3.2) (which is a sum of zeros in this case).

Next, for the inductive step, we assume that (3.2) holds for ranks smaller that $p$. (1.4) gives:

$$ \left\{ \begin{array}{l} \partial_Y^2 U_{p,\ell} = -(\partial_x^2 + k_1^2) U_{p-2,\ell} = -\sum_{n=0}^{\infty} (\partial_x^2 + k_1^2)^{n+1} \partial_y u_{(p-2)-1-2n,\ell}(x, 0) \cdot \mathcal{U}_n(Y) \quad \text{in } \Lambda \\ \partial_Y U_{p,\ell} = \frac{\mu_0}{\mu_1} \partial_y u_{p-1,\ell} \quad \text{on } \Gamma \\ U_{p,\ell} = 0 \quad \text{on } \Sigma_\Lambda \end{array} \right. $$

It is easy to see that $\sum_{n=0}^{\infty} (\partial_x^2 + k_1^2)^n \partial_y u_{p-1-2n,\ell}(x, 0) \cdot \mathcal{U}_n(Y)$ is the only solution of this. $\square$

We saw in Section 2 that far and corner fields possess singularities when $r \to 0$, resp. $r \to \infty$. So the usual variational frameworks are not sufficient to build these fields and we need to design new frameworks. It is done in Sections 3.1 and 3.2. In both sections we start by introducing the natural space $H_{\text{var}}$ in which an ad hoc variational problem is well-posed. Then we define a bigger space
**Lemma 3.1: expression of the layer fields**

Let us assume that the fields $u_{p,\ell}$ and $U_{p,\ell}$ are regular enough (we will check later that they are). (1.4) implies for any $(p, \ell) \in \mathbb{P} \times \mathbb{N}$ and $(x, Y) \in \Lambda$:

$$ U_{p,\ell}(x, Y) = \sum_{n=0}^{\infty} (\partial_x^2 + k_1^2)^n \partial_y u_{p-1-2n,\ell}(x, 0) \cdot \mathcal{U}_n(Y). \quad (3.2) $$

where this sum has a finite number of non-zero terms by the convention: $\forall p \in \mathbb{R} \setminus \mathbb{P}, \forall \ell \in \mathbb{N}, u_{p,\ell} := 0$.

**Proof:** There exists an increasing sequence $(p_m)$ s.t. $\mathbb{P} = \{p_m | m \in \mathbb{N}\}$. So we can prove the result by induction on $p \in \mathbb{P}$. For $p = 0$, (1.4) states:

$$ \left\{ \begin{array}{ll} \partial_Y^2 U_{0,\ell} = 0 & \text{in } \Lambda \\ \partial_Y U_{0,\ell} = 0 & \text{on } \Gamma \\ U_{0,\ell} = 0 & \text{on } \Sigma_{\Lambda} \end{array} \right. $$

so $U_{0,\ell} = 0$ for any $\ell$. It is coherent (3.2) (which is a sum of zeros in this case).

Next, for the inductive step, we assume that (3.2) holds for ranks smaller that $p$. (1.4) gives:

$$ \left\{ \begin{array}{ll} \partial_Y^2 U_{p,\ell} = -(\partial_x^2 + k_1^2)U_{p-2,\ell} = & -\sum_{n=0}^{\infty} (\partial_x^2 + k_1^2)^{n+1} \partial_y u_{(p-2)-1-2n,\ell}(x, 0) \cdot \mathcal{U}_n(Y) \\ & = -\sum_{n=1}^{\infty} (\partial_x^2 + k_1^2)^n \partial_y u_{p-1-2n,\ell}(x, 0) \cdot \mathcal{U}_{n-1}(Y) & \text{in } \Lambda \\ \partial_Y U_{p,\ell} = \frac{\mu_0}{\mu_1} \partial_y u_{p-1,\ell} & \text{on } \Gamma \\ U_{p,\ell} = 0 & \text{on } \Sigma_{\Lambda} \end{array} \right. $$

It is easy to see that $\sum_{n=0}^{\infty} (\partial_x^2 + k_1^2)^n \partial_y u_{p-1-2n,\ell}(x, 0) \cdot \mathcal{U}_n(Y)$ is the only solution of this. $\square$

We saw in Section 2 that far and corner fields possess singularities when $r \to 0$, resp. $r \to \infty$. So the usual variational frameworks are not sufficient to build these fields and we need to design new frameworks. It is done in Sections 3.1 and 3.2. In both sections we start by introducing the natural space $H_{\text{var}}$ in which an ad hoc variational problem is well-posed. Then we define a bigger space $H := H_{\text{var}} + \chi A(D) = \{u + \chi\varphi | u \in H_{\text{var}}, \varphi \in A(D)\}$ that contains the singularities, where $\chi$ is a $C^\infty$ truncation function in the vicinity of 0 (for $u_{p,\ell}$) or infinity (for $S_{p,\ell}$), and $D \in \{\Omega, \Pi\}$. Next we determine the elements of $H_{\text{var}} \cap \chi A(D)$, which allows us to define on $H$ the linear forms $\sigma_d$ associated to the singularities. Finally we show that some model problems are well-posed in $H$.

Before we start, the following lemma is a tool to estimate the behavior at 0 and $\infty$ of functions of $\mathcal{A}$.

**Lemma 3.2:** Let $a < b$ in $\mathbb{R}$, $n \in \mathbb{N}^*$, $(d_i, \ell_i)_{i \in [1,n]}$ be $n$ distinct elements of $\mathbb{R} \times \mathbb{N}$, $(f_i)_{i \in [1,n]} \in (C([a,b], \mathbb{C}) \setminus \{0\})^n$ and:

$$ \varphi : (r, \theta) \in \mathbb{R}_+^* \times [a, b] \mapsto \sum_{i=1}^{n} r^{d_i} \ln^{\ell_i} r \cdot f_i(\theta). $$

Then there is an interval $I \subset [a, b]$ with non-empty interior, $c \in \mathbb{R}_+^*$ and $r_1, r_2 \in \mathbb{R}$ such that:

$$ \forall r \in (0, r_1), \forall \theta \in I, \quad |\varphi(r, \theta)| > c r^{\min_j d_j} \quad \text{and} \quad \forall r \in (r_2, \infty), \forall \theta \in I, \quad |\varphi(r, \theta)| > c r^{\max_j d_j}. $$

**Proof:** Let $j \in [1, n]$ be s.t. $(d_j, \ell_j)$ is maximal for the lexicographic order. Let $I \subset [a, b]$ be a non trivial interval on which $|f_j|$ is greater than a positive constant. Since $\sum_{i \neq j} r^{d_i} \ln^{\ell_i} r \cdot f_i(\theta) = o(r^{d_j} \ln^{\ell_j} r)$, we have when $r \to \infty$ and $\theta \in I: |\varphi(r, \theta)| \gtrsim r^{d_j} \ln^{\ell_j} r \gtrsim r^{d_j}$. And we can similarly treat the vicinity of 0. $\square$

We can apply Lemma 3.2 to any $\varphi \in \mathcal{A}_d(\Omega)$ with $(a, b) := (0, \Theta)$, or to any $\varphi \in \mathcal{A}_d(\Lambda)$ with $(a, b) := (-1, 0)$ (replacing the variables $(r, \theta)$ by $(x, Y)$). We can also apply it to $\partial_r\varphi$ and $\partial_\theta\varphi$ when $\varphi \in \mathcal{A}_d(\Omega)$ and to $\partial_x\varphi$ and $\partial_Y\varphi$ when $\varphi \in \mathcal{A}_d(\Lambda)$.
$H := H_{\text{var}} + \chi \mathcal{A}(D) = \{u + \chi\varphi \mid u \in H_{\text{var}}, \varphi \in \mathcal{A}(D)\}$ that contains the singularities, where $\chi$ is a $\mathcal{C}^{\infty}$ truncation function in the vicinity of 0 (for $u_{p,l}$) or infinity (for $S_{p,l}$), and $D \in \{\Omega, \Pi\}$. Next we determine the elements of $H_{\text{var}} \cap \chi \mathcal{A}(D)$, which allows us to define on $H$ the linear forms $\sigma_d$ associated to the singularities. Finally we show that some model problems are well-posed in $H$.

Before we start, the following lemma is a tool to estimate the behavior at 0 and $\infty$ of functions of $\mathcal{A}$.

**Lemma 3.2:** Let $a < b$ in $\mathbb{R}$, $n \in \mathbb{N}^*$, $(d_i, \ell_i)_{i \in [1, n]}$ be $n$ distinct elements of $\mathbb{R} \times \mathbb{N}$, $(f_i)_{i \in [1, n]} \in (\mathcal{C}([a, b], \mathbb{C}) \setminus \{0\})^n$ and:

$$ \varphi : (r, \theta) \in \mathbb{R}_+^* \times [a, b] \mapsto \sum_{i=1}^{n} r^{d_i} \ln^{\ell_i} r \cdot f_i(\theta). $$

Then there is an interval $I \subset [a, b]$ with non-empty interior, $c \in \mathbb{R}_+^*$ and $r_1, r_2 \in \mathbb{R}$ such that:

$$ \forall r \in (0, r_1), \forall \theta \in I, \quad |\varphi(r, \theta)| > c r^{\min_j d_j} \quad \text{and} \quad \forall r \in (r_2, \infty), \forall \theta \in I, \quad |\varphi(r, \theta)| > c r^{\max_j d_j}. $$

*Proof:* Let $j \in [1, n]$ be s.t. $(d_j, \ell_j)$ is maximal for the lexicographic order. Let $I \subset [a, b]$ be a non trivial interval on which $|f_j|$ is greater than a positive constant. Since $\sum_{i \neq j} r^{d_i} \ln^{\ell_i} r \cdot f_i(\theta) = o(r^{d_j} \ln^{\ell_j} r)$, we have when $r \to \infty$ and $\theta \in I$: $|\varphi(r, \theta)| \gtrsim r^{d_j} \ln^{\ell_j} r \gtrsim r^{d_j}$. And we can similarly treat the vicinity of 0. $\square$

We can apply Lemma 3.2 to any $\varphi \in \mathcal{A}_d(\Omega)$ with $(a, b) := (0, \Theta)$, or to any $\varphi \in \mathcal{A}_d(\Lambda)$ with $(a, b) := (-1, 0)$ (replacing the variables $(r, \theta)$ by $(x, Y)$). We can also apply it to $\partial_r\varphi$ and $\partial_\theta\varphi$ when $\varphi \in \mathcal{A}_d(\Omega)$ and to $\partial_x\varphi$ and $\partial_y\varphi$ when $\varphi \in \mathcal{A}_d(\Lambda)$.

**Definition:** For any $D \in \{\Pi, \Omega, \Gamma, \Lambda\}$ and $\varphi \in \mathcal{A}(D)$, we denote $\text{deg}_{\min} \varphi := \sup\{d \in \mathbb{R} \mid \varphi \in \sum_{q \ge d} \mathcal{A}_q(D)\}$ and $\text{deg}_{\max} \varphi := \inf\{d \in \mathbb{R} \mid \varphi \in \sum_{q \le d} \mathcal{A}_q(D)\}$.

## 3.1 Existence and uniqueness for far fields-like problems

We denote $H_{00}^{1/2}(\Gamma)$ the set of functions of $H^{1/2}(\Gamma)$ whose extension by 0 to $\partial\Omega = \Gamma \cup \{(0, 0)\} \cup \Sigma_{\Omega}$ is in $H^{1/2}(\partial\Omega)$. Using Lax-Milgram theorem, it is easy to prove the following lemma.

**Lemma 3.3:** the Helmholtz problem in $H^1(\Omega)$

Let $f \in (H^1(\Omega))'$ and $g \in H_{00}^{1/2}(\Gamma)$. The following system has a unique solution in $H^1(\Omega)$.

$$ \left\{ \begin{array}{ll} \mu_0 \Delta u + \omega^2 \rho_0 u = f & \text{in } \Omega \\ u = g & \text{on } \Gamma \\ u = 0 & \text{on } \Sigma_{\Omega} \end{array} \right. $$

Let $\chi_0$ be a radial function of $\mathcal{C}^{\infty}(\mathbb{R}^2)$ equal to 1 in the vicinity of 0 and to 0 in the vicinity of infinity. The appropriate space to build the far fields is $H^1(\Omega) + \chi_0 \mathcal{A}(\Omega)$. One can check that it does not depend on the choice of $\chi_0$.

**Lemma 3.4:** $H^1(\Omega) \cap \chi_0 \mathcal{A}(\Omega) = \chi_0 \sum_{d>0} \mathcal{A}_d(\Omega)$.

*Proof:* The inclusion $\supseteq$ is easy to check, so we focus on $\subset$. Let $\varphi \in \mathcal{A}(\Omega) \setminus \{0\}$ be s.t. $\chi_0\varphi \in H^1(\Omega)$. There is $d \in \mathbb{R}$, $\varphi_1 \in \mathcal{A}_d(\Omega) \setminus \{0\}$ and $\varphi_2 \in \sum_{q>d} \mathcal{A}_q(\Omega)$ s.t. $\varphi = \varphi_1 + \varphi_2$. If $\partial_\theta\varphi_1$ were null everywhere, then would so too $\varphi_1$ because $\varphi_{1|\Sigma_\Omega} = 0$. But we assumed the contrary, so $\partial_\theta\varphi_1 \neq 0$. Thus Lemma 3.2 implies that there is a non-trivial interval $I \subset [0, \Theta]$ s.t. $|\partial_\theta\varphi_1(r, \theta)| \gtrsim r^d$ when $r \to 0$ and $\theta \in I$. Finally $r^{-1}\partial_\theta(\chi_0\varphi) \in L^2(\Omega)$ implies that $d > 0$. $\square$
**Definition:** For any $D \in \{\Pi, \Omega, \Gamma, \Lambda\}$ and $\varphi \in \mathcal{A}(D)$, we denote $\deg_{\min} \varphi := \sup\{d \in \mathbb{R} \mid \varphi \in \sum_{q \ge d} \mathcal{A}_q(D)\}$ and $\deg_{\max} \varphi := \inf\{d \in \mathbb{R} \mid \varphi \in \sum_{q \le d} \mathcal{A}_q(D)\}$.

## 3.1 Existence and uniqueness for far fields-like problems

We denote $H_{00}^{1/2}(\Gamma)$ the set of functions of $H^{1/2}(\Gamma)$ whose extension by 0 to $\partial\Omega = \Gamma \cup \{(0,0)\} \cup \Sigma_{\Omega}$ is in $H^{1/2}(\partial\Omega)$. Using Lax-Milgram theorem, it is easy to prove the following lemma.

**Lemma 3.3: the Helmholtz problem in $H^1(\Omega)$**

Let $f \in (H^1(\Omega))'$ and $g \in H_{00}^{1/2}(\Gamma)$. The following system has a unique solution in $H^1(\Omega)$.

$$ \left\{ \begin{array}{ll} \mu_0 \Delta u + \omega^2 \rho_0 u = f & \text{in } \Omega \\ u = g & \text{on } \Gamma \\ u = 0 & \text{on } \Sigma_{\Omega} \end{array} \right. $$

Let $\chi_0$ be a radial function of $C^{\infty}(\mathbb{R}^2)$ equal to 1 in the vicinity of 0 and to 0 in the vicinity of infinity. The appropriate space to build the far fields is $H^1(\Omega) + \chi_0\mathcal{A}(\Omega)$. One can check that it does not depend on the choice of $\chi_0$.

**Lemma 3.4:** $H^1(\Omega) \cap \chi_0\mathcal{A}(\Omega) = \chi_0 \sum_{d>0} \mathcal{A}_d(\Omega)$.

*Proof:* The inclusion $\supseteq$ is easy to check, so we focus on $\subset$. Let $\varphi \in \mathcal{A}(\Omega) \setminus \{0\}$ be s.t. $\chi_0\varphi \in H^1(\Omega)$. There is $d \in \mathbb{R}$, $\varphi_1 \in \mathcal{A}_d(\Omega) \setminus \{0\}$ and $\varphi_2 \in \sum_{q>d} \mathcal{A}_q(\Omega)$ s.t. $\varphi = \varphi_1 + \varphi_2$. If $\partial_{\theta}\varphi_1$ were null everywhere, then would so too $\varphi_1$ because $\varphi_{1|\Sigma_{\Omega}} = 0$. But we assumed the contrary, so $\partial_{\theta}\varphi_1 \ne 0$. Thus Lemma 3.2 implies that there is a non-trivial interval $I \subset [0, \Theta]$ s.t. $|\partial_{\theta}\varphi_1(r, \theta)| \gtrsim r^d$ when $r \to 0$ and $\theta \in I$. Finally $r^{-1}\partial_{\theta}(\chi_0\varphi) \in L^2(\Omega)$ implies that $d > 0$. $\square$

**Définition 3.5:** For any $u \in H^1(\Omega) + \chi_0\mathcal{A}(\Omega)$ and $d \in -\frac{\pi}{\Theta}\mathbb{N}^*$, we denote $\sigma_d(u) := \sigma_d(\varphi)$ where $\varphi \in \mathcal{A}(\Omega)$ is s.t. $u - \chi_0\varphi \in H^1(\Omega)$ (see Definition 2.13 for $\sigma_d(\varphi)$). It does not depend on the choice of $\varphi$ thanks to Lemma 3.4.

**Theorem 3.6: existence and uniqueness for a far fields-like model problem**

Let $f \in (H^1(\Omega))'$, $g \in H_{00}^{1/2}(\Gamma) + \chi_0\mathcal{A}(\Gamma)$ and $(s_d) \in \mathbb{C}^{-\frac{\pi}{\Theta}\mathbb{N}^*}$ with finite support. The following system has a unique solution in $H^1(\Omega) + \chi_0\mathcal{A}(\Omega)$.

$$ \left\{ \begin{array}{ll} \mu_0 \Delta u + \omega^2 \rho_0 u = f & \text{in } \Omega \\ u = g & \text{on } \Gamma \\ u = 0 & \text{on } \Sigma_{\Omega} \\ \sigma_d(u) = s_d & \forall d \in -\frac{\pi}{\Theta}\mathbb{N}^* \end{array} \right. $$

*Proof:* Let us show the existence first, and then the uniqueness.

**Existence:** Let $\tilde{g} \in H_{00}^{1/2}(\Gamma)$ and $\varphi \in \mathcal{A}(\Gamma)$ be s.t. $g = \tilde{g} + \chi_0\varphi$. We look for the solution in the form $u = \tilde{u} + \chi_0\psi$ with $\tilde{u} \in H^1(\Omega)$ and $\psi \in \mathcal{A}(\Omega)$. Let

$$ \psi^+ := \sum_{n=0}^{\infty} (-k_0^2 R_{\Delta}^{\Omega})^n \left( R_{\mathrm{D}}^{\Omega}(\varphi) + \sum_{d \in -\frac{\pi}{\Theta}\mathbb{N}^*} s_d \phi_d^{\Omega} \right) \in \mathcal{A}^+(\Omega) $$

(where $\phi_d^{\Omega} := \phi_{d|\Omega}$) and $\psi := T_{\le 2}(\psi^+)$. Using Proposition 2.15 one can check by calculus the first system below (see also Lemma 2.18 for a similar result). Then, the second system below derives from $\psi \in \mathcal{A}(\Omega)$,
**Définition 3.5:** For any $u \in H^1(\Omega) + \chi_0 A(\Omega)$ and $d \in -\frac{\pi}{\Theta}N^*$, we denote $\sigma_d(u) := \sigma_d(\varphi)$ where $\varphi \in A(\Omega)$ is s.t. $u - \chi_0\varphi \in H^1(\Omega)$ (see Definition 2.13 for $\sigma_d(\varphi)$). It does not depend on the choice of $\varphi$ thanks to Lemma 3.4.

**Theorem 3.6: existence and uniqueness for a far fields-like model problem**

Let $f \in (H^1(\Omega))'$, $g \in H_{00}^{1/2}(\Gamma) + \chi_0 A(\Gamma)$ and $(s_d) \in \mathbb{C}^{-\frac{\pi}{\Theta}N^*}$ with finite support. The following system has a unique solution in $H^1(\Omega) + \chi_0 A(\Omega)$.

$$ 
\left\{ 
\begin{array}{ll}
\mu_0 \Delta u + \omega^2 \rho_0 u = f & \text{in } \Omega \\
u = g & \text{on } \Gamma \\
u = 0 & \text{on } \Sigma_{\Omega} \\
\sigma_d(u) = s_d & \forall d \in -\frac{\pi}{\Theta}N^*
\end{array}
\right. 
$$

**Proof:** Let us show the existence first, and then the uniqueness.

**Existence:** Let $\tilde{g} \in H_{00}^{1/2}(\Gamma)$ and $\varphi \in A(\Gamma)$ be s.t. $g = \tilde{g} + \chi_0\varphi$. We look for the solution in the form $u = \tilde{u} + \chi_0\psi$ with $\tilde{u} \in H^1(\Omega)$ and $\psi \in A(\Omega)$. Let

$$ \psi^+ := \sum_{n=0}^{\infty} (-k_0^2 R_{\Delta}^{\Omega})^n \left( R_{\mathrm{D}}^{\Omega}(\varphi) + \sum_{d \in -\frac{\pi}{\Theta}N^*} s_d \phi_d^{\Omega} \right) \in A^{+}(\Omega) $$

(where $\phi_d^\Omega := \phi_{d|\Omega}$) and $\psi := T_{\le 2}(\psi^+)$. Using Proposition 2.15 one can check by calculus the first system below (see also Lemma 2.18 for a similar result). Then, the second system below derives from $\psi \in A(\Omega)$, $\psi = \psi^+ - (\text{id} - T_{\le 2})(\psi^+)$ and $\deg \Delta = -2$ (by Lemma 2.16).

$$ 
\left\{ 
\begin{array}{ll}
(\mu_0 \Delta + \omega^2 \rho_0) \psi^+ = 0 & \text{in } \Omega \\
\psi^+ = \varphi & \text{on } \Gamma \\
\psi^+ = 0 & \text{on } \Sigma_{\Omega} \\
\sigma_d(\psi^+) = s_d & \forall d \in -\frac{\pi}{\Theta}N^*
\end{array}
\right. 
\quad \text{so} 
\quad 
\left\{ 
\begin{array}{l}
(\mu_0 \Delta + \omega^2 \rho_0) \psi \in A(\Omega) \cap \sum_{d>0} A_d(\Omega) = \sum_{d>0} A_d(\Omega) \\
\psi|_{\Gamma} - \varphi \in A(\Gamma) \cap \sum_{d>2} A_d(\Gamma) = \sum_{d>2} A_d(\Gamma) \\
\psi|_{\Sigma_{\Omega}} = 0 \\
\sigma_q(\psi) = s_d \quad \forall d \in -\frac{\pi}{\Theta}N^*
\end{array}
\right. 
$$

Therefore $(\mu_0\Delta + \omega^2\rho_0)(\chi_0\psi) \in L^2(\Omega)$ and $\chi_0(\psi|_{\Gamma} - \varphi) \in H_{00}^{1/2}(\Gamma)$. Finally, thanks to Lemma 3.3, we can take $\tilde{u}$ as the unique solution in $H^1(\Omega)$ of:

$$ 
\left\{ 
\begin{array}{ll}
\mu_0 \Delta \tilde{u} + \omega^2 \rho_0 \tilde{u} = f - (\mu_0 \Delta + \omega^2 \rho_0)(\chi_0 \psi) & \text{in } \Omega \\
\tilde{u} = \tilde{g} - \chi_0 (\psi - \varphi) & \text{on } \Gamma \\
\tilde{u} = 0 & \text{on } \Sigma_{\Omega}
\end{array}
\right. 
$$

**Uniqueness:** Let $u$ be a homogeneous solution. There is $\tilde{u} \in H^1(\Omega)$ and $\psi \in A(\Omega)$ such that $u = \tilde{u} + \chi_0\psi$. Let $\varphi := \bar{\Delta}\psi + k_0^2\psi$. By Lemma 2.16, $\varphi \in A(\Omega)$. Let us show that $\deg_{\min} \varphi > -2$.

We denote $d := \deg_{\min} \varphi$ and $\tilde{\varphi} := (\Delta + k_0^2)(\chi_0\psi)$. By Lemma 3.2, there is a non-trivial interval $I \subset [0, \Theta]$ s.t. when $r \to 0$ and $\theta \in I$: $\tilde{\varphi}(r, \theta) = \varphi(r, \theta) \ge r^d$. Besides, $\tilde{\varphi} = -(\Delta + k_0^2)\tilde{u} \in (H_0^1(\Omega))'$. Let us test it with $\zeta_q : (r, \theta) \mapsto \tilde{\varphi}(r, \theta) \cdot r^{q-2d-2} (1 - \chi_0(2^{1/q}r)) \chi(\theta)$ where $q > 0$ and $\chi \in D(0, \Theta) \setminus \{0\}$ is everywhere non negative. Since $\zeta_q \in D(\Omega)$:

$$ \langle \tilde{\varphi}, \zeta_q \rangle := \int_0^\infty \int_0^\Theta \tilde{\varphi} \cdot \zeta_q \cdot r d\theta dr \lesssim \|\zeta_q\|_{H^1(\Omega)} \quad \forall q > 0 $$

Let us assume by contradiction that $d \le -2$. It is not difficult to check that when $q \to 0^+$:

$$ \langle \tilde{\varphi}, \zeta_q \rangle \gtrsim \int_{c_1 2^{-\frac{1}{q}}}^{c_2} r^d r^{q-d-2} r dr \gtrsim \frac{1}{q} \quad \text{and} \quad \|\zeta_q\|_{H^1} \lesssim \|\nabla \zeta_q\|_{L^2} \lesssim \left( \int_{c_3 2^{-\frac{1}{q}}}^{c_4} r^{2(q-d-3)} r dr \right)^{\frac{1}{2}} \lesssim \frac{1}{\sqrt{q}} $$

with some constants $c_i$ and Poincaré's inequality. But it contradicts $\langle \tilde{\varphi}, \zeta_q \rangle \lesssim \|\zeta_q\|_{H^1}$. So $d > -2$.

Now let us show that $\deg_{\min} \psi > 0$.
Let us assume the contrary. Then there is $d \le 0$, $\psi_1 \in \mathcal{A}_d(\Omega) \setminus \{0\}$ and $\psi_2 \in \sum_{q>d} \mathcal{A}_q(\Omega)$ s.t. $\psi = \psi_1 + \psi_2$. Lemma 2.16 states that $\deg \Delta = -2$, so $\Delta\psi_1 \in \mathcal{A}_{d-2}(\Omega)$ and $\Delta\psi_2 \in \sum_{q>d-2} \mathcal{A}_q(\Omega)$. In addition, $\varphi \in \sum_{q>-2} \mathcal{A}_q(\Omega)$ and $\mathcal{A}(\Omega) = \bigoplus_{q \in \mathbb{R}} \mathcal{A}_q(\Omega)$ according to Lemma 2.8. So identifying the coordinate of $\Delta\psi + k_0^2\psi = \varphi$ in $\mathcal{A}_{d-2}(\Omega)$ gives $\Delta\psi_1 = 0$.

Moreover, $\psi|_{\Gamma} = -\tilde{u}|_{\Gamma} \in H^{1/2}(\Gamma)$, so $\psi|_{\Gamma} \in \sum_{q>0} \mathcal{A}_q(\Gamma)$. Hence:

$$ \begin{cases} \Delta\psi_1 = 0 & \text{in } \Omega \\ \psi_1 = 0 & \text{on } \Sigma_{\Omega} \cup \Gamma \end{cases} $$

By Proposition 2.12, it implies $d \in -\frac{\pi}{\Theta}\mathbb{N}^*$ and $\psi_1 = \sigma_d(\psi_1) \phi_d^\Omega$. However, $\sigma_d(\psi_1) = \sigma_d(\psi) = \sigma_d(u) = 0$, which contradicts $\psi_1 \ne 0$.

Finally, Lemma 3.4 implies $\chi_0\psi \in H^1(\Omega)$, so $u \in H^1(\Omega)$. This means that $u = 0$ by Lemma 3.3. $\square$

## 3.2 Existence and uniqueness for corner fields-like problems

**Definition 3.7: the variational space V**

We define $V := \{v \in H_{\text{loc}}^1(\Omega_1) \mid \nabla v \in L^2(\Omega_1) \text{ and } v|_{\partial\Omega_1} = 0\}$ and the norm $\|v\|_V := \|\nabla v\|_{L^2(\Omega_1)}$.

Here “$v|_{\partial\Omega_1} = 0$” means that $\chi_0(\dot{\frac{\cdot}{R}}) v \in H_0^1(\Omega_1)$ for any $R > 0$, where $\chi_0(\dot{\frac{\cdot}{R}}) : (X, Y) \mapsto \chi_0(\frac{X}{R}, \frac{Y}{R})$.

**Lemma 3.7:** Any $v \in V$ satisfies $\|v\|_{H^1(\Lambda \cap \{X > R_c\})} \lesssim \|v\|_V$ and $\left\| \frac{v}{1+r} \right\|_{L^2(\Omega_1)} \lesssim \|v\|_V$.

*Proof:* We denote $A := \Lambda \cap \{X > R_c\}$ and $B := B(0, R_c)$, and we recall that $R_c > 0$ is s.t. $\Omega_1 \setminus B = \Pi \setminus B$. Poincaré's inequality gives for a.e. $X > R_c$: $\|v(X, \cdot)\|_{L^2(-1,0)} \lesssim \|\partial_Y v(X, \cdot)\|_{L^2(-1,0)}$. Integrating w.r.t. $X$ the square of this then gives: $\|v\|_{L^2(A)} \lesssim \|\partial_Y v\|_{L^2(A)} \lesssim \|v\|_V$. So $\|v\|_{H^1(A)} \lesssim \|v\|_V$. Next, a generalized Poincaré inequality on circular slices of $\Omega \setminus B$ gives for a.e. $r > R_c$: $\|v(r, \cdot)\|_{L^2(0,\pi)} \lesssim \|\partial_\theta v(r, \cdot)\|_{L^2(0,\pi)} + |v(r, \theta = 0)|$. Then integrating w.r.t. $r$ the square of this times $\frac{1}{1+r}$ gives:

$$ \int_{\Omega \setminus B} \frac{v^2}{(1+r)r} r d\theta dr \lesssim \int_{\Omega \setminus B} \frac{(\partial_\theta v)^2}{(1+r)r} r d\theta dr + \int_{\Gamma \setminus B} \frac{v^2}{1+r} dr. $$

Therefore $\|\frac{v}{1+r}\|_{L^2(\Omega \setminus B)} \lesssim \|\nabla v\|_{L^2(\Omega \setminus B)}^2 + \|v\|_{L^2(\Gamma \setminus B)}^2$. But we know that $\|v\|_{L^2(\Gamma \setminus B)} \lesssim \|v\|_{H^1(A)} \lesssim \|v\|_V$, so $\|\frac{v}{1+r}\|_{L^2(\Pi \setminus B)} \lesssim \|v\|_V$. Finally Poincaré's inequality in $B$ also gives $\|\frac{v}{1+r}\|_{L^2(\Omega_1 \cap B)} \lesssim \|v\|_V$. $\square$

**Lemma 3.8: the Poisson problem in $\Omega_1$**

Let $F : \Omega_1 \to \mathbb{C}$ be s.t. $(1+r)F \in L^2(\Omega_1)$, and $g \in L^2(\Gamma \cap \{X > R_c\})$. The following system is well-posed in $V$.

$$ \left\{ \begin{array}{ll} \text{div}(\mu\nabla S) = F & \text{in } \Omega_1 \setminus (\Gamma \cap \{X > R_c\}) \\ S = 0 & \text{on } \partial\Omega_1 \\ S|_{Y=0^+} - S|_{Y=0^-} = 0 & \text{on } \Gamma \cap \{X > R_c\} \\ \mu_0\partial_Y S|_{Y=0^+} - \mu_1\partial_Y S|_{Y=0^-} = g & \text{on } \Gamma \cap \{X > R_c\} \end{array} \right. $$

*Proof:* The variational formulation of this problem is:

$$ \forall v \in V, \quad \int_{\Omega_1} \mu \nabla S \cdot \nabla v = - \int_{\Omega_1} F v + \int_{\Gamma \cap \{X > R_c\}} gv. $$

The left-hand side is coercive by definition of $V$. Lemma 3.7 ensures that the right-hand side is continuous. Moreover it is easy to see that $V$ is complete. So we can conclude using the Lax-Milgram theorem. $\square$

Let $\chi_\infty$ be a function of $C^\infty(\mathbb{R}^2)$ equal to 0 on $B(0, R_c)$ and 1 in a vicinity of infinity. The proper space to build the corner fields is $V + \chi_\infty A(\Pi)$. One can check that it does not depend on the choice of $\chi_\infty$.
**Proof:** We denote $A := \Lambda \cap \{X > R_c\}$ and $B := B(0, R_c)$, and we recall that $R_c > 0$ is s.t. $\Omega_1 \setminus B = \Pi \setminus B$. Poincaré's inequality gives for a.e. $X > R_c$: $\|v(X, \cdot)\|_{L^2(-1,0)} \lesssim \|\partial_Y v(X, \cdot)\|_{L^2(-1,0)}$. Integrating w.r.t. $X$ the square of this then gives: $\|v\|_{L^2(A)} \lesssim \|\partial_Y v\|_{L^2(A)} \lesssim \|v\|_V$. So $\|v\|_{H^1(A)} \lesssim \|v\|_V$. Next, a generalized Poincaré inequality on circular slices of $\Omega \setminus B$ gives for a.e. $r > R_c$: $\|v(r, \cdot)\|_{L^2(0,\pi)} \lesssim \|\partial_\theta v(r, \cdot)\|_{L^2(0,\pi)} + |v(r, \theta = 0)|$. Then integrating w.r.t. $r$ the square of this times $\frac{1}{1+r}$ gives:

$$ \int_{\Omega \setminus B} \frac{v^2}{(1+r)r} r d\theta dr \lesssim \int_{\Omega \setminus B} \frac{(\partial_\theta v)^2}{(1+r)r} r d\theta dr + \int_{\Gamma \setminus B} \frac{v^2}{1+r} dr. $$

Therefore $\|\frac{v}{1+r}\|_{L^2(\Omega \setminus B)} \lesssim \|\nabla v\|_{L^2(\Omega \setminus B)}^2 + \|v\|_{L^2(\Gamma \setminus B)}^2$. But we know that $\|v\|_{L^2(\Gamma \setminus B)} \lesssim \|v\|_{H^1(A)} \lesssim \|v\|_V$, so $\|\frac{v}{1+r}\|_{L^2(\Pi \setminus B)} \lesssim \|v\|_V$. Finally Poincaré's inequality in $B$ also gives $\|\frac{v}{1+r}\|_{L^2(\Omega_1 \cap B)} \lesssim \|v\|_V$. $\square$

**Lemma 3.8: the Poisson problem in $\Omega_1$**

Let $F: \Omega_1 \to \mathbb{C}$ be s.t. $(1+r)F \in L^2(\Omega_1)$, and $g \in L^2(\Gamma \cap \{X > R_c\})$. The following system is well-posed in $V$.

$$ \left\{ \begin{array}{ll} \text{div}(\mu \nabla S) = F & \text{in } \Omega_1 \setminus (\Gamma \cap \{X > R_c\}) \\ S = 0 & \text{on } \partial \Omega_1 \\ S|_{Y=0^+} - S|_{Y=0^-} = 0 & \text{on } \Gamma \cap \{X > R_c\} \\ \mu_0 \partial_Y S|_{Y=0^+} - \mu_1 \partial_Y S|_{Y=0^-} = g & \text{on } \Gamma \cap \{X > R_c\} \end{array} \right. $$

**Proof:** The variational formulation of this problem is:

$$ \forall v \in V, \quad \int_{\Omega_1} \mu \nabla S \cdot \nabla v = - \int_{\Omega_1} Fv + \int_{\Gamma \cap \{X > R_c\}} gv. $$

The left-hand side is coercive by definition of $V$. Lemma 3.7 ensures that the right-hand side is continuous. Moreover it is easy to see that $V$ is complete. So we can conclude using the Lax-Milgram theorem. $\square$

Let $\chi_\infty$ be a function of $C^\infty(\mathbb{R}^2)$ equal to 0 on $B(0, R_c)$ and 1 in a vicinity of infinity. The proper space to build the corner fields is $V + \chi_\infty A(\Pi)$. One can check that it does not depend on the choice of $\chi_\infty$.

**Lemma 3.9:** $V \cap \chi_\infty A(\Pi) = \{\chi_\infty \varphi | \varphi \in A(\Pi), \deg_{\max}(\varphi|_\Omega) < 0 \text{ and } \deg_{\max}(\varphi|_\Lambda) < -\frac{1}{2}\}$.

**Proof:** The inclusion $\supset$ is easy to check, so we focus on $\subset$. Let $\varphi \in A(\Pi)$ be s.t. $\chi_\infty \varphi \in V$. The same method as Lemma 3.4 shows that $\deg_{\max} \varphi|_\Omega < 0$. And using that $\partial_Y(\chi_\infty \varphi) \in L^2(\Lambda)$ we likewise get $\deg_{\max}(\varphi|_\Lambda) < -\frac{1}{2}$. $\square$

**Definition 3.10:** For any $S \in V + \chi_\infty A(\Pi)$ and $d \in \pi_\Theta \mathbb{N}^*$, we denote $\sigma_d(S) := \sigma_d(\varphi)$ where $\varphi \in A(\Pi)$ is s.t. $S - \chi_\infty \varphi \in V$. It does not depend on the choice of $\varphi$ thanks to Lemma 3.9.

**Theorem 3.11: existence and uniqueness for a corner fields-like model problem**

Let $f: \Omega_1 \to \mathbb{C}$ be s.t. $(1+r)F \in L^2(\Omega_1)$, $\varphi: \Pi \to \mathbb{C}$ be s.t. $\varphi|_\Omega \in A(\Omega)$ and $\varphi|_\Lambda \in A(\Lambda)$, and $(s_d) \in C_{\pi_\Theta \mathbb{N}^*}$ with finite support. The following system has a unique solution in $V + \chi_\infty A(\Pi)$.

$$ \left\{ \begin{array}{ll} \text{div}(\mu \nabla S) = F + \chi_\infty \varphi & \text{in } \Omega_1 \\ S = 0 & \text{on } \partial \Omega_1 \\ \sigma_d(S) = s_d & \text{for } d \in \pi_\Theta \mathbb{N}^* \end{array} \right. $$

**Remark:** $\varphi$ is not just an element of $A(\Pi)$ because it may be discontinuous on $\Gamma$.
**Lemma 3.9:** $V \cap \chi_{\infty}A(\Pi) = \{\chi_{\infty}\varphi | \varphi \in A(\Pi), \deg_{\max}(\varphi|_{\Omega}) < 0 \text{ and } \deg_{\max}(\varphi|_{\Lambda}) < -\frac{1}{2}\}$.

*Proof:* The inclusion $\supset$ is easy to check, so we focus on $\subset$. Let $\varphi \in A(\Pi)$ be s.t. $\chi_{\infty}\varphi \in V$. The same method as Lemma 3.4 shows that $\deg_{\max}(\varphi|_{\Omega}) < 0$. And using that $\partial_Y(\chi_{\infty}\varphi) \in L^2(\Lambda)$ we likewise get $\deg_{\max}(\varphi|_{\Lambda}) < -\frac{1}{2}$. $\square$

**Definition 3.10:** For any $S \in V + \chi_{\infty}A(\Pi)$ and $d \in \frac{\pi}{\Theta}N^*$, we denote $\sigma_d(S) := \sigma_d(\varphi)$ where $\varphi \in A(\Pi)$ is s.t. $S - \chi_{\infty}\varphi \in V$. It does not depend on the choice of $\varphi$ thanks to Lemma 3.9.

**Theorem 3.11: existence and uniqueness for a corner fields-like model problem**

Let $F : \Omega_1 \to \mathbb{C}$ be s.t. $(1+r)F \in L^2(\Omega_1)$, $\varphi : \Pi \to \mathbb{C}$ be s.t. $\varphi|_{\Omega} \in A(\Omega)$ and $\varphi|_{\Lambda} \in A(\Lambda)$, and $(s_d) \in C_{\frac{\pi}{\Theta}N^*}$ with finite support. The following system has a unique solution in $V + \chi_{\infty}A(\Pi)$.

$$ 
\left\{ 
\begin{array}{ll}
\mathrm{div}(\mu\nabla S) = F + \chi_{\infty}\varphi & \text{in } \Omega_1 \\
S = 0 & \text{on } \partial\Omega_1 \\
\sigma_d(S) = s_d & \text{for } d \in \frac{\pi}{\Theta}N^*
\end{array}
\right.
$$

**Remark:** $\varphi$ is not just an element of $A(\Pi)$ because it may be discontinuous on $\Gamma$.

*Proof:* This proof is similar to Theorem 3.6. Let us show the existence first, and then the uniqueness.

*Existence:* We look for the solution in the form $S = \tilde{S} + \chi_{\infty}\psi$ with $\tilde{S} \in V$ and $\psi \in A(\Pi)$. More precisely we denote (using notation $\langle \cdot, \cdot \rangle$ from Definition 2.3)

$$ \varphi^{-} := \left\langle -R_{\partial_{Y}^{2}} \circ \partial_{X|\Lambda}^{2}, \frac{\mu_{0}}{\mu_{1}} R_{N} \circ \partial_{Y|\Gamma, Y=0^{+}} \right\rangle \left( \frac{1}{\mu_{0}} R_{\Delta}(\varphi|_{\Omega}) + \frac{1}{\mu_{1}} R_{\partial_{Y}^{2}}(\varphi|_{\Lambda}) + \sum_{d \in \frac{\pi}{\Theta}N^{*}} s_{d} \phi_{d} \right) \in A^{-}(\Pi) $$

and $\psi := T_{\ge -2}(\varphi^{-})$. By Lemma 2.19, we have the first system below. Then, similarly to the proof of Theorem 3.6, one can check that it implies the second system below.

$$ 
\left\{ 
\begin{array}{ll}
\mu_0 \Delta \varphi^- = \varphi & \text{in } \Omega \\
\mu_1 \Delta \varphi^- = \varphi & \text{in } \Lambda \\
[\varphi^-]_\Gamma = 0 & \text{on } \Gamma \\
[\mu \partial_Y \varphi^-]_\Gamma = 0 & \text{on } \Gamma \\
\varphi^- = 0 & \text{on } \Sigma_\Omega \cup \Sigma_\Lambda \\
\sigma_d(\varphi^-) = s_d & \text{for } d \in \frac{\pi}{\Theta}N^*
\end{array}
\right.
\quad \text{so} \quad
\left\{ 
\begin{array}{ll}
(\mu_0 \Delta \psi - \varphi)|_{\Omega} \in A(\Omega) \cap \sum_{d<-4} A_d(\Omega) = \sum_{d<-4} A_d(\Omega) \\
(\mu_1 \Delta \psi - \varphi)|_{\Lambda} \in A(\Lambda) \cap \sum_{d<-2} A_d(\Lambda) = \sum_{d<-2} A_d(\Lambda) \\
[\psi]_\Gamma = 0 \\
[\mu \partial_Y \psi]_\Gamma \in A(\Gamma) \cap \sum_{d<-2} A_d(\Gamma) = \sum_{d<-2} A_d(\Gamma) \\
\psi|_{\Sigma_\Omega \cup \Sigma_\Lambda} = 0 \\
\sigma_q(\psi) = s_d \quad \forall d \in \frac{\pi}{\Theta}N^*
\end{array}
\right.
$$

where $[\dots]_\Gamma$ stands for the jump on $\Gamma$. Therefore $(1+r)[\mathrm{div}(\mu\nabla(\chi_{\infty}\psi)) - \chi_{\infty}\varphi] \in L^2(\Omega_1)$ and $[\mu\partial_Y(\chi_{\infty}\psi)]_\Gamma \in L^2(\Gamma \cap \{X > R_c\})$. Finally, thanks to Lemma 3.8, we take $\tilde{S}$ as the unique solution in $V$ of:

$$ 
\left\{ 
\begin{array}{ll}
\mathrm{div}(\mu\nabla\tilde{S}) = F + \chi_{\infty}\varphi - \mathrm{div}(\mu\nabla(\chi_{\infty}\psi)) & \text{in } \Omega_1 \setminus (\Gamma \cap \{X > R_c\}) \\
\tilde{S} = 0 & \text{on } \partial\Omega_1 \\
[\mu\partial_Y\tilde{S}]_\Gamma = -[\mu\partial_Y(\chi_{\infty}\psi)]_\Gamma & \text{on } \Gamma \cap \{X > R_c\}
\end{array}
\right.
$$

*Uniqueness:* Let $S$ be a homogeneous solution and let us show that $S = 0$. There is $\tilde{S} \in V$ and $\psi \in A(\Omega)$ s.t. $S = \tilde{S} + \chi_{\infty}\psi$. The proof decomposes into the following steps, all proven by contradiction.

1. $\deg_{\max}(\Delta\psi|_{\Omega}) < -2$: Like in step 1 of the uniqueness proof of Theorem 3.6, we test the inequality:

$$ \int_{\Omega} \Delta(\chi_{\infty}\psi) \cdot \zeta = - \int_{\Omega} \Delta \tilde{S} \cdot \zeta = \int_{\Omega} \nabla \tilde{S} \cdot \nabla \zeta \lesssim \|\zeta\|_{V} \quad \forall \zeta \in \mathcal{D}(\Omega) $$

with $\zeta_q : (r, \theta) \mapsto \Delta(\chi_{\infty}\psi)(r, \theta) \cdot r^{-q-2d-2} \chi_0(2^{-1/q}r) \chi(\theta)$ when $q \to 0^+$, where $d := \deg_{\max}(\Delta\psi|_{\Omega})$ and $\chi \in \mathcal{D}(0, \Theta) \setminus \{0\}$ is everywhere non negative.
**Proof:** This proof is similar to Theorem 3.6. Let us show the existence first, and then the uniqueness.

**Existence:** We look for the solution in the form $S = \tilde{S} + \chi_{\infty}\psi$ with $\tilde{S} \in V$ and $\psi \in A(\Pi)$. More precisely we denote (using notation $\langle \cdot, \cdot \rangle$ from Definition 2.3)

$$ \varphi^{-} := \left\langle -R_{\partial_{Y}^{2}} \circ \partial_{X|\Lambda}^{2}, \frac{\mu_{0}}{\mu_{1}} R_{N} \circ \partial_{Y|\Gamma, Y=0^{+}} \right\rangle \left( \frac{1}{\mu_{0}} R_{\Delta}(\varphi|_{\Omega}) + \frac{1}{\mu_{1}} R_{\partial_{Y}^{2}}(\varphi|_{\Lambda}) + \sum_{d \in \frac{\pi}{\Theta}N^{*}} s_{d} \phi_{d} \right) \in A^{-}(\Pi) $$

and $\psi := T_{\ge -2}(\varphi^{-})$. By Lemma 2.19, we have the first system below. Then, similarly to the proof of Theorem 3.6, one can check that it implies the second system below.

$$ \left\{ \begin{array}{ll} \mu_0 \Delta \varphi^{-} = \varphi & \text{in } \Omega \\ \mu_1 \Delta \varphi^{-} = \varphi & \text{in } \Lambda \\ [\varphi^{-}]_{\Gamma} = 0 & \text{on } \Gamma \\ [\mu \partial_Y \varphi^{-}]_{\Gamma} = 0 & \text{on } \Gamma \\ \varphi^{-} = 0 & \text{on } \Sigma_{\Omega} \cup \Sigma_{\Lambda} \\ \sigma_d(\varphi^{-}) = s_d & \forall d \in \frac{\pi}{\Theta}N^{*} \end{array} \right. \quad \text{so} \quad \left\{ \begin{array}{ll} (\mu_0 \Delta \psi - \varphi)|_{\Omega} \in A(\Omega) \cap \sum_{d<-4} A_d(\Omega) = \sum_{d<-4} A_d(\Omega) \\ (\mu_1 \Delta \psi - \varphi)|_{\Lambda} \in A(\Lambda) \cap \sum_{d<-2} A_d(\Lambda) = \sum_{d<-2} A_d(\Lambda) \\ [\psi]_{\Gamma} = 0 \\ [\mu \partial_Y \psi]_{\Gamma} \in A(\Gamma) \cap \sum_{d<-2} A_d(\Gamma) = \sum_{d<-2} A_d(\Gamma) \\ \psi|_{\Sigma_{\Omega} \cup \Sigma_{\Lambda}} = 0 \\ \sigma_q(\psi) = s_d \quad \forall d \in \frac{\pi}{\Theta}N^{*} \end{array} \right. $$

where $[\dots]_{\Gamma}$ stands for the jump on $\Gamma$. Therefore $(1+r)[\text{div}(\mu\nabla(\chi_{\infty}\psi)) - \chi_{\infty}\varphi] \in L^2(\Omega_1)$ and $[\mu\partial_Y(\chi_{\infty}\psi)]_{\Gamma} \in L^2(\Gamma \cap \{X > R_c\})$. Finally, thanks to Lemma 3.8, we take $\tilde{S}$ as the unique solution in $V$ of:

$$ \left\{ \begin{array}{ll} \text{div}(\mu\nabla\tilde{S}) = F + \chi_{\infty}\varphi - \text{div}(\mu\nabla(\chi_{\infty}\psi)) & \text{in } \Omega_1 \setminus (\Gamma \cap \{X > R_c\}) \\ \tilde{S} = 0 & \text{on } \partial\Omega_1 \\ [\mu\partial_Y\tilde{S}]_{\Gamma} = -[\mu\partial_Y(\chi_{\infty}\psi)]_{\Gamma} & \text{on } \Gamma \cap \{X > R_c\} \end{array} \right. $$

**Uniqueness:** Let $S$ be a homogeneous solution and let us show that $S = 0$. There is $\tilde{S} \in V$ and $\psi \in A(\Omega)$ s.t. $S = \tilde{S} + \chi_{\infty}\psi$. The proof decomposes into the following steps, all proven by contradiction.

1. $\text{deg}_{\max}(\Delta\psi|_{\Omega}) < -2$: Like in step 1 of the uniqueness proof of Theorem 3.6, we test the inequality:

$$ \int_{\Omega} \Delta(\chi_{\infty}\psi) \cdot \zeta = - \int_{\Omega} \Delta \tilde{S} \cdot \zeta = \int_{\Omega} \nabla \tilde{S} \cdot \nabla \zeta \lesssim \|\zeta\|_{V} \quad \forall \zeta \in \mathcal{D}(\Omega) $$

with $\zeta_q : (r, \theta) \mapsto \Delta(\chi_{\infty}\psi)(r, \theta) \cdot r^{-q-2d-2} \chi_0(2^{-1/q}r) \chi(\theta)$ when $q \to 0^+$, where $d := \text{deg}_{\max}(\Delta\psi|_{\Omega})$ and $\chi \in \mathcal{D}(0, \Theta) \setminus \{0\}$ is everywhere non negative.

2. $\text{deg}_{\max}(\Delta\psi|_{\Lambda}) < -\frac{1}{2}$: This step works like the previous one.

3. $\text{deg}_{\max}[\mu\partial_Y\psi]_{\Gamma} < -\frac{1}{2}$: Let $d := \text{deg}_{\max}[\mu\partial_Y\psi]_{\Gamma}$ and $\varphi := [\mu\partial_Y(\chi_{\infty}\psi)]_{\Gamma}$. We have $\nabla\tilde{S} \in L^2(\Omega_1)$ and, by steps 1 and 2, $\text{div}(\mu\nabla\tilde{S}) = -\text{div}(\mu\nabla(\chi_{\infty}\psi)) \in L^2(\Omega_1)$. Thus $\varphi = -[\mu\partial_Y\tilde{S}]_{\Gamma} \in H^{-1/2}(\Gamma)$. Like previously, we test it with $\zeta_q: x \mapsto \varphi(x) \cdot x^{-q-2d-1} \chi_0(2^{-1/q}x)$ when $q \to 0^+$. If $d \ge -\frac{1}{2}$, we get $\langle\varphi, \zeta_q\rangle_{\Gamma} \ge \frac{1}{q}$ and $\|\zeta_q\|_{H^{1/2}(\Gamma)} \lesssim \|\zeta_q\|_{H^1(\Gamma)} \lesssim \frac{1}{\sqrt{q}}$, which contradicts $\langle\varphi, \zeta_q\rangle_{\Gamma} \lesssim \|\zeta_q\|_{H^{1/2}(\Gamma)}$.

4. $\text{deg}_{\max}(\psi|_{\Lambda}) < -\frac{1}{2}$: Let $d := \text{deg}_{\max}(\psi|_{\Lambda})$. There is $\psi_1 \in A_d(\Lambda) \setminus \{0\}$ and $\psi_2 \in \sum_{q<d} A_q(\Lambda)$ s.t. $\psi|_{\Lambda} = \psi_1 + \psi_2$. By Lemma 2.16, $\text{deg}(\partial_{x|\Lambda}^2) = -2$, $\text{deg}(\partial_{Y|\Gamma, Y=0^+}) = -1$ and $\text{deg}(\partial_{Y|\Lambda}^2) = \text{deg}(\partial_{Y|\Gamma, Y=0^-}) = 0$. In addition, Proposition 2.8 states that $A(D) = \bigoplus_{q \in \mathbb{R}} A_q(D)$ for any $D \in \{\Lambda, \Gamma\}$. So by taking the coordinates of $\Delta\psi|_{\Lambda}$ in $A_d(\Lambda)$ and of $[\mu\partial_Y\psi]_{\Gamma}$ in $A_d(\Gamma)$, we get if $d \ge -\frac{1}{2}$:

$$ \left\{ \begin{array}{ll} \partial_{Y}^{2} \psi_{1} = 0 & \text{in } \Lambda \\ \partial_{Y|\Gamma, Y=0^{-}} \psi_{1} = 0 & \text{on } \Gamma \end{array} \right. $$

But since $\psi|_{\Sigma_{\Lambda}} = 0$, it implies $\psi_1 = 0$, which is contradictory.

5. $\text{deg}_{\max}(\psi|_{\Omega}) < 0$: We denote $d := \text{deg}_{\max}(\psi|_{\Omega})$. There is $\psi_1 \in A_d(\Omega) \setminus \{0\}$ and $\psi_2 \in \sum_{q<d} A_q(\Omega)$ s.t. $\psi|_{\Omega} = \psi_1 + \psi_2$. Since $\text{deg}(\Delta|_{\Omega}) = -2$, taking the coordinate of $\Delta\psi|_{\Omega}$ in $A_{d-2}(\Omega)$ and of $\psi|_{\Gamma}$ in $A_d(\Lambda)$ gives if $d \ge 0$:

$$ \left\{ \begin{array}{ll} \Delta\psi_1 = 0 & \text{in } \Omega \\ \psi_1 = 0 & \text{on } \Gamma \end{array} \right. $$

By Proposition 2.12, it implies $d \in \frac{\pi}{\Theta}N^*$ and $\psi_1 = \sigma_d(\psi_1)\phi_q$. But $\sigma_d(\psi_1) = \sigma_d(\psi) = \sigma_d(S) = 0$.

Finally, Lemma 3.9 implies $\chi_{\infty}\psi \in V$, so $S \in V$. So the uniqueness in Lemma 3.8 implies $S = 0$. $\square$
2. $\deg_{\max}(\Delta\psi|_{\Lambda}) < -\frac{1}{2}$: This step works like the previous one.

3. $\deg_{\max}[\mu\partial_Y\psi]_{\Gamma} < -\frac{1}{2}$: Let $d := \deg_{\max}[\mu\partial_Y\psi]_{\Gamma}$ and $\varphi := [\mu\partial_Y(\chi_{\infty}\psi)]_{\Gamma}$. We have $\nabla\tilde{S} \in L^2(\Omega_1)$ and, by steps 1 and 2, $\text{div}(\mu\nabla\tilde{S}) = -\text{div}(\mu\nabla(\chi_{\infty}\psi)) \in L^2(\Omega_1)$. Thus $\varphi = -[\mu\partial_Y\tilde{S}]_{\Gamma} \in H^{-1/2}(\Gamma)$. Like previously, we test it with $\zeta_q: x \mapsto \varphi(x) \cdot x^{-q-2d-1}\chi_0(2^{-1/q}x)$ when $q \to 0^+$. If $d \ge -\frac{1}{2}$, we get $\langle\varphi, \zeta_q\rangle_{\Gamma} \gtrsim \frac{1}{q}$ and $\|\zeta_q\|_{H^{1/2}(\Gamma)} \lesssim \|\zeta_q\|_{H^1(\Gamma)} \lesssim \frac{1}{\sqrt{q}}$, which contradicts $\langle\varphi, \zeta_q\rangle_{\Gamma} \lesssim \|\zeta_q\|_{H^{1/2}(\Gamma)}$.

4. $\deg_{\max}(\psi|_{\Lambda}) < -\frac{1}{2}$: Let $d := \deg_{\max}(\psi|_{\Lambda})$. There is $\psi_1 \in \mathcal{A}_d(\Lambda) \setminus \{0\}$ and $\psi_2 \in \sum_{q<d} \mathcal{A}_q(\Lambda)$ s.t. $\psi|_{\Lambda} = \psi_1 + \psi_2$. By Lemma 2.16, $\deg(\partial_x^2|_{\Lambda}) = -2$, $\deg(\partial_{Y|\Gamma, Y=0^+}) = -1$ and $\deg(\partial_{Y|\Lambda}^2) = \deg(\partial_{Y|\Gamma, Y=0^-}) = 0$. In addition, Proposition 2.8 states that $\mathcal{A}(D) = \bigoplus_{q \in \mathbb{R}} \mathcal{A}_q(D)$ for any $D \in \{\Lambda, \Gamma\}$. So by taking the coordinates of $\Delta\psi|_{\Lambda}$ in $\mathcal{A}_d(\Lambda)$ and of $[\mu\partial_Y\psi]_{\Gamma}$ in $\mathcal{A}_d(\Gamma)$, we get if $d \ge -\frac{1}{2}$:

$$ \begin{cases} \partial_Y^2 \psi_1 = 0 & \text{in } \Lambda \\ \partial_{Y|\Gamma, Y=0^-} \psi_1 = 0 & \text{on } \Gamma \end{cases} $$

But since $\psi|_{\Sigma_{\Lambda}} = 0$, it implies $\psi_1 = 0$, which is contradictory.

5. $\deg_{\max}(\psi|_{\Omega}) < 0$: We denote $d := \deg_{\max}(\psi|_{\Omega})$. There is $\psi_1 \in \mathcal{A}_d(\Omega) \setminus \{0\}$ and $\psi_2 \in \sum_{q<d} \mathcal{A}_q(\Omega)$ s.t. $\psi|_{\Omega} = \psi_1 + \psi_2$. Since $\deg(\Delta|_{\Omega}) = -2$, taking the coordinate of $\Delta\psi|_{\Omega}$ in $\mathcal{A}_{d-2}(\Omega)$ and of $\psi|_{\Gamma}$ in $\mathcal{A}_d(\Lambda)$ gives if $d \ge 0$:

$$ \begin{cases} \Delta\psi_1 = 0 & \text{in } \Omega \\ \psi_1 = 0 & \text{on } \Gamma \end{cases} $$

By Proposition 2.12, it implies $d \in \frac{\pi}{\Theta}\mathbb{N}^*$ and $\psi_1 = \sigma_d(\psi_1)\phi_q$. But $\sigma_d(\psi_1) = \sigma_d(\psi) = \sigma_d(S) = 0$.

Finally, Lemma 3.9 implies $\chi_{\infty}\psi \in V$, so $S \in V$. So the uniqueness in Lemma 3.8 implies $S = 0$. $\square$

## 3.3 Construction of the fields

**Definition 3.12:** For any $(p, \ell) \in (\mathbb{R} \setminus \mathbb{P}) \times \mathbb{N}$ we denote by convention $u_{p,\ell} = 0$, $U_{p,\ell} = 0$ and $S_{p,\ell} = 0$. We define by induction on $p \in \mathbb{P}$ that for any $\ell \in \mathbb{N}$:

* $U_{p,\ell}: (x, Y) \in \Lambda \mapsto \sum_{n=0}^{\infty} (\partial_x^2 + k_1^2)^n \partial_y u_{p-1-2n,\ell}(x, 0) \cdot \mathcal{U}_n(Y). \quad (3.3)$

* $u_{p,\ell}$ is the unique solution in $H^1(\Omega) + \chi_0\mathcal{A}(\Omega)$ of:

$$ \left\{ \begin{array}{ll} \mu_0 \Delta u_{p,\ell} + \omega^2 \rho_0 u_{p,\ell} = f_s \delta_{p,0} \delta_{\ell,0} & \text{in } \Omega \\ u_{p,\ell} = 0 & \text{on } \Sigma_{\Omega} \\ u_{p,\ell} = U_{p,\ell} & \text{on } \Gamma \\ \sigma_d(u_{p,\ell}) = \sum_{p' \in [0, p+d]} \sum_{d' \in \frac{\pi}{\Theta}Z^*} \sum_{\ell'=0}^{\ell} c_{d, d', p-p', \ell-\ell'}^{u \leftarrow S} \cdot \sigma_{d'}(S_{p', \ell'}^{\infty}) & \forall d \in -\frac{\pi}{\Theta}N^* \end{array} \right. \quad (3.4) $$

* $S_{p,\ell}$ is the unique solution in $V + \chi_{\infty}\mathcal{A}(\Pi)$ of:

$$ \left\{ \begin{array}{ll} \text{div}(\mu\nabla S_{p,\ell}) = -\omega^2 \rho S_{p-2,\ell} & \text{in } \Omega_1 \\ S_{p,\ell} = 0 & \text{on } \partial\Omega_1 \\ \sigma_d(S_{p,\ell}) = \sum_{p' \in [0, p-d]} \sum_{d' \in \frac{\pi}{\Theta}Z^*} \sum_{\ell'=0}^{\ell} c_{d, d', p-p', \ell-\ell'}^{S \leftarrow u} \cdot \sigma_{d'}(u_{p', \ell'}^0) & \forall d \in -\frac{\pi}{\Theta}N^* \end{array} \right. \quad (3.5) $$

We will show that these fields are well-defined at the same time as the following proposition.

**Proposition 3.13:** For any $(p, \ell) \in \mathbb{P} \times \mathbb{N}$ there exist $u_{p,\ell}^0 \in \mathcal{A}^+(\Pi)$ and $S_{p,\ell}^\infty \in \mathcal{A}^-(\Pi)$ s.t., for any $d \in \mathbb{R}$, $u_{p,\ell} = T_{\le d}(u_{p,\ell}^0) + o_{\partial}(\mathbf{r}^d)$ when $\mathbf{r} \to 0$ and $S_{p,\ell} = T_{\ge d}(S_{p,\ell}^\infty) + o_{\partial}(\mathbf{r}^d)$ when $\mathbf{r} \to \infty$.
## 3.3 Construction of the fields

**Definition 3.12:** For any $(p, \ell) \in (\mathbb{R} \setminus \mathbb{P}) \times \mathbb{N}$ we denote by convention $u_{p,\ell} = 0$, $U_{p,\ell} = 0$ and $S_{p,\ell} = 0$. We define by induction on $p \in \mathbb{P}$ that for any $\ell \in \mathbb{N}$:

$$ \bullet \quad U_{p,\ell} : (x, Y) \in \Lambda \mapsto \sum_{n=0}^{\infty} (\partial_x^2 + k_1^2)^n \partial_y u_{p-1-2n,\ell}(x, 0) \cdot \mathcal{U}_n(Y). \quad (3.3) $$

• $u_{p,\ell}$ is the unique solution in $H^1(\Omega) + \chi_0\mathcal{A}(\Omega)$ of:

$$ \left\{ \begin{array}{ll} \mu_0 \Delta u_{p,\ell} + \omega^2 \rho_0 u_{p,\ell} = f_s \delta_{p,0} \delta_{\ell,0} & \text{in } \Omega \\ u_{p,\ell} = 0 & \text{on } \Sigma_{\Omega} \\ u_{p,\ell} = U_{p,\ell} & \text{on } \Gamma \\ \sigma_d(u_{p,\ell}) = \sum_{p' \in [0,p+d]_{\mathbb{P},N}} \sum_{d' \in \frac{\pi}{\Theta}Z^*} \sum_{\ell'=0}^{\ell} c_{d,d',p-p',\ell-\ell'}^{u \leftarrow S} \sigma_{d'}(S_{p',\ell'}^{\infty}) & \forall d \in -\frac{\pi}{\Theta}N^* \end{array} \right. \quad (3.4) $$

• $S_{p,\ell}$ is the unique solution in $V + \chi_\infty\mathcal{A}(\Pi)$ of:

$$ \left\{ \begin{array}{ll} \mathrm{div}(\mu \nabla S_{p,\ell}) = -\omega^2 \rho S_{p-2,\ell} & \text{in } \Omega_1 \\ S_{p,\ell} = 0 & \text{on } \partial \Omega_1 \\ \sigma_d(S_{p,\ell}) = \sum_{p' \in [0,p-d]_{\mathbb{P},N}} \sum_{d' \in \frac{\pi}{\Theta}Z^*} \sum_{\ell'=0}^{\ell} c_{d,d',p-p',\ell-\ell'}^{S \leftarrow u} \sigma_{d'}(u_{p',\ell'}^0) & \forall d \in \frac{\pi}{\Theta}N^* \end{array} \right. \quad (3.5) $$

We will show that these fields are well-defined at the same time as the following proposition.

**Proposition 3.13:** For any $(p, \ell) \in \mathbb{P} \times \mathbb{N}$ there exist $u_{p,\ell}^0 \in \mathcal{A}^+(\Pi)$ and $S_{p,\ell}^\infty \in \mathcal{A}^-(\Pi)$ s.t., for any $d \in \mathbb{R}$, $u_{p,\ell} = T_{\le d}(u_{p,\ell}^0) + o_\partial(\mathbf{r}^d)$ when $\mathbf{r} \to 0$ and $S_{p,\ell} = T_{\ge d}(S_{p,\ell}^\infty) + o_\partial(\mathbf{r}^d)$ when $\mathbf{r} \to \infty$.

**Justification of Definition 3.12 and Proposition 3.13:** Let us show by induction on $p$ that for any $\ell$:

(H1) $u_{p,\ell}$, $U_{p,\ell}$ and $S_{p,\ell}$ exist and are unique,

(H2) Proposition 3.13 is true at rank $(p, \ell)$,

(H3) $\forall m \in \mathbb{N}$, $\partial_x^m \partial_y u_{p,\ell|\Gamma} \in H_0^1(\Gamma) + \chi_0\mathcal{A}(\Gamma)$ (additional property that will be useful during the proof).

Since all fields are null for $p < 0$, the initial case is trivial and only the inductive step remains to prove. Let $p \in \mathbb{P}$. Let us assume (H1)-(H3) at any rank $p' < p$ and prove it at rank $p$.

1. Existence and uniqueness of the fields:

• $U_{p,\ell}$: By (H3), the $\partial_y u_{p',\ell|\Gamma}$ with $p' < p$ are all in $H_{\mathrm{loc}}^m(\Gamma)$ for any $m \in \mathbb{N}$. So the functions $(\partial_x^2 + k_1^2)^n \partial_y u_{p-1-2n,\ell|\Gamma}$ are continuous and Formula (3.3) is well-defined at any point of $\Lambda$.

• $u_{p,\ell}$: By (H2), $S_{p',\ell'}^\infty$ exists for any $p' < p$ and $\ell'$, so the coefficients $\sigma_{d'}(S_{p',\ell'}^\infty)$ are well-defined. Thus, (3.4) defines well $\sigma_d(u_{p,\ell}) \neq 0 \Rightarrow [0, p+d]_{\mathbb{P},N} \neq \emptyset \Rightarrow d \ge -p$, so $(\sigma_d(u_{p,\ell}))_{d \in -\frac{\pi}{\Theta}N^*}$ has finite support. Next, we apply Theorem 3.6. Its hypotheses are satisfied, because (H3) implies:

$$ U_{p,\ell|\Gamma} = \sum_{n=0}^{\infty} (\partial_x^2 + k_1^2)^n \partial_y u_{p-1-2n,\ell}(x, 0) \cdot \mathcal{U}_n(0) \in H_0^1(\Gamma) + \chi_0 \mathcal{A}(\Gamma) \subset H_{00}^{1/2}(\Gamma) + \chi_0 \mathcal{A}(\Gamma) $$

• $S_{p,\ell}$: Similarly, (H2) implies that $(\sigma_d(S_{p,\ell}))_{d \in \frac{\pi}{\Theta}N^*}$ is well defined and has finite support. Next, we apply Theorem 2.23. We need to check its hypotheses, i.e. there is $f: \Omega_1 \to \mathbb{C}$ and $\varphi: \Pi \to \mathbb{C}$ s.t. $\omega^2 \rho S_{p-2,\ell} = F + \chi_\infty \varphi$, $(1+r)F \in L^2(\Omega_1)$, $\varphi|_{\Omega} \in \mathcal{A}(\Omega)$ and $\varphi|_{\Lambda} \in \mathcal{A}(\Lambda)$. By (H2), there is $\psi \in \mathcal{A}(\Pi)$ s.t. $S_{p-2,\ell} = \psi + o_\partial(\mathbf{r}^{-4})$. So it suffices to set $\varphi := \omega^2 \rho \psi$ and $F := \omega^2 \rho S_{p-2,\ell} - \chi_\infty \varphi$.

2. Asymptotic expansions:
**Justification of Definition 3.12 and Proposition 3.13:** Let us show by induction on $p$ that for any $\ell$:

(H1) $u_{p,\ell}$, $U_{p,\ell}$ and $S_{p,\ell}$ exist and are unique,

(H2) Proposition 3.13 is true at rank $(p, \ell)$,

(H3) $\forall m \in \mathbb{N}$, $\partial_x^m \partial_y u_{p,\ell|\Gamma} \in H_0^1(\Gamma) + \chi_0 A(\Gamma)$ (additional property that will be useful during the proof).

Since all fields are null for $p < 0$, the initial case is trivial and only the inductive step remains to prove. Let $p \in \mathbb{P}$. Let us assume (H1)-(H3) at any rank $p' < p$ and prove it at rank $p$.

1. Existence and uniqueness of the fields:

* $U_{p,\ell}$: By (H3), the $\partial_y u_{p',\ell|\Gamma}$ with $p' < p$ are all in $H_{\text{loc}}^m(\Gamma)$ for any $m \in \mathbb{N}$. So the functions $(\partial_x^2 + k_1^2)^n \partial_y u_{p-1-2n,\ell|\Gamma}$ are continuous and Formula (3.3) is well-defined at any point of $\Lambda$.

* $u_{p,\ell}$: By (H2), $S_{p',\ell}^\infty$ exists for any $p' < p$ and $\ell'$, so the coefficients $\sigma_{d'}(S_{p',\ell}^\infty)$ are well-defined. Thus, (3.4) defines well $\sigma_d(u_{p,\ell})$. And we have: $\sigma_d(u_{p,\ell}) \neq 0 \Rightarrow [0, p+d]_{\mathbb{P},\mathbb{N}} \neq \emptyset \Rightarrow d \ge -p$, so $(\sigma_d(u_{p,\ell}))_{d \in -\frac{\pi}{\Theta}\mathbb{N}^*}$ has finite support. Next, we apply Theorem 3.6. Its hypotheses are satisfied, because (H3) implies:

$$U_{p,\ell|\Gamma} = \sum_{n=0}^{\infty} (\partial_x^2 + k_1^2)^n \partial_y u_{p-1-2n,\ell}(x, 0) \cdot \mathcal{U}_n(0) \in H_0^1(\Gamma) + \chi_0 A(\Gamma) \subset H_{00}^{1/2}(\Gamma) + \chi_0 A(\Gamma)$$

* $S_{p,\ell}$: Similarly, (H2) implies that $(\sigma_d(S_{p,\ell}))_{d \in -\frac{\pi}{\Theta}\mathbb{N}^*}$ is well defined and has finite support. Next, we apply Theorem 2.23. We need to check its hypotheses, i.e. there is $f: \Omega_1 \to \mathbb{C}$ and $\varphi: \Pi \to \mathbb{C}$ s.t. $\omega^2 \rho S_{p-2,\ell} = F + \chi_{\infty} \varphi$, $(1+r)F \in L^2(\Omega_1)$, $\varphi|_{\Omega} \in A(\Omega)$ and $\varphi|_{\Lambda} \in A(\Lambda)$. By (H2), there is $\psi \in A(\Pi)$ s.t. $S_{p-2,\ell} = \psi + o_\partial(\mathbf{r}^{-4})$. So it suffices to set $\varphi := \omega^2 \rho \psi$ and $F := \omega^2 \rho S_{p-2,\ell} - \chi_\infty \varphi$.

2. Asymptotic expansions:

* $u_{p,\ell}$: We use Theorem 2.22. To do so, we must check that there is $g_\Gamma \in A^+(\Gamma)$ and $g_\Lambda \in A^+(\Lambda)$ s.t.:
$$ \forall d \in \mathbb{R}, \quad \left\{ \begin{array}{ll} \partial_Y^2 U_{p,\ell} = T_{\le d}(g_\Lambda) + o_\partial(x^d) & \text{in } \Lambda \\ \partial_Y U_{p,\ell|Y=0^{-}} = T_{\le d}(g_\Gamma) + o_\partial(x^d) & \text{on } \Gamma \end{array} \right. $$

Given the definition of $U_{p,\ell}$ (3.3), it suffices to show that, for any $p' < p$ there is $h \in A^+(\Gamma)$ s.t.: $\forall d \in \mathbb{R}$, $\partial_n u_{p',\ell|\Gamma} = T_{\le d}(h) + o_\partial(x^d)$. But it derives from (H2).

* $S_{p,\ell}$: Similarly, we use Theorem 2.23 thanks to (H2).

3. Let $m \in \mathbb{N}$. The asymptotic expansion of $u_{p,\ell}$ implies: $\exists h \in A(\Gamma)$, $\partial_x^m \partial_y u_{p,\ell|\Gamma} = h + o_\partial(x^1)$. So there is $x_0 \in \mathbb{R}_+^*$ s.t. $\partial_x^m \partial_y u_{p,\ell|\Gamma} - h$ is $H^1$ on $\Gamma \cap \{x < x_0\} = (0, x_0) \times \{0\}$ and it vanishes at 0. To prove (H3) at rank $p$, it remains to show that $\partial_x^m \partial_y u_{p,\ell} \in H^1((x_0/2, \infty) \times \{0\})$. To do so, it suffices to get $u \in H^{m+3}((x_0/2, \infty) \times (0, \delta/2))$ with $\delta := \text{dist}(\text{supp}(f), \Gamma)$. But it follows from classical elliptic regularity because on one side $\mu_0 \Delta u_{p,\ell} + \omega^2 \rho_0 u_{p,\ell} = 0$ on $(x_0/4, \infty) \times (0, \delta)$, and on the other $u_{p,\ell|\Gamma} = U_{p,\ell|\Gamma} \in H^{m+3}((x_0/4, \infty) \times \{0\})$ by (3.3) and (H3). $\square$

**Proposition 3.14:** $\forall p \in \mathbb{P}, \exists n_p \in \mathbb{N}, \forall \ell > n_p$, $(u_{p,\ell} = 0$ and $U_{p,\ell} = 0$ and $S_{p,\ell} = 0)$.

*Proof:* For any $(d, d', p) \in (\frac{\pi}{\Theta}\mathbb{Z}^2)^2 \times \mathbb{P}$, one has $H_\varepsilon^{-1} \circ \langle \mathcal{R}_\varepsilon^\vee \rangle(\phi_{d'}) \in \mathcal{A}_\varepsilon^+(\Pi)$, so by definition of $\mathcal{A}_\varepsilon^+(\Pi)$ there is $n$ (depending of $d, d'$, $p$) s.t. for any $\ell > n$: $c_{d,d',p,\ell}^{\mathbf{u} \leftarrow S} := \sigma_d \circ \tau_{p,\ell} \circ H_\varepsilon^{-1} \circ \langle \mathcal{R}_\varepsilon^\vee \rangle(\phi_{d'}) = 0$. The same is true for the coefficients $c_{d,d',p,\ell}^{\mathbf{u} \leftarrow \mathbf{u}}$. Finally the result follows by induction from Definition 3.12. $\square$

**Proposition 3.15:** Ansatz 2.25 and the matching condition of Definition 2.28 are satisfied.

*Proof:* First let us note that for any $(p, \ell)$

$$ \sigma_d(u_{p,\ell}) = \sigma_d(\mathbf{u}_{p,\ell}^0) \text{ when } d \in -\frac{\pi}{\Theta}\mathbb{N}^* \quad \text{and} \quad \sigma_d(S_{p,\ell}) = \sigma_d(S_{p,\ell}^\infty) \text{ when } d \in \frac{\pi}{\Theta}\mathbb{N}^*. \quad (3.6) $$

where $\sigma_d(u_{p,\ell})$, resp. $\sigma_d(S_{p,\ell})$, is set by Definition 3.5, resp. 3.10, whereas $\sigma_d(\mathbf{u}_{p,\ell}^0)$ and $\sigma_d(S_{p,\ell}^\infty)$ rest on the definition of $\sigma_d$ on $A^\pm(\Pi)$ at page 15. Indeed we have $u_{p,\ell} - T_{\le 1}(\mathbf{u}_{p,\ell}^0)|_{\Omega} = o_\partial(r^1)$, so $\chi_0 \cdot (u_{p,\ell} -$
• $\mathbf{u}_{p,\ell}$: We use Theorem 2.22. To do so, we must check that there is $g_{\Gamma} \in \mathcal{A}^{+}(\Gamma)$ and $g_{\Lambda} \in \mathcal{A}^{+}(\Lambda)$ s.t.:

$$ \forall d \in \mathbb{R}, \quad \left\{ \begin{array}{ll} \partial_Y^2 U_{p,\ell} = T_{\le d}(g_{\Lambda}) + o_{\partial}(x^d) & \text{in } \Lambda \\ \partial_Y U_{p,\ell|\Gamma=0^{-}} = T_{\le d}(g_{\Gamma}) + o_{\partial}(x^d) & \text{on } \Gamma \end{array} \right. $$

Given the definition of $U_{p,\ell}$ (3.3), it suffices to show that, for any $p' < p$ there is $h \in \mathcal{A}^{+}(\Gamma)$ s.t.: $\forall d \in \mathbb{R}$, $\partial_n u_{p',\ell|\Gamma} = T_{\le d}(h) + o_{\partial}(x^d)$. But it derives from (H2).

• $S_{p,\ell}$: Similarly, we use Theorem 2.23 thanks to (H2).

3. Let $m \in \mathbb{N}$. The asymptotic expansion of $\mathbf{u}_{p,\ell}$ implies: $\exists h \in \mathcal{A}(\Gamma)$, $\partial_x^m \partial_y u_{p,\ell|\Gamma} = h + o_{\partial}(x^1)$. So there is $x_0 \in \mathbb{R}_+^*$ s.t. $\partial_x^m \partial_y u_{p,\ell|\Gamma} - h$ is $H^1$ on $\Gamma \cap \{x < x_0\} = (0, x_0) \times \{0\}$ and it vanishes at 0. To prove (H3) at rank $p$, it remains to show that $\partial_x^m \partial_y u_{p,\ell} \in H^1((\frac{x_0}{2}, \infty) \times \{0\})$. To do so, it suffices to get $u \in H^{m+3}((\frac{x_0}{2}, \infty) \times (0, \frac{\delta}{2}))$ with $\delta := \text{dist}(\text{supp}(f), \Gamma)$. But it follows from classical elliptic regularity because on one side $\mu_0 \Delta u_{p,\ell} + \omega^2 \rho_0 u_{p,\ell} = 0$ on $(\frac{x_0}{4}, \infty) \times (0, \delta)$, and on the other $u_{p,\ell|\Gamma} = U_{p,\ell|\Gamma} \in H^{m+3}((\frac{x_0}{4}, \infty) \times \{0\})$ by (3.3) and (H3). $\square$

**Proposition 3.14:** $\forall p \in \mathbb{P}, \exists n_p \in \mathbb{N}, \forall \ell > n_p$, ($u_{p,\ell} = 0$ and $U_{p,\ell} = 0$ and $S_{p,\ell} = 0$).

*Proof:* For any $(d, d', p) \in (\frac{\pi}{\Theta}\mathbb{Z}^*)^2 \times \mathbb{P}$, one has $(\text{id} - \langle \pi_{\varepsilon}^+ | \mathcal{R}_{\varepsilon}^+ \rangle)(\phi_{d'}) \in \mathcal{A}_{\varepsilon}^+(\Pi)$, so by definition of $\mathcal{A}_{\varepsilon}^+(\Pi)$ there is $n$ (depending of $d, d', p$) s.t. for any $\ell > n$: $c_{d,d',p,\ell}^{u \leftarrow S} := \sigma_d \circ \tau_{p+d',\ell} \circ (\text{id} - \langle \pi_{\varepsilon}^+ | \mathcal{R}_{\varepsilon}^+ \rangle)(\phi_{d'}) = 0$. The same is true for the coefficients $c_{d,d',p,\ell}^{S \leftarrow u}$. Finally the result follows by induction from Definition 3.12. $\square$

**Proposition 3.15:** Ansatz 2.25 and the matching condition of Definition 2.27 is satisfied.

*Proof:* First let us note that for any $(p, \ell)$

$$ \sigma_d(u_{p,\ell}) = \sigma_d(\mathbf{u}_{p,\ell}^0) \text{ when } d \in -\frac{\pi}{\Theta}\mathbb{N}^* \quad \text{and} \quad \sigma_d(S_{p,\ell}) = \sigma_d(S_{p,\ell}^\infty) \text{ when } d \in \frac{\pi}{\Theta}\mathbb{N}^*. \quad (3.6) $$

where $\sigma_d(u_{p,\ell})$, resp. $\sigma_d(S_{p,\ell})$, is set by Definition 3.5, resp. 3.10, whereas $\sigma_d(\mathbf{u}_{p,\ell}^0)$ and $\sigma_d(S_{p,\ell}^\infty)$ rest on the definition of $\sigma_d$ on $\mathcal{A}^\pm(\Pi)$ at page 15. Indeed we have $u_{p,\ell} - T_{\le 1}(\mathbf{u}_{p,\ell}^0)|_{\Omega} = o_{\partial}(r^1)$, so $\chi_0 \cdot (u_{p,\ell} - T_{\le 1}(\mathbf{u}_{p,\ell}^0)|_{\Omega}) \in H^1(\Omega)$ which implies by Definition 3.5 that $\sigma_d(u_{p,\ell}) = \sigma_d(T_{\le 1}(\mathbf{u}_{p,\ell}^0)|_{\Omega}) = \sigma_d(\mathbf{u}_{p,\ell}^0)$. Likewise for $S_{p,\ell}$.

Now, given Proposition 3.14, to prove Ansatz 2.25 it suffices to check that, for any $(p, \ell)$, $\mathbf{u}_{p,\ell}^0 \in \sum_{d \in \mathbb{P}-p} \mathcal{A}_d(\Pi)$ and $S_{p,\ell}^\infty \in \sum_{d \in \mathbb{P}-p} \mathcal{A}_d(\Pi)$. Let us show it for $\mathbf{u}_{p,\ell}^0$ only, using induction.

Since all fields vanish for $p < 0$, only the inductive step is non-trivial. (2.25) states that

$$ \mathbf{u}_{p,\ell}^0 = \sum_{n=0}^{\infty} (-k_0^2 R_{\Delta})^n \left( \underbrace{R_{\partial_Y^2} \circ (\partial_{x|\Lambda}^2 + k_1^2)(\mathbf{u}_{p-2,\ell}^0) + \frac{\mu_0}{\mu_1} R_{\mathbb{N}} \circ \partial_{y|\Gamma, y=0^+}}_{:=A} + \sum_{d \in \frac{\pi}{\Theta}\mathbb{Z}^*} \sigma_d(\mathbf{u}_{p,\ell}^0) \phi_d \right). $$

Since deg $R_{\Delta} \in \mathbb{N}$, it suffices to show that the big brackets belong to $\sum_{d \in \mathbb{P}-p} \mathcal{A}_d(\Pi)$. This is true for $A$ using the induction hypothesis and deg $R_{\partial_Y^2} = 0$, deg($\partial_{x|\Lambda}^2$) = -2 and deg($R_{\mathbb{N}} \circ \partial_{y|\Gamma, y=0^+}$) = -1 (see Proposition 2.14 and Lemma 2.16). It remains to show that $\forall d \in \frac{\pi}{\Theta}\mathbb{Z}^*$, $\sigma_d(\mathbf{u}_{p,\ell}^0) \ne 0 \Rightarrow d \in \mathbb{P}-p$. This last assertion holds because on one hand $\frac{\pi}{\Theta}\mathbb{N}^* \subset \mathbb{P}-p$, and on the other (3.6) and (3.4) imply: $\forall d \in -\frac{\pi}{\Theta}\mathbb{N}^*, \sigma_d(\mathbf{u}_{p,\ell}^0) \ne 0 \Leftrightarrow \sigma_d(u_{p,\ell}) \ne 0 \Rightarrow [0, p+d]_{\mathbb{P},\mathbb{N}} \ne \emptyset \Rightarrow d \in \mathbb{P}-p$.

Finally (3.4)-(3.5) and (3.6) show that the matching relations of Theorem 2.31 are satisfied, and we can apply Theorem 2.31 thanks to Ansatz 2.25. Thus the matching condition of Definition 2.27 is satisfied. $\square$

## 3.4 Practical way to build the far fields

This section shows how to build directly the far fields without computing the layer and corner fields. Thanks to the explicit expression of the layer fields in (3.3), the layer is replaced by boundary conditions on $\Gamma$, while the corner fields are replaced by corner conditions depending on corner profiles.
$T_{\le 1}(\mathbf{u}_{p,\ell}^0)|_{\Omega} \in H^1(\Omega)$ which implies by Definition 3.5 that $\sigma_d(u_{p,\ell}) = \sigma_d(T_{\le 1}(\mathbf{u}_{p,\ell}^0)|_{\Omega}) = \sigma_d(\mathbf{u}_{p,\ell}^0)$. Likewise for $S_{p,\ell}$.

Now, given Proposition 3.14, to prove Ansatz 2.25 it suffices to check that, for any $(p, \ell)$, $\mathbf{u}_{p,\ell}^0 \in \sum_{d \in \mathbb{P}-p} \mathcal{A}_d(\Pi)$ and $S_{p,\ell}^\infty \in \sum_{d \in \mathbb{P}-p} \mathcal{A}_d(\Pi)$. Let us show it for $\mathbf{u}_{p,\ell}^0$ only, using induction.

Since all fields vanish for $p < 0$, only the inductive step is non-trivial. (2.25) states that

$$ \mathbf{u}_{p,\ell}^0 = \sum_{n=0}^{\infty} (-k_0^2 R_{\Delta})^n \left( \underbrace{R_{\partial_Y^2} \circ (\partial_{x|\Lambda}^2 + k_1^2)(\mathbf{u}_{p-2,\ell}^0) + \frac{\mu_0}{\mu_1} R_N \circ \partial_{y|\Gamma,y=0^+}(\mathbf{u}_{p-1,\ell}^0)}_{:=A} + \sum_{d \in \frac{\pi}{\Theta}\mathbb{Z}^*} \sigma_d(\mathbf{u}_{p,\ell}^0) \phi_d \right). $$

Since deg $R_{\Delta} \in \mathbb{N}$, it suffices to show that the big brackets belong to $\sum_{d \in \mathbb{P}-p} \mathcal{A}_d(\Pi)$. This is true for $A$ using the induction hypothesis and deg $R_{\partial_Y^2} = 0$, deg$(\partial_{x|\Lambda}^2) = -2$ and deg$(R_N \circ \partial_{y|\Gamma,y=0^+}) = -1$ (see Proposition 2.14 and Lemma 2.16). It remains to show that $\forall d \in \frac{\pi}{\Theta}\mathbb{Z}^*$, $\sigma_d(\mathbf{u}_{p,\ell}^0) \ne 0 \Rightarrow d \in \mathbb{P}-p$. This last assertion holds because on one hand $\frac{\pi}{\Theta}\mathbb{N}^* \subset \mathbb{P}-p$, and on the other (3.6) and (3.4) imply: $\forall d \in -\frac{\pi}{\Theta}\mathbb{N}^*$, $\sigma_d(\mathbf{u}_{p,\ell}^0) \ne 0 \Leftrightarrow \sigma_d(u_{p,\ell}) \ne 0 \Rightarrow [0, p+d]_{\mathbb{P},N} \ne \emptyset \Rightarrow d \in \mathbb{P}-p$.

Finally (3.4)-(3.5) and (3.6) show that the matching relations of Theorem 2.30 are satisfied, and we can apply Theorem 2.30 thanks to Ansatz 2.25. Thus the matching condition of Definition 2.28 is satisfied. $\square$

## 3.4 Practical way to build the far fields

This section shows how to build directly the far fields without computing the layer and corner fields. Thanks to the explicit expression of the layer fields in (3.3), the layer is replaced by boundary conditions on $\Gamma$, while the corner fields are replaced by corner conditions depending on corner profiles.

**Definition 3.16: corner profiles**

Let $d \in \frac{\pi}{\Theta}\mathbb{N}^*$. We denote $(S_{d,n})_{n \in \mathbb{N}}$ the unique sequence of $V + \chi_\infty\mathcal{A}(\Pi)$ s.t. for any $n \in \mathbb{N}^*$:

$$ \left\{ \begin{array}{ll} \text{div}(\mu \nabla S_{d,0}) = 0 & \text{in } \Omega_1 \\ S_{d,0} = 0 & \text{on } \partial\Omega_1 \\ \sigma_q(S_{d,0}) = \delta_{d,q} & \forall q \in \frac{\pi}{\Theta}\mathbb{N}^* \end{array} \right. \quad \text{and} \quad \left\{ \begin{array}{ll} \text{div}(\mu \nabla S_{d,n}) = -\omega^2 \rho S_{d,n-1} & \text{in } \Omega_1 \\ S_{d,n} = 0 & \text{on } \partial\Omega_1 \\ \sigma_q(S_{d,n}) = 0 & \forall q \in \frac{\pi}{\Theta}\mathbb{N}^* \end{array} \right. $$

And for any $(d, n)$ we denote $S_{d,n}^\infty$ the element of $\mathcal{A}^-(\Pi)$ s.t. $\forall d \in \mathbb{R}$, $S_{d,n} = T_{\ge d}(S_{d,n}^\infty) + o_\partial(\mathbf{r}^d)$.

These objects are well-defined thanks to Theorems 3.11 and 2.23. The proof is the same as for Definition 3.12 and Proposition 3.13.

Using (3.5) and the uniqueness in Theorem 3.11, one can easily show by induction on $p$:

$$ \forall (p, \ell) \in \mathbb{P} \times \mathbb{N}, \quad S_{p,\ell} = \sum_{n=0}^{\infty} \sum_{d \in \frac{\pi}{\Theta}\mathbb{N}^* \cap (p-2n-\mathbb{P})} \sigma_d(S_{p-2n,\ell}) S_{d,n} \quad (3.7) $$

Thus, the same holds replacing $S_{p,\ell}$ and $S_{d,n}$ by resp. $S_{p,\ell}^\infty$ and $S_{d,n}^\infty$.

For any $(d, d', p, \ell) \in (-\frac{\pi}{\Theta}\mathbb{N}^*) \times \frac{\pi}{\Theta}\mathbb{Z}^* \times \mathbb{P} \times \mathbb{N}$, we introduce the corner coefficient $c_{d,d',p,\ell}^{\mathbf{u} \leftarrow \mathbf{u}}$, we have

$$ c_{d,d',p,\ell}^{\mathbf{u} \leftarrow \mathbf{u}} := \sum_{\substack{(p_1,p_2) \in (\mathbb{N}-d) \times (\mathbb{N}+d') \\ n \in \mathbb{N}, p_1+p_2+2n=p}} \sum_{p_1+d_1 \in \mathbb{N}} \sum_{d_1 \in \frac{\pi}{\Theta}\mathbb{Z}^*} \sum_{d_2 \in \frac{\pi}{\Theta}\mathbb{N}^*} \sum_{\substack{(\ell_1, \ell_2) \in \mathbb{N}^2 \\ \ell_1+\ell_2=\ell}} c_{d,d_1,p_1,\ell_1}^{\mathbf{u} \leftarrow S} \cdot \sigma_{d_1}(S_{d_2,n}^\infty) \cdot c_{d_2,d',p_2,\ell_2}^{S \leftarrow \mathbf{u}} \quad (3.8a) $$

$$ = \sum_{\substack{(p_1,p_2) \in (\mathbb{N}-d) \times (\mathbb{N}+d') \\ n \in \mathbb{N}, p_1+p_2+2n=p}} \sum_{p_1+d_1 \in \mathbb{N}} \sum_{d_2 \in \frac{\pi}{\Theta}\mathbb{N}^*} \sum_{\substack{(\ell_1, \ell_2) \in \mathbb{N}^2 \\ \ell_1+\ell_2=\ell}} c_{d,d_1,p_1,\ell_1}^{\mathbf{u} \leftarrow S} \cdot \sigma_{d_1}(S_{d_2,n}^\infty) \cdot c_{d_2,d',p_2,\ell_2}^{S \leftarrow \mathbf{u}} \quad (3.8b) $$

$$ + \sum_{\substack{(p_1,p_2) \in (\mathbb{N}-d) \times (\mathbb{N}+d') \\ p_1+p_2=p}} \sum_{p_1+d_1 \in \mathbb{N}, p_2-d_1 \in \mathbb{N}} \sum_{\substack{(\ell_1, \ell_2) \in \mathbb{N}^2 \\ \ell_1+\ell_2=\ell}} c_{d,d_1,p_1,\ell_1}^{\mathbf{u} \leftarrow S} \cdot c_{d_1,d',p_2,\ell_2}^{S \leftarrow \mathbf{u}} \quad (3.8c) $$

(the two given formulas are equal because, when $d_1 > 0$, $\sigma_{d_1}(S_{d_2,n}^\infty) = \sigma_{d_1}(S_{d_2,n}) = \delta_{d_1,d_2} \delta_{n,0}$). Like in Remark 2.31, we have $d - d_1, d_2 - d' \in \mathbb{Z} \cap \frac{\pi}{\Theta}\mathbb{Z}$ in (3.8a). Thus, if $d - d' \notin \mathbb{Z} \cap \frac{\pi}{\Theta}\mathbb{Z}$, line (3.8c)
**Definition 3.16: corner profiles**

Let $d \in \frac{\pi}{\Theta} \mathbb{N}^*$. We denote $(S_{d,n})_{n \in \mathbb{N}}$ the unique sequence of $V + \chi_\infty \mathcal{A}(\Pi)$ s.t. for any $n \in \mathbb{N}^*$:

$$ egin{cases} 
\operatorname{div}(\mu \nabla S_{d,0}) = 0 & \text{in } \Omega_1 \\
S_{d,0} = 0 & \text{on } \partial\Omega_1 \\
\sigma_q(S_{d,0}) = \delta_{d,q} & \forall q \in \frac{\pi}{\Theta} \mathbb{N}^* 
\end{cases} 
\quad \text{and} 
\quad 
\begin{cases} 
\operatorname{div}(\mu \nabla S_{d,n}) = -\omega^2 \rho S_{d,n-1} & \text{in } \Omega_1 \\
S_{d,n} = 0 & \text{on } \partial\Omega_1 \\
\sigma_q(S_{d,n}) = 0 & \forall q \in \frac{\pi}{\Theta} \mathbb{N}^* 
\end{cases} $$

And for any $(d, n)$ we denote $S_{d,n}^\infty$ the element of $\mathcal{A}^-(\Pi)$ s.t. $\forall d \in \mathbb{R}$, $S_{d,n} = T_{\ge d}(S_{d,n}^\infty) + o_\partial(\mathbf{r}^d)$.

These objects are well-defined thanks to Theorems 3.11 and 2.23. The proof is the same as for Definition 3.12 and Proposition 3.13.

Using (3.5) and the uniqueness in Theorem 3.11, one can easily show by induction on $p$:

$$ \forall (p, \ell) \in \mathbb{P} \times \mathbb{N}, \quad S_{p,\ell} = \sum_{n=0}^{\infty} \sum_{d \in \frac{\pi}{\Theta} \mathbb{N}^* \cap (p-2n-\mathbb{P})} \sigma_d(S_{p-2n,\ell}) S_{d,n} \quad (3.7) $$

Thus, the same holds replacing $S_{p,\ell}$ and $S_{d,n}$ by resp. $S_{p,\ell}^\infty$ and $S_{d,n}^\infty$.

For any $(d, d', p, \ell) \in (-\frac{\pi}{\Theta} \mathbb{N}^*) \times \frac{\pi}{\Theta} \mathbb{Z}^* \times \mathbb{P} \times \mathbb{N}$, we introduce the corner coefficient $c_{d,d',p,\ell}^{u \leftarrow u}$, we have

$$ c_{d,d',p,\ell}^{u \leftarrow u} := \sum_{\substack{(p_1,p_2) \in (\mathbb{N}-d) \times (\mathbb{N}+d') \\ n \in \mathbb{N}, p_1+p_2+2n=p}} \sum_{p_1+d_1 \in \mathbb{N}} \sum_{d_2 \in \frac{\pi}{\Theta} \mathbb{N}^*} \sum_{\substack{\ell_1, \ell_2 \in \mathbb{N}^2 \\ \ell_1+\ell_2=\ell}} c_{d,d_1,p_1,\ell_1}^{u \leftarrow S} \cdot \sigma_{d_1}(S_{d_2,n}^\infty) \cdot c_{d_2,d',p_2,\ell_2}^{S \leftarrow u} \quad (3.8a) $$

$$ = \sum_{\substack{(p_1,p_2) \in (\mathbb{N}-d) \times (\mathbb{N}+d') \\ n \in \mathbb{N}, p_1+p_2+2n=p}} \sum_{p_1+d_1 \in \mathbb{N}} \sum_{d_2 \in \frac{\pi}{\Theta} \mathbb{N}^*} \sum_{\substack{\ell_1, \ell_2 \in \mathbb{N}^2 \\ \ell_1+\ell_2=\ell}} c_{d,d_1,p_1,\ell_1}^{u \leftarrow S} \cdot \sigma_{d_1}(S_{d_2,n}^\infty) \cdot c_{d_2,d',p_2,\ell_2}^{S \leftarrow u} \quad (3.8b) $$

$$ + \sum_{\substack{(p_1,p_2) \in (\mathbb{N}-d) \times (\mathbb{N}+d') \\ p_1+p_2=p}} \sum_{p_1+d_1 \in \mathbb{N}, p_2-d_1 \in \mathbb{N}} \sum_{\substack{\ell_1, \ell_2 \in \mathbb{N}^2 \\ \ell_1+\ell_2=\ell}} c_{d,d_1,p_1,\ell_1}^{u \leftarrow S} \cdot c_{d_1,d',p_2,\ell_2}^{S \leftarrow u} \quad (3.8c) $$

(the two given formulas are equal because, when $d_1 > 0$, $\sigma_{d_1}(S_{d_2,n}^\infty) = \sigma_{d_1}(S_{d_2,n}) = \delta_{d_1,d_2} \delta_{n,0}$). Like in Remark 2.32, we have $d - d_1, d_2 - d' \in \mathbb{Z} \cap \frac{\pi}{\Theta} \mathbb{Z}$ in (3.8a). Thus, if $d - d' \notin \mathbb{Z} \cap \frac{\pi}{\Theta} \mathbb{Z}$, line (3.8c) vanishes. Moreover, for any $(p, \ell)$, we denote $u_{p,\ell}^0 := u_{p,\ell|\Omega}^0 \in \mathcal{A}^+(\Omega)$, which satisfies: $\forall d \in \mathbb{R}$, $u_{p,\ell} = \underset{r \to 0}{T_d(u_{p,\ell}^0) + o_\partial(r^d)}$.

**Theorem 3.17: direct construction of the far fields**

Let $(T_n)_{n \in \mathbb{N}}$ be the sequence of Taylor coefficients of the tangent: $\forall t \in (-\frac{\pi}{2}, \frac{\pi}{2})$, $\tan t = \sum_{n=0}^\infty T_n t^{2n+1}$. $(u_{p,\ell})_{p \in \mathbb{P}, \ell \in \mathbb{N}}$ is the unique family of $H^1(\Omega) + \chi_0 \mathcal{A}(\Omega)$ s.t. for any $(p, \ell) \in \mathbb{P} \times \mathbb{N}$:

$$ 
\left\{ 
\begin{array}{ll}
\mu_0 \Delta u_{p,\ell} + \omega^2 \rho_0 u_{p,\ell} = f_s \delta_{p,0} \delta_{\ell,0} & \text{in } \Omega \\
u_{p,\ell} = 0 & \text{on } \Sigma_{\Omega} \\
u_{p,\ell} = \sum_{n=0}^{\infty} \frac{\mu_0}{\mu_1} T_n \cdot (\partial_x^2 + k_1^2)^n \partial_y u_{p-1-2n,\ell} & \text{on } \Gamma \\
\sigma_d(u_{p,\ell}) = \sum_{p' \in \mathbb{P}} \sum_{d' \in \frac{\pi}{\Theta} \mathbb{Z}^*} \sum_{\ell'=0}^{\ell} c_{d,d',p-p',\ell-\ell'}^{u \leftarrow u} \cdot \sigma_{d'}(u_{p',\ell'}) & \forall d \in -\frac{\pi}{\Theta} \mathbb{N}^*
\end{array}
\right. 
\quad \begin{array}{l}
p-p' \in \mathbb{P} + \frac{2\pi}{\Theta} \quad p'+d' \in [0,p+d]_{\mathbb{P},\mathbb{N}}
\end{array} $$

**Proof:** Uniqueness follows from the uniqueness in Theorem 3.6. So it suffices to prove that the far fields satisfy the equations above.

**Boundary condition:** By (3.3) and (3.4), we have

$$ u_{p,\ell}(x, 0) = U_{p,\ell}(x, 0) = \sum_{n=0}^{\infty} (\partial_x^2 + k_1^2)^n \partial_y u_{p-1-2n,\ell}(x, 0) \cdot \mathcal{U}_n(0). $$
vanishes. Moreover, for any $(p, \ell)$, we denote $u_{p,\ell}^0 := \mathbf{u}_{p,\ell|\Omega}^0 \in \mathcal{A}^+(\Omega)$, which satisfies: $\forall d \in \mathbb{R}$, $u_{p,\ell} \underset{r \to 0}{=} T_{\le d}(u_{p,\ell}^0) + o_{\partial}(r^d)$.

**Theorem 3.17: direct construction of the far fields**

Let $(T_n)_{n \in \mathbb{N}}$ be the sequence of Taylor coefficients of the tangent: $\forall t \in (-\frac{\pi}{2}, \frac{\pi}{2})$, $\tan t = \sum_{n=0}^{\infty} T_n t^{2n+1}$. $(u_{p,\ell})_{p \in \mathbb{P}, \ell \in \mathbb{N}}$ is the unique family of $H^1(\Omega) + \chi_0 \mathcal{A}(\Omega)$ s.t. for any $(p, \ell) \in \mathbb{P} \times \mathbb{N}$:

$$
\left\{
\begin{array}{ll}
\mu_0 \Delta u_{p,\ell} + \omega^2 \rho_0 u_{p,\ell} = f_s \delta_{p,0} \delta_{\ell,0} & \text{in } \Omega \\
u_{p,\ell} = 0 & \text{on } \Sigma_{\Omega} \\
u_{p,\ell} = \sum_{n=0}^{\infty} \frac{\mu_0}{\mu_1} T_n \cdot (\partial_x^2 + k_1^2)^n \partial_y u_{p-1-2n,\ell} & \text{on } \Gamma \\
\sigma_d(u_{p,\ell}) = \sum_{\substack{p' \in \mathbb{P} \\ p-p' \in \mathbb{P} + \frac{2\pi}{\Theta}}} \sum_{\substack{d' \in \bar{\Theta} Z^* \\ p'+d' \in [0,p+d]_{\mathbb{P},\mathbb{N}}}} \sum_{\ell'=0}^{\ell} c_{d,d',p-p',\ell-\ell'}^{\mathbf{u} \leftarrow \mathbf{u}} \cdot \sigma_{d'}(u_{p',\ell'}^0) & \forall d \in -\frac{\pi}{\Theta} \mathbb{N}^*
\end{array}
\right.
$$

**Proof:** Uniqueness follows from the uniqueness in Theorem 3.6. So it suffices to prove that the far fields satisfy the equations above.

**Boundary condition:** By (3.3) and (3.4), we have

$$ u_{p,\ell}(x,0) = U_{p,\ell}(x,0) = \sum_{n=0}^{\infty} (\partial_x^2 + k_1^2)^n \partial_y u_{p-1-2n,\ell}(x,0) \cdot \mathcal{U}_n(0). $$

So we need to calculate $\mathcal{U}_n(0)$. Let $U: (Y, t) \mapsto \sum_{n=0}^{\infty} \mathcal{U}_n(Y) t^{2n+1}$. The definition of $(\mathcal{U}_n)$ (reminded in (3.9) below) formally implies a differential equation on $U$ given in (3.10).

$$
\forall n \in \mathbb{N}, \quad \left\{
\begin{array}{l}
\mathcal{U}_n'' = -\mathcal{U}_{n-1} \\
\mathcal{U}_n'(0) = \frac{\mu_0}{\mu_1} \delta_{n,0} \\
\mathcal{U}_n(-1) = 0
\end{array}
\right.
\qquad (3.9) \qquad \left\{
\begin{array}{l}
\partial_Y^2 U = -t^2 U \\
\partial_Y U|_{Y=0} = \mathcal{U}_0'(0) \cdot t = \frac{\mu_0}{\mu_1} t \\
U|_{Y=-1} = 0
\end{array}
\right.
\qquad (3.10)
$$

Thus $U(Y, t) = \frac{\mu_0 \sin(t(Y+1))}{\mu_1 \cos(t)} := \varphi(Y, t)$. This is formal, as we do not know whether the series $U$ converges. However, there is a sequence of polynomial functions $(\Phi_n)$ s.t., for any $(Y, t) \in (-1, 0) \times (-\frac{\pi}{2}, \frac{\pi}{2})$, $\varphi(Y, t) = \sum_{n=0}^{\infty} \Phi_n(Y) t^{2n+1}$ (because $\varphi$ is odd w.r.t. $t$). Since $\varphi$ satisfies (3.10), $(\Phi_n)$ is a solution of (3.9). But this solution is unique, so $(\Phi_n) = (\mathcal{U}_n)$ and $U = \varphi$. Hence $\forall t \in (-\frac{\pi}{2}, \frac{\pi}{2})$, $\sum_{n=0}^{\infty} \mathcal{U}_n(0) t^{2n+1} = \frac{\mu_0}{\mu_1} \tan(t)$. That is to say $\forall n \in \mathbb{N}$, $\mathcal{U}_n(0) = \frac{\mu_0}{\mu_1} T_n$.

**Corner condition:** For any $(d, p, \ell) \in \frac{\pi}{\Theta} \mathbb{N}^* \times \mathbb{R} \times \mathbb{N}$, let $\tilde{S}_{d,p,\ell}^{\infty} := S_{d,p/2}^{\infty}$ if $\frac{p}{2} \in \mathbb{N}$ and $\ell = 0$ and $\tilde{S}_{d,p,\ell}^{\infty} := 0$ otherwise. For any family $(x_{p,\ell})$, we denote $x_{\bullet,\bullet} := (x_{p,\ell})$ the family itself. We also denote $*$ the convolution product w.r.t. $(p, \ell)$. Let $d \in -\frac{\pi}{\Theta} \mathbb{N}^*$. We have

$$ \sigma_d(u_{\bullet, \bullet}) = \sum_{d_1 \in \frac{\pi}{\Theta} Z^*} c_{d, d_1, \bullet, \bullet}^{\mathbf{u} \leftarrow S} * \sigma_{d_1}(S_{\bullet, \bullet}^{\infty}) \quad \text{by (3.4)} $$

$$ = \sum_{d_1 \in \frac{\pi}{\Theta} Z^*} c_{d, d_1, \bullet, \bullet}^{\mathbf{u} \leftarrow S} * \sum_{d_2 \in \frac{\pi}{\Theta} \mathbb{N}^*} \sigma_{d_1}(\tilde{S}_{d_2, \bullet, \bullet}^{\infty}) * \sigma_{d_2}(S_{\bullet, \bullet}) \quad \text{by (3.7)} $$

$$ = \sum_{d_1 \in \frac{\pi}{\Theta} Z^*} c_{d, d_1, \bullet, \bullet}^{\mathbf{u} \leftarrow S} * \sum_{d_2 \in \frac{\pi}{\Theta} \mathbb{N}^*} \sigma_{d_1}(\tilde{S}_{d_2, \bullet, \bullet}^{\infty}) * \sum_{d' \in \frac{\pi}{\Theta} Z^*} c_{d_2, d', \bullet, \bullet}^{S \leftarrow \mathbf{u}} * \sigma_{d'}(\mathbf{u}_{\bullet, \bullet}^0) \quad \text{by (3.5)} $$

$$ = \sum_{d' \in \frac{\pi}{\Theta} Z^*} \left( \sum_{d_1 \in \frac{\pi}{\Theta} Z^*} \sum_{d_2 \in \frac{\pi}{\Theta} \mathbb{N}^*} c_{d, d_1, \bullet, \bullet}^{\mathbf{u} \leftarrow S} * \sigma_{d_1}(\tilde{S}_{d_2, \bullet, \bullet}^{\infty}) * c_{d_2, d', \bullet, \bullet}^{S \leftarrow \mathbf{u}} \right) * \sigma_{d'}(u_{\bullet, \bullet}^0) $$

$$ = \sum_{d' \in \frac{\pi}{\Theta} Z^*} \tilde{c}_{d, d', \bullet, \bullet}^{\mathbf{u} \leftarrow \mathbf{u}} * \sigma_{d'}(u_{\bullet, \bullet}^0) $$

$$ \text{with } \tilde{c}_{d, d', p, \ell}^{\mathbf{u} \leftarrow \mathbf{u}} := \sum_{(p_1, p_2), n \in \mathbb{N}} \sum_{p_1+p_2+2n=p} \sum_{d_1 \in \frac{\pi}{\Theta} Z^*} \sum_{d_2 \in \frac{\pi}{\Theta} \mathbb{N}^*} \sum_{(\ell_1, \ell_2) \in \mathbb{N}^2} c_{d, d_1, p_1, \ell_1}^{\mathbf{u} \leftarrow S} * \sigma_{d_1}(S_{d_2, n}^{\infty}) * c_{d_2, d', p_2, \ell_2}^{S \leftarrow \mathbf{u}}. $$
So we need to calculate $\mathcal{U}_n(0)$. Let $U : (Y, t) \mapsto \sum_{n=0}^{\infty} \mathcal{U}_n(Y) t^{2n+1}$. The definition of $(\mathcal{U}_n)$ (reminded in (3.9) below) formally implies a differential equation on $U$ given in (3.10).

$$ \forall n \in \mathbb{N}, \quad \left\{ \begin{array}{l} U_n'' = -U_{n-1} \\ U_n'(0) = \frac{\mu_0}{\mu_1} \delta_{n,0} \\ U_n(-1) = 0 \end{array} \right. \qquad (3.9) \qquad \left\{ \begin{array}{l} \partial_Y^2 U = -t^2 U \\ \partial_Y U|_{Y=0} = U_0'(0) \cdot t = \frac{\mu_0}{\mu_1} t \\ U|_{Y=-1} = 0 \end{array} \right. \qquad (3.10) $$

Thus $U(Y, t) = \frac{\mu_0}{\mu_1} \frac{\sin(t(Y+1))}{\cos(t)} := \varphi(Y, t)$. This is formal, as we do not know whether the series $U$ converges. However, there is a sequence of polynomial functions $(\Phi_n)$ s.t., for any $(Y, t) \in (-1, 0) \times (-\frac{\pi}{2}, \frac{\pi}{2})$, $\varphi(Y, t) = \sum_{n=0}^{\infty} \Phi_n(Y) t^{2n+1}$ (because $\varphi$ is odd w.r.t. $t$). Since $\varphi$ satisfies (3.10), $(\Phi_n)$ is a solution of (3.9). But this solution is unique, so $(\Phi_n) = (\mathcal{U}_n)$ and $U = \varphi$. Hence $\forall t \in (-\frac{\pi}{2}, \frac{\pi}{2})$, $\sum_{n=0}^{\infty} \mathcal{U}_n(0) t^{2n+1} = \frac{\mu_0}{\mu_1} \tan(t)$. That is to say $\forall n \in \mathbb{N}$, $\mathcal{U}_n(0) = \frac{\mu_0}{\mu_1} T_n$.

**Corner condition:** For any $(d, p, \ell) \in \frac{\pi}{\Theta}\mathbb{N}^* \times \mathbb{R} \times \mathbb{N}$, let $\tilde{S}_{d,p,\ell}^{\infty} := S_{d,p/2}^{\infty}$ if $\frac{p}{2} \in \mathbb{N}$ and $\ell = 0$ and $\tilde{S}_{d,p,\ell}^{\infty} := 0$ otherwise. For any family $(x_{p,\ell})$, we denote $x_{\bullet,\bullet} := (x_{p,\ell})$ the family itself. We also denote $*$ the convolution product w.r.t. $(p, \ell)$. Let $d \in -\frac{\pi}{\Theta}\mathbb{N}^*$. We have

$$ \sigma_d(u_{\bullet, \bullet}) = \sum_{d_1 \in \frac{\pi}{\Theta}Z^*} c_{d, d_1, \bullet, \bullet}^{u \leftarrow S} * \sigma_{d_1}(S_{\bullet, \bullet}^{\infty}) \qquad \text{by (3.4)} $$

$$ = \sum_{d_1 \in \frac{\pi}{\Theta}Z^*} c_{d, d_1, \bullet, \bullet}^{u \leftarrow S} * \sum_{d_2 \in \frac{\pi}{\Theta}\mathbb{N}^*} \sigma_{d_1}(\tilde{S}_{d_2, \bullet, \bullet}^{\infty}) * \sigma_{d_2}(S_{\bullet, \bullet}) \qquad \text{by (3.7)} $$

$$ = \sum_{d_1 \in \frac{\pi}{\Theta}Z^*} c_{d, d_1, \bullet, \bullet}^{u \leftarrow S} * \sum_{d_2 \in \frac{\pi}{\Theta}\mathbb{N}^*} \sigma_{d_1}(\tilde{S}_{d_2, \bullet, \bullet}^{\infty}) * \sum_{d' \in \frac{\pi}{\Theta}Z^*} c_{d_2, d', \bullet, \bullet}^{S \leftarrow u} * \sigma_{d'}(u_{\bullet, \bullet}^0) \qquad \text{by (3.5)} $$

$$ = \sum_{d' \in \frac{\pi}{\Theta}Z^*} \left( \sum_{d_1 \in \frac{\pi}{\Theta}Z^*} \sum_{d_2 \in \frac{\pi}{\Theta}\mathbb{N}^*} c_{d, d_1, \bullet, \bullet}^{u \leftarrow S} * \sigma_{d_1}(\tilde{S}_{d_2, \bullet, \bullet}^{\infty}) * c_{d_2, d', \bullet, \bullet}^{S \leftarrow u} \right) * \sigma_{d'}(u_{\bullet, \bullet}^0) $$

$$ = \sum_{d' \in \frac{\pi}{\Theta}Z^*} \tilde{c}_{d, d', \bullet, \bullet}^{u \leftarrow u} * \sigma_{d'}(u_{\bullet, \bullet}^0) $$

$$ \text{with } \tilde{c}_{d, d', p, \ell}^{u \leftarrow u} := \sum_{(p_1, p_2), n \in \mathbb{N}} \sum_{p_1+p_2+2n=p} \sum_{d_1 \in \frac{\pi}{\Theta}Z^*} \sum_{d_2 \in \frac{\pi}{\Theta}\mathbb{N}^*} \sum_{\ell_1, \ell_2=\ell} c_{d, d_1, p_1, \ell_1}^{u \leftarrow S} \cdot \sigma_{d_1}(S_{d_2, n}^{\infty}) \cdot c_{d_2, d', p_2, \ell_2}^{S \leftarrow u} $$

Moreover, for any $R \in \mathcal{R}_{\varepsilon}^{\pm}$, we have $\deg_{\varepsilon} R \in \mathbb{N}$ and $\deg_{\varepsilon} R \pm \deg_{\mathcal{A}} R \in \mathbb{N}$ (see Figure 9). So by definition of the coefficients $c_{\dots}^{u \leftarrow S}$ and $c_{\dots}^{S \leftarrow u}$:

$$ \bullet \quad c_{d, d_1, p_1, \ell_1}^{u \leftarrow S} \neq 0 \Rightarrow p_1 + d \in \mathbb{N} \text{ and } p_1 + d_1 \in \mathbb{N}, $$

$$ \bullet \quad c_{d_2, d', p_2, \ell_2}^{S \leftarrow u} \neq 0 \Rightarrow p_2 - d_2 \in \mathbb{N} \text{ and } p_2 - d' \in \mathbb{N}. $$

This implies that $\tilde{c}_{d, d', p, \ell}^{u \leftarrow u} = c_{d, d', p, \ell}^{u \leftarrow u}$ for any $(d, d', p, \ell)$.

Furthermore, those conditions on $(p_1, p_2, d_1, d_2)$ imply that, if $c_{d, d', p, \ell}^{u \leftarrow u} \neq 0$, then:

$$ \bullet \quad p = (p_1 + d) + 2n + (p_2 - d_2) - d + d_2 \in \mathbb{N} + \frac{\pi}{\Theta}\mathbb{N}^* + \frac{\pi}{\Theta}\mathbb{N}^* \subset \mathbb{P} + \frac{2\pi}{\Theta} $$

$$ \bullet \quad \text{and } p + d - d' = (p_1 + d) + 2n + (p_2 - d') \in \mathbb{N}. $$

This and the property $(\sigma_{d'}(u_{p',\ell'}) \neq 0 \Rightarrow p' \in \mathbb{P} \text{ and } p' + d' \in \mathbb{P})$ explain the sum indexes of the formula given for $\sigma_d(u_{p,\ell})$ in Theorem 3.17. $\square$

**Example:** Using Theorem 3.17, one can check that $u_{\pi/\Theta, \ell}$ vanishes for any $\ell$. In addition $\frac{2\pi}{\Theta} > 1$, so the first non-zero far fields are $u_{0,0}$ and $u_{1,0}$ and they satisfy

$$ \left\{ \begin{array}{ll} \mu_0 \Delta u_{0,0} + \omega^2 \rho_0 u_{0,0} = f_s & \text{in } \Omega \\ u_{0,0} = 0 & \text{on } \Sigma_{\Omega} \\ u_{0,0} = 0 & \text{on } \Gamma \\ \sigma_d(u_{p,\ell}) = 0 & \forall d \in -\frac{\pi}{\Theta}\mathbb{N}^* \end{array} \right. \quad \text{and} \quad \left\{ \begin{array}{ll} \mu_0 \Delta u_{1,0} + \omega^2 \rho_0 u_{1,0} = 0 & \text{in } \Omega \\ u_{1,0} = 0 & \text{on } \Sigma_{\Omega} \\ u_{1,0} = \frac{\mu_0}{\mu_1} \partial_y u_{0,0} & \text{on } \Gamma \\ \sigma_d(u_{1,0}) = 0 & \forall d \in -\frac{\pi}{\Theta}\mathbb{N}^* \end{array} \right. $$
Moreover, for any $R \in \mathbb{R}_{\varepsilon}^{\pm}$, we have $\deg_{\varepsilon} R \in \mathbb{N}$ and $\deg_{\varepsilon} R \pm \deg_{A} R \in \mathbb{N}$ (see Figure 9). So by definition of the coefficients $c_{\dots, \dots, \dots}^{\mathbf{u} \leftarrow \mathbf{u}}$ and $c_{\dots, \dots, \dots}^{\mathbf{u} \leftarrow \mathbf{u}}$:

* $c_{d,d_1,p_1,\ell_1}^{\mathbf{u} \leftarrow \mathbf{u}} \neq 0 \Rightarrow p_1 + d \in \mathbb{N}$ and $p_1 + d_1 \in \mathbb{N}$,

* $c_{d_2,d',p_2,\ell_2}^{\mathbf{u} \leftarrow \mathbf{u}} \neq 0 \Rightarrow p_2 - d_2 \in \mathbb{N}$ and $p_2 - d' \in \mathbb{N}$.

This implies that $\tilde{c}_{d,d',p,\ell}^{\mathbf{u} \leftarrow \mathbf{u}} = c_{d,d',p,\ell}^{\mathbf{u} \leftarrow \mathbf{u}}$ for any $(d, d', p, \ell)$.

Furthermore, those conditions on $(p_1, p_2, d_1, d_2)$ imply that, if $c_{d,d',p,\ell}^{\mathbf{u} \leftarrow \mathbf{u}} \neq 0$, then:

* $p = (p_1 + d) + 2n + (p_2 - d_2) - d + d_2 \in \mathbb{N} + \frac{\pi}{\Theta}\mathbb{N}^* + \frac{\pi}{\Theta}\mathbb{N}^* \subset \mathbb{P} + \frac{2\pi}{\Theta}$

* and $p + d - d' = (p_1 + d) + 2n + (p_2 - d') \in \mathbb{N}$.

This and the property $(\sigma_{d'}(u_{p',\ell'}) \neq 0 \Rightarrow p' \in \mathbb{P}$ and $p' + d' \in \mathbb{P})$ explain the sum indexes of the formula given for $\sigma_d(u_{p,\ell})$ in Theorem 3.17. $\square$

**Example:** Using Theorem 3.17, one can check that $u_{\pi/\Theta,\ell}$ vanishes for any $\ell$. In addition $\frac{2\pi}{\Theta} > 1$, so the first non-zero far fields are $u_{0,0}$ and $u_{1,0}$ and they satisfy

$$ \left\{ \begin{array}{ll} \mu_0 \Delta u_{0,0} + \omega^2 \rho_0 u_{0,0} = f_s & \text{in } \Omega \\ u_{0,0} = 0 & \text{on } \Sigma_{\Omega} \\ u_{0,0} = 0 & \text{on } \Gamma \\ \sigma_d(u_{p,\ell}) = 0 & \forall d \in -\frac{\pi}{\Theta} \mathbb{N}^* \end{array} \right. $$

and

$$ \left\{ \begin{array}{ll} \mu_0 \Delta u_{1,0} + \omega^2 \rho_0 u_{1,0} = 0 & \text{in } \Omega \\ u_{1,0} = 0 & \text{on } \Sigma_{\Omega} \\ u_{1,0} = \frac{\mu_0}{\mu_1} \partial_y u_{0,0} & \text{on } \Gamma \\ \sigma_d(u_{1,0}) = 0 & \forall d \in -\frac{\pi}{\Theta} \mathbb{N}^* \end{array} \right. $$

# 4 Error estimates

Let $\chi \in C^{\infty}(\mathbb{R}^2)$ be equal to 1 on $B(0, 1)$ and 0 outside $B(0, 2)$, and, for any $\eta > 0$, $\chi_{\eta} : x \mapsto \chi(\frac{x}{\eta})$. We denote $\Lambda_{\varepsilon} := (-\varepsilon, 1) \times \mathbb{R}_{+}^*$ and $\Pi_{\varepsilon} := \Omega \sqcup \Gamma \sqcup \Lambda_{\varepsilon}$ (defined similarly as $\Pi$ at page 6). We define on $\Pi_{\varepsilon}$ the following variant of the far-and-layer fields $u_{p,\ell}$

$$ \mathbf{u}_{p,\ell}^{\varepsilon}(x, y) := \begin{cases} u_{p,\ell}(x, y) & \text{in } \Omega \\ U_{p,\ell}(x, \frac{y}{\varepsilon}) & \text{in } \Lambda_{\varepsilon} \end{cases} $$

We also denote, for any $p \in \mathbb{P}$, $n_p := \max\{\ell \in \mathbb{N} \mid u_{p,\ell} \neq 0 \text{ or } U_{p,\ell} \neq 0 \text{ or } S_{p,\ell} \neq 0\}$.

Let us define the approximate global field at order $P \in \mathbb{R}_+$ as follows for any $\varepsilon$ small enough

$$ \forall (x, y) \in \Omega_{\varepsilon}, \quad u_{\varepsilon,P}(x, y) := (1 - \chi_{\eta}(x, y)) \sum_{\substack{p \in \mathbb{P} \cap [0, P] \\ \ell \in [0, n_p]}} \varepsilon^p \ln^\ell \varepsilon \mathbf{u}_{p,\ell}^{\varepsilon}(x, y) + \chi_{\eta}(x, y) \sum_{\substack{p \in \mathbb{P} \cap [0, P] \\ \ell \in [0, n_p]}} \varepsilon^p \ln^\ell \varepsilon S_{p,\ell}(\frac{x}{\varepsilon}, \frac{y}{\varepsilon}) $$

where $\eta := \sqrt{\varepsilon}$. Note that $(1 - \chi_{\eta}(x, y))\mathbf{u}_{p,\ell}^{\varepsilon}(x, y)$ is well-defined on $\Omega_{\varepsilon}$ when $\eta > \varepsilon R_c$ since $\Pi_{\varepsilon} \setminus B(0, \varepsilon R_c) = \Omega_{\varepsilon} \setminus B(0, \varepsilon R_c)$. We will see as a consequence of Theorem 4.2 that $u_{\varepsilon,P} \in H^1(\Omega_{\varepsilon})$.

The matching zone is $\Omega_{\varepsilon} \cap C_{\eta}$ where $C_{\eta}$ is the annulus $C_{\eta} := B(0, 2\eta) \setminus B(0, \eta)$. Letting $\eta = \sqrt{\varepsilon}$ makes the matching zone tend to 0 w.r.t. the far fields (because $\eta \to 0$ when $\varepsilon \to 0$) and to infinity w.r.t. the corner fields (because $\frac{\eta}{\varepsilon} \to \infty$). Thanks to the matching assumption, we can state the following first error estimate concerning the error in the matching zone. We use the symbol $\lesssim$ for majorations valid up to a constant independent of $\varepsilon$.

**Lemma 4.1:** Let $P \in \mathbb{R}_+$. For $\varepsilon$ small enough, we have

$$ \left\| \sum_{p \in \mathbb{P} \cap [0, P]} \sum_{\ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon \left( \mathbf{u}_{p,\ell}^\varepsilon(x, y) - S_{p,\ell}\left(\frac{x}{\varepsilon}, \frac{y}{\varepsilon}\right) \right) \right\|_{H^1(\Omega_\varepsilon \cap C_\eta)} \lesssim \varepsilon^{\frac{P}{2}-1}. $$

**Proof:** We will compare $u_{p,\ell}^{\varepsilon}$ and $S_{p,\ell}$ in $C_{\eta}$ to their asymptotic expansions at 0, resp $\infty$. Let us denote:

* $u_{\varepsilon,P} := \sum_{p \in \mathbb{P} \cap [0, P]} \sum_{\ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon u_{p,\ell}^{\varepsilon}$ in $\Pi_{\varepsilon}$
# 4 Error estimates

Let $\chi \in C^\infty(\mathbb{R}^2)$ be equal to 1 on $B(0, 1)$ and 0 outside $B(0, 2)$, and, for any $\eta > 0$, $\chi_\eta : x \mapsto \chi(\frac{x}{\eta})$. We denote $\Lambda_\varepsilon := (-\varepsilon, 1) \times \mathbb{R}_+$ and $\Pi_\varepsilon := \Omega \sqcup \Gamma \sqcup \Lambda_\varepsilon$ (defined similarly as $\Pi$ at page 6). We define on $\Pi_\varepsilon$ the following variant of the far-and-layer fields $\mathbf{u}_{p,\ell}$

$$ \mathbf{u}_{p,\ell}^{\varepsilon}(x, y) := \begin{cases} u_{p,\ell}(x, y) & \text{in } \Omega \\ U_{p,\ell}(x, \frac{y}{\varepsilon}) & \text{in } \Lambda_{\varepsilon} \end{cases} $$

We also denote, for any $p \in \mathbb{P}$, $n_p := \max\{\ell \in \mathbb{N} | u_{p,\ell} \neq 0 \text{ or } U_{p,\ell} \neq 0 \text{ or } S_{p,\ell} \neq 0\}$.

Let us define the approximate global field at order $P \in \mathbb{R}_+$ as follows for any $\varepsilon$ small enough

$$ \forall (x, y) \in \Omega_{\varepsilon}, \quad u_{\varepsilon,P}(x, y) := (1 - \chi_{\eta}(x, y)) \sum_{\substack{p \in \mathbb{P} \cap [0, P] \\ \ell \in [0, n_p]}} \varepsilon^p \ln^\ell \varepsilon \mathbf{u}_{p, \ell}^\varepsilon(x, y) + \chi_\eta(x, y) \sum_{\substack{p \in \mathbb{P} \cap [0, P] \\ \ell \in [0, n_p]}} \varepsilon^p \ln^\ell \varepsilon S_{p, \ell}(\frac{x}{\varepsilon}, \frac{y}{\varepsilon}) $$

where $\eta := \sqrt{\varepsilon}$. Note that $(1 - \chi_\eta(x, y))\mathbf{u}_{p,\ell}^\varepsilon(x, y)$ is well-defined on $\Omega_\varepsilon$ when $\eta > \varepsilon R_c$ since $\Pi_\varepsilon \setminus B(0, \varepsilon R_c) = \Omega_\varepsilon \setminus B(0, \varepsilon R_c)$. We will see as a consequence of Theorem 4.2 that $u_{\varepsilon,P} \in H^1(\Omega_\varepsilon)$.

The matching zone is $\Omega_\varepsilon \cap C_\eta$ where $C_\eta$ is the annulus $C_\eta := B(0, 2\eta) \setminus B(0, \eta)$. Letting $\eta = \sqrt{\varepsilon}$ makes the matching zone tend to 0 w.r.t. the far fields (because $\eta \to 0$ when $\varepsilon \to 0$) and to infinity w.r.t. the corner fields (because $\frac{\eta}{\varepsilon} \to \infty$). Thanks to the matching assumption, we can state the following first error estimate concerning the error in the matching zone. We use the symbol $\lesssim$ for majorations valid up to a constant independent of $\varepsilon$.

**Lemma 4.1:** Let $P \in \mathbb{R}_+$. For $\varepsilon$ small enough, we have

$$ \left\| \sum_{p \in \mathbb{P} \cap [0, P]} \sum_{\ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon \left( \mathbf{u}_{p, \ell}^\varepsilon(x, y) - S_{p, \ell}\left(\frac{x}{\varepsilon}, \frac{y}{\varepsilon}\right) \right) \right\|_{H^1(\Omega_\varepsilon \cap C_\eta)} \lesssim \varepsilon^{\frac{P}{2}-1}. $$

**Proof:** We will compare $\mathbf{u}_{p,\ell}^\varepsilon$ and $S_{p,\ell}$ in $C_\eta$ to their asymptotic expansions at 0, resp $\infty$. Let us denote:

$$ \bullet \quad \mathbf{u}_{\varepsilon,P} := \sum_{p \in \mathbb{P} \cap [0, P]} \sum_{\ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon \mathbf{u}_{p,\ell}^\varepsilon \text{ in } \Pi_\varepsilon $$

$$ \bullet \quad S_{\varepsilon,P} := \sum_{p \in \mathbb{P} \cap [0, P]} \sum_{\ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon S_{p,\ell}\left(\frac{x}{\varepsilon}, \frac{y}{\varepsilon}\right) \text{ in } \Omega_\varepsilon $$

$$ \bullet \quad \mathbf{u}_{\varepsilon,P}^A := \sum_{p \in \mathbb{P} \cap [0, P]} \sum_{\ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon \cdot \begin{cases} T \leq_{P-p} (\mathbf{u}_{p,\ell}^0)(x, y) & \text{in } \Omega \\ T \leq_{P-p} (\mathbf{u}_{p,\ell}^0)(x, \frac{y}{\varepsilon}) & \text{in } \Lambda_\varepsilon \end{cases} $$

$$ \bullet \quad S_{\varepsilon,P}^A := \sum_{p \in \mathbb{P} \cap [0, P]} \sum_{\ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon \cdot T_{\geq p-P} (S_{p,\ell}^\infty)\left(\frac{x}{\varepsilon}, \frac{y}{\varepsilon}\right) \text{ in } \Pi_\varepsilon. $$

(see Definition 2.21 for $T_\bullet$). We split the estimate into three parts that we will majorize separately:

$$ \|\mathbf{u}_{\varepsilon,P} - S_{\varepsilon,P}\|_{H^1(\Omega_\varepsilon \cap C_\eta)} \le \|\mathbf{u}_{\varepsilon,P} - \mathbf{u}_{\varepsilon,P}^A\|_{H^1(\Omega_\varepsilon \cap C_\eta)} + \|\mathbf{u}_{\varepsilon,P}^A - S_{\varepsilon,P}^A\|_{H^1(\Omega_\varepsilon \cap C_\eta)} + \|S_{\varepsilon,P}^A - S_{\varepsilon,P}\|_{H^1(\Omega_\varepsilon \cap C_\eta)} $$

In addition, we will split some of the norms $\|\dots\|_{H^1(\Omega_\varepsilon \cap C_\eta)}$ into $\|\dots\|_{H^1(\Omega_\varepsilon \cap C_\eta)} + \|\dots\|_{H^1(\Lambda_\varepsilon \cap C_\eta)}$.

1. $\|\mathbf{u}_{\varepsilon,P} - \mathbf{u}_{\varepsilon,P}^A\|_{H^1(\Omega_\varepsilon \cap C_\eta)}$: By Proposition 3.13, for any $(p, \ell)$ we have $u_{p,\ell} - T_{\leq P-p}(\mathbf{u}_{p,\ell}^0) = o_\partial(r^{P-p})$ in $\Omega$ when $r \to 0$. By Definition 2.20 of $o_\partial$, it implies $u_{p,\ell} - T_{\leq P-p}(\mathbf{u}_{p,\ell}^0) = O(r^{P-p}) = O(r^{P-p-1})$ and $\nabla[u_{p,\ell} - T_{\leq P-p}(\mathbf{u}_{p,\ell}^0)] = O(r^{P-p-1})$ uniformly in $\theta$. Thus:

$$ \begin{align*} \|\mathbf{u}_{\varepsilon,P} - \mathbf{u}_{\varepsilon,P}^A\|_{H^1(\Omega \cap C_\eta)} &\le \sum_{p \in \mathbb{P} \cap [0, P]} \sum_{\ell=0}^{n_p} \varepsilon^p \ln^\ell \varepsilon \cdot \|u_{p,\ell} - T_{\le P-p}(\mathbf{u}_{p,\ell}^0)\|_{H^1(\Omega \cap C_\eta)} \\ &\lesssim \sum_{p \in \mathbb{P} \cap [0, P]} \sum_{\ell=0}^{n_p} \varepsilon^p \ln^\ell \varepsilon \cdot \left( \int_{B(0,2\eta)} r^{2(P-p-1)} dr r d\theta \right)^{1/2} \\ &\lesssim \sum_{p \in \mathbb{P} \cap [0, P]} \sum_{\ell=0}^{n_p} \varepsilon^p \ln^\ell \varepsilon \cdot \eta^{P-p} \\ &\lesssim \eta^{P-2} \quad \text{because } \varepsilon^p \lesssim \eta^p \text{ and } \ln^\ell \varepsilon \lesssim \eta^{-2} \end{align*} $$
• $S_{\varepsilon,P} := \sum_{p \in \mathbb{P} \cap [0,P]} \sum_{\ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon S_{p,\ell}(\frac{x}{\varepsilon}, \frac{y}{\varepsilon})$ in $\Omega_\varepsilon$

• $\mathbf{u}_{\varepsilon,P}^A := \sum_{p \in \mathbb{P} \cap [0,P]} \sum_{\ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon \cdot \begin{cases} T_{\le P-p}(\mathbf{u}_{p,\ell}^0)(x,y) & \text{in } \Omega \\ T_{\le P-p}(\mathbf{u}_{p,\ell}^0)(x, \frac{y}{\varepsilon}) & \text{in } \Lambda_\varepsilon \end{cases}$

• $S_{\varepsilon,P}^A := \sum_{p \in \mathbb{P} \cap [0,P]} \sum_{\ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon \cdot T_{\ge p-P}(S_{p,\ell}^\infty)(\frac{x}{\varepsilon}, \frac{y}{\varepsilon})$ in $\Pi_\varepsilon$.

(see Definition 2.21 for $T_\bullet$). We split the estimate into three parts that we will majorize separately:

$$ \|\mathbf{u}_{\varepsilon,P} - S_{\varepsilon,P}\|_{H^1(\Omega_{\varepsilon} \cap C_{\eta})} \le \|\mathbf{u}_{\varepsilon,P} - \mathbf{u}_{\varepsilon,P}^A\|_{H^1(\Omega_{\varepsilon} \cap C_{\eta})} + \|\mathbf{u}_{\varepsilon,P}^A - S_{\varepsilon,P}\|_{H^1(\Omega_{\varepsilon} \cap C_{\eta})} + \|S_{\varepsilon,P}^A - S_{\varepsilon,P}\|_{H^1(\Omega_{\varepsilon} \cap C_{\eta})} $$

In addition, we will split some of the norms $\|\dots\|_{H^1(\Omega_{\varepsilon} \cap C_{\eta})}$ into $\|\dots\|_{H^1(\Omega_{\varepsilon} \cap C_{\eta})}$ and $\|\dots\|_{H^1(\Lambda_{\varepsilon} \cap C_{\eta})}$.

1. $\|\mathbf{u}_{\varepsilon,P} - \mathbf{u}_{\varepsilon,P}^A\|_{H^1(\Omega_{\varepsilon} \cap C_{\eta})}$: By Proposition 3.13, for any $(p, \ell)$ we have $u_{p,\ell} - T_{\le P-p}(\mathbf{u}_{p,\ell}^0) = o_\partial(r^{P-p})$ in $\Omega$ when $r \to 0$. By Definition 2.20 of $o_\partial$, it implies $u_{p,\ell} - T_{\le P-p}(\mathbf{u}_{p,\ell}^0) = O(r^{P-p}) = O(r^{P-p-1})$ and $\nabla[u_{p,\ell} - T_{\le P-p}(\mathbf{u}_{p,\ell}^0)] = O(r^{P-p-1})$ uniformly in $\theta$. Thus:

$$ \begin{aligned} \|\mathbf{u}_{\varepsilon,P} - \mathbf{u}_{\varepsilon,P}^A\|_{H^1(\Omega \cap C_{\eta})} &\le \sum_{p \in \mathbb{P} \cap [0,P]} \sum_{\ell=0}^{n_p} \varepsilon^p \ln^\ell \varepsilon \cdot \|u_{p,\ell} - T_{\le P-p}(\mathbf{u}_{p,\ell}^0)\|_{H^1(\Omega \cap C_{\eta})} \\ &\lesssim \sum_{p \in \mathbb{P} \cap [0,P]} \sum_{\ell=0}^{n_p} \varepsilon^p \ln^\ell \varepsilon \cdot \left( \int_{B(0,2\eta)} r^{2(P-p-1)} dr \, r d\theta \right)^{1/2} \\ &\lesssim \sum_{p \in \mathbb{P} \cap [0,P]} \sum_{\ell=0}^{n_p} \varepsilon^p \ln^\ell \varepsilon \cdot \eta^{P-p} \\ &\lesssim \eta^{P-2} \quad \text{because } \varepsilon^p \lesssim \eta^p \text{ and } \ln^\ell \varepsilon \lesssim \eta^{-2} \end{aligned} $$

2. $\|\mathbf{u}_{\varepsilon,P} - \mathbf{u}_{\varepsilon,P}^A\|_{H^1(\Lambda_{\varepsilon} \cap C_{\eta})}$: Similarly, $U_{p,\ell} - T_{\le P-p}(\mathbf{u}_{p,\ell}^0) = o_\partial(x^{P-p})$ in $\Lambda$ when $x \to \infty$. Thus

• $U_{p,\ell} - T_{\le P-p}(\mathbf{u}_{p,\ell}^0) = O(x^{P-p})$ so $U_{p,\ell}(x, \frac{y}{\varepsilon}) - T_{\le P-p}(\mathbf{u}_{p,\ell}^0)(x, \frac{y}{\varepsilon}) = O(x^{P-p})$

• $\partial_x[U_{p,\ell} - T_{\le P-p}(\mathbf{u}_{p,\ell}^0)] = O(x^{P-p-1})$ so $\partial_x[U_{p,\ell}(x, \frac{y}{\varepsilon}) - T_{\le P-p}(\mathbf{u}_{p,\ell}^0)(x, \frac{y}{\varepsilon})] = O(\varepsilon^{-1}x^{P-p-1})$

• $\partial_Y[U_{p,\ell} - T_{\le P-p}(\mathbf{u}_{p,\ell}^0)] = O(x^{P-p})$ so $\partial_Y[U_{p,\ell}(x, \frac{y}{\varepsilon}) - T_{\le P-p}(\mathbf{u}_{p,\ell}^0)(x, \frac{y}{\varepsilon})] = O(\varepsilon^{-1}x^{P-p})$

which are all $O(\varepsilon^{-1}x^{P-p-1})$ (uniformly in $Y$). Hence:

$$ \begin{aligned} \|\mathbf{u}_{\varepsilon,P} - \mathbf{u}_{\varepsilon,P}^A\|_{H^1(\Lambda_{\varepsilon} \cap C_{\eta})} &\lesssim \sum_{p \in \mathbb{P} \cap [0,P]} \sum_{\ell=0}^{n_p} \varepsilon^p \ln^\ell \varepsilon \cdot \varepsilon^{-1} \left( \int_{[0,2\eta] \times [-\varepsilon,0]} x^{2(P-p-1)} \, dx \, dy \right)^{1/2} \\ &\lesssim \sum_{p \in \mathbb{P} \cap [0,P]} \sum_{\ell=0}^{n_p} \varepsilon^p \ln^\ell \varepsilon \cdot \varepsilon^{-1} \cdot \varepsilon^{\frac{1}{2}} \eta^{P-p-\frac{1}{2}} \\ &\lesssim \eta^{P-2} \quad \text{because } \varepsilon = \eta^2 \text{ and } \ln^\ell \varepsilon \lesssim \eta^{-\frac{1}{2}} \end{aligned} $$

3. $\|S_{\varepsilon,P} - S_{\varepsilon,P}^A\|_{H^1(\Omega_{\varepsilon} \cap C_{\eta})}$: Similarly $S_{p,\ell} - T_{\ge p-P}(S_{p,\ell}^\infty) = o_\partial(r^{p-P})$ in $\Omega$ when $r \to \infty$, so $S_{p,\ell} - T_{\ge p-P}(S_{p,\ell}^\infty) = O(r^{p-P})$ and $\nabla_{(X,Y)}[S_{p,\ell} - T_{\ge p-P}(S_{p,\ell}^\infty)] = O(r^{p-P-1}) = O(r^{p-P})$ uniformly in $\theta$. Since $\nabla_{(x,y)} = \varepsilon^{-1}\nabla_{(X,Y)}$, we deduce:

$$ \begin{aligned} \|S_{\varepsilon,P} - S_{\varepsilon,P}^A\|_{H^1(\Omega \cap C_{\eta})} &\le \sum_{p \in \mathbb{P} \cap [0,P]} \sum_{\ell=0}^{n_p} \varepsilon^p \ln^\ell \varepsilon \cdot \left\| [S_{p,\ell} - T_{\ge p-P}(S_{p,\ell}^\infty)](\frac{x}{\varepsilon}, \frac{y}{\varepsilon}) \right\|_{H^1(\Omega \cap C_{\eta})} \\ &\lesssim \sum_{p \in \mathbb{P} \cap [0,P]} \sum_{\ell=0}^{n_p} \varepsilon^p \ln^\ell \varepsilon \cdot \varepsilon^{-1} \left( \int_{\mathbb{R}^2 \setminus B(0,\eta)} \left(\frac{r}{\varepsilon}\right)^{2(p-P)} dr \, r d\theta \right)^{1/2} \\ &\lesssim \sum_{p \in \mathbb{P} \cap [0,P]} \sum_{\ell=0}^{n_p} \varepsilon^p \ln^\ell \varepsilon \cdot \varepsilon^{P-p-1} \cdot \eta^{p-P+1} \\ &\lesssim \eta^{P-2} \quad \text{because } \varepsilon = \eta^2 \text{ and } \ln^\ell \varepsilon \lesssim \eta^{-1} \end{aligned} $$

4. $\|S_{\varepsilon,P} - S_{\varepsilon,P}^A\|_{H^1(\Lambda_{\varepsilon} \cap C_{\eta})}$: $S_{p,\ell} - T_{\ge p-P}(S_{p,\ell}^\infty) = o_\partial(X^{p-P})$ in $\Lambda$ when $x \to 0$, so:
2. $\|\mathbf{u}_{\varepsilon,P} - \mathbf{u}_{\varepsilon,P}^A\|_{H^1(\Lambda_{\varepsilon} \cap C_{\eta})}$: Similarly, $U_{p,\ell} - T_{\leqslant P-p}(\mathbf{u}_{p,\ell}^0) = o_{\partial}(x^{P-p})$ in $\Lambda$ when $x \to \infty$. Thus

• $U_{p,\ell} - T_{\leqslant P-p}(\mathbf{u}_{p,\ell}^0) = O(x^{P-p})$ so $U_{p,\ell}(x, \frac{y}{\varepsilon}) - T_{\leqslant P-p}(\mathbf{u}_{p,\ell}^0)(x, \frac{y}{\varepsilon}) = O(x^{P-p})$

• $\partial_x [U_{p,\ell} - T_{\leqslant P-p}(\mathbf{u}_{p,\ell}^0)] = O(x^{P-p-1})$ so $\partial_x [U_{p,\ell}(x, \frac{y}{\varepsilon}) - T_{\leqslant P-p}(\mathbf{u}_{p,\ell}^0)(x, \frac{y}{\varepsilon})] = O(\varepsilon^{-1}x^{P-p-1})$

• $\partial_Y [U_{p,\ell} - T_{\leqslant P-p}(\mathbf{u}_{p,\ell}^0)] = O(x^{P-p})$ so $\partial_Y [U_{p,\ell}(x, \frac{y}{\varepsilon}) - T_{\leqslant P-p}(\mathbf{u}_{p,\ell}^0)(x, \frac{y}{\varepsilon})] = O(\varepsilon^{-1}x^{P-p})$

which are all $O(\varepsilon^{-1}x^{P-p-1})$ (uniformly in $Y$). Hence:

$$ \begin{aligned} \|\mathbf{u}_{\varepsilon,P} - \mathbf{u}_{\varepsilon,P}^A\|_{H^1(\Lambda_{\varepsilon} \cap C_{\eta})} & \lesssim \sum_{p \in \mathbb{P} \cap [0,P]} \sum_{\ell=0}^{n_p} \varepsilon^p \ln^\ell \varepsilon \cdot \varepsilon^{-1} \left( \int_{[0,2\eta] \times [-\varepsilon,0]} x^{2(P-p-1)} \, dx \, dy \right)^{1/2} \\ & \lesssim \sum_{p \in \mathbb{P} \cap [0,P]} \sum_{\ell=0}^{n_p} \varepsilon^p \ln^\ell \varepsilon \cdot \varepsilon^{-1} \cdot \varepsilon^{\frac{1}{2}} \eta^{P-p-\frac{1}{2}} \\ & \lesssim \eta^{P-2} \quad \text{because } \varepsilon = \eta^2 \text{ and } \ln^\ell \varepsilon \lesssim \eta^{-\frac{1}{2}} \end{aligned} $$

3. $\|S_{\varepsilon,P} - S_{\varepsilon,P}^A\|_{H^1(\Omega \cap C_{\eta})}$: Similarly $S_{p,\ell} - T_{\geqslant p-P}(S_{p,\ell}^{\infty}) = o_{\partial}(r^{p-P})$ in $\Omega$ when $r \to \infty$, so $S_{p,\ell} - T_{\geqslant p-P}(S_{p,\ell}^{\infty}) = O(r^{p-P})$ and $\nabla_{(X,Y)}[S_{p,\ell} - T_{\geqslant p-P}(S_{p,\ell}^{\infty})] = O(r^{p-P-1}) = O(r^{p-P})$ uniformly in $\theta$. Since $\nabla_{(x,y)} = \varepsilon^{-1}\nabla_{(X,Y)}$, we deduce:

$$ \begin{aligned} \|S_{\varepsilon,P} - S_{\varepsilon,P}^A\|_{H^1(\Omega \cap C_{\eta})} & \lesssim \sum_{p \in \mathbb{P} \cap [0,P]} \sum_{\ell=0}^{n_p} \varepsilon^p \ln^\ell \varepsilon \cdot \left\| [S_{p,\ell} - T_{\geqslant p-P}(S_{p,\ell}^{\infty})] (\frac{x}{\varepsilon}, \frac{y}{\varepsilon}) \right\|_{H^1(\Omega \cap C_{\eta})} \\ & \lesssim \sum_{p \in \mathbb{P} \cap [0,P]} \sum_{\ell=0}^{n_p} \varepsilon^p \ln^\ell \varepsilon \cdot \varepsilon^{-1} \left( \int_{\mathbb{R}^2 \setminus B(0,\eta)} \left(\frac{r}{\varepsilon}\right)^{2(p-P)} dr \, r d\theta \right)^{1/2} \\ & \lesssim \sum_{p \in \mathbb{P} \cap [0,P]} \sum_{\ell=0}^{n_p} \varepsilon^p \ln^\ell \varepsilon \cdot \varepsilon^{P-p-1} \cdot \eta^{p-P+1} \\ & \lesssim \eta^{P-2} \quad \text{because } \varepsilon = \eta^2 \text{ and } \ln^\ell \varepsilon \lesssim \eta^{-1} \end{aligned} $$

4. $\|S_{\varepsilon,P} - S_{\varepsilon,P}^A\|_{H^1(\Lambda_{\varepsilon} \cap C_{\eta})}$: $S_{p,\ell} - T_{\geqslant p-P}(S_{p,\ell}^{\infty}) = o_{\partial}(X^{p-P})$ in $\Lambda$ when $x \to 0$, so:

• $S_{p,\ell} - T_{\geqslant p-P}(\mathbf{u}_{p,\ell}^0) = O(X^{p-P})$

• $\partial_X [S_{p,\ell} - T_{\geqslant p-P}(S_{p,\ell}^{\infty})] = O(X^{p-P-1}) = O(X^{p-P})$

• $\partial_Y [S_{p,\ell} - T_{\geqslant p-P}(S_{p,\ell}^{\infty})] = O(X^{p-P})$

uniformity in $Y$. Hence:

$$ \begin{aligned} \|S_{\varepsilon,P} - S_{\varepsilon,P}^A\|_{H^1(\Lambda_{\varepsilon} \cap C_{\eta})} & \lesssim \sum_{p \in \mathbb{P} \cap [0,P]} \sum_{\ell=0}^{n_p} \varepsilon^p \ln^\ell \varepsilon \left\| [S_{p,\ell} - T_{\geqslant p-P}(S_{p,\ell}^{\infty})] (\frac{x}{\varepsilon}, \frac{y}{\varepsilon}) \right\|_{H^1(\Lambda_{\varepsilon} \cap C_{\eta})} \\ & \lesssim \sum_{p \in \mathbb{P} \cap [0,P]} \sum_{\ell=0}^{n_p} \varepsilon^p \ln^\ell \varepsilon \cdot \varepsilon^{-1} \left( \int_{[\eta, \infty) \times [-\varepsilon,0]} \left(\frac{x}{\varepsilon}\right)^{2(p-P)} dx \, dy \right)^{1/2} \\ & \lesssim \sum_{p \in \mathbb{P} \cap [0,P]} \sum_{\ell=0}^{n_p} \varepsilon^p \ln^\ell \varepsilon \cdot \varepsilon^{P-p-1} \cdot \varepsilon^{\frac{1}{2}} \eta^{p-P+\frac{1}{2}} \\ & \lesssim \eta^{P-2} \quad \text{because } \varepsilon = \eta^2 \text{ and } \ln^\ell \varepsilon \lesssim \eta^{-\frac{3}{2}} \end{aligned} $$

5. $\|\mathbf{u}_{\varepsilon,P}^A - S_{\varepsilon,P}^A\|_{H^1(\Omega_{\varepsilon} \cap C_{\eta})}$: Let us show that this norm vanishes. It suffices to prove that in $\mathcal{A}_{\varepsilon}^+(\Pi)$:

$$ \sum_{p \in \mathbb{P} \cap [0,P]} \sum_{\ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon \cdot T_{\leqslant P-p}(\mathbf{u}_{p,\ell}^0) = \mathcal{H}_{\varepsilon}^{-1} \left( \sum_{p \in \mathbb{P} \cap [0,P]} \sum_{\ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon \cdot T_{\geqslant p-P}(S_{p,\ell}^{\infty}) \right) \quad (4.1) $$

where $\mathcal{H}_{\varepsilon}$ is defined in (2.29)-(2.31), and $\varepsilon$ and $\ln \varepsilon$ denote the algebraic indeterminates of $\mathcal{A}_{\varepsilon}^+(\Pi)$ (we denote them differently from Sections 2.1 and 2.5 to avoid confusion with the real number $\varepsilon$).
• $S_{p,l} - T_{\ge p-P}(\mathbf{u}_{p,l}^0) = O(X^{p-P})$

• $\partial_X [S_{p,l} - T_{\ge p-P}(S_{p,l}^\infty)] = O(X^{p-P-1}) = O(X^{p-P})$

• $\partial_Y [S_{p,l} - T_{\ge p-P}(S_{p,l}^\infty)] = O(X^{p-P})$

uniformly in Y. Hence:

$$
\begin{align*}
\|S_{\varepsilon,P} - S_{\varepsilon,P}^A\|_{H^1(\Lambda_\varepsilon \cap C_\eta)} &\le \sum_{p \in \mathbb{P} \cap [0,P]} \sum_{\ell=0}^{n_p} \varepsilon^p \ln^\ell \varepsilon \left\| [S_{p,\ell} - T_{\ge p-P}(S_{p,\ell}^\infty)] \left(\frac{x}{\varepsilon}, \frac{y}{\varepsilon}\right) \right\|_{H^1(\Lambda_\varepsilon \cap C_\eta)} \\
&\lesssim \sum_{p \in \mathbb{P} \cap [0,P]} \sum_{\ell=0}^{n_p} \varepsilon^p \ln^\ell \varepsilon \cdot \varepsilon^{-1} \left( \int_{[\eta, \infty \times [-\varepsilon, 0]} \left(\frac{x}{\varepsilon}\right)^{2(p-P)} dx dy \right)^{1/2} \\
&\lesssim \sum_{p \in \mathbb{P} \cap [0,P]} \sum_{\ell=0}^{n_p} \varepsilon^p \ln^\ell \varepsilon \cdot \varepsilon^{P-p-1} \cdot \varepsilon^{\frac{1}{2}} \eta^{p-P+\frac{1}{2}} \\
&\lesssim \eta^{P-2} \quad \text{because } \varepsilon = \eta^2 \text{ and } \ln^\ell \varepsilon \lesssim \eta^{-\frac{3}{2}}
\end{align*}
$$

5. $\|\mathbf{u}_{\varepsilon,P}^A - S_{\varepsilon,P}^A\|_{H^1(\Omega_\varepsilon \cap C_\eta)}$: Let us show that this norm vanishes. It suffices to prove that in $A_\varepsilon^+(\Pi)$:

$$
\sum_{p \in \mathbb{P} \cap [0, P]} \sum_{\ell \in \mathbb{N}} \epsilon^p \ln^\ell \epsilon \cdot T_{\le P-p}(\mathbf{u}_{p,\ell}^0) = \mathcal{H}_\epsilon^{-1} \left( \sum_{p \in \mathbb{P} \cap [0, P]} \sum_{\ell \in \mathbb{N}} \epsilon^p \ln^\ell \epsilon \cdot T_{\ge p-P}(S_{p,\ell}^\infty) \right) \quad (4.1)
$$

where $\mathcal{H}_\epsilon$ is defined in (2.30)–(2.32), and $\epsilon$ and $\ln \epsilon$ denote the algebraic indeterminates of $A_\epsilon^+(\Pi)$ (we denote them differently from Sections 2.1 and 2.5 to avoid confusion with the real number $\epsilon$). For any $(p, d)$ let $\tilde{\mathbf{u}}_{p,d}$ be the coordinate of $\sum_{p,l} \epsilon^p \ln^l \epsilon \mathbf{u}_{p,l}^0$ in $\mathcal{A}_d(\Pi)[\ln \epsilon]$ and $\tilde{S}_{p,d}$ be the one of $\sum_{p,l} \epsilon^p \ln^l \epsilon S_{p,l}^\infty$. Moreover, for any $d$ and $\varphi \in \mathcal{A}_d(\Pi)[\ln \epsilon]$, let $\tilde{\mathcal{H}}_\epsilon^{-1}(\varphi) := \epsilon^d \mathcal{H}_\epsilon^{-1}(\varphi) \in \mathcal{A}_d(\Pi)[\ln \epsilon]$. Then:

$$
\begin{align}
(4.1) \Leftrightarrow & \sum_{p \le P} \sum_{d \le P-p} \epsilon^p \tilde{\mathbf{u}}_{p,d} = \mathcal{H}_\epsilon^{-1} \left( \sum_{p \le P} \sum_{d \ge p-P} \epsilon^p \tilde{S}_{p,d} \right) \nonumber \\
\Leftrightarrow & \sum_{p \le P} \sum_{p+d \le P} \epsilon^p \tilde{\mathbf{u}}_{p,d} = \sum_{p \le P} \sum_{p-d \le P} \epsilon^{p-d} \tilde{\mathcal{H}}_\epsilon^{-1} (\tilde{S}_{p,d}) \nonumber \\
\Leftrightarrow & \sum_{p \le P} \sum_{p+d \le P} \epsilon^p \tilde{\mathbf{u}}_{p,d} = \sum_{p+d \le P} \sum_{p \le P} \epsilon^p \tilde{\mathcal{H}}_\epsilon^{-1} (\tilde{S}_{p+d,d}) \nonumber \\
\Leftrightarrow & \forall (p, d) \text{ s.t. } p \le P \text{ and } p+d \le P, \quad \tilde{\mathbf{u}}_{p,d} = \tilde{\mathcal{H}}_\epsilon^{-1} (\tilde{S}_{p+d,d}) \tag{4.2}
\end{align}
$$

because two formal series coincide iff their coordinates coincide one by one.

But Proposition 3.15 shows the matching condition $\sum_{p,d} \epsilon^p \tilde{\mathbf{u}}_{p,d} = \mathcal{H}_\epsilon^{-1} \left( \sum_{p,d} \epsilon^p \tilde{S}_{p,d} \right) = \sum_{p,d} \epsilon^p \tilde{\mathcal{H}}_\epsilon^{-1} (\tilde{S}_{p+d,d})$, which is equivalent to: $\forall (p,d)$, $\tilde{\mathbf{u}}_{p,d} = \tilde{\mathcal{H}}_\epsilon^{-1} (\tilde{S}_{p+d,d})$. Thus we get (4.2), and then (4.1). $\square$

**Theorem 4.2: global error estimate**

For any $P \in \mathbb{R}_+$ we have $\|u_{\varepsilon} - u_{\varepsilon,P}\|_{H^1(\Omega_{\varepsilon})} = o(\varepsilon^{\frac{P}{2}-2})$.

**Proof:** Let $r_{\varepsilon,P} := u_{\varepsilon,P} - u_{\varepsilon}$. It satisfies, for some functions $f_\varepsilon$ and $g_\varepsilon$,

$$
\left\{
\begin{array}{ll}
\text{div}(\mu_{\varepsilon} \nabla r_{\varepsilon,P}) + \omega^2 \rho_{\varepsilon} r_{\varepsilon,P} = f_{\varepsilon} & \text{in } \Omega_{\varepsilon} \\
r_{\varepsilon,P} = 0 & \text{on } \partial\Omega_{\varepsilon} \\
r_{\varepsilon,P|_{y=0^+}} - r_{\varepsilon,P|_{y=0^-}} = 0 & \text{on } \Gamma \cap \{x > \eta\} \\
\mu_0 \partial_y r_{\varepsilon,P|_{y=0^+}} - \mu_1 \partial_y r_{\varepsilon,P|_{y=0^-}} = g_{\varepsilon} & \text{on } \Gamma \cap \{x > \eta\}
\end{array}
\right.
$$

As for (1.2), this problem is well-posed with a stability constant independent of $\varepsilon$:

$$
\|r_{\varepsilon,P}\|_{H^1(\Omega_{\varepsilon})} \lesssim \|f_{\varepsilon}\|_{L^2(\Omega_{\varepsilon})} + \|g_{\varepsilon}\|_{L^2(\Gamma \cap \{x>\eta\})}.
$$

To get our error estimate, it suffices to show that $\|f_{\varepsilon}\|_{L^2} + \|g_{\varepsilon}\|_{L^2} \lesssim \varepsilon^{\frac{P}{2}-2} = \eta^{P-4}$.

Estimate of $\|f_{\varepsilon}\|_{L^2}$: We denote:
For any $(p, d)$ let $\tilde{\mathbf{u}}_{p,d}$ be the coordinate of $\sum_{p, \ell} \epsilon^p \ln^\ell \epsilon \mathbf{u}_{p, \ell}^0$ in $\mathcal{A}_d(\Pi)[\ln \epsilon]$ and $\tilde{S}_{p,d}$ be the one of $\sum_{p, \ell} \epsilon^p \ln^\ell \epsilon S_{p, \ell}^\infty$. Moreover, for any $d$ and $\varphi \in \mathcal{A}_d(\Pi)[\ln \epsilon]$, let $\tilde{\mathcal{H}}_\epsilon^{-1}(\varphi) := \epsilon^d \mathcal{H}_\epsilon^{-1}(\varphi) \in \mathcal{A}_d(\Pi)[\ln \epsilon]$. Then:

$$ (4.1) \iff \sum_{p \le P} \sum_{d \le P-p} \epsilon^p \tilde{\mathbf{u}}_{p,d} = \mathcal{H}_\epsilon^{-1} \left( \sum_{p \le P} \sum_{d \ge p-P} \epsilon^p \tilde{S}_{p,d} \right) \\ \iff \sum_{p \le P} \sum_{p+d \le P} \epsilon^p \tilde{\mathbf{u}}_{p,d} = \sum_{p \le P} \sum_{p-d \le P} \epsilon^{p-d} \tilde{\mathcal{H}}_\epsilon^{-1} (\tilde{S}_{p,d}) \\ \iff \sum_{p \le P} \sum_{p+d \le P} \epsilon^p \tilde{\mathbf{u}}_{p,d} = \sum_{p+d \le P} \sum_{p \le P} \epsilon^p \tilde{\mathcal{H}}_\epsilon^{-1} (\tilde{S}_{p+d,d}) \\ \iff \forall (p, d) \text{ s.t. } p \le P \text{ and } p+d \le P, \quad \tilde{\mathbf{u}}_{p,d} = \tilde{\mathcal{H}}_\epsilon^{-1} (\tilde{S}_{p+d,d}) \quad (4.2) $$

because two formal series coincide iff their coordinates coincide one by one.

But Proposition 3.15 shows the matching condition $\sum_{p,d} \epsilon^p \tilde{\mathbf{u}}_{p,d} = \mathcal{H}_\epsilon^{-1} \left( \sum_{p,d} \epsilon^p \tilde{S}_{p,d} \right) = \sum_{p,d} \epsilon^p \tilde{\mathcal{H}}_\epsilon^{-1} (\tilde{S}_{p+d,d})$, which is equivalent to: $\forall (p, d)$, $\tilde{\mathbf{u}}_{p,d} = \tilde{\mathcal{H}}_\epsilon^{-1} (\tilde{S}_{p+d,d})$. Thus we get (4.2), and then (4.1). $\square$

**Theorem 4.2: global error estimate**

For any $P \in \mathbb{R}_+$ we have $\|u_\epsilon - u_{\epsilon, P}\|_{H^1(\Omega_\epsilon)} = o(\epsilon^{\frac{P}{2}-2})$.

**Proof:** Let $r_{\epsilon,P} := u_{\epsilon,P} - u_{\epsilon}$. It satisfies, for some functions $f_\epsilon$ and $g_\epsilon$,

$$ \left\{ \begin{array}{ll} \operatorname{div}(\mu_{\epsilon} \nabla r_{\epsilon, P}) + \omega^2 \rho_{\epsilon} r_{\epsilon, P} = f_{\epsilon} & \text{in } \Omega_{\epsilon} \\ r_{\epsilon, P} = 0 & \text{on } \partial \Omega_{\epsilon} \\ r_{\epsilon, P|_{y=0^+}} - r_{\epsilon, P|_{y=0^-}} = 0 & \text{on } \Gamma \cap \{x > \eta\} \\ \mu_0 \partial_y r_{\epsilon, P|_{y=0^+}} - \mu_1 \partial_y r_{\epsilon, P|_{y=0^-}} = g_{\epsilon} & \text{on } \Gamma \cap \{x > \eta\} \end{array} \right. $$

As for (1.2), this problem is well-posed with a stability constant independent of $\epsilon$:

$$ \|r_{\epsilon,P}\|_{H^1(\Omega_{\epsilon})} \lesssim \|f_{\epsilon}\|_{L^2(\Omega_{\epsilon})} + \|g_{\epsilon}\|_{L^2(\Gamma \cap \{x>\eta\})}. $$

To get our error estimate, it suffices to show that $\|f_\epsilon\|_{L^2} + \|g_\epsilon\|_{L^2} \lesssim \epsilon^{\frac{P}{2}-2} = \eta^{P-4}$.

Estimate of $\|f_\epsilon\|_{L^2}$: We denote:

• $\mathbf{u}_{\epsilon,P} := \sum_{p \in \mathbb{P} \cap [0, P]} \sum_{\ell \in \mathbb{N}} \epsilon^p \ln^\ell \epsilon \mathbf{u}_{p, \ell}^\epsilon$ in $\Pi_\epsilon$

• $S_{\epsilon,P} := \sum_{p \in \mathbb{P} \cap [0, P]} \sum_{\ell \in \mathbb{N}} \epsilon^p \ln^\ell \epsilon S_{p, \ell} (\frac{x}{\epsilon}, \frac{y}{\epsilon})$ in $\Omega_\epsilon$

• and $\mathcal{D}_\epsilon: u \mapsto \operatorname{div}(\mu_\epsilon \nabla u) + \omega^2 \rho_\epsilon u$ the differential operator of Helmholtz's equation.

Since $\mathcal{D}_\epsilon u_\epsilon = f$ by definition of $u_\epsilon$, we have

$$ \begin{aligned} f_{\epsilon} &= \mathcal{D}_{\epsilon} r_{\epsilon, P} \\ &= \mathcal{D}_{\epsilon}((1 - \chi_{\eta})\mathbf{u}_{\epsilon, P} + \chi_{\eta}S_{\epsilon, P} - u_{\epsilon}) \\ &= ((1 - \chi_{\eta})\mathcal{D}_{\epsilon}\mathbf{u}_{\epsilon, P} - f) + \chi_{\eta}\mathcal{D}_{\epsilon}S_{\epsilon, P} + [\mathcal{D}_{\epsilon}, \chi_{\eta}](S_{\epsilon, P} - \mathbf{u}_{\epsilon, P}) \end{aligned} $$

where $[.,]$ is the commutator. Let us estimate these terms one by one.

1. For $\epsilon$ small enough, $1 - \chi_\eta$ is equal to 0 in $B(0, \epsilon R_c)$ and 1 in $\text{supp}(f)$, so $(1 - \chi_\eta)\mathcal{D}_\epsilon \mathbf{u}_{\epsilon, P}|_{\Omega} = (1 - \chi_\eta)(\mu_0 \Delta + \omega^2 \rho_0)\mathbf{u}_{\epsilon, P}|_{\Omega} = (1 - \chi_\eta)f = f$.

Moreover, using that $\mu_1 \partial_y^2 U_{p, \ell} = -(\mu_1 \partial_x^2 + \omega^2 \rho_1)U_{p-2, \ell}$ for any $(p, \ell)$, we get in $\Lambda_\epsilon \setminus B(0, \epsilon R_c)$:

$$ \begin{aligned} \mathcal{D}_{\epsilon} \mathbf{u}_{\epsilon, P} &= (\mu_1 \Delta + \omega^2 \rho_1) \mathbf{u}_{\epsilon, P} \\ &= \sum_{p \in \mathbb{P} \cap (P-2, P]} \sum_{\ell \in \mathbb{N}} \epsilon^p \ln^\ell \epsilon (\mu_1 \partial_x^2 + \omega^2 \rho_1) U_{p, \ell} \\ &= \sum_{p \in \mathbb{P} \cap (P-2, P]} \sum_{\ell \in \mathbb{N}} \epsilon^{p+\frac{1}{2}} \ln^\ell \epsilon \sum_{n \in \mathbb{N}} \mu_1 (\partial_x^2 + k_1^2)^{n+1} \partial_y u_{p-1-2n, \ell}(x, 0) \cdot \mathcal{U}_n(\frac{y}{\epsilon}) \quad \text{by (3.3)} \end{aligned} $$
• $\mathbf{u}_{\varepsilon,P} := \sum_{p \in \mathbb{P} \cap [0,P]} \sum_{\ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon \mathbf{u}_{p,\ell}^\varepsilon$ in $\Pi_\varepsilon$

• $S_{\varepsilon,P} := \sum_{p \in \mathbb{P} \cap [0,P]} \sum_{\ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon S_{p,\ell}(\frac{x}{\varepsilon}, \frac{y}{\varepsilon})$ in $\Omega_\varepsilon$

• and $\mathcal{D}_\varepsilon: u \mapsto \text{div}(\mu_\varepsilon \nabla u) + \omega^2 \rho_\varepsilon u$ the differential operator of Helmholtz's equation.

Since $\mathcal{D}_\varepsilon u_\varepsilon = f$ by definition of $u_\varepsilon$, we have

$$
\begin{align*}
f_\varepsilon &= \mathcal{D}_\varepsilon r_{\varepsilon,P} \\
&= \mathcal{D}_\varepsilon \left( (1 - \chi_\eta) \mathbf{u}_{\varepsilon,P} + \chi_\eta S_{\varepsilon,P} - u_\varepsilon \right) \\
&= \left( (1 - \chi_\eta) \mathcal{D}_\varepsilon \mathbf{u}_{\varepsilon,P} - f \right) + \chi_\eta \mathcal{D}_\varepsilon S_{\varepsilon,P} + [\mathcal{D}_\varepsilon, \chi_\eta](S_{\varepsilon,P} - \mathbf{u}_{\varepsilon,P})
\end{align*}
$$

where [., .] is the commutator. Let us estimate these terms one by one.

1. For $\varepsilon$ small enough, $1 - \chi_\eta$ is equal to 0 in $B(0, \varepsilon R_c)$ and 1 in $\text{supp}(f)$, so $(1 - \chi_\eta)\mathcal{D}_\varepsilon \mathbf{u}_{\varepsilon,P}|_{\Omega} = (1 - \chi_\eta)(\mu_0\Delta + \omega^2\rho_0)\mathbf{u}_{\varepsilon,P}|_{\Omega} = (1 - \chi_\eta)f = f$.

Moreover, using that $\mu_1 \partial_y^2 U_{p,\ell} = -(\mu_1 \partial_x^2 + \omega^2 \rho_1) U_{p-2,\ell}$ for any $(p, \ell)$, we get in $\Lambda_\varepsilon \setminus B(0, \varepsilon R_c)$:

$$
\begin{align*}
\mathcal{D}_\varepsilon \mathbf{u}_{\varepsilon,P} &= (\mu_1 \Delta + \omega^2 \rho_1) \mathbf{u}_{\varepsilon,P} \\
&= \sum_{p \in \mathbb{P} \cap (P-2,P]} \sum_{\ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon (\mu_1 \partial_x^2 + \omega^2 \rho_1) U_{p,\ell} \\
&= \sum_{p \in \mathbb{P} \cap (P-2,P]} \sum_{\ell \in \mathbb{N}} \varepsilon^{p+\frac{1}{2}} \ln^\ell \varepsilon \sum_{n \in \mathbb{N}} \mu_1 (\partial_x^2 + k_1^2)^{n+1} \partial_y u_{p-1-2n,\ell}(x,0) \cdot \mathcal{U}_n(\frac{y}{\varepsilon}) \quad \text{by (3.3)}
\end{align*}
$$

Since $\mathcal{U}_n \in L^\infty(-1,0)$ for any $n$ (see (3.1)), we deduce:

$$
\|(1 - \chi_{\eta}) \mathcal{D}_{\varepsilon} \mathbf{u}_{\varepsilon, P} \|_{L^2(\Lambda_{\varepsilon})} \lesssim \sum_{p \in \mathbb{P} \cap (P-2,P]} \sum_{\ell=0}^{n_p} \sum_{n=0}^{\lfloor(p-1)/2\rfloor} \varepsilon^{p+\frac{1}{2}} \ln^{\ell} \varepsilon \cdot \|(\partial_x^2 + k_1^2)^{n+1} \partial_y u_{p-1-2n, \ell} \|_{L^2(\Gamma \cap \{x>\eta\})}.
$$

Let us show that, for any $p, \ell, m$, $\|\partial_x^m \partial_y u_{p,\ell|\Gamma} \|_{L^2(\Gamma \cap \{x>\eta\})} \lesssim \eta^{-p-m-1}$ (4.3).

• The proof of Proposition 3.13 shows that $\partial_x^m \partial_y u_{p,\ell} \in H_0^1(\Gamma) + \chi_0 A(\Gamma)$ for any $p, \ell, m$. So $\|\partial_x^m \partial_y u_{p,\ell|\Gamma} \|_{L^2(\Gamma \cap \{x>1\})} < \infty$.

• By Proposition 3.13, $\partial_y u_{p,\ell|\Gamma} = \partial_y T_{\leq -p-\frac{1}{2}}(\mathbf{u}_{p,\ell}^0) + o_\partial(x^{-p-\frac{3}{2}})$ when $x \to 0$, and Ansatz 2.25 is satisfied, thus $T_{\leq -p-\frac{1}{2}}(\mathbf{u}_{p,\ell}^0) = 0$. So: $\forall m \in \mathbb{N}$, $\partial_x^m \partial_y u_{p,\ell|\Gamma} = O(x^{-p-\frac{3}{2}-m})$ when $x \to 0$. It implies $\|\partial_x^m \partial_y u_{p,\ell|\Gamma} \|_{L^2(\Gamma \cap \{\eta<x<1\})} \lesssim \eta^{-p-m-1}$.

Hence:

$$
\begin{align*}
\|(1 - \chi_{\eta}) \mathcal{D}_{\varepsilon} \mathbf{u}_{\varepsilon, P} \|_{L^2(\Lambda_{\varepsilon})} &\lesssim \sum_{p \in \mathbb{P} \cap (P-2, P]} \sum_{\ell=0}^{n_p} \varepsilon^{p+\frac{1}{2}} \ln^{\ell} \varepsilon \sum_{n=0}^{\lfloor(p-1)/2\rfloor} \eta^{-(p-1-2n)-2(n+1)-1} \\
&\lesssim \sum_{p \in \mathbb{P} \cap (P-2, P]} \sum_{\ell=0}^{n_p} \varepsilon^{p+\frac{1}{2}} \ln^{\ell} \varepsilon \cdot \eta^{-p-2} \\
&\lesssim \eta^{P-4} \quad \text{because } \ln^{\ell} \varepsilon \lesssim \eta^{-1}
\end{align*}
$$

So $\|(1 - \chi_\eta) \mathcal{D}_\varepsilon \mathbf{u}_{\varepsilon,P} - f\|_{L^2(\Omega_\varepsilon)} \lesssim \eta^{P-4}$.

2. For the second term of $f_\varepsilon$, (3.5) implies that $\mathcal{D}_\varepsilon S_{\varepsilon,P} = \sum_{p \in \mathbb{P} \cap (P-2,P]} \sum_{\ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon \omega^2 \rho S_{p,\ell}$. But by Proposition 3.13 and Ansatz 2.25: $\forall (p, \ell)$, $S_{p,\ell} = O(r^p)$. Thus, using that $\rho$ is bounded and $\|S_{p,\ell}(\frac{x}{\varepsilon}, \frac{y}{\varepsilon})\|_{L^2(\Omega_\varepsilon \cap B(0,2\eta))} = \varepsilon \|S_{p,\ell}\|_{L^2(\Omega_1 \cap B(0,2\eta/\varepsilon))}$, we get

$$
\begin{align*}
\|\chi_\eta \mathcal{D}_\varepsilon S_{\varepsilon,P}\|_{L^2(\Omega_\varepsilon)} &\lesssim \sum_{p \in \mathbb{P} \cap (P-2,P]} \sum_{\ell=0}^{n_p} \varepsilon^p \ln^\ell \varepsilon \cdot \varepsilon \|S_{p,\ell}\|_{L^2(\Omega_1 \cap B(0,2\eta/\varepsilon))} \\
&\lesssim \sum_{p \in \mathbb{P} \cap (P-2,P]} \sum_{\ell=0}^{n_p} \varepsilon^{p+1} \ln^\ell \varepsilon \cdot \left(\frac{\eta}{\varepsilon}\right)^{p+1} \\
&\lesssim \eta^{P-4} \quad \text{because } \ln^\ell \varepsilon \lesssim \eta^{-3}
\end{align*}
$$
Since $\mathcal{U}_n \in L^\infty(-1, 0)$ for any $n$ (see (3.1)), we deduce:

$$ \|(1 - \chi_\eta) \mathcal{D}_\varepsilon \mathbf{u}_{\varepsilon,P} \|_{L^2(\Lambda_\varepsilon)} \lesssim \sum_{p \in \mathbb{P} \cap (P-2,P]} \sum_{\ell=0}^{n_p} \sum_{n=0}^{\lfloor(p-1)/2\rfloor} \varepsilon^{p+\frac{1}{2}} \ln^\ell \varepsilon \cdot \|(\partial_x^2 + k_1^2)^{n+1} \partial_y u_{p-1-2n,\ell} \|_{L^2(\Gamma \cap \{x>\eta\})} $$

Let us show that, for any $p, \ell, m$, $\|\partial_x^m \partial_y u_{p,\ell|\Gamma} \|_{L^2(\Gamma \cap \{x>\eta\})} \lesssim \eta^{-p-m-1}$ (4.3).

* The proof of Proposition 3.13 shows that $\partial_x^m \partial_y u_{p,\ell} \in H_0^1(\Gamma) + \chi_0 A(\Gamma)$ for any $p, \ell, m$. So $\|\partial_x^m \partial_y u_{p,\ell|\Gamma} \|_{L^2(\Gamma \cap \{x>1\})} < \infty$.

* By Proposition 3.13, $\partial_y u_{p,\ell|\Gamma} = \partial_y T_{\le -p-\frac{1}{2}}(\mathbf{u}_{p,\ell}^0) + o_\partial(x^{-p-\frac{3}{2}})$ when $x \to 0$, and Ansatz 2.25 is satisfied, thus $T_{\le -p-\frac{1}{2}}(\mathbf{u}_{p,\ell}^0) = 0$. So: $\forall m \in \mathbb{N}$, $\partial_x^m \partial_y u_{p,\ell|\Gamma} = O(x^{-p-\frac{3}{2}-m})$ when $x \to 0$. It implies $\|\partial_x^m \partial_y u_{p,\ell|\Gamma} \|_{L^2(\Gamma \cap \{\eta<x<1\})} \lesssim \eta^{-p-m-1}$.

Hence:

$$ \begin{aligned} \|(1 - \chi_\eta) \mathcal{D}_\varepsilon \mathbf{u}_{\varepsilon,P} \|_{L^2(\Lambda_\varepsilon)} & \lesssim \sum_{p \in \mathbb{P} \cap (P-2,P]} \sum_{\ell=0}^{n_p} \varepsilon^{p+\frac{1}{2}} \ln^\ell \varepsilon \sum_{n=0}^{\lfloor(p-1)/2\rfloor} \eta^{-(p-1-2n)-2(n+1)-1} \\ & \lesssim \sum_{p \in \mathbb{P} \cap (P-2,P]} \sum_{\ell=0}^{n_p} \varepsilon^{p+\frac{1}{2}} \ln^\ell \varepsilon \cdot \eta^{-p-2} \\ & \lesssim \eta^{P-4} \quad \text{because } \ln^\ell \varepsilon \lesssim \eta^{-1} \end{aligned} $$

So $\|(1 - \chi_\eta) \mathcal{D}_\varepsilon \mathbf{u}_{\varepsilon,P} - f\|_{L^2(\Omega_\varepsilon)} \lesssim \eta^{P-4}$.

2. For the second term of $f_\varepsilon$, (3.5) implies that $\mathcal{D}_\varepsilon S_{\varepsilon,P} = \sum_{p \in \mathbb{P} \cap (P-2,P]} \sum_{\ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon \omega^2 \rho S_{p,\ell}$. But by Proposition 3.13 and Ansatz 2.25: $\forall (p, \ell)$, $S_{p,\ell} = O(r^p)$. Thus, using that $\rho$ is bounded and $\|S_{p,\ell}(\frac{x}{\varepsilon}, \frac{y}{\varepsilon})\|_{L^2(\Omega_\varepsilon \cap B(0,2\eta))} = \varepsilon \|S_{p,\ell}\|_{L^2(\Omega_1 \cap B(0,2\eta/\varepsilon))}$, we get

$$ \begin{aligned} \|\chi_\eta \mathcal{D}_\varepsilon S_{\varepsilon,P}\|_{L^2(\Omega_\varepsilon)} & \lesssim \sum_{p \in \mathbb{P} \cap (P-2,P]} \sum_{\ell=0}^{n_p} \varepsilon^p \ln^\ell \varepsilon \cdot \varepsilon \|S_{p,\ell}\|_{L^2(\Omega_1 \cap B(0,2\eta/\varepsilon))} \\ & \lesssim \sum_{p \in \mathbb{P} \cap (P-2,P]} \sum_{\ell=0}^{n_p} \varepsilon^{p+1} \ln^\ell \varepsilon \cdot \left(\frac{\eta}{\varepsilon}\right)^{p+1} \\ & \lesssim \eta^{P-4} \quad \text{because } \ln^\ell \varepsilon \lesssim \eta^{-3} \end{aligned} $$

3. Using that $\|\Delta\chi_\eta\|_{L^2} + \|\nabla\chi_\eta\|_{L^2} = O(\eta^{-2})$, the last term of $f_\varepsilon$ satisfies

$$ \begin{aligned} \|[\mathcal{D}_\varepsilon, \chi_\eta](S_{\varepsilon,P} - \mathbf{u}_{\varepsilon,P})\|_{L^2(\Omega_\varepsilon)} & \lesssim \|\Delta\chi_\eta \cdot (S_{\varepsilon,P} - \mathbf{u}_{\varepsilon,P}) + 2\nabla\chi_\eta \cdot \nabla(S_{\varepsilon,P} - \mathbf{u}_{\varepsilon,P})\|_{L^2(\Omega_\varepsilon)} \\ & \lesssim \eta^{-2} \|S_{\varepsilon,P} - \mathbf{u}_{\varepsilon,P}\|_{H^1(\Omega_\varepsilon \cap B(0,2\eta) \setminus B(0,\eta))} \\ & \lesssim \eta^{P-4} \quad \text{by Lemma 4.1.} \end{aligned} $$

Estimate of $\|g_\varepsilon\|_{L^2}$: Without loss of generality, we assume $\partial_y \chi_\eta|\Gamma = 0$. (3.5) implies that $\mu_0 \partial_Y S_{p,\ell|Y=0^+} = \mu_1 \partial_Y S_{p,\ell|Y=0^-}$ on $\Gamma \cap \{X > R_c\}$ for any $(p, \ell)$. Thus we have on $\Gamma \cap \{x > \eta\}$:

$$ \begin{aligned} g_\varepsilon &= \mu_0 \partial_y r_{\varepsilon,P|Y=0^+} - \mu_1 \partial_y r_{\varepsilon,P|Y=0^-} \\ &= \chi_\eta \cdot (\mu_0 \partial_y \mathbf{u}_{\varepsilon,P|Y=0^+} - \mu_1 \partial_y \mathbf{u}_{\varepsilon,P|Y=0^-}) \\ &= \chi_\eta \sum_{p \in \mathbb{P} \cap (P-1,P]} \sum_{\ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon \cdot \mu_0 \partial_y u_{p,\ell|Y=0^+} \end{aligned} $$

By (4.3), we have that $\|\partial_y u_{p,\ell|\Gamma} \|_{L^2(\Gamma \cap \{x>\eta\})} \lesssim \eta^{-p-1}$. So $\|g_\varepsilon\|_{L^2(\Gamma \cap \{x>\eta\})} \lesssim \eta^{P-4}$ since $\ln^\ell \varepsilon \lesssim \eta^{-2}$. $\square$

**Remark:** Theorem 4.2 can be improved to $\|u_\varepsilon - u_{\varepsilon,P}\|_{H^1(\Omega_\varepsilon)} = o(\varepsilon^{P/2})$. Indeed with the same notations as the above proof, one can show that, for any $(p, \ell) \in \mathbb{P} \times \mathbb{N}$, $\|(1 - \chi_\eta)\mathbf{u}_{\varepsilon,P,\ell}^\varepsilon\|_{H^1(\Omega_\varepsilon)} \lesssim \eta^{-p}$ and $\|\chi_\eta S_{p,\ell}\|_{H^1(\Omega_\varepsilon)} \lesssim \eta^{-p}$. Therefore:

$$ \|r_{\varepsilon,P}\|_{H^1(\Omega_\varepsilon)} \lesssim \|r_{\varepsilon,P+4}\|_{H^1(\Omega_\varepsilon)} + \sum_{p \in \mathbb{P} \cap (P,P+4]} \sum_{\ell \in \mathbb{N}} \varepsilon^p \ln^\ell \varepsilon \cdot \left( \|(1 - \chi_\eta)\mathbf{u}_{\varepsilon,P,\ell}^\varepsilon\|_{H^1(\Omega_\varepsilon)} + \|\chi_\eta S_{p,\ell}\|_{H^1(\Omega_\varepsilon)} \right) = o(\varepsilon^{P/2}). $$
3. Using that $||\Delta\chi_\eta||_{L^2} + ||\nabla\chi_\eta||_{L^2} = O(\eta^{-2})$, the last term of $f_\epsilon$ satisfies

$$ \begin{aligned} \|[D, \chi_\eta](S_{\epsilon,P} - \mathbf{u}_{\epsilon,P})\|_{L^2(\Omega_\epsilon)} &\lesssim ||\Delta\chi_\eta \cdot (S_{\epsilon,P} - \mathbf{u}_{\epsilon,P}) + 2\nabla\chi_\eta \cdot \nabla(S_{\epsilon,P} - \mathbf{u}_{\epsilon,P})\|_{L^2(\Omega_\epsilon)} \\ &\lesssim \eta^{-2} \|S_{\epsilon,P} - \mathbf{u}_{\epsilon,P}\|_{H^1(\Omega_\epsilon \cap B(0,2\eta)\setminus B(0,\eta))} \\ &\lesssim \eta^{P-4} \quad \text{by Lemma 4.1.} \end{aligned} $$

Estimate of $||g_\epsilon||_{L^2}$: Without loss of generality, we assume $\partial_y\chi_\eta|_{\Gamma} = 0$. (3.5) implies that $\mu_0\partial_y S_{p,\ell}|_{Y=0^+} = \mu_1\partial_y S_{p,\ell}|_{Y=0^-}$ on $\Gamma \cap \{X > R_c\}$ for any $(p, \ell)$. Thus we have on $\Gamma \cap \{x > \eta\}$:

$$ \begin{aligned} g_\epsilon &= \mu_0 \partial_y r_{\epsilon,P}|_{y=0^+} - \mu_1 \partial_y r_{\epsilon,P}|_{y=0^-} \\ &= \chi_\eta \cdot (\mu_0 \partial_y \mathbf{u}_{\epsilon,P}|_{y=0^+} - \mu_1 \partial_y \mathbf{u}_{\epsilon,P}|_{y=0^-}) \\ &= \chi_\eta \sum_{p \in \mathbb{P} \cap (P-1,P]} \sum_{\ell \in \mathbb{N}} \epsilon^p \ln^\ell \epsilon \cdot \mu_0 \partial_y u_{p,\ell}|_{y=0^+} \end{aligned} $$

By (4.3), we have that $||\partial_y u_{p,\ell}|_{\Gamma}||_{L^2(\Gamma \cap \{x>\eta\})} \lesssim \eta^{-p-1}$. So $||g_\epsilon||_{L^2(\Gamma \cap \{x>\eta\})} \lesssim \eta^{P-4}$ since $\ln^\ell \epsilon \lesssim \eta^{-2}$. $\square$

**Remark:** Theorem 4.2 can be improved to $||u_\epsilon - u_{\epsilon,P}||_{H^1(\Omega_\epsilon)} = o(\epsilon^{P/2})$. Indeed with the same notations as the above proof, one can show that, for any $(p, \ell) \in \mathbb{P} \times \mathbb{N}$, $||(1 - \chi_\eta)\mathbf{u}_{p,\ell}^\epsilon||_{H^1(\Omega_\epsilon)} \lesssim \eta^{-p}$ and $||\chi_\eta S_{p,\ell}||_{H^1(\Omega_\epsilon)} \lesssim \eta^{-p}$. Therefore:

$$ ||r_{\epsilon,P}||_{H^1(\Omega_\epsilon)} \lesssim ||r_{\epsilon,P+4}||_{H^1(\Omega_\epsilon)} + \sum_{p \in \mathbb{P} \cap (P,P+4]} \sum_{\ell \in \mathbb{N}} \epsilon^p \ln^\ell \epsilon \cdot (||(1 - \chi_\eta)\mathbf{u}_{p,\ell}^\epsilon||_{H^1(\Omega_\epsilon)} + ||\chi_\eta S_{p,\ell}||_{H^1(\Omega_\epsilon)}) = o(\epsilon^{P/2}). $$

**Proof of Theorem 1.1:** It follows from Theorem 4.2 applied at order $2P+4$ and from the fact that, for small enough $\epsilon$, $\sum_{p \in \mathbb{P} \cap [0,2P]} \sum_{\ell \in \mathbb{N}} \epsilon^p \ln^\ell \epsilon u_{p,\ell}$ and $u_{\epsilon,2P}$ coincide in $\Omega \setminus B(0,\delta)$:

$$ \begin{aligned} \left\| u_{\epsilon} - \sum_{p \in \mathbb{P} \cap [0, P]} \sum_{\ell \in \mathbb{N}} \epsilon^p \ln^\ell \epsilon u_{p, \ell} \right\|_{H^1(\Omega \setminus B(0, \delta))} &\lesssim ||u_{\epsilon} - u_{\epsilon, 2P+4}||_{H^1(\Omega_{\epsilon})} + \sum_{p \in \mathbb{P} \cap (P, 2P+4]} \sum_{\ell \in \mathbb{N}} \epsilon^p \ln^\ell \epsilon ||u_{p, \ell}||_{H^1(\Omega \setminus B(0, \delta))} \\ &= o(\epsilon^P) + o(\epsilon^P). \end{aligned} \quad \square $$

**Remark:** An alternative way of stating the error estimate is:

$$ \left\| u_{\epsilon} - \sum_{p \in \mathbb{P} \cap [0, P]} \sum_{\ell \in \mathbb{N}} \epsilon^p \ln^\ell \epsilon u_{p, \ell} \right\|_{H^1(\Omega \setminus B(0, \delta))} = O(\epsilon^P \ln^n \epsilon) \quad \text{with } n := \max\{\ell \in \mathbb{N} \mid u_{p, \ell} \neq 0\}. $$

# A Appendix: proof of Lemma 2.8

By Definition 2.6, we already know that the formulas to prove are true replacing $\bigoplus$ by $\bigoplus$. So it suffices to show that those sums are direct.

1. Let us show that $\mathcal{A}(\Omega) = \bigoplus_{d \in \mathbb{R}} \mathcal{A}_d(\Omega)$. Let $n \in \mathbb{N}$ and, for any $j \in [1, n]$, $d_j \in \mathbb{R}$ and $\varphi_j \in \mathcal{A}_{d_j}(\Omega)$. We assume that $d_1 < d_2 < \dots < d_n$ and $\sum_{j=1}^n \varphi_j = 0$. Let us show that $\forall j \in [1, n]$, $\varphi_j = 0$. Let $\theta \in (0, \Theta)$. By definition of $\mathcal{A}_{d_j}(\Omega)$, for any $j \in [1, n]$, there is $P_{\theta,j} \in \mathbb{C}[T]$ s.t.: $\forall r \in \mathbb{R}_+^*$, $\varphi_j(r, \theta) = r^{d_j} P_{\theta,j}(\ln r)$. Let us assume by contradiction that: $\exists j \in [1, n]$, $P_{\theta,j} \neq 0$. Let $j_0 := \max\{j \in [1, n] \mid P_{\theta,j} \neq 0\}$. Then $0 = \sum_{i=1}^n \lambda_i \varphi_i(r, \theta) \sim r^{d_{j_0}} P_{\theta,j_0}(\ln r)$ when $r \to \infty$, so $P_{\theta,j_0} = 0$. This is contradictory, so: $\forall \theta \in (0, \Theta), \forall j \in [1, n], \forall r \in \mathbb{R}_+^*, \varphi_j(r, \theta) = 0$.

2. By the same method, one can show that $\mathcal{A}(D) = \bigoplus_{d \in \mathbb{R}} \mathcal{A}_d(D)$ for any $D \in \{\Pi, \Gamma, \Lambda\}$.

3. Now we will prove (2.7). Let $d \in \mathbb{R}$ and $I := \{(q, k) \in \mathbb{R} \times \mathbb{N} \mid q + k = d\}$. Separating real and imaginary parts, it is enough to show that:

$$ \mathcal{A}_d(\Omega) \cap \mathcal{C}^0(\Omega, \mathbb{R}) = \bigoplus_{(q,k) \in I} \{z \mapsto \Im[(\alpha z)^q \bar{\alpha} \bar{z}^k P(\log(\alpha z))] \mid P \in \mathbb{R}[T] \text{ and } \mathcal{P}(q, k, P)\}. $$
**Proof of Theorem 1.1:** It follows from Theorem 4.2 applied at order $2P+4$ and from the fact that, for small enough $\epsilon$, $\sum_{p \in \mathbb{P} \cap [0,2P]} \sum_{\ell \in \mathbb{N}} \epsilon^p \ln^\ell \epsilon u_{p,\ell}$ and $u_{\epsilon,2P}$ coincide in $\Omega \setminus B(0, \delta)$:

$$ \left\| u_{\epsilon} - \sum_{p \in \mathbb{P} \cap [0, P]} \sum_{\ell \in \mathbb{N}} \epsilon^p \ln^{\ell} \epsilon u_{p, \ell} \right\|_{H^1(\Omega \setminus B(0, \delta))} \lesssim \|u_{\epsilon} - u_{\epsilon, 2P+4}\|_{H^1(\Omega_{\epsilon})} + \sum_{p \in \mathbb{P} \cap (P, 2P+4]} \sum_{\ell \in \mathbb{N}} \epsilon^p \ln^{\ell} \epsilon \|u_{p, \ell}\|_{H^1(\Omega \setminus B(0, \delta))} \\ = o(\epsilon^P) + o(\epsilon^P). $$

□

**Remark:** An alternative way of stating the error estimate is:

$$ \left\| u_{\epsilon} - \sum_{p \in \mathbb{P} \cap [0, P)} \sum_{\ell \in \mathbb{N}} \epsilon^p \ln^{\ell} \epsilon u_{p, \ell} \right\|_{H^1(\Omega \setminus B(0, \delta))} = O(\epsilon^P \ln^n \epsilon) \quad \text{with } n := \max\{\ell \in \mathbb{N} \mid u_{P, \ell} \neq 0\}. $$

# A Appendix: proof of Lemma 2.8

By Definition 2.6, we already know that the formulas to prove are true replacing $\bigoplus$ by $\bigoplus$. So it suffices to show that those sums are direct.

1. Let us show that $\mathcal{A}(\Omega) = \bigoplus_{d \in \mathbb{R}} \mathcal{A}_d(\Omega)$. Let $n \in \mathbb{N}$ and, for any $j \in [1, n]$, $d_j \in \mathbb{R}$ and $\varphi_j \in \mathcal{A}_{d_j}(\Omega)$. We assume that $d_1 < d_2 < \cdots < d_n$ and $\sum_{j=1}^n \varphi_j = 0$. Let us show that $\forall j \in [1, n]$, $\varphi_j = 0$. Let $\theta \in (0, \Theta)$. By definition of $\mathcal{A}_{d_j}(\Omega)$, for any $j \in [1, n]$, there is $P_{\theta,j} \in \mathbb{C}[T]$ s.t.: $\forall r \in \mathbb{R}_+^*$, $\varphi_j(r, \theta) = r^{d_j} P_{\theta,j}(\ln r)$. Let us assume by contradiction that: $\exists j \in [1, n]$, $P_{\theta,j} \neq 0$. Let $j_0 := \max\{j \in [1, n] \mid P_{\theta,j} \neq 0\}$. Then $0 = \sum_{i=1}^n \lambda_i \varphi_i(r, \theta) \sim r^{d_{j_0}} P_{\theta,j_0}(\ln r)$ when $r \to \infty$, so $P_{\theta,j_0} = 0$. This is contradictory, so: $\forall \theta \in (0, \Theta), \forall j \in [1, n], \forall r \in \mathbb{R}_+^*, \varphi_j(r, \theta) = 0$.

2. By the same method, one can show that $\mathcal{A}(D) = \bigoplus_{d \in \mathbb{R}} \mathcal{A}_d(D)$ for any $D \in \{\Pi, \Gamma, \Lambda\}$.

3. Now we will prove (2.7). Let $d \in \mathbb{R}$ and $I := \{(q, k) \in \mathbb{R} \times \mathbb{N} \mid q + k = d\}$. Separating real and imaginary parts, it is enough to show that:

$$ \mathcal{A}_d(\Omega) \cap \mathcal{C}^0(\Omega, \mathbb{R}) = \bigoplus_{(q,k) \in I} \{z \mapsto \Im[(\alpha z)^q \bar{\alpha} \bar{z}^k P(\log(\alpha z))] \mid P \in \mathbb{R}[T] \text{ and } \mathcal{P}(q,k,P)\}. $$

Let, for any $(q, k) \in I$, $P_{q,k} \in \mathbb{R}[T]$ and $\varphi_{q,k} : z \mapsto \Im[(\alpha z)^q \bar{\alpha} \bar{z}^k P_{q,k}(\log(\alpha z))]$. We assume that $\mathcal{P}(q, k, P_{q,k})$ holds for any $(q, k) \in I$, that the $P_{q,k}$ are all null except for a finite number, and that $\sum_{(q,k) \in I} \varphi_{q,k} = 0$. Let us show by induction on $m := \max_{(q,k) \in I} \deg P_{q,k}$ that: $\forall (q, k) \in I$, $P_{q,k} = 0$ (which implies in turn: $\forall (q, k), \varphi_{q,k} = 0$). We initialize at $m = -\infty$, i.e. $(\forall (q, k), P_{q,k} = 0)$, which is trivial. Thus, only the inductive step $(m \in \mathbb{N})$ remains to prove. To do so, we first note that in $\Omega$:

$$ 0 = r^{-d} \sum_{(q,k) \in I} \varphi_{q,k}(r, \theta) = \sum_{(q,k) \in I} \Im[e^{i(q-k)(\theta-\Theta)} P_{q,k}(\ln r + i(\theta-\Theta))] \quad (A.1) $$

By applying $r \partial_r$, we deduce: $0 = \sum_{(q,k) \in I} \Im[e^{i(q-k)(\theta-\Theta)} P'_{q,k}(\ln r + i(\theta-\Theta))]$.

For any $(q, k) \in I$, let us define (note the switch of indexes at line 2):

$$ Q_{q,k} := \begin{cases} P'_{q,k} & \text{if } q \notin \mathbb{N} \\ P'_{q,k} - P'_{q,k}(0) & \text{if } q \in \mathbb{N} \text{ and } q \le k \\ P'_{q,k} - P'_{k,q}(0) & \text{if } q \in \mathbb{N} \text{ and } q > k \end{cases} $$

Then $(\forall (q, k) \in I, \mathcal{P}(q, k, Q_{q,k}))$. Since $\Im[(\alpha z)^q \bar{\alpha} \bar{z}^k] = -\Im[(\alpha z)^k \bar{\alpha} \bar{z}^q]$ (and it is null when $q = k$), the previous equality rewrites as

$$ 0 = \sum_{(q,k) \in I} \Im[(\alpha z)^q \bar{\alpha} \bar{z}^k Q_{q,k}(\log(\alpha z))] $$

Then, by induction hypothesis: $\forall (q, k) \in I$, $Q_{q,k} = 0$. This means for $P_{q,k}$ that:
Let, for any $(q, k) \in I$, $P_{q,k} \in \mathbb{R}[T]$ and $\varphi_{q,k} : z \mapsto \mathfrak{F}[(\alpha z)^q \bar{\alpha} \bar{z}^k P_{q,k}(\log(\alpha z))]$. We assume that $\mathcal{P}(q, k, P_{q,k})$ holds for any $(q, k) \in I$, that the $P_{q,k}$ are all null except for a finite number, and that $\sum_{(q,k) \in I} \varphi_{q,k} = 0$. Let us show by induction on $m := \max_{(q,k) \in I} \deg P_{q,k}$ that: $\forall (q, k) \in I$, $P_{q,k} = 0$ (which implies in turn: $\forall (q, k)$, $\varphi_{q,k} = 0$). We initialize at $m = -\infty$, i.e. $(\forall (q, k), P_{q,k} = 0)$, which is trivial. Thus, only the inductive step $(m \in \mathbb{N})$ remains to prove.

To do so, we first note that in $\Omega$:

$$0 = r^{-d} \sum_{(q,k) \in I} \varphi_{q,k}(r, \theta) = \sum_{(q,k) \in I} \mathfrak{F}[e^{i(q-k)(\theta-\Theta)} P_{q,k}(\ln r + i(\theta-\Theta))] \quad (A.1)$$

$$\text{By applying } r \partial_r, \text{ we deduce: } 0 = \sum_{(q,k) \in I} \mathfrak{F}[e^{i(q-k)(\theta-\Theta)} P'_{q,k}(\ln r + i(\theta-\Theta))].$$

For any $(q, k) \in I$, let us define (note the switch of indexes at line 2):

$$Q_{q,k} := \begin{cases} P'_{q,k} & \text{if } q \notin \mathbb{N} \\ P'_{q,k} - P'_{q,k}(0) & \text{if } q \in \mathbb{N} \text{ and } q \le k \\ P'_{q,k} - P'_{k,q}(0) & \text{if } q \in \mathbb{N} \text{ and } q > k \end{cases}$$

Then $(\forall (q, k) \in I, \mathcal{P}(q, k, Q_{q,k}))$. Since $\mathfrak{F}[(\alpha z)^q \bar{\alpha} \bar{z}^k] = -\mathfrak{F}[(\alpha z)^k \bar{\alpha} \bar{z}^q]$ (and it is null when $q = k$), the previous equality rewrites as

$$0 = \sum_{(q,k) \in I} \mathfrak{F}[(\alpha z)^q \bar{\alpha} \bar{z}^k Q_{q,k}(\log(\alpha z))].$$

Then, by induction hypothesis: $\forall (q, k) \in I$, $Q_{q,k} = 0$. This means for $P_{q,k}$ that:

* If $q \notin \mathbb{N}$, then $P'_{q,k} = 0$.

* If $q \in \mathbb{N}$ and $q \neq k$, then $P''_{q,k} = 0$ and $P'_{q,k}(0) - P'_{k,q}(0) = 0$ (A.2).

* If $q \in \mathbb{N}$ and $q = k$, then $P''_{q,k} = 0$.

So (A.1) reduces to:

$$
\begin{align*}
0 &= \sum_{(q,k) \in I} \mathfrak{F}[e^{i(q-k)(\theta-\Theta)} P_{q,k}(0)] + \mathbf{1}_{q \in \mathbb{N}} \cdot \mathfrak{F}[e^{i(q-k)(\theta-\Theta)} P'_{q,k}(0)(\ln r + i(\theta-\Theta))] \\
&= \sum_{(q,k) \in I} \mathbf{1}_{(q \notin \mathbb{N} \text{ or } q > k)} \cdot \mathfrak{F}[e^{i(q-k)(\theta-\Theta)} P_{q,k}(0)] && \text{by } \mathcal{P} \\
&\quad + \mathbf{1}_{(q \in \mathbb{N} \text{ and } q=k)} \cdot \mathfrak{F}[e^{i(q-k)(\theta-\Theta)} P'_{q,k}(0)(\ln r + i(\theta-\Theta))] \\
&\quad + \mathbf{1}_{(q \in \mathbb{N} \text{ and } q > k)} \cdot \mathfrak{F}[e^{i(q-k)(\theta-\Theta)} P'_{q,k}(0) 2i(\theta-\Theta)] && \text{by (A.2)} \\
&= \sum_{(q,k) \in I} \mathbf{1}_{(q \notin \mathbb{N} \text{ or } q > k)} \cdot P_{q,k}(0) \sin((q-k)(\theta-\Theta)) \\
&\quad + \mathbf{1}_{(q \in \mathbb{N} \text{ and } q=k)} \cdot P'_{q,k}(0) \cdot (\theta-\Theta) \\
&\quad + \mathbf{1}_{(q \in \mathbb{N} \text{ and } q > k)} \cdot 2P'_{q,k}(0) \cdot \sin((q-k)(\theta-\Theta) + \frac{\pi}{2}) \cdot (\theta-\Theta)
\end{align*}
$$

The functions of $\theta$ present here are linearly independent. So the coefficients $P_{q,k}(0)$ and $P'_{q,k}(0)$ are all zero. This concludes the proof. $\square$

# B Appendix: asymptotic behaviors w.r.t. r

In this section, we prove Theorems 2.22 and 2.23. We will use Sections 3.1 and 3.2 which are after Theorems 2.22 and 2.23 in the paper, but are independent of them.

## B.1 Proof of Theorem 2.23: asymptotic behavior for corner fields-like problems

The proof relies on the Kondrat'ev theory, usually used to analyse singularities of solutions of elliptic equations, see [30, 31, 33, 12, 14, 32]. We use it in a way that gives an expansion in $\mathcal{A}^+(\Pi)$.
* If $q \notin \mathbb{N}$, then $P'_{q,k} = 0$.

* If $q \in \mathbb{N}$ and $q \neq k$, then $P''_{q,k} = 0$ and $P'_{q,k}(0) - P'_{k,q}(0) = 0$ (A.2).

* If $q \in \mathbb{N}$ and $q = k$, then $P''_{q,k} = 0$.

So (A.1) reduces to :

$$
\begin{align*}
0 &= \sum_{(q,k) \in I} \mathfrak{F}[e^{i(q-k)(\theta-\Theta)} P_{q,k}(0)] + \mathbf{1}_{q \in \mathbb{N}} \cdot \mathfrak{F}[e^{i(q-k)(\theta-\Theta)} P'_{q,k}(0)(\ln r + i(\theta - \Theta))] \\
&= \sum_{(q,k) \in I} \mathbf{1}_{(q \notin \mathbb{N} \text{ or } q > k)} \cdot \mathfrak{F}[e^{i(q-k)(\theta-\Theta)} P_{q,k}(0)] & \text{by } \mathcal{P} \\
&\quad + \mathbf{1}_{(q \in \mathbb{N} \text{ and } q=k)} \cdot \mathfrak{F}[e^{i(q-k)(\theta-\Theta)} P'_{q,k}(0)(\ln r + i(\theta - \Theta))] \\
&\quad + \mathbf{1}_{(q \in \mathbb{N} \text{ and } q > k)} \cdot \mathfrak{F}[e^{i(q-k)(\theta-\Theta)} P'_{q,k}(0) 2i(\theta - \Theta)] & \text{by (A.2)} \\
&= \sum_{(q,k) \in I} \mathbf{1}_{(q \notin \mathbb{N} \text{ or } q > k)} \cdot P_{q,k}(0) \sin((q-k)(\theta - \Theta)) \\
&\quad + \mathbf{1}_{(q \in \mathbb{N} \text{ and } q=k)} \cdot P'_{q,k}(0) \cdot (\theta - \Theta) \\
&\quad + \mathbf{1}_{(q \in \mathbb{N} \text{ and } q > k)} \cdot 2P'_{q,k}(0) \cdot \sin((q-k)(\theta - \Theta) + \frac{\pi}{2}) \cdot (\theta - \Theta)
\end{align*}
$$

The functions of $\theta$ present here are linearly independent. So the coefficients $P_{q,k}(0)$ and $P'_{q,k}(0)$ are all zero. This concludes the proof. $\square$

# B Appendix: asymptotic behaviors w.r.t. r

In this section, we prove Theorems 2.22 and 2.23. We will use Sections 3.1 and 3.2 which are after Theorems 2.22 and 2.23 in the paper, but are independent of them.

## B.1 Proof of Theorem 2.23: asymptotic behavior for corner fields-like problems

The proof relies on the Kondratiev theory, usually used to analyse singularities of solutions of elliptic equations, see [25, 27, 11, 13, 26]. We use it in a way that gives an expansion in $A^+(\Pi)$.

First we introduce the variables $(t, \theta)$, defined as $(\ln r, \theta)$ in $\Omega$ and $(\ln x, Y)$ in $\Lambda$. The pair $(t, \theta)$ lies in $\bar{\Pi} := \mathbb{R} \times (-1, \Theta)$. Moreover, we denote $\bar{\Omega}$, $\bar{\Lambda}$, $\bar{\Sigma}_{\Omega}$, $\bar{\Gamma}$ and $\bar{\Sigma}_{\Lambda}$ the images of $\Omega$, $\Lambda$, $\Sigma_{\Omega}$, $\Gamma$ and $\Sigma_{\Lambda}$ by the change of variable $(x, y) \rightsquigarrow (t, \theta)$, see Figure 10. The notation $\dots$ is intended to remind the strip shape of $\bar{\Pi}$. Finally, for any $u: \Pi \to \mathbb{C}$, we denote

$$ \bar{u}: (t, \theta) \in \bar{\Pi} \mapsto \begin{cases} u(r = e^t, \theta) & \text{in } \bar{\Omega} \\ u(x = e^t, Y = \theta) & \text{in } \bar{\Lambda} \end{cases} $$

<figure><img src="image_8.png" /><figcaption>Figure 10: The change of variables $(x, y) \rightsquigarrow (t, \theta)$ and the associated domains.</figcaption></figure>
First we introduce the variables $(t, \theta)$, defined as $(\ln r, \theta)$ in $\Omega$ and $(\ln x, Y)$ in $\Lambda$. The pair $(t, \theta)$ lies in $\bar{\Pi} := \mathbb{R} \times (-1, \Theta)$. Moreover, we denote $\bar{\Omega}$, $\bar{\Lambda}$, $\bar{\Sigma}_{\Omega}$, $\bar{\Gamma}$ and $\bar{\Sigma}_{\Lambda}$ the images of $\Omega$, $\Lambda$, $\Sigma_{\Omega}$, $\Gamma$ and $\Sigma_{\Lambda}$ by the change of variable $(x, y) \rightsquigarrow (t, \theta)$, see Figure 10. The notation $\dots$ is intended to remind the strip shape of $\bar{\Pi}$. Finally, for any $u: \Pi \to \mathbb{C}$, we denote

$$ \bar{u} : (t, \theta) \in \bar{\Pi} \mapsto \begin{cases} u(r = e^t, \theta) & \text{in } \bar{\Omega} \\ u(x = e^t, Y = \theta) & \text{in } \bar{\Lambda} \end{cases} $$

<figure><img src="image_11.png" /><figcaption>Figure 10: The change of variables $(x, y) \rightsquigarrow (t, \theta)$ and the associated domains.</figcaption></figure>

**Definition B.1: Kondrat'ev spaces**

Let $H$ be a Hilbert space and $(s, \beta) \in \mathbb{R}^2$. We define $K_{\beta}^s(\mathbb{R}, H) := \{t \mapsto e^{\beta t} u(t) \mid u \in H^s(\mathbb{R}, H)\}$, equipped with the norm $\|u\|_{K_{\beta}^s(\mathbb{R}, H)} := \|t \mapsto e^{-\beta t} u(t)\|_{H^s(\mathbb{R}, H)}$. We also denote $K_{\beta}^{\infty}(\mathbb{R}, H) := \bigcap_{s \in \mathbb{R}} K_{\beta}^s(\mathbb{R}, H)$, and $K_{\beta}^s(\bar{\Gamma}) := K_{\beta}^s(\mathbb{R}, \mathbb{C})$ (identifying the line $\bar{\Gamma}$ with $\mathbb{R}$).

**Remarks:**

* If $s = m \in \mathbb{N}$, then $K_{\beta}^m(\mathbb{R}, H) = \{u \in H_{\text{loc}}^m(\mathbb{R}, H) \mid \forall k \in [0, m], e^{-\beta t} \partial_t^k u(t) \in L^2(\mathbb{R}, H)\}$.

* Note that if $u \in K_{\beta}^s(\mathbb{R}, H)$ and $u$ is zero in a vicinity of $-\infty$, then: $\forall \beta' \ge \beta$, $u \in K_{\beta'}^s(\mathbb{R}, H)$.

* Kondrat'ev's spaces are linked to $o_{\partial}$ (see Definition 2.20) in the following way. Let $\chi \in C^{\infty}(\mathbb{R})$ be equal to 0 in a vicinity of $-\infty$ and 1 in a vicinity of $+\infty$. Then for any $u: \Omega \to \mathbb{C}$:

$$ \begin{cases} \forall \beta \in \mathbb{R}, \quad \chi \bar{u} \in \bigcap_{m \in \mathbb{N}} K_{\beta}^{\infty}(\mathbb{R}, H^m(0, \Theta)) \Rightarrow \forall d > \beta, \quad u = o_{\partial}(r^d) \\ \forall d \in \mathbb{R}, \quad u = o_{\partial}(r^d) \Rightarrow \exists t_0 \in \mathbb{R}, \forall \beta > d, \quad \chi(\cdot + t_0) \bar{u} \in \bigcap_{m \in \mathbb{N}} K_{\beta}^{\infty}(\mathbb{R}, H^m(0, \Theta)) \end{cases} $$

And there are similar implications in $\Lambda$ and $\Gamma$.

Let $H$ be a Hilbert space, $\beta \in \mathbb{R}$ and $\varphi \in K_{\beta}^0(\mathbb{R}, H)$. For any $\lambda \in \mathbb{C}$ s.t. $\Re(\lambda) = \beta$ we define the (bilateral) Laplace transform of $\varphi$ at $\lambda$ as

$$ \hat{\varphi}(\lambda) := \int_{-\infty}^{+\infty} e^{-\lambda t} \varphi(t) dt = \mathcal{F}[t \mapsto e^{-\beta t} \varphi(t)](\mathfrak{F}(\lambda)) \quad (B.1) $$

where $\mathcal{F}$ denotes the Fourier transform. By properties of $\mathcal{F}$, we have $\hat{\varphi} \in L^2(\{\lambda \in \mathbb{C} \mid \Re(\lambda) = \beta\})$. If $\varphi$ depends on $t$ and $\theta$, $\hat{\varphi}$ implies that we see $\varphi$ as a function from $\mathbb{R}$ to a space of functions of $\theta$.

To introduce the method, let us use the Laplace transform on $\bar{S}$. Let $\chi_{\infty}$ be the truncation function introduced in Section 3.2 and $s := \chi_{\infty}S$. Since $S \in V + \chi_{\infty}A(\Pi)$, there is $\beta \in \mathbb{R}$ s.t. $s \in K_{\beta}^0(\mathbb{R}, L^2(-1, \Theta))$, so $\hat{s}(\lambda, \theta)$ is well-defined for any $\lambda \in \mathbb{C}$ s.t. $\Re(\lambda) > \beta$. In addition we have for any $f \in L^2(\Omega)$, $g \in L^2(\Lambda)$, $h \in H^{-1/2}(\Gamma)$ and $u \in H_{\text{loc}}^1(\Pi)$:

$$ \left\{ \begin{array}{ll} -\mu_0 \Delta u = f & \text{in } \Omega \\ -\mu_1 \Delta u = g & \text{in } \Lambda \\ u|_{y=0^+} - u|_{y=0^-} = 0 & \text{on } \Gamma \\ \mu_0 \partial_y u|_{y=0^+} - \mu_1 \partial_y u|_{y=0^-} = h & \text{on } \Gamma \\ u = 0 & \text{on } \Sigma_{\Omega} \cup \Sigma_{\Lambda} \end{array} \right. \iff \left\{ \begin{array}{ll} -e^{-2t} \mu_0 \Delta \bar{u} = \bar{f} & \text{in } \bar{\Omega} \\ -\mu_1 (e^{-2t}(\partial_t^2 - \partial_t) + \partial_{\theta}^2) \bar{u} = \bar{g} & \text{in } \bar{\Lambda} \\ \bar{u}|_{\theta=0^+} - \bar{u}|_{\theta=0^-} = 0 & \text{on } \bar{\Gamma} \\ e^{-t} \mu_0 \partial_{\theta} \bar{u}|_{\theta=0^+} - \mu_1 \partial_{\theta} \bar{u}|_{\theta=0^-} = \bar{h} & \text{on } \bar{\Gamma} \\ \bar{u} = 0 & \text{on } \bar{\Sigma}_{\Omega} \cup \bar{\Sigma}_{\Lambda} \end{array} \right. \quad (B.2) $$
**Definition B.1: Kondratiev spaces**

Let $H$ be a Hilbert space and $(s, \beta) \in \mathbb{R}^2$. We define $K_{\beta}^s(\mathbb{R}, H) := \{t \mapsto e^{\beta t}u(t) \mid u \in H^s(\mathbb{R}, H)\}$,
equipped with the norm $\|u\|_{K_{\beta}^s(\mathbb{R}, H)} := \|t \mapsto e^{-\beta t}u(t)\|_{H^s(\mathbb{R}, H)}$. We also denote $K_{\beta}^{\infty}(\mathbb{R}, H) := \bigcap_{s \in \mathbb{R}} K_{\beta}^s(\mathbb{R}, H)$, and $K_{\beta}^s(\bar{\Gamma}) := K_{\beta}^s(\mathbb{R}, \mathbb{C})$ (identifying the line $\bar{\Gamma}$ with $\mathbb{R}$).

**Remarks:**

* If $s = m \in \mathbb{N}$, then $K_{\beta}^{m}(\mathbb{R}, H) = \{u \in H_{\text{loc}}^{m}(\mathbb{R}, H) \mid \forall k \in [0, m], e^{-\beta t} \partial_{t}^{k} u(t) \in L^{2}(\mathbb{R}, H)\}$.

* Note that if $u \in K_{\beta}^s(\mathbb{R}, H)$ and $u$ is zero in a vicinity of $-\infty$, then: $\forall \beta' \ge \beta$, $u \in K_{\beta'}^s(\mathbb{R}, H)$.

* Kondratiev's spaces are linked to $o_{\partial}$ (see Definition 2.20) in the following way. Let $\chi \in C^{\infty}(\mathbb{R})$ be equal to 0 in a vicinity of $-\infty$ and 1 in a vicinity of $+\infty$. Then for any $u : \Omega \to \mathbb{C}$:

$$ \left\{ \begin{array}{ll} \forall \beta \in \mathbb{R}, & \chi \bar{u} \in \bigcap_{m \in \mathbb{N}} K_{\beta}^{\infty}(\mathbb{R}, H^m(0, \Theta)) \Rightarrow \forall d > \beta, u = \underset{r \to \infty}{o_{\partial}(r^d)} \\ \\ \forall d \in \mathbb{R}, & u = \underset{r \to \infty}{o_{\partial}(r^d)} \Rightarrow \exists t_0 \in \mathbb{R}, \forall \beta > d, \chi(\cdot + t_0) \bar{u} \in \bigcap_{m \in \mathbb{N}} K_{\beta}^{\infty}(\mathbb{R}, H^m(0, \Theta)) \end{array} \right. $$

And there are similar implications in $\Lambda$ and $\Gamma$.

Let $H$ be a Hilbert space, $\beta \in \mathbb{R}$ and $\varphi \in K_{\beta}^{0}(\mathbb{R}, H)$. For any $\lambda \in \mathbb{C}$ s.t. $\Re(\lambda) = \beta$ we define the (bilateral) Laplace transform of $\varphi$ at $\lambda$ as

$$ \hat{\varphi}(\lambda) := \int_{-\infty}^{+\infty} e^{-\lambda t} \varphi(t) \, dt = \mathcal{F}[t \mapsto e^{-\beta t} \varphi(t)](\mathfrak{F}(\lambda)) \quad (B.1) $$

where $\mathcal{F}$ denotes the Fourier transform. By properties of $\mathcal{F}$, we have $\hat{\varphi} \in L^2(\{\lambda \in \mathbb{C} \mid \Re(\lambda) = \beta\})$. If $\varphi$ depends on $t$ and $\theta$, $\hat{\varphi}$ implies that we see $\varphi$ as a function from $\mathbb{R}$ to a space of functions of $\theta$.

To introduce the method, let us use the Laplace transform on $\bar{S}$. Let $\chi_{\infty}$ be the truncation function introduced in Section 3.2 and $s := \chi_{\infty}\bar{S}$. Since $S \in V + \chi_{\infty}\mathcal{A}(\Pi)$, there is $\beta \in \mathbb{R}$ s.t. $s \in K_{\beta}^{0}(\mathbb{R}, L^2(-1, \Theta))$, so $\hat{s}(\lambda, \theta)$ is well-defined for any $\lambda \in \mathbb{C}$ s.t. $\Re(\lambda) > \beta$. In addition we have for any $f \in L^2(\Omega)$, $g \in L^2(\Lambda)$, $h \in H^{-1/2}(\Gamma)$ and $u \in H_{\text{loc}}^{1}(\Pi)$:

$$ \left\{ \begin{array}{ll} -\mu_0 \Delta u = f & \text{in } \Omega \\ -\mu_1 \Delta u = g & \text{in } \Lambda \\ u|_{y=0^+} - u|_{y=0^-} = 0 & \text{on } \Gamma \\ \mu_0 \partial_y u|_{y=0^+} - \mu_1 \partial_y u|_{y=0^-} = h & \text{on } \Gamma \\ u=0 & \text{on } \Sigma_{\Omega} \cup \Sigma_{\Lambda} \end{array} \right. \iff \left\{ \begin{array}{ll} -e^{-2t} \mu_0 \Delta \bar{u} = \bar{f} & \text{in } \bar{\Omega} \\ -\mu_1 (e^{-2t}(\partial_t^2 - \partial_t) + \partial_\theta^2) \bar{u} = \bar{g} & \text{in } \bar{\Lambda} \\ \bar{u}|_{\theta=0^+} - \bar{u}|_{\theta=0^-} = 0 & \text{on } \bar{\Gamma} \\ e^{-t} \mu_0 \partial_\theta \bar{u}|_{\theta=0^+} - \mu_1 \partial_\theta \bar{u}|_{\theta=0^-} = \bar{h} & \text{on } \bar{\Gamma} \\ \bar{u}=0 & \text{on } \bar{\Sigma}_{\Omega} \cup \bar{\Sigma}_{\Lambda} \end{array} \right. \quad (B.2) $$

Taking $u := \chi_{\infty}S$ in (B.2) and applying the Laplace transform yield that $\hat{s}$ satisfies:

$$ \left\{ \begin{array}{ll} (\partial_{\theta}^2 + \lambda^2) \hat{s}(\lambda, \theta) = f & \text{if } \theta \in (0, \Theta) \\ \partial_{\theta}^2 \hat{s}(\lambda, \theta) = g - [(\lambda + 2)^2 - (\lambda + 2)] \hat{s}(\lambda + 2, \theta) & \text{if } \theta \in (-1, 0) \\ \hat{s}(\lambda, 0^+) - \hat{s}(\lambda, 0^-) = 0 & \\ \mu_1 \partial_{\theta} \hat{s}(\lambda, 0^-) = \mu_0 \partial_{\theta} \hat{s}(\lambda + 1, 0^+) & \\ \hat{s}(\lambda, \Theta) = \hat{s}(\lambda, -1) = 0 & \end{array} \right. \quad (B.3) $$

for some functions $f$ and $g$ depending on $F$ and $\chi_{\infty}$. Solving this system w.r.t. $\theta$ allows us to extend $\hat{s}$ w.r.t. $\lambda$ further to the left in the complex plane, except at the $\lambda$ for which (B.3) is ill-posed. These $\lambda$ are poles of $\hat{s}$ and they will be used in Proposition B.3 to identify terms of the asymptotic expansion of $S$.

For any $m \in \mathbb{N}^*$, let us define the Hilbert spaces $\mathcal{H}^m := \{u \in H_0^1(-1, \Theta) \mid u|_{(0,\Theta)} \in H^m(0, \Theta)$ and $u|_{(-1,0)} \in H^m(-1, 0)\}$ with the norm $\|u\|_{\mathcal{H}^m}^2 := \|u\|_{H^m(-1,0)}^2 + \|u\|_{H^m(0,\Theta)}^2$, and $\mathcal{H}_{\lambda}^m := H^m(0, \Theta) \times H^m(-1, 0) \times \mathbb{C}$.
We need to solve problems of the following form with $u \in \mathcal{H}^{m+2}$ and $(f,g,a) \in \mathcal{H}_{\lambda}^m$, $m \in \mathbb{N}$:

$$ \left\{ \begin{array}{ll} u'' + \lambda^2 u = f & \text{on } (0, \Theta) \\ u'' = g & \text{on } (-1, 0) \\ u(0^+) - u(0^-) = 0 & \\ u'(0^-) = a & \\ u(\Theta) = u(-1) = 0 & \end{array} \right. \quad (B.4) $$
Taking $u := \chi_{\infty}S$ in (B.2) and applying the Laplace transform yield that $\hat{s}$ satisfies:

$$ 
\left\{
\begin{array}{ll}
(\partial_{\theta}^2 + \lambda^2)\hat{s}(\lambda, \theta) = f & \text{if } \theta \in (0, \Theta) \\
\partial_{\theta}^2\hat{s}(\lambda, \theta) = g - [(\lambda + 2)^2 - (\lambda + 2)]\hat{s}(\lambda + 2, \theta) & \text{if } \theta \in (-1, 0) \\
\hat{s}(\lambda, 0^+) - \hat{s}(\lambda, 0^-) = 0 & \\
\mu_1\partial_{\theta}\hat{s}(\lambda, 0^-) = \mu_0\partial_{\theta}\hat{s}(\lambda + 1, 0^+) & \\
\hat{s}(\lambda, \Theta) = \hat{s}(\lambda, -1) = 0 &
\end{array}
\right.
\qquad (B.3) $$

for some functions $f$ and $g$ depending on $F$ and $\chi_{\infty}$. Solving this system w.r.t. $\theta$ allows us to extend $\hat{s}$ w.r.t. $\lambda$ further to the left in the complex plane, except at the $\lambda$ for which (B.3) is ill-posed. These $\lambda$ are poles of $\hat{s}$ and they will be used in Proposition B.3 to identify terms of the asymptotic expansion of $S$.

For any $m \in \mathbb{N}^*$, let us define the Hilbert spaces $\mathcal{H}^m := \{u \in H_0^1(-1, \Theta) | u|_{(0,\Theta)} \in H^m(0, \Theta)$ and $u|_{(-1,0)} \in H^m(-1, 0)\}$ with the norm $\|u\|_{\mathcal{H}^m} := \|u\|_{H^m(-1,0)}^2 + \|u\|_{H^m(0,\Theta)}^2$, and $\mathcal{H}_\chi^m := H^m(0, \Theta) \times H^m(-1, 0) \times \mathbb{C}$. We need to solve problems of the following form with $u \in \mathcal{H}^{m+2}$ and $(f,g,a) \in \mathcal{H}_\chi^m, m \in \mathbb{N}$:

$$ 
\left\{
\begin{array}{ll}
u'' + \lambda^2 u = f & \text{on } (0, \Theta) \\
u'' = g & \text{on } (-1, 0) \\
u(0^+) - u(0^-) = 0 & \\
u'(0^-) = a & \\
u(\Theta) = u(-1) = 0 &
\end{array}
\right.
\qquad (B.4) $$

Denoting $A(\lambda) : u \mapsto ((u'' + \lambda^2 u)|_{(0,\Theta)}, u''|_{(-1,0)}, u'(0^-))$, (B.4) is equivalent to $A(\lambda)(u) = (f,g,a)$. Note that for any $m \in \mathbb{N}$, $A(\lambda) \in \mathcal{L}(\mathcal{H}^{m+2}, \mathcal{H}_\chi^m)$ where $\mathcal{L}(\cdot)$ denotes the space of continuous linear maps between two normed vector spaces.

**Lemma B.2:** Let $m \in \mathbb{N}$.

1. $A(\lambda) : \mathcal{H}^{m+2} \to \mathcal{H}_\chi^m$ is invertible iff $\lambda \in \mathbb{C} \setminus \frac{\pi}{\Theta}\mathbb{Z}^*$.

2. $\lambda \mapsto A(\lambda)^{-1}$ is meromorphic from $\mathbb{C}$ to $\mathcal{L}(\mathcal{H}_\chi^m, \mathcal{H}^{m+2})$. Its poles are in $\frac{\pi}{\Theta}\mathbb{Z}^*$ and are simple. Moreover, for any $q \in \frac{\pi}{\Theta}\mathbb{Z}^*$ and $v \in \mathcal{H}_\chi^m$, the residue $(\text{Res}_{\lambda=q} A(\lambda)^{-1})(v)$ is proportional to the function $\theta \in [-1, \Theta] \mapsto \mathbf{1}_{[0,\Theta]}(\theta) \cdot \sin(q\theta)$.

3. Let $\beta_1 < \beta_2$ be some reals. There is $C > 0$ depending only on $(m, \beta_1, \beta_2)$ s.t., for any $\lambda \in \mathbb{C}$ satisfying $\beta_1 < \Re(\lambda) < \beta_2$ and $|\Im(\lambda)| > 1$, we have: $\|A(\lambda)^{-1}\|_{\mathcal{L}(\mathcal{H}_\chi^m, \mathcal{H}^{m+2})} \le C |\Im(\lambda)|^{m+2}$ (B.5).

**Proof:**

1. Let $\lambda \in \mathbb{C}$. An easy calculation gives that any element of $\text{Ker } A(\lambda)$ must be proportional to $\theta \in [-1, \Theta] \mapsto \mathbf{1}_{[0,\Theta]}(\theta) \cdot \sin(\lambda\theta)$. This function belongs to $\mathcal{H}^m \setminus \{0\}$ iff $\lambda \in \frac{\pi}{\Theta}\mathbb{Z}^*$. Therefore $\text{Ker } A(\lambda) \ne 0 \Leftrightarrow \lambda \in \frac{\pi}{\Theta}\mathbb{Z}^*$.

Moreover, it is easy to see that, for any $(f,g,a) \in \mathcal{H}_\chi^m$, (B.4) with $\lambda := 0$ has a unique solution in $\mathcal{H}^{m+2}$. Thus $A(0) : \mathcal{H}^{m+2} \to \mathcal{H}_\chi^m$ is invertible. Now, for any $\lambda \in \mathbb{C}$, we have $A(\lambda) = A(0) + \lambda^2 B$ with $B : u \mapsto (u|_{(0,\Theta)}, 0, 0)$. $B$ is a compact operator from $\mathcal{H}^{m+2}$ to $\mathcal{H}_\chi^m$, so the Fredholm alternative holds for $A(\lambda)$. Therefore $A(\lambda) : \mathcal{H}^{m+2} \to \mathcal{H}_\chi^m$ is invertible iff $\lambda \in \mathbb{C} \setminus \frac{\pi}{\Theta}\mathbb{Z}^*$.

2. Since $\lambda \mapsto A(\lambda)$ is holomorphic on $\mathbb{C} \setminus \frac{\pi}{\Theta}\mathbb{Z}^*$, so too is $\lambda \mapsto A(\lambda)^{-1}$. Let us describe its behavior near the points $q \in \frac{\pi}{\Theta}\mathbb{Z}^*$ using [31, Theorem 5.1.1, p.147]. It depends on the “Jordan chains” of $A(q)$, which are the sequences $(u_0, \dots, u_n) \in (\mathcal{H}^m)^{n+1}$, $n \in \mathbb{N}$, s.t.

$$ \forall k \in [0, n], \quad \sum_{j=0}^{k} \frac{1}{j!} \left. \frac{d^j A}{d\lambda^j} \right|_{\lambda=q} (u_{k-j}) = 0 \qquad (B.6) $$

Let us compute these chains. Taking $k := 0$ in (B.6) gives $A(q)(u_0) = 0$, so $u_0$ is proportional to $\theta \mapsto \mathbf{1}_{[0,\Theta]}(\theta) \cdot \sin(q\theta)$ by step 1. In addition, if $n \ge 1$, taking $k := 1$ gives $A(q)(u_1) + \frac{dA}{d\lambda}(q)(u_0) = 0$. This implies on one hand $u_1'' + q^2 u_1 + 2qu_0 = 0$ in $(0, \Theta)$. On the other we get $u_1''|_{(-1,0)} = 0$ and
Denoting $A(\lambda) : u \mapsto ((u'' + \lambda^2 u)|_{(0,\Theta)}, u''|_{(-1,0)}, u'(0^-))$, (B.4) is equivalent to $A(\lambda)(u) = (f,g,a)$. Note that for any $m \in \mathbb{N}$, $A(\lambda) \in \mathcal{L}(\mathcal{H}^{m+2}, \mathcal{H}_\times^m)$ where $\mathcal{L}(\dots)$ denotes the space of continuous linear maps between two normed vector spaces.

**Lemma B.2:** Let $m \in \mathbb{N}$.

1. $A(\lambda) : \mathcal{H}^{m+2} \to \mathcal{H}_\times^m$ is invertible iff $\lambda \in \mathbb{C} \setminus \frac{\pi}{\Theta}\mathbb{Z}^*$.

2. $\lambda \mapsto A(\lambda)^{-1}$ is meromorphic from $\mathbb{C}$ to $\mathcal{L}(\mathcal{H}_\times^m, \mathcal{H}_\times^{m+2})$. Its poles are in $\frac{\pi}{\Theta}\mathbb{Z}^*$ and are simple. Moreover, for any $q \in \frac{\pi}{\Theta}\mathbb{Z}^*$ and $v \in \mathcal{H}_\times^m$, the residue $(\mathrm{Res}_{\lambda=q} A(\lambda)^{-1})(v)$ is proportional to the function $\theta \in [-1, \Theta] \mapsto \mathbf{1}_{[0,\Theta]}(\theta) \cdot \sin(q\theta)$.

3. Let $\beta_1 < \beta_2$ be some reals. There is $C > 0$ depending only on $(m, \beta_1, \beta_2)$ s.t., for any $\lambda \in \mathbb{C}$ satisfying $\beta_1 < \Re(\lambda) < \beta_2$ and $|\Im(\lambda)| > 1$, we have: $\|A(\lambda)^{-1}\|_{\mathcal{L}(\mathcal{H}_\times^m, \mathcal{H}^{m+2})} \le C|\Im(\lambda)|^{m+2}$ (B.5).

**Proof:**

1. Let $\lambda \in \mathbb{C}$. An easy calculation gives that any element of $\mathrm{Ker} A(\lambda)$ must be proportional to $\theta \in [-1, \Theta] \mapsto \mathbf{1}_{[0,\Theta]}(\theta) \cdot \sin(\lambda\theta)$. This function belongs to $\mathcal{H}^m \setminus \{0\}$ iff $\lambda \in \frac{\pi}{\Theta}\mathbb{Z}^*$. Therefore $\mathrm{Ker} A(\lambda) \ne 0 \Leftrightarrow \lambda \in \frac{\pi}{\Theta}\mathbb{Z}^*$.

Moreover, it is easy to see that, for any $(f,g,a) \in \mathcal{H}_\times^m$, (B.4) with $\lambda := 0$ has a unique solution in $\mathcal{H}^{m+2}$. Thus $A(0) : \mathcal{H}^{m+2} \to \mathcal{H}_\times^m$ is invertible. Now, for any $\lambda \in \mathbb{C}$, we have $A(\lambda) = A(0) + \lambda^2 B$ with $B : u \mapsto (u|_{(0,\Theta)}, 0, 0)$. $B$ is a compact operator from $\mathcal{H}^{m+2}$ to $\mathcal{H}_\times^m$, so the Fredholm alternative holds for $A(\lambda)$. Therefore $A(\lambda) : \mathcal{H}^{m+2} \to \mathcal{H}_\times^m$ is invertible iff $\lambda \in \mathbb{C} \setminus \frac{\pi}{\Theta}\mathbb{Z}^*$.

2. Since $\lambda \mapsto A(\lambda)$ is holomorphic on $\mathbb{C} \setminus \frac{\pi}{\Theta}\mathbb{Z}^*$, so too is $\lambda \mapsto A(\lambda)^{-1}$. Let us describe its behavior near the points $q \in \frac{\pi}{\Theta}\mathbb{Z}^*$ using [25, Theorem 5.1.1, p.147]. It depends on the “Jordan chains” of $A(q)$, which are the sequences $(u_0, \dots, u_n) \in (\mathcal{H}^m)^{n+1}$, $n \in \mathbb{N}$, s.t.

$$ \forall k \in [0, n], \quad \sum_{j=0}^{k} \frac{1}{j!} \frac{d^j A}{d\lambda^j} \Big|_{\lambda=q} (u_{k-j}) = 0 \qquad (B.6) $$

Let us compute these chains. Taking $k := 0$ in (B.6) gives $A(q)(u_0) = 0$, so $u_0$ is proportional to $\theta \mapsto \mathbf{1}_{[0,\Theta]}(\theta) \cdot \sin(q\theta)$ by step 1. In addition, if $n \ge 1$, taking $k := 1$ gives $A(q)(u_1) + \frac{dA}{d\lambda}(q)(u_0) = 0$. This implies on one hand $u_1'' + q^2 u_1 + 2qu_0 = 0$ in $(0, \Theta)$. On the other we get $u_1''|_{(-1,0)} = 0$ and $u_1'(0^-) = 0$, so $u_1|_{(-1,0)} = 0$, which gives $u_1(0) = u_1(\Theta) = 0$. Therefore

$$ 0 \ne \int_0^\Theta 2q|u_0|^2 = \int_0^\Theta -(u_1'' + q^2 u_1) \cdot \overline{u_0} = \int_0^\Theta -u_1 \cdot \underbrace{(u_0'' + q^2 u_0)}_{=0} = 0. $$

This is absurd, so we must have $n = 0$ for any Jordan chain of $A(q)$. Therefore, [25, Theorem 5.1.1, p.147] states that $A(\lambda)^{-1}$ has a simple pole at $q$, and $\mathrm{Im}(\mathrm{Res}_{\lambda=q} A(\lambda)^{-1}) = \mathbb{C}u_0$.

3. Let $(f,g,a) \in \mathcal{H}_\times^m$ and $u := A(\lambda)^{-1}(f,g,a) \in \mathcal{H}^{m+2}$. We will write $\lesssim$ for inequalities valid up to a constant that depends on $(m, \beta_1, \beta_2)$ but not on $(\lambda, f,g,a)$. By Poincaré's inequality and integration by parts, we have

$$ \|u\|_{H^1(-1,0)}^2 \lesssim \int_{-1}^{0} |u'|^2 = - \int_{0}^{-1} g\bar{u} + a\bar{u}(0) \lesssim (\|g\|_{L^2(-1,0)} + |a|) \cdot \|u\|_{H^1(-1,0)} $$

thus $\|u\|_{H^1(-1,0)} \lesssim \|g\|_{L^2(-1,0)} + |a|$. Since $u''|_{(-1,0)} = g$, we deduce $\|u\|_{H^{m+2}(-1,0)} \lesssim \|g\|_{H^m} + |a|$.

Then, let $u_1 : \theta \in [0, \Theta] \mapsto u(0)(1 - \frac{\theta}{\Theta})$, $v := u - u_1$ and $f_\lambda := f - \lambda^2 u_1$. We have $v(0) = v(\Theta) = 0$ and $v'' + \lambda^2 v = f_\lambda$, so

$$ \int_0^\Theta (-|v'|^2 + \lambda^2 |v|^2) = \int_0^\Theta f_\lambda \bar{v} $$

Dividing by $\lambda$ and taking the absolute value of the imaginary part, we get: $\frac{|\Im(\lambda)|}{|\lambda|^2} \|v'\|_{L^2}^2 + |\Im(\lambda)| \|v\|_{L^2}^2 \le \frac{1}{|\lambda|} \|f_\lambda\|_{L^2} \|v\|_{L^2}$. Now we assume $|\Im(\lambda)| > 1$ and $\beta_1 < \Re(\lambda) < \beta_2$, so $|\lambda| \lesssim |\Im(\lambda)|$. Thus $\|v'\|_{L^2}^2 + |\lambda|^2 \|v\|_{L^2}^2 \lesssim \|f_\lambda\|_{L^2} \|v\|_{L^2}$, which yields $\|v\|_{L^2} \lesssim |\lambda|^{-2} \|f_\lambda\|_{L^2}$ and then $\|v'\|_{L^2} \lesssim |\lambda|^{-1} \|f_\lambda\|_{L^2}$. Now from $v'' = f_\lambda - \lambda^2 v$, one can easily derive by induction on $m$ that $\|v\|_{H^{m+2}} \lesssim |\lambda|^m \|f_\lambda\|_{H^m}$. This implies $\|u\|_{H^{m+2}(0,\Theta)} \lesssim |\lambda|^m \|f\|_{H^m} + |\lambda|^{m+2} |u(0)|$, with $|u(0)| \lesssim \|u\|_{H^{m+2}(-1,0)}$. So finally $\|u\|_{H^{m+2}} \lesssim |\Im(\lambda)|^{m+2} \|(f,g,a)\|_{\mathcal{H}_\times^m}$. $\square$
$u_1'(0^-) = 0$, so $u_{1|(-1,0)} = 0$, which gives $u_1(0) = u_1(\Theta) = 0$. Therefore

$$0 \neq \int_{0}^{\Theta} 2q|u_0|^2 = \int_{0}^{\Theta} -(u_1'' + q^2 u_1) \cdot \overline{u_0} = \int_{0}^{\Theta} -u_1 \cdot \overbrace{(u_0'' + q^2 u_0)}^{=0} = 0.$$

This is absurd, so we must have $n = 0$ for any Jordan chain of $A(q)$. Therefore, [31, Theorem 5.1.1, p.147] states that $A(\lambda)^{-1}$ has a simple pole at $q$, and $\mathrm{Im}(\mathrm{Res}_{\lambda=q} A(\lambda)^{-1}) = \mathbb{C}u_0$.

3. Let $(f, g, a) \in \mathcal{H}_{\times}^m$ and $u := A(\lambda)^{-1}(f, g, a) \in \mathcal{H}^{m+2}$. We will write $\lesssim$ for inequalities valid up to a constant that depends on $(m, \beta_1, \beta_2)$ but not on $(\lambda, f, g, a)$. By Poincaré's inequality and integration by parts, we have

$$\|u\|_{H^1(-1,0)}^2 \lesssim \int_{-1}^{0} |u'|^2 = - \int_{0}^{-1} g \bar{u} + a \bar{u}(0) \lesssim (\|g\|_{L^2(-1,0)} + |a|) \cdot \|u\|_{H^1(-1,0)}$$

thus $\|u\|_{H^1(-1,0)} \lesssim \|g\|_{L^2(-1,0)} + |a|$. Since $u''|_{(-1,0)} = g$, we deduce $\|u\|_{H^{m+2}(-1,0)} \lesssim \|g\|_{H^m} + |a|$.

Then, let $u_1: \theta \in [0, \Theta] \mapsto u(0)(1-\frac{\theta}{\Theta})$, $v := u - u_1$ and $f_{\lambda} := f - \lambda^2 u_1$. We have $v(0) = v(\Theta) = 0$ and $v'' + \lambda^2 v = f_{\lambda}$, so

$$\int_{0}^{\Theta} (-|v'|^2 + \lambda^2 |v|^2) = \int_{0}^{\Theta} f_{\lambda} \bar{v}$$

Dividing by $\lambda$ and taking the absolute value of the imaginary part, we get: $\frac{|\Im(\lambda)|}{|\lambda|^2} \|v'\|_{L^2}^2 + |\Im(\lambda)| \|v\|_{L^2}^2 \le \frac{1}{|\lambda|} \|f_{\lambda}\|_{L^2} \|v\|_{L^2}$. Now we assume $|\Im(\lambda)| > 1$ and $\beta_1 < \Re(\lambda) < \beta_2$, so $|\lambda| \lesssim |\Im(\lambda)|$. Thus $\|v'\|_{L^2}^2 + |\lambda|^2 \|v\|_{L^2}^2 \lesssim \|f_{\lambda}\|_{L^2} \|v\|_{L^2}$, which yields $\|v\|_{L^2} \lesssim |\lambda|^{-2} \|f_{\lambda}\|_{L^2}$ and then $\|v'\|_{L^2} \lesssim |\lambda|^{-1} \|f_{\lambda}\|_{L^2}$. Now from $v'' = f_{\lambda} - \lambda^2 v$, one can easily derive by induction on $m$ that $\|v\|_{H^{m+2}} \lesssim |\lambda|^m \|f_{\lambda}\|_{H^m}$. This implies $\|u\|_{H^{m+2}(0,\Theta)} \lesssim |\lambda|^m \|f\|_{H^m} + |\lambda|^{m+2} |u(0)|$, with $|u(0)| \lesssim \|u\|_{H^{m+2}(-1,0)}$.
So finally $\|u\|_{H^{m+2}} \lesssim |\Im(\lambda)|^{m+2} \|(f,g,a)\|_{\mathcal{H}_{\times}^m}$. $\square$

**Proposition B.3** is the base step of the asymptotic expansion. It will be applied to $s$ the rest of the asymptotic expansion at a given order, and the function $s_r$ below will be the rest at the next order. Iterating this process provides an asymptotic expansion of $S$ at any order. Since Theorem 2.23 implies infinite regularity on $S$, we work in the spaces $K_{\beta}^{\infty}$ and $\mathcal{H}^m$ for any $m$.

**Proposition B.3:** Let $d \in \mathbb{R}$, $m \in \mathbb{N} \setminus \{0,1\}$, $s \in \bigcap_{\beta>d} K_{\beta}^{\infty}(\mathbb{R}, \mathcal{H}^m)$, $f \in \bigcap_{\beta>d-1} K_{\beta}^{\infty}(\mathbb{R}, H^{m-2}(0, \Theta))$, $g \in \bigcap_{\beta>d-1} K_{\beta}^{\infty}(\mathbb{R}, H^{m-2}(-1, 0))$, $h \in \bigcap_{\beta>d-1} K_{\beta}^{\infty}(\bar{\Gamma})$ and $d_{\min} := \min(\frac{\pi}{\Theta}\mathbb{Z}^* \cap (d-1, d])$. We assume:

$$\left\{ \begin{array}{ll} \mu_0 \Delta s = f & \text{in } \bar{\Omega} \\ \mu_1 (e^{-2t}(\partial_t^2 - \partial_t) + \partial_\theta^2)s = g & \text{in } \bar{\Lambda} \\ s|_{\theta=0^+} - s|_{\theta=0^-} = 0 & \text{on } \bar{\Gamma} \\ e^{-t}\mu_0 \partial_\theta s|_{\theta=0^+} - \mu_1 \partial_\theta s|_{\theta=0^-} = h & \text{on } \bar{\Gamma} \\ s = 0 & \text{on } \bar{\Sigma}_{\bar{\Omega}} \cup \bar{\Sigma}_{\bar{\Lambda}} \end{array} \right.$$

Then there is $s_r \in \bigcap_{d-1<\beta<d_{\min}} K_{\beta}^{\infty}(\mathbb{R}, \mathcal{H}^m)$ and $(c_q) \in \mathbb{C}^{\frac{\pi}{\Theta}\mathbb{Z}^* \cap (d-1, d]}$ s.t. $s = \sum_{q \in \frac{\pi}{\Theta}\mathbb{Z}^* \cap (d-1, d]} c_q \bar{\phi}_q + s_r$.

**Proof:** For any $\beta \in \mathbb{R}$, let $\mathbb{C}_{\mathbb{R}>\beta} := \{\lambda \in \mathbb{C} | \Re(\lambda) > \beta\}$. The Laplace transform of $s$ is well-defined and holomorphic from $\mathbb{C}_{\mathbb{R}>d}$ to $\mathcal{H}^m$. We will show that $\hat{s}$ has a meromorphic extension on $\mathbb{C}_{\mathbb{R}>d-1}$, with poles belonging in $\mathbb{P} := \frac{\pi}{\Theta}\mathbb{Z}^* \cap (d-1, d]$, then we will apply the residue theorem on a rectangle surrounding these poles. The functions $\phi_q$ will appear in the residues.

**Step 1:** $\hat{s}$ satisfies in $\mathbb{C}_{\mathbb{R}>d}$:

$$\left\{ \begin{array}{ll} \mu_0(\partial_\theta^2 + \lambda^2)\hat{s}(\lambda, \theta) = \hat{f}(\lambda, \theta) & \text{if } \theta \in (0, \Theta) \\ \mu_1 \partial_\theta^2 \hat{s}(\lambda, \theta) = \hat{g}(\lambda, \theta) - \mu_1 [(\lambda + 2)^2 - (\lambda + 2)] \hat{s}(\lambda + 2, \theta) := \bar{g}(\lambda, \theta) & \text{if } \theta \in (-1, 0) \\ \hat{s}(\lambda, 0^+) - \hat{s}(\lambda, 0^-) = 0 & \\ \mu_1 \partial_\theta \hat{s}(\lambda, 0^-) = \hat{h}(\lambda) + \mu_0 \partial_\theta \hat{s}(\lambda + 1, 0^+) := \bar{h}(\lambda) & \\ \hat{s}(\lambda, \Theta) = \hat{s}(\lambda, -1) = 0 & \end{array} \right.$$
**Proposition B.3** is the base step of the asymptotic expansion. It will be applied to $s$ the rest of the at a given order, and the function $s_r$ below will be the rest at the next order. Iterating this process provides an asymptotic expansion of $S$ at any order. Since Theorem 2.23 implies infinite regularity on $S$, we work in the spaces $K_\beta^\infty$ and $H^m$ for any $m$.

**Proposition B.3:** Let $d \in \mathbb{R}$, $m \in \mathbb{N} \setminus \{0, 1\}$, $s \in \bigcap_{\beta > d} K_\beta^\infty(\mathbb{R}, \mathcal{H}^m)$, $f \in \bigcap_{\beta > d-1} K_\beta^\infty(\mathbb{R}, H^{m-2}(0, \Theta))$, $g \in \bigcap_{\beta > d-1} K_\beta^\infty(\mathbb{R}, H^{m-2}(-1, 0))$, $h \in \bigcap_{\beta > d-1} K_\beta^\infty(\bar{\Gamma})$ and $d_{\min} := \min(\frac{\pi}{\Theta}\mathbb{Z}^* \cap (d-1, d])$. We assume:

$$ egin{cases} 
\mu_0 \Delta s = f & \text{in } \bar{\Omega} \\
\mu_1 (e^{-2t}(\partial_t^2 - \partial_t) + \partial_\theta^2)s = g & \text{in } \bar{\Lambda} \\
s|_{\theta=0^+} - s|_{\theta=0^-} = 0 & \text{on } \bar{\Gamma} \\
e^{-t}\mu_0 \partial_\theta s|_{\theta=0^+} - \mu_1 \partial_\theta s|_{\theta=0^-} = h & \text{on } \bar{\Gamma} \\
s = 0 & \text{on } \bar{\Sigma}_\Omega \cup \bar{\Sigma}_\Lambda
\end{cases} $$

Then there is $s_r \in \bigcap_{d-1 < \beta < d_{\min}} K_\beta^\infty(\mathbb{R}, \mathcal{H}^m)$ and $(c_q) \in \mathbb{C}^{\frac{\pi}{\Theta}\mathbb{Z}^* \cap (d-1, d]}$ s.t. $s = \sum_{q \in \frac{\pi}{\Theta}\mathbb{Z}^* \cap (d-1, d]} c_q \bar{\phi}_q + s_r$.

**Proof:** For any $\beta \in \mathbb{R}$, let $\mathbb{C}_{\mathbb{R}>\beta} := \{\lambda \in \mathbb{C} \mid \Re(\lambda) > \beta\}$. The Laplace transform of $s$ is well-defined and holomorphic from $\mathbb{C}_{\mathbb{R}>d}$ to $\mathcal{H}^m$. We will show that $\hat{s}$ has a meromorphic extension on $\mathbb{C}_{\mathbb{R}>d-1}$, with poles belonging in $\mathfrak{P} := \frac{\pi}{\Theta}\mathbb{Z}^* \cap (d-1, d]$, then we will apply the residue theorem on a rectangle surrounding these poles. The functions $\phi_q$ will appear in the residues.

**Step 1:** $\hat{s}$ satisfies in $\mathbb{C}_{\mathbb{R}>d}$:

$$ egin{cases} 
\mu_0(\partial_\theta^2 + \lambda^2)\hat{s}(\lambda, \theta) = \hat{f}(\lambda, \theta) & \text{if } \theta \in (0, \Theta) \\
\mu_1 \partial_\theta^2 \hat{s}(\lambda, \theta) = \hat{g}(\lambda, \theta) - \mu_1 [(\lambda + 2)^2 - (\lambda + 2)] \hat{s}(\lambda + 2, \theta) := \tilde{g}(\lambda, \theta) & \text{if } \theta \in (-1, 0) \\
\hat{s}(\lambda, 0^+) - \hat{s}(\lambda, 0^-) = 0 \\
\mu_1 \partial_\theta \hat{s}(\lambda, 0^-) = \hat{h}(\lambda) + \mu_0 \partial_\theta \hat{s}(\lambda + 1, 0^+) := \tilde{h}(\lambda) \\
\hat{s}(\lambda, \Theta) = \hat{s}(\lambda, -1) = 0
\end{cases} $$

For any $\lambda \in \mathbb{C}_{\mathbb{R}>d-1} \setminus \frac{\pi}{\Theta}\mathbb{Z}^*$, let $\tilde{s}(\lambda) := A(\lambda)^{-1}(\frac{1}{\mu_0}\hat{f}(\lambda), \frac{1}{\mu_1}\tilde{g}(\lambda), \frac{1}{\mu_1}\tilde{h}(\lambda))$. Then $\tilde{s}$ is also solution of the above system, so $\tilde{s}$ and $\hat{s}$ coincide on $\mathbb{C}_{\mathbb{R}>d} \setminus \frac{\pi}{\Theta}\mathbb{Z}^*$ by Lemma B.2. Hence $\tilde{s}$ is an extension of $\hat{s}$ on $\mathbb{C}_{\mathbb{R}>d-1} \setminus \mathfrak{P}$, that we will still denote $\hat{s}$. Lemma B.2 implies that this extension is meromorphic with simple poles and: $\forall q \in \mathfrak{P}$, $\mathrm{Res}_q \hat{s} = (\mathrm{Res}_{\lambda=q} A(\lambda)^{-1})(\frac{1}{\mu_0}\hat{f}(q), \frac{1}{\mu_1}\tilde{g}(q), \frac{1}{\mu_1}\tilde{h}(q))$.

**Step 2:** Let $t \in \mathbb{R}$, $b \in (d-1, d_{\min})$ and $k \in \mathbb{R}_+^*$. The residue theorem applied to $\lambda \mapsto e^{\lambda t}\hat{s}(\lambda)$ on the rectangle $[b, d+1] \times [-k, k] \subset \mathbb{C}$ gives the following equality in $\mathcal{H}^m$.

$$ 2i\pi \sum_{q \in \mathfrak{P}} \mathrm{Res}_q(\lambda \mapsto e^{\lambda t} \hat{s}(\lambda)) = \int_{-k}^{k} e^{(d+1+i\gamma)t} \hat{s}(d+1+i\gamma) \, d\gamma - \int_{-k}^{k} e^{(b+i\gamma)t} \hat{s}(b+i\gamma) \, d\gamma \\ + \int_{b}^{d+1} e^{(\beta-ik)t} \hat{s}(\beta-ik) \, d\beta - \int_{b}^{d+1} e^{(\beta+ik)t} \hat{s}(\beta+ik) \, d\beta $$

Note that when $k \to \infty$ the first integral tends to the inverse Laplace transform of $s$ (up to a constant). We will show that the last two integrals tend to 0, by proving the following:

$$ \forall n \in \mathbb{N}, \quad \sup_{\beta \in [b, d+1]} \int_{\mathbb{R} \setminus [-1, 1]} \gamma^n \|\hat{s}(\beta + i\gamma)\|_{\mathcal{H}^m} \, d\gamma < \infty \quad (B.7) $$

$$ \left| \int_{\mathbb{R} \setminus [-1, 1]} \gamma^n \|\hat{s}(\beta + i\gamma)\|_{\mathcal{H}^m} d\gamma \le \int_{\mathbb{R} \setminus [-1, 1]} \gamma^n \|A(\lambda)^{-1}\|_{\mathcal{L}(\mathcal{H}_\lambda^{m-2}, \mathcal{H}^m)} \left\| \left( \frac{1}{\mu_0} \hat{f}(\beta + i\gamma), \frac{1}{\mu_1} \tilde{g}(\beta + i\gamma), \frac{1}{\mu_1} \tilde{h}(\beta + i\gamma) \right) \right\|_{\mathcal{H}_\lambda^{m-2}} d\gamma \\ \lesssim \int_{\mathbb{R} \setminus [-1, 1]} \gamma^{n+m+2} \left( \|\hat{f}(\beta + i\gamma)\|_{H^{m-2}} + \|\hat{g}(\beta + i\gamma)\|_{H^{m-2}} + |\hat{h}(\beta + i\gamma)| \right. \\ \left. + \|\hat{s}(\beta + i\gamma + 1)\|_{\mathcal{H}^m} + \gamma^2 \|\hat{s}(\beta + i\gamma + 2)\|_{\mathcal{H}^m} \right) d\gamma $$
For any $\lambda \in \mathbb{C}_{\mathbb{R}>d-1} \setminus \frac{\pi}{\Theta}\mathbb{Z}^*$, let $\tilde{s}(\lambda) := A(\lambda)^{-1}(\frac{1}{\mu_0}\hat{f}(\lambda), \frac{1}{\mu_1}\tilde{g}(\lambda), \frac{1}{\mu_1}\tilde{h}(\lambda))$. Then $\tilde{s}$ is also solution of the above system, so $\tilde{s}$ and $\hat{s}$ coincide on $\mathbb{C}_{\mathbb{R}>d} \setminus \frac{\pi}{\Theta}\mathbb{Z}^*$ by Lemma B.2. Hence $\tilde{s}$ is an extension of $\hat{s}$ on $\mathbb{C}_{\mathbb{R}>d-1} \setminus \mathfrak{P}$, that we will still denote $\hat{s}$. Lemma B.2 implies that this extension is meromorphic with simple poles and: $\forall q \in \mathfrak{P}$, $\mathrm{Res}_q \hat{s} = (\mathrm{Res}_{\lambda=q} A(\lambda)^{-1})(\frac{1}{\mu_0}\hat{f}(q), \frac{1}{\mu_1}\tilde{g}(q), \frac{1}{\mu_1}\tilde{h}(q))$.

**Step 2:** Let $t \in \mathbb{R}$, $b \in (d-1, d_{\min})$ and $k \in \mathbb{R}_+^*$. The residue theorem applied to $\lambda \mapsto e^{\lambda t} \hat{s}(\lambda)$ on the rectangle $[b, d+1] \times [-k, k] \subset \mathbb{C}$ gives the following equality in $\mathcal{H}^m$.

$$2i\pi \sum_{q \in \mathfrak{P}} \mathrm{Res}_q(\lambda \mapsto e^{\lambda t} \hat{s}(\lambda)) = \int_{-k}^{k} e^{(d+1+i\gamma)t} \hat{s}(d+1+i\gamma) \, d\gamma - \int_{-k}^{k} e^{(b+i\gamma)t} \hat{s}(b+i\gamma) \, d\gamma \\ + \int_{b}^{d+1} e^{(\beta-ik)t} \hat{s}(\beta-ik) \, d\beta - \int_{b}^{d+1} e^{(\beta+ik)t} \hat{s}(\beta+ik) \, d\beta$$

Note that when $k \to \infty$ the first integral tends to the inverse Laplace transform of $s$ (up to a constant). We will show that the last two integrals tend to 0, by proving the following:

$$\forall n \in \mathbb{N}, \quad \sup_{\beta \in [b, d+1]} \int_{\mathbb{R} \setminus [-1, 1]} \gamma^n \|\hat{s}(\beta + i\gamma)\|_{\mathcal{H}^m} d\gamma < \infty \quad (B.7)$$

Let $n \in \mathbb{N}$ and $\beta \in [b, d+1]$. (B.5) and the expressions of $\tilde{g}$ and $\tilde{h}$ yield

$$\int_{\mathbb{R}\setminus[-1,1]} \gamma^n \|\hat{s}(\beta + i\gamma)\|_{\mathcal{H}^m} d\gamma \le \int_{\mathbb{R}\setminus[-1,1]} \gamma^n \|A(\lambda)^{-1}\|_{\mathcal{L}(\mathcal{H}_\times^{m-2}, \mathcal{H}^m)} \|(\frac{1}{\mu_0}\hat{f}(\beta + i\gamma), \frac{1}{\mu_1}\tilde{g}(\beta + i\gamma), \frac{1}{\mu_1}\tilde{h}(\beta + i\gamma))\|_{\mathcal{H}_\times^{m-2}} d\gamma \\ \lesssim \int_{\mathbb{R}\setminus[-1,1]} \gamma^{n+m+2} \left( \|\hat{f}(\beta + i\gamma)\|_{\mathcal{H}^{m-2}} + \|\hat{g}(\beta + i\gamma)\|_{\mathcal{H}^{m-2}} + |\hat{h}(\beta + i\gamma)| \right. \\ \left. + \|\hat{s}(\beta + i\gamma + 1)\|_{\mathcal{H}^m} + \gamma^2 \|\hat{s}(\beta + i\gamma + 2)\|_{\mathcal{H}^m} \right) d\gamma$$

Let us treat the term with $f$, given that the others are similar.

$$\int_{\mathbb{R}\setminus[-1,1]} \gamma^{n+m+2} \|\hat{f}(\beta + i\gamma)\|_{\mathcal{H}^{m-2}} d\gamma \quad \begin{aligned}[t] &\le \|\gamma^{-1}\|_{L^2(\mathbb{R}\setminus[-1,1])} \cdot \|\gamma^{n+m+3} \|_{\mathcal{H}^{m-2}} \|\hat{f}(\beta + i\gamma)\|_{L^2(\mathbb{R}\setminus[-1,1])} \\ &\lesssim \|e^{-\beta t} \partial_t^{n+m+3} f\|_{L^2(\mathbb{R}, \mathcal{H}^{m-2}(0,\Theta))} \\ &\lesssim \max_{\beta' \in \{b, d+1\}} \|e^{-\beta' t} \partial_t^{n+m+3} f\|_{L^2(\mathbb{R}, \mathcal{H}^{m-2}(0,\Theta))} \\ &= \max_{\beta' \in \{b, d+1\}} \|f\|_{K_{\beta'}^{n+m+3}(\mathbb{R}, \mathcal{H}^{m-2}(0,\Theta))} \end{aligned}$$

thanks to an interpolation between $b$ and $d+1$. We have majorized by a finite constant independent of $\beta$, so the supremum on $\beta$ is finite. Thus (B.7) is proven.

$$\text{Therefore: } \int_{1}^{\infty} \left\| \int_{b}^{d+1} e^{(\beta \pm i k)t} \hat{s}(\beta \pm i k) d\beta \right\|_{\mathcal{H}^m} dk \lesssim \int_{1}^{\infty} \int_{b}^{d+1} \|\hat{s}(\beta \pm i k)\|_{\mathcal{H}^m} d\beta dk < +\infty.$$

So by taking arbitrarily large $k$, we can make the terms $\int_{b}^{d+1} e^{(\beta \pm i\gamma)t} \hat{s}(\beta \pm i\gamma) d\beta$ tend to 0 in $\mathcal{H}^m$. Hence:

$$\sum_{q \in \mathfrak{P}} \mathrm{Res}_q(\lambda \mapsto e^{\lambda t} \hat{s}(\lambda)) = \frac{1}{2i\pi} \int_{\mathbb{R}} e^{(d+1+i\gamma)t} \hat{s}(d+1+i\gamma) \, d\gamma - \frac{1}{2i\pi} \int_{\mathbb{R}} e^{(b+i\gamma)t} \hat{s}(b+i\gamma) \, d\gamma.$$

Now, since $\hat{s}$ has only simple poles, the terms of the sum are equal to

$$e^{qt} \mathrm{Res}_q \hat{s} = e^{qt} (\mathrm{Res}_{\lambda=q} A(\lambda)^{-1}) (\frac{1}{\mu_0} \hat{f}(q), \frac{1}{\mu_1} \tilde{g}(q), \frac{1}{\mu_1} \tilde{h}(q)) = e^{qt} c_q \mathbf{1}_{[0,\Theta]}(\theta) \sin(d\theta) = c_q \bar{\phi}_q(t, \theta)$$

for some constant $c_q \in \mathbb{C}$ by point 2 of Lemma B.2. The first integral is equal to $s(t)$ by inverse Laplace transform (because $s \in K_{d+1}^{\infty}(\mathbb{R}, \mathcal{H}^m)$). We define $s_r(t)$ to be equal to the last integral. Thus we get the desired formula, and (B.7) shows that $s_r \in K_b^{\infty}(\mathbb{R}, \mathcal{H}^m)$. $\square$

Let $S$ be the function set in Theorem 2.23 and $(\sigma_d(S))_{d \in \mathbb{N}^*}$ the coefficients of Definition 3.10, which vanish for $d$ big enough. With the $\sigma_d(S)$, the non-variational part of $S$ (denoted $S^A$ below) can be explicitly computed, so it remains to get an asymptotic expansion of its variational part (denoted $S^V$
$$
\left[
\begin{array}{l}
\text{Let us treat the term with } f, \text{ given that the others are similar.} \\
\\
\int_{\mathbb{R}\setminus[-1,1]} \gamma^{n+m+2} \|\hat{f}(\beta + i\gamma)\|_{H^{m-2}} d\gamma \\
\le \underset{\text{C.S.}}{\|\gamma^{-1}\|_{L^2(\mathbb{R}\setminus[-1,1])} \cdot \|\gamma^{n+m+3}\|_{H^{m-2}}\|\hat{f}(\beta + i\gamma)\|_{H^{m-2}}\|\|_{L^2(\mathbb{R}\setminus[-1,1])}} \\
\lesssim \|e^{-\beta t} \partial_t^{n+m+3} f\|_{L^2(\mathbb{R}, H^{m-2}(0,\Theta))} \\
\lesssim \max_{\beta' \in \{b,d+1\}} \|e^{-\beta' t} \partial_t^{n+m+3} f\|_{L^2(\mathbb{R}, H^{m-2}(0,\Theta))} \\
= \max_{\beta' \in \{b,d+1\}} \|f\|_{K_{\beta'}^{n+m+3}(\mathbb{R}, H^{m-2}(0,\Theta))}
\end{array}
\right.
$$

thanks to an interpolation between $b$ and $d+1$. We have majorized by a finite constant independent of $\beta$, so the supremum on $\beta$ is finite. Thus (B.7) is proven.

$$
\text{Therefore: } \int_{1}^{\infty} \left\| \int_{b}^{d+1} e^{(\beta \pm i k)t} \hat{s}(\beta \pm i k) \, d\beta \right\|_{H^m} \, dk \lesssim \int_{1}^{\infty} \int_{b}^{d+1} \| \hat{s}(\beta \pm i k) \|_{H^m} \, d\beta \, dk < +\infty.
$$

So by taking arbitrarily large $k$, we can make the terms $\int_b^{d+1} e^{(\beta \pm i\gamma)t} \hat{s}(\beta \pm i\gamma) d\beta$ tend to 0 in $H^m$. Hence:

$$
\sum_{q \in \mathfrak{P}} \mathrm{Res}_q(\lambda \mapsto e^{\lambda t} \hat{s}(\lambda)) = \frac{1}{2i\pi} \int_{\mathbb{R}} e^{(d+1+i\gamma)t} \hat{s}(d+1+i\gamma) \, d\gamma - \frac{1}{2i\pi} \int_{\mathbb{R}} e^{(b+i\gamma)t} \hat{s}(b+i\gamma) \, d\gamma.
$$

Now, since $\hat{s}$ has only simple poles, the terms of the sum are equal to

$$
e^{qt} \mathrm{Res}_q \hat{s} = e^{qt} (\mathrm{Res}_{\lambda=q} A(\lambda)^{-1}) \left( \frac{1}{\mu_0} \hat{f}(q), \frac{1}{\mu_1} \tilde{g}(q), \frac{1}{\mu_1} \tilde{h}(q) \right) = e^{qt} c_q \mathbf{1}_{[0,\Theta]}(\theta) \sin(d\theta) = c_q \bar{\phi}_q(t, \theta)
$$

for some constant $c_q \in \mathbb{C}$ by point 2 of Lemma B.2. The first integral is equal to $s(t)$ by inverse Laplace transform (because $s \in K_{d+1}^{\infty}(\mathbb{R}, \mathcal{H}^m)$). We define $s_r(t)$ to be equal to the last integral. Thus we get the desired formula, and (B.7) shows that $s_r \in K_b^{\infty}(\mathbb{R}, \mathcal{H}^m)$. $\square$

Let $S$ be the function set in Theorem 2.23 and $(\sigma_d(S))_{d \in \frac{\pi}{\Theta}N^*}$ the coefficients of Definition 3.10, which vanish for $d$ big enough. With the $\sigma_d(S)$, the non-variational part of $S$ (denoted $S^A$ below) can be explicitly computed, so it remains to get an asymptotic expansion of its variational part (denoted $S^V$ below). In order to apply Proposition B.3, we need that $\overline{\chi_\infty S^V}$ belongs to some space $K_\beta^\infty$ (since $\Omega_1 \setminus B(0, R_c) = \Pi \setminus B(0, R_c)$, $\chi_\infty S^V$ can be seen as a function defined on $\Pi$, which allows us to consider $\overline{\chi_\infty S^V}$). We will use again the notations $\langle \cdot, \cdot \rangle$ and $T_{\ge d}$ of Definitions 2.3 and 2.21.

**Proposition B.4:** Let $d \in \mathbb{R}_-$,

$$
S^A := T_{\ge d} \left[ \left\langle -R_{\partial_Y^2} \circ \partial_X^2|_{\Lambda}, \frac{\mu_0}{\mu_1} R_N \circ \partial_{Y|\Gamma, Y=0+} \right\rangle \left( \frac{1}{\mu_0} R_{\Delta}(F_{\Omega}^{\infty}) + \frac{1}{\mu_1} R_{\partial_Y^2}(F_{\Lambda}^{\infty}) + \sum_{d \in \frac{\pi}{\Theta}N^*} \sigma_d(S) \phi_d \right) \right]
$$

in $\mathcal{A}(\Pi)$ and $S^V := S - \chi_\infty S^A$. There is $d \in \mathbb{R}_-$ s.t. $S^V \in V$ and: $\forall m \in \mathbb{N}$, $\overline{\chi_\infty S^V} \in K_{1/2}^\infty(\mathbb{R}, \mathcal{H}^m)$.

**Proof:** We consider any $d \in \mathbb{R}_-$ and we will fix it later. The proof has three steps: showing that $S^V \in V$, showing that $S^V$ is regular w.r.t $t$, and deducing that it is regular w.r.t. $\theta$. For any $r_1, r_2 \in \mathbb{R}$ s.t. $R_c < r_1 < r_2$, we will denote $\chi \in C^\infty(\mathbb{R}^2)$ a radial function equal to 0 on $B(0, r_1)$ and 1 on $\mathbb{R}^2 \setminus B(0, r_2)$ ($r_1, r_2$ are implicit in this notation).

Step 1: Let $\varphi : \Pi \to \mathbb{C}$ be equal to $T_{\le d}(F_D^\infty)$ on $D$ for any $D \in \{\Omega, \Lambda\}$, and $F_\nu = F - \chi_\infty \varphi$. If $d < -2$ then $(1+r)F_\nu \in L^2(\Omega_1)$. So the same construction than in Theorem 3.11 shows that for $d$ small enough, the problem

$$
\left\{
\begin{array}{ll}
\mathrm{div}(\mu\nabla S^V) = f_1 := F_\nu + \chi_\infty\varphi - \mathrm{div}(\mu\nabla(\chi_\infty \cdot S^A)) & \text{in } \Omega_1 \setminus (\Gamma \cap \{X > R_c\}) \\
[\mu\partial_Y S^V]_\Gamma = g_1 := -[\mu\partial_Y(\chi_\infty S^A)]_\Gamma & \text{on } \Gamma \cap \{X > R_c\} \\
S^V = 0 & \text{on } \partial\Omega_1
\end{array}
\right.
\qquad (B.8)
$$

has a unique solution in $V$ and that $S = S^V + \chi_\infty S^A$. In addition, it also gives that $(\mu\Delta S^A - \varphi)|_D \in \sum_{d'<d} A_{d'}(D)$ for any $D \in \{\Omega, \Lambda\}$, and $[\mu\partial_Y S^A]_\Gamma \in \sum_{d'<d} A_{d'}(\Gamma)$. And we have $F_\nu = o_\partial(\mathbf{r}^d)$ by