\[lem:smp\] (iii) with $\Omega\equiv(p^1+r_{\ast}\Omega_1)\cup (p^2+r_{\ast}\Omega_2)$.

\[rem:lim\] Let the assumptions of Lemma \[lem:lim\] hold. Let $\tilde{u}$ be the unique solution in $C^{1,\alpha}(\mathrm{cl}\Omega^o)$ of $$\left\{
\begin{array}{ll}
\Delta u(x)=0 & \forall x \in \Omega^o\,,\\
u(x)=g(x) & \forall x \in \partial \Omega^o\, .
\end{array}
\right.$$ Then $\tilde{u}= v^+[\partial \Omega^o,\tilde{\theta}^{o}]+\tilde{\xi}$.

We are now ready to analyze equation around the degenerate pair $(\varrho_1, \varrho_2)=(0, r_\ast)$.

\[prop:ansol\] Let $\alpha\in]0,1[$. Let $\Omega^i_1$, $\Omega^i_2$, $\Omega^o$ be as in . Let $p^1$, $p^2$ be as in . Let $r_\ast \in [0,+\infty[$. Let hold. Let $\tilde{\mathcal{U}}$ be as in . Let $f_1 \in C^{0,\alpha}(\partial \Omega^i_1)$, $f_2 \in C^{0,\alpha}(\partial \Omega^i_2)$, $g \in C^{1,\alpha}(\partial \Omega^o)$. Let $(\tilde{\theta}^{i}_1,\tilde{\theta}^{i}_2, \tilde{\theta}^{o}, \tilde{\xi})$ be as in Lemma \[lem:lim\]. Then there exist an open neighborhood $\mathcal{U}$ of $(0,r_{\ast})$ in $\mathbb{R}^2$ and a real analytic map $(\Theta^{i}_1,\Theta^{i}_2, \Theta^{o}, \Xi)$ from $\mathcal{U}$ to $C^{0,\alpha}(\partial \Omega^i_1)\times C^{0,\alpha}(\partial \Omega^i_2)\times C^{0,\alpha}(\partial \Omega^o)_0\times \mathbb{R}$ such that $$\mathcal{U} \subseteq \tilde{\mathcal{U}}\, ,$$ and that $$\begin{split}
(\theta^{i}_1[\varrho_1,\varrho_2],\theta^{i}_2[\varrho_1,\varrho_2], \theta^{o}[\varrho_1,\varrho_2], \xi[\varrho_1,\varrho_2])=(\Theta^{i}_1[\varrho_1,\varrho_2],\Theta^{i}_2&[ \varrho_1,\varrho_2], \Theta^{o}[ \varrho_1,\varrho_2], \Xi[ \varrho_1,\varrho_2])\\ &  \forall (\varrho_1,\varrho_2) \in \mathcal{U} \cap ]0,+\infty[^2\, ,
\end{split}$$ and that $$(\tilde{\theta}^{i}_1,\tilde{\theta}^{i}_2, \tilde{\theta}^{o}, \tilde{\xi})=(\Theta^{i}_1[0,r_\ast], \Theta^{i}_2[0,r_\ast], \Theta^{o}[0,r_\ast], \Xi[0,r_\ast])\, .$$

By standard properties of integral operators with real analytic kernels and with no singularity, and by classical mapping properties of layer potentials (cf. Miranda [@Mi65], 