$$\gamma(x)\equiv g(x)\qquad \forall x \in \partial \Omega^o\, .$$ Then the conclusion follows by Theorem \[thm:ex\].

By Proposition \[prop:finteq\], we are reduced to analyze equation around the case $(\varrho_1,\varrho_2)=(0,r_\ast)$. As a first step, in the following lemma we analyze the system which we obtain by taking $(\varrho_1,\varrho_2)=(0,r_\ast)$ in equation .

\[lem:lim\] 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 $r_\ast \in [0,+\infty[$ be such that holds. Let $f_1 \in C^{0,\alpha}(\partial \Omega^i_1)$, $f_2 \in C^{0,\alpha}(\partial \Omega^i_2)$, $g \in C^{1,\alpha}(\partial \Omega^o)$. Then the system of equations $$\label{inteqlim0}
\begin{split}
&\frac{1}{2}\theta^{i}_1(t)+\int_{\partial \Omega^i_1}DS_n(t-s)\nu_{\Omega^i_1}(t)\theta^{i}_1(s)\, d\sigma_s\\ 
& \quad +r_\ast^{n-1}\int_{\partial \Omega^i_2}DS_n\big((p^1-p^2)+r_\ast (t-s)\big)\nu_{\Omega^i_1}(t)\theta^{i}_2(s)\, d\sigma_s\\
& \quad -\int_{\partial \Omega^o}DS_n(y)\nu_{\Omega^i_1}(t)\theta^{o}(y)\, d\sigma_y -f_1(t)=0\qquad \forall t \in \partial \Omega^i_1\, ,
\end{split}$$ $$\label{inteqlim1}
\begin{split}
&\frac{1}{2}\theta^{i}_2(t)+\int_{\partial \Omega^i_2}DS_n(t-s)\nu_{\Omega^i_2}(t)\theta^{i}_2(s)\, d\sigma_s\\ 
& \quad +r_\ast^{n-1}\int_{\partial \Omega^i_1}DS_n\big((p^2-p^1)+r_\ast(t-s)\big)\nu_{\Omega^i_2}(t)\theta^{i}_1(s)\, d\sigma_s\\
& \quad -\int_{\partial \Omega^o}DS_n(y)\nu_{\Omega^i_2}(t)\theta^{o}(y)\, d\sigma_y -f_2(t)=0\qquad \forall t \in \partial \Omega^i_2\, ,
\end{split}$$ $$\label{inteqlim2}
\begin{split}
\int_{\partial \Omega^o}S_n(x-y)\theta^{o}(y)\, d\sigma_y +\xi-g(x)=0 \qquad \forall x \in \partial \Omega^o\, ,
\end{split}$$ has a unique solution $(\theta^{i}_1, \theta^{i}_2, \theta^{o}, \xi)$ in $C^{0,\alpha}(\partial \Omega^i_1)\times C^{0,\alpha}(\partial \Omega^i_2)\times C^{0,\alpha}(\partial \Omega^o)_0\times \mathbb{R}$, which we denote by $(\tilde{\theta}^{i}_1, \tilde{\theta}^{i}_2, \tilde{\theta}^{o}, \tilde{\xi})$.

By Lemma \[lem:smp\] (ii), equation has a unique solution $(\tilde{\theta}^{o}, \tilde{\xi})$ in the space $C^{0,\alpha}(\partial \Omega^o)_0\times \mathbb{R}$. Then we consider equations , 