notation. 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 $f_1 \in C^{0,\alpha}(\partial \Omega^i_1)$, $f_2 \in C^{0,\alpha}(\partial \Omega^i_2)$, $g \in C^{1,\alpha}(\partial \Omega^o)$. Then we introduce the map $\Lambda=(\Lambda_1,\Lambda_2,\Lambda_3)$ from $\tilde{\mathcal{U}}\times C^{0,\alpha}(\partial \Omega^i_1)\times C^{0,\alpha}(\partial \Omega^i_2)\times C^{0,\alpha}(\partial \Omega^o)_0\times \mathbb{R}$ to $C^{0,\alpha}(\partial \Omega^i_1)\times C^{0,\alpha}(\partial \Omega^i_2)\times C^{1,\alpha}(\partial \Omega^o)$ defined by $$\begin{split}
\Lambda_1[\varrho_1,&\varrho_2,\theta^{i}_1,\theta^{i}_2,\theta^{o},\xi](t)\equiv\frac{1}{2}\theta^{i}_1(t)+\int_{\partial \Omega^i_1}DS_n(t-s)\nu_{\Omega^i_1}(t)\theta^{i}_1(s)\, d\sigma_s\\ 
& \quad +\varrho_2^{n-1}\int_{\partial \Omega^i_2}DS_n\bigg((p^1-p^2)+\varrho_2(t-s)\bigg)\nu_{\Omega^i_1}(t)\theta^{i}_2(s)\, d\sigma_s\\
& \quad +\int_{\partial \Omega^o}DS_n\big(\varrho_1p^1+\varrho_1\varrho_2 t-y\big)\nu_{\Omega^i_1}(t)\theta^{o}(y)\, d\sigma_y -f_1(t)\qquad \forall t \in \partial \Omega^i_1\, , \\
\Lambda_2[\varrho_1,&\varrho_2,\theta^{i}_1,\theta^{i}_2,\theta^{o},\xi](t)\equiv\frac{1}{2}\theta^{i}_2(t)+\int_{\partial \Omega^i_2}DS_n(t-s)\nu_{\Omega^i_2}(t)\theta^{i}_2(s)\, d\sigma_s\\ 
& \quad +\varrho_2^{n-1}\int_{\partial \Omega^i_1}DS_n\bigg((p^2-p^1)+\varrho_2(t-s)\bigg)\nu_{\Omega^i_2}(t)\theta^{i}_1(s)\, d\sigma_s\\
& \quad +\int_{\partial \Omega^o}DS_n\big(\varrho_1p^2+\varrho_1\varrho_2 t-y\big)\nu_{\Omega^i_2}(t)\theta^{o}(y)\, d\sigma_y -f_2(t)\qquad \forall t \in \partial \Omega^i_2\, ,\\
\Lambda_3[\varrho_1,&\varrho_2,\theta^{i}_1,\theta^{i}_2,\theta^{o},\xi](x)\equiv(\varrho_1 \varrho_2)^{n-1}\sum_{j=1}^2 \int_{\partial \Omega^i_j}S_n(x-\varrho_1p^j-\varrho_1 \varrho_2 s)\theta^{i}_j(s)\, d\sigma_s\\&+\int_{\partial \Omega^o}S_n(x-y)\theta^{o}(y)\, d\sigma_y +\xi-g(x) \qquad \forall x \in \partial \Omega^o\, ,
\end{split}$$ for all $(\varrho_1,\varrho_2,\theta^{i}_1,\theta^{i}_2,\theta^{o},\xi) \in \tilde{\mathcal{U}}\times C^{0,\alpha}(\partial \Omega^i_1)\times C^{0,\alpha}(\partial \Omega^i_2)\times C^{0,\alpha}(\partial \Omega^o)_0\times \mathbb{R}$.

In the following proposition, we describe the link between the map $\Lambda$ and problem .

\[prop:finteq\] 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 $f_1 \in C^{0,\alpha}(\partial \Omega^i_1)$, $f_2 \in C^{0,\alpha}(\partial \Omega^i_2)$, $g \in C^{1,\alpha}(\partial \Omega^o)$. Let $(\varrho_1, \varrho_2) \in \tilde{\mathcal{U}} \cap ]0,+\infty[^2$. Then the unique solution $u[\varrho_1,\varrho_2]$ in 