Lanza de Cristoforis and Rossi [@LaRo04 Thm. 3.1], Lanza de Cristoforis and the second-named author [@LaMu13 §4]), we conclude that $\Lambda$ is real analytic. Now we plan to apply the Implicit Function Theorem to equation $\Lambda[\varrho_1,\varrho_2,\theta^{i}_1,\theta^{i}_2,\theta^{o},\xi]=0$ around the point $(0, r_\ast,\tilde{\theta}^{i}_1,\tilde{\theta}^{i}_2,\tilde{\theta}^{o},\tilde{\xi})$. By definition of $(\tilde{\theta}^{i}_1,\tilde{\theta}^{i}_2,\tilde{\theta}^{o},\tilde{\xi})$, we have $\Lambda[0, r_\ast, \tilde{\theta}^{i}_1,\tilde{\theta}^{i}_2,\tilde{\theta}^{o},\tilde{\xi}]=0$. By standard calculus in Banach spaces, the differential of $\Lambda$ at $(0, r_\ast, \tilde{\theta}^{i}_1,\tilde{\theta}^{i}_2,\tilde{\theta}^{o},\tilde{\xi})$ with respect to the variables $(\theta^{i}_1,\theta^{i}_2,\theta^{o},\xi)$ is delivered by the formulas $$\begin{split}
\partial_{(\theta^{i}_1, \theta^{i}_2,\theta^{o},\xi)}\Lambda_1[0,r_\ast,\tilde{\theta}^{i}_1,\tilde{\theta}^{i}_2,\tilde{\theta}^{o},&\tilde{\xi}](\bar{\theta}^{i}_1,\bar{\theta}^{i}_2,\bar{\theta}^{o},\bar{\xi})(t)\\
&\equiv\frac{1}{2}\bar{\theta}^{i}_1(t)+\int_{\partial \Omega^i_1}DS_n(t-s)\nu_{\Omega^i_1}(t)\bar{\theta}^{i}_1(s)\, d\sigma_s\\ 
&  +r_\ast^{n-1}\int_{\partial \Omega^i_2}DS_n\big((p^1-p^2)+r_\ast (t-s)\big)\nu_{\Omega^i_1}(t)\bar{\theta}^{i}_2(s)\, d\sigma_s\\
&  -\int_{\partial \Omega^o}DS_n(y)\nu_{\Omega^i_1}(t)\bar{\theta}^{o}(y)\, d\sigma_y \qquad \forall t \in \partial \Omega^i_1\, ,
\end{split}$$ $$\begin{split}
\partial_{(\theta^{i}_1, \theta^{i}_2,\theta^{o},\xi)}\Lambda_2[0,r_\ast,\tilde{\theta}^{i}_1,\tilde{\theta}^{i}_2,\tilde{\theta}^{o},&\tilde{\xi}](\bar{\theta}^{i}_1,\bar{\theta}^{i}_2,\bar{\theta}^{o},\bar{\xi})(t)\\&\equiv\frac{1}{2}\bar{\theta}^{i}_2(t)+\int_{\partial \Omega^i_2}DS_n(t-s)\nu_{\Omega^i_2}(t)\bar{\theta}^{i}_2(s)\, d\sigma_s\\ 
&  +r_\ast^{n-1}\int_{\partial \Omega^i_1}DS_n\big((p^2-p^1)+r_\ast(t-s)\big)\nu_{\Omega^i_2}(t)\bar{\theta}^{i}_1(s)\, d\sigma_s\\
&  -\int_{\partial \Omega^o}DS_n(y)\nu_{\Omega^i_2}(t)\bar{\theta}^{o}(y)\, d\sigma_y\qquad \forall t \in \partial \Omega^i_2\, ,
\end{split}$$ $$\begin{split}
\partial_{(\theta^{i}_1, \theta^{i}_2,\theta^{o},\xi)}\Lambda_3[0,r_\ast,\tilde{\theta}^{i}_1,\tilde{\theta}^{i}_2,\tilde{\theta}^{o},\tilde{\xi}](\bar{\theta}^{i}_1,\bar{\theta}^{i}_2,\bar{\theta}^{o},\bar{\xi})(x)\equiv\int_{\partial \Omega^o}S_n(x-y)\bar{\theta}^{o}(y)\, d\sigma_y &+\bar{\xi}\\ &\forall x \in \partial \Omega^o\, ,
\end{split}$$ for all $(\bar{\theta}^{i}_1,\bar{\theta}^{i}_2,\bar{\theta}^{o},\bar{\xi}) \in C^{0,\alpha}(\partial \Omega^i_1)\times C^{0,\alpha}(\partial \Omega^i_2)\times C^{0,\alpha}(\partial \Omega^o)_0\times \mathbb{R}$. Then, by arguing as in the proof of Lemma \[lem:lim\], by classical potential theory, and by standard calculus in Banach spaces, one can show that $\partial_{(\theta^{i}_1,\theta^{i}_2,\theta^{o},\xi)}\Lambda[0,r_\ast,\tilde{\theta}^{i}_1,\tilde{\theta}^{i}_2,\tilde{\theta}^{o},\tilde{\xi}]$ is a linear homeomorphism from $ 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}$ onto $ C^{0,\alpha}(\partial \Omega^i_1)\times C^{0,\alpha}(\partial \Omega^i_2)\times C^{1,\alpha}(\partial \Omega^o)$. Then by the Implicit Function Theorem for real analytic maps in 