\hat{J}_2[\mu_1,\mu_2,\xi](x)\equiv &\int_{\partial \mathcal{O}^o}S_n(x-y)\mu_2(y)\, d\sigma_y+\xi \qquad \forall x \in \partial \mathcal{O}^o\, ,
\end{split}$$ for all $(\mu_1,\mu_2,\xi)\in C^{0,\alpha}(\partial \mathcal{O}^i)\times C^{0,\alpha}(\partial \mathcal{O}^o)_0\times \mathbb{R}$. By Lemma \[lem:smp\] (ii), (iii) one can show that $\hat{J}$ is a linear homeomorphism. Then let $\tilde{J}\equiv(\tilde{J}_1,\tilde{J}_2)$ be the operator from $C^{0,\alpha}(\partial \mathcal{O}^i)\times C^{0,\alpha}(\partial \mathcal{O}^o)_0\times \mathbb{R}$ to $C^{0,\alpha}(\partial \mathcal{O}^i)\times C^{1,\alpha}(\partial \mathcal{O}^o)$ defined by $$\begin{split}
\tilde{J}_1[\mu_1,\mu_2,\xi](x)\equiv& \int_{\partial \mathcal{O}^o}DS_n(x-y)\nu_{\mathcal{O}^i}(x)\mu_2(y)\, d\sigma_y \quad \forall x \in \partial \mathcal{O}^i\, ,\\
\tilde{J}_2[\mu_1,\mu_2,\xi](x)\equiv &\int_{\partial \mathcal{O}^i}S_n(x-y)\mu_1(y)\, d\sigma_y \qquad \forall x \in \partial \mathcal{O}^o\, ,
\end{split}$$ for all $(\mu_1,\mu_2,\xi)\in C^{0,\alpha}(\partial \mathcal{O}^i)\times C^{0,\alpha}(\partial \mathcal{O}^o)_0\times \mathbb{R}$. By classical potential theory and standard calculus in Schauder spaces, one can show that $\tilde{J}$ is a compact operator. Since $J=\hat{J}+\tilde{J}$, we deduce that $J$ is a Fredholm operator of index $0$. As a consequence, in order to prove that $J$ is a linear homeormorphism, it suffices to show that it is injective. So let $(\mu_1,\mu_2,\xi)\in C^{0,\alpha}(\partial \mathcal{O}^i)\times C^{0,\alpha}(\partial \mathcal{O}^o)_0\times \mathbb{R}$ be such that $J[\mu_1,\mu_2,\xi]=(0,0)$. Then by classical potential theory, the function $v\equiv v[\partial \mathcal{O}^i,\mu_1]_{|\mathrm{cl}\mathcal{O}^o \setminus \mathcal{O}^i}+v[\partial \mathcal{O}^o,\mu_2]_{|\mathrm{cl}\mathcal{O}^o \setminus \mathcal{O}^i}+ \xi$ is a solution in $C^{1,\alpha}(\mathrm{cl}\mathcal{O}^o \setminus \mathcal{O}^i)$ of problem . Accordingly, $v[\partial \mathcal{O}^i,\mu_1]_{|\mathrm{cl}\mathcal{O}^o \setminus \mathcal{O}^i}+v[\partial \mathcal{O}^o,\mu_2]_{|\mathrm{cl}\mathcal{O}^o \setminus \mathcal{O}^i}+ \xi=0$ in $\mathrm{cl}\mathcal{O}^o \setminus \mathcal{O}^i$, and so $$\label{v-=-v+-xi}
v^-[\partial \mathcal{O}^i,\mu_1]=-v^+[\partial \mathcal{O}^o,\mu_2]- \xi\qquad\textrm{in $\mathrm{cl}\mathcal{O}^o \setminus \mathcal{O}^i$}\, .$$ 