linear homeomorphism.

3.  Let $\mathbb{R}^n \setminus \mathrm{cl}\Omega$ be connected. Then the map from $C^{0,\alpha}(\partial \Omega)$ to $C^{0,\alpha}(\partial \Omega)$ which takes $\mu$ to the function $$\frac{1}{2}\mu(x)+\int_{\partial \Omega}DS_n(x-y)\nu_{\Omega}(x)\mu(y)\, d\sigma_y$$ of the variable $x \in \partial \Omega$, is a linear homeomorphism.

We now introduce and study an integral operator which we use in order to solve a mixed problem by means of simple layer potentials.

\[prop:J\] Let $\alpha \in ]0,1[$. Let $\mathcal{O}^i$, $\mathcal{O}^o$ be bounded open subsets of $\mathbb{R}^n$ of class $C^{1,\alpha}$ such that $\mathcal{O}^o$, $\mathbb{R}^n \setminus \mathrm{cl}\mathcal{O}^i$, and $\mathbb{R}^n \setminus \mathrm{cl}\mathcal{O}^o$ are connected and that $\mathrm{cl}\mathcal{O}^i \subseteq \mathcal{O}^o$. Let $J\equiv(J_1,J_2)$ 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 $$\begin{split}
J_1[\mu_1,\mu_2,\xi](x)\equiv& \frac{1}{2}\mu_1(x)+\int_{\partial \mathcal{O}^i}DS_n(x-y)\nu_{\mathcal{O}^i}(x)\mu_1(y)\, d\sigma_y\\ &+\int_{\partial \mathcal{O}^o}DS_n(x-y)\nu_{\mathcal{O}^i}(x)\mu_2(y)\, d\sigma_y \qquad \forall x \in \partial \mathcal{O}^i\, ,\\
J_2[\mu_1,\mu_2,\xi](x)\equiv &\int_{\partial \mathcal{O}^i}S_n(x-y)\mu_1(y)\, d\sigma_y+\int_{\partial \mathcal{O}^o}S_n(x-y)\mu_2(y)\, d\sigma_y\\
&+\xi \qquad \forall x \in \partial \mathcal{O}^o\, ,
\end{split}$$ for all $(\mu_1,\mu_2,\xi)\in C^{0,\alpha}(\partial \mathcal{O}^i)\times C^{0,\alpha}(\partial \mathcal{O}^o)_0\times \mathbb{R}$. Then $J$ is a linear homeomorphism.

We first prove that $J$ is a Fredholm operator of index $0$. Let $\hat{J}\equiv(\hat{J}_1,\hat{J}_2)$ 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}
\hat{J}_1[\mu_1,\mu_2,\xi](x)\equiv& \frac{1}{2}\mu_1(x)+\int_{\partial \mathcal{O}^i}DS_n(x-y)\nu_{\mathcal{O}^i}(x)\mu_1(y)\, d\sigma_y \quad \forall x \in \partial \mathcal{O}^i\, ,\\
