Also, $v[\partial \mathcal{O}^i,\mu_1]=-v[\partial \mathcal{O}^o,\mu_2]- \xi$ on $\partial \mathcal{O}^i$ and by uniqueness of the solution of the Dirichlet problem for the Laplace operator, we deduce $$\label{v+=-v+-xi}
v^+[\partial \mathcal{O}^i,\mu_1]=-v^+[\partial \mathcal{O}^o,\mu_2]- \xi\qquad\textrm{in $\mathrm{cl}\mathcal{O}^i$}\, .$$ As a consequence, $v[\partial \mathcal{O}^i,\mu_1]=-v^+[\partial \mathcal{O}^o,\mu_2]- \xi$ on the whole of $\mathrm{cl} \mathcal{O}^o$. Since $v^+[\partial \mathcal{O}^o,\mu_2]$ is in $C^{1,\alpha}(\mathrm{cl} \mathcal{O}^o)$ (cf. Lemma \[lem:smp\]), we have $$\label{-dv++dv+=0}
-\frac{\partial}{\partial \nu_{\mathcal{O}^i}}v^+[\partial \mathcal{O}^o,\mu_2]_{|\mathrm{cl}\mathcal{O}^o \setminus \mathcal{O}^i}+\frac{\partial}{\partial \nu_{\mathcal{O}^i}}v^+[\partial \mathcal{O}^o,\mu_2]_{|\mathrm{cl}\mathcal{O}^i}=0\qquad\textrm{on $\partial\mathcal{O}^i$}\,.$$ By equalities and , and by standard jump properties of the single layer potential, the expression on the left hand side of equals $$\label{dv--dv+=mu}
\frac{\partial}{\partial \nu_{\mathcal{O}^i}}v^-[\partial \mathcal{O}^i,\mu_1]-\frac{\partial}{\partial \nu_{\mathcal{O}^i}}v^+[\partial \mathcal{O}^i,\mu_1]=\mu_1\qquad\textrm{on $\partial\mathcal{O}^i$}\, .$$ Hence, by and it follows that $\mu_1=0$. Thus $v[\partial \mathcal{O}^o,\mu_2]+ \xi=0$ on $\partial \mathcal{O}^o$ (cf. ). Accordingly, Lemma \[lem:smp\] (ii) implies that $(\mu_2,\xi)=(0,0)$, and so the proof is complete.

By Propositions \[prop:uniq\] and \[prop:J\] and by the jump properties of the single layer potential, we deduce the validity of the following theorem on the solution of a mixed problem.

\[thm:ex\] 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 $J$ be as in Proposition \[prop:J\]. Let $(\phi,\gamma) \in C^{0,\alpha}(\partial 