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 $v \in C^{1,\alpha}(\mathrm{cl}\mathcal{O}^o \setminus \mathcal{O}^i)$ be such that $$\label{eq:uniq}
\left\{
\begin{array}{ll}
\Delta v(x)=0 & \forall x \in \mathcal{O}^o\setminus \mathrm{cl}\mathcal{O}^i\,,\\
\frac{\partial }{\partial \nu_{\mathcal{O}^i}}v(x)=0 & \forall x \in  \partial  \mathcal{O}^i\, ,\\
v(x)=0 & \forall x \in \partial \mathcal{O}^o\, .
\end{array}
\right.$$ Then $v=0$ in $\mathrm{cl}\mathcal{O}^o \setminus \mathcal{O}^i$.

In the following lemma, we collect some well-known results of classical potential theory (cf. Folland [@Fo95 Ch. 3], Lanza de Cristoforis and Rossi [@LaRo04 Thm. 3.1], Miranda [@Mi65 Thm 5.I]).

\[lem:smp\] Let $\alpha \in ]0,1[$. Let $\Omega$ be a bounded open subset of $\mathbb{R}^n$ of class $C^{1,\alpha}$. Then the following statements hold.

1.  The map from $C^{0,\alpha}(\partial \Omega)$ to $C^{1,\alpha}(\mathrm{cl}\Omega)$ which takes $\mu$ to $v^+[\partial \Omega, \mu]$ is linear and continuous. Similarly, if $\tilde{\Omega}$ is a bounded open subset of $\mathbb{R}^n\setminus\mathrm{cl}\Omega$, then the map from $C^{0,\alpha}(\partial \Omega)$ to $C^{1,\alpha}(\mathrm{cl}\tilde\Omega)$ which takes $\mu$ to $v^-[\partial \Omega, \mu]_{|\mathrm{cl}\tilde\Omega}$ is linear and continuous.

2.  Let $\Omega$ be connected. The map from $C^{0,\alpha}(\partial\Omega)_0\times \mathbb{R}$ to $C^{1,\alpha}(\partial\Omega)$ which takes $(\mu,\xi)$ to $v[\partial \Omega, \mu]_{|\partial\Omega}+\xi$ is a 