Then we introduce the perforated domain $$\Omega(\varrho_1,\varrho_2)\equiv \Omega^o \setminus  \bigcup_{j=1}^2 \bigg (\varrho_1p^j+\varrho_1 \varrho_2\mathrm{cl}\Omega^i_j\bigg) \qquad \forall (\varrho_1,\varrho_2) \in \tilde{\mathcal{U}}\, .$$ Next we take a function $f_1 \in C^{0,\alpha}(\partial \Omega^i_1)$, a function $f_2  \in C^{0,\alpha}(\partial \Omega^i_2)$, a function $g$ in $C^{1,\alpha}(\partial \Omega^o)$, and for each pair $(\varrho_1,\varrho_2) \in \tilde{\mathcal{U}} \cap ]0,+\infty[^2$ we consider the following mixed problem $$\label{bvprho}
\left\{
\begin{array}{ll}
\Delta u(x)=0 & \forall x \in \Omega(\varrho_1,\varrho_2)\,,\\
\frac{\partial }{\partial \nu_{\varrho_1p^j+\varrho_1 \varrho_2\Omega^i_j}}u(x)=f_j\Big(\big(x-\varrho_1p^j\big)/(\varrho_1 \varrho_2)\Big) & \forall x \in   \varrho_1p^j+\varrho_1 \varrho_2\partial\Omega^i_j,  \forall j\in \{1,2\} ,\\
u(x)=g(x) & \forall x \in \partial \Omega^o\, ,
\end{array}
\right.$$ where $\nu_{\varrho_1p^j+\varrho_1 \varrho_2\Omega^i_j}$ denotes the outward unit normal to $\varrho_1p^j+\varrho_1 \varrho_2\partial\Omega^i_j$ for $j\in \{1,2\}$. If $(\varrho_1, \varrho_2) \in \tilde{\mathcal{U}} \cap ]0,+\infty[^2$, problem has a unique solution in $C^{1,\alpha}(\mathrm{cl}\Omega(\varrho_1,\varrho_2))$ and we denote such a solution by $u[\varrho_1,\varrho_2]$. Clearly, if $\eta$, $r_\ast$ are as in and if $\epsilon_{0}$ is such that $(\eta(\epsilon), \epsilon/ \eta(\epsilon)) \in \tilde{\mathcal{U}} \cap ]0,+\infty[^2$ for all $\epsilon \in ]0,\epsilon_0[$, then $$\Omega_{\epsilon}=\Omega(\eta(\epsilon), \epsilon/ \eta(\epsilon)) \qquad \text{and} \qquad u_{\epsilon}=u[\eta(\epsilon), \epsilon/ \eta(\epsilon)] \, ,$$ for all $\epsilon \in ]0,\epsilon_0[$.

Preliminaries {#prel}
=============

In this section we collect some preliminary results concerning mixed problems for the Laplace operator.

First of all, by the Divergence Theorem, we deduce the following uniqueness result.

\[prop:uniq\] 