and small enough, we want to introduce a mixed problem for the Laplace operator in $\Omega_\epsilon$. Namely, we consider a Dirichlet condition on $\partial \Omega^o$ and Neumann conditions on the boundary of the holes. Thus, 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 $\epsilon \in ]0,\epsilon_0[$ we consider the following mixed problem: $$\label{bvpe}
\left\{
\begin{array}{ll}
\Delta u(x)=0 & \forall x \in \Omega_{\epsilon}\,,\\
\frac{\partial }{\partial \nu_{\eta(\epsilon)p^j+\epsilon\Omega^i_j}}u(x)=f_j\Big(\big(x-\eta(\epsilon)p^j\big)/\epsilon\Big) & \forall x \in  \eta(\epsilon)p^j+\epsilon\partial\Omega^i_j\, ,  \forall j\in \{1,2\}\, ,\\
u(x)=g(x) & \forall x \in \partial \Omega^o\, ,
\end{array}
\right.$$ where $\nu_{\eta(\epsilon)p^j+\epsilon\Omega^i_j}$ denotes the outward unit normal to $ \eta(\epsilon)p^j+\epsilon\partial\Omega^i_j$ for $j\in \{1,2\}$.

Then, if $\epsilon \in ]0,\epsilon_0[$, problem has a unique solution in $C^{1,\alpha}(\mathrm{cl}\Omega_{\epsilon})$ and we denote such a solution by $u_{\epsilon}$. We are interested in studying the behavior of $u_\epsilon$ as $\epsilon \to 0$ and thus we pose the following questions.

1.  Let $x$ be a fixed point in $\Omega^o\setminus\{0\}$. What can be said of the map $\epsilon\mapsto u_{\epsilon}(x)$ when $\epsilon$ is close to $0$ and positive?

2.  Let $t$ be a fixed point in ${\mathbb{R}}^{n}\setminus 
    \cup_{j=1}^2(p^j +r_\ast \Omega^i_j)$. What can be said of the map $\epsilon\mapsto u_{\epsilon}(\eta(\epsilon) 