we set $$v[\partial\Omega,\mu](x)\equiv
\int_{\partial\Omega}S_{n}(x-y)\mu(y)\,d\sigma_{y}
\qquad\forall x\in {\mathbb{R}}^{n}\,.$$ As is well-known, if $\mu\in C^{0}(\partial{\Omega})$, then $v[\partial\Omega,\mu]$ is continuous in ${\mathbb{R}}^{n}$. Moreover, if $\mu\in C^{0,\alpha}(\partial\Omega)$, then the function $v^{+}[\partial\Omega,\mu]\equiv v[\partial\Omega,\mu]_{|{\mathrm{cl}}\Omega}$ belongs to $C^{1,\alpha}({\mathrm{cl}}\Omega)$, and the function $v^{-}[\partial\Omega,\mu]\equiv v[\partial\Omega,\mu]_{|\mathbb{R}^n \setminus \Omega}$ belongs to $C^{1,\alpha}_{\mathrm{loc}}
(\mathbb{R}^n \setminus \Omega)$.

A more general formulation {#for}
==========================

In this section, we formulate a more general version of the problem we are interested in. Then, by the analysis of such a new problem, we are able to deduce our results concerning the behavior of the solution $u_\epsilon$ for $\epsilon$ close to $0$. In a sense, what we are going to do it is to replace $\eta(\epsilon)$ by $\varrho_1$ and $\epsilon/\eta(\epsilon)$ by $\varrho_2$, and to analyze the dependence of the solution of the problem upon $\varrho_1$ and $\varrho_2$, which we think as two independent variables.

Let $\alpha\in]0,1[$. Let $\Omega^i_1$, $\Omega^i_2$, $\Omega^o$ be as in . Let $p^1$, $p^2$ be as in . Let $r_\ast \in [0,+\infty[$ be such that assumption holds. Then we fix an open neighborhood $\tilde{\mathcal{U}}$ of $(0,r_\ast)$ in $\mathbb{R}^2$, such that $$\label{tildeU}
\begin{split}
&\left(p^1+\varrho_2\mathrm{cl}\Omega^i_1\right)\cap\left(p^2+\varrho_2\mathrm{cl}\Omega^i_2\right)= \emptyset  \qquad \forall (\varrho_1,\varrho_2) \in \tilde{\mathcal{U}}\, ,\\
&\bigg (\varrho_1 p^1+\varrho_1 \varrho_2 \mathrm{cl}\Omega^i_1\bigg)\cup\bigg(\varrho_1 p^2+\varrho_1 \varrho_2\mathrm{cl}\Omega^i_2\bigg) \subseteq \Omega^o \qquad \forall (\varrho_1,\varrho_2)\in \tilde{\mathcal{U}}\, .
\end{split}$$ 