\mathcal{O}^i)\times C^{1,\alpha}(\partial \mathcal{O}^o)$. Then problem $$\left\{
\begin{array}{ll}
\Delta u(x)=0 & \forall x \in \mathcal{O}^o\setminus \mathrm{cl}\mathcal{O}^i\,,\\
\frac{\partial }{\partial \nu_{\mathcal{O}^i}}u(x)=\phi(x) & \forall x \in  \partial  \mathcal{O}^i\, ,\\
u(x)=\gamma(x) & \forall x \in \partial \mathcal{O}^o\, ,
\end{array}
\right.$$ has a unique solution $u$ in $C^{1,\alpha}(\mathrm{cl}\mathcal{O}^o \setminus \mathcal{O}^i)$. The solution $u$ is delivered by $$u(x)\equiv v[\partial \mathcal{O}^i,\mu_1](x)+v[\partial \mathcal{O}^o,\mu_2](x)+\xi \qquad \forall x \in \mathrm{cl}\mathcal{O}^o \setminus \mathcal{O}^i\, ,$$ where $(\mu_1,\mu_2,\xi)$ is the unique triple in $C^{0,\alpha}(\partial \mathcal{O}^i)\times C^{0,\alpha}(\partial \mathcal{O}^o)_0\times \mathbb{R}$ such that $$J[\mu_1,\mu_2,\xi]=(\phi,\gamma)\, .$$

Formulation of problem in terms of integral equations {#finteq}
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In this section, we formulate problem in terms of integral equations on $\partial \Omega^i_1$, $\partial\Omega^i_2$, and $\partial \Omega^o$, by exploiting Theorem \[thm:ex\] and the rule of change of variables in integrals. Indeed, if $(\varrho_1, \varrho_2) \in \tilde{\mathcal{U}} \cap ]0,+\infty[^2$, by Theorem \[thm:ex\], one can convert problem into a system of integral equations which include an equation defined on $\partial \Omega^o$ and two equations defined on the $(\varrho_1,\varrho_2)$-dependent domains $\partial(\varrho_1p^1+\varrho_1 \varrho_2  \Omega^i_1)$ and $\partial(\varrho_1 p^2+\varrho_1 \varrho_2  \Omega^i_2)$. Then, by exploiting an appropriate change of variable, one can obtain an equivalent system of integral equations defined on the fixed domains $\partial \Omega^i_1$, $\partial \Omega^i_2$, and $\partial \Omega^o$.

We find convenient to introduce the following 