$C^{1,\alpha}(\mathrm{cl}\Omega(\varrho_1,\varrho_2))$ of problem is delivered by $$\label{eq:finteq1}
\begin{split}
u[\varrho_1,\varrho_2](x)\equiv &(\varrho_1 \varrho_2)^{n-1}\sum_{j=1}^2 \int_{\partial \Omega^i_j}S_n(x-\varrho_1p^j-\varrho_1\varrho_2 s)\theta^{i}_j[\varrho_1,\varrho_2](s)\, d\sigma_s \\& +\int_{\partial \Omega^o}S_n(x-y)\theta^{o}[\varrho_1,\varrho_2](y)\, d\sigma_y +\xi[\varrho_1,\varrho_2]\qquad \forall x \in \mathrm{cl}\Omega(\varrho_1,\varrho_2)\,,
\end{split}$$ where $(\theta^{i}_1[\varrho_1,\varrho_2],\theta^{i}_2[\varrho_1,\varrho_2],\theta^{o}[\varrho_1,\varrho_2],\xi[\varrho_1,\varrho_2])$ is the unique quadruple $(\theta^{i}_1,\theta^{i}_2,\theta^{o},\xi)$ in $C^{0,\alpha}(\partial \Omega^i_1)\times C^{0,\alpha}(\partial \Omega^i_2) \times C^{0,\alpha}(\partial \Omega^o)_0\times \mathbb{R}$ such that $$\label{inteq2a}
\Lambda [\varrho_1,\varrho_2,\theta^{i}_1,\theta^{i}_2,\theta^{o},\xi]=0\, .$$

Let $J$ be as in Proposition \[prop:J\] with $$\mathcal{O}^i\equiv \bigg (\varrho_1p^1+\varrho_1\varrho_2 \Omega^i_1\bigg)\cup\bigg(\varrho_1p^2+\varrho_1 \varrho_2\Omega^i_2\bigg) \, , \qquad \mathcal{O}^o\equiv \Omega^o\, .$$ Then by the definition of $\Lambda$ and the rule of change of variables in integrals one verifies that the quadruple $(\theta^{i}_1,\theta^{i}_2,\theta^{o},\xi)$ in $C^{0,\alpha}(\partial \Omega^i_1)\times C^{0,\alpha}(\partial \Omega^i_2) \times C^{0,\alpha}(\partial \Omega^o)_0\times \mathbb{R}$ is a solution of equation if and only if the triple $(\mu_1,\mu_2,\xi)$ in $C^{0,\alpha}(\partial \mathcal{O}^i)\times C^{0,\alpha}(\partial \mathcal{O}^o)_0\times \mathbb{R}$ with $\mu_1$ and $\mu_2$ defined by $$\mu_1(x)\equiv
\left\{
\begin{array}{ll}
\theta^i_1\big((x-\varrho_1p^1)/(\varrho_1 \varrho_2)\big) & \forall x \in \varrho_1p^1+\varrho_1 \varrho_2 \partial \Omega_1\, ,   
\\
\theta^i_2\big((x-\varrho_1p^2)/(\varrho_1 \varrho_2)\big) & \forall x \in \varrho_1p^2+\varrho_1 \varrho_2 \partial \Omega_2\, , \end{array}
\right.$$ $$\mu_2(x)\equiv \theta^o(x)\qquad \forall x \in \partial \Omega^o\, ,$$ is a solution of $$J[\mu_1,\mu_2,\xi]=(\phi,\gamma)\, ,$$ with $\phi$ and $\gamma$ defined by $$\phi(x)\equiv
\left\{
\begin{array}{ll}
f_1\big((x-\varrho_1 p^1)/(\varrho_1 \varrho_2)\big) & \forall x \in \varrho_1 p^1+\varrho_1 \varrho_2 \partial \Omega_1\, ,   
\\
f_2\big((x-\varrho_1 p^2)/(\varrho_1 \varrho_2)\big) & \forall x \in \varrho_1 p^2+\varrho_1 \varrho_2 \partial \Omega_2\, , \end{array}
\right.$$ 