Asymptotic behavior of the solutions of a nonlinear Robin problemfor the Laplace operator in a domain with a small hole. A functionalanalytic approach

We consider a bounded open subset ${\mathbb{I}}^{o}$ of ${\mathbb{R}}^{n}$ with outward unit normal $\nu^{o}$ and with $0\in{\mathbb{I}}^{o}$, and we assume that the boundary value problem \[ \Delta u=0 \ {\mathrm{in}}\ {\mathbb{I}}^{o}, \qquad \frac{\partial u}{\partial\nu^{o}}(t)=G^{o}(t,u(t)) \ \ \forall t\in\partial {\mathbb{I}}^{o} \] has a solution $\tilde{u}$. Here $G^{o}$ is a function of $\partial {\mathbb{I}}^{o}\times {\mathbb{R}}$ to ${\mathbb{R}}$. Then we consider another bounded open subset ${\mathbb{I}}^{i}$ of ${\mathbb{R}}^{n}$ with outward unit normal $\nu^{i}$ and with $0\in{\mathbb{I}}^{i}$ and we consider the boundary value problem \[ \Delta u=0 \ {\mathrm{in}}\ {\mathbb{I}}^{o}\setminus\epsilon{\mathrm{cl}}{\mathbb{I}}^{i}, \quad -\frac{\partial u}{\partial\nu^{i}} =0 \ {\mathrm{on}}\ \epsilon\partial{\mathbb{I}}^{i}\,, \quad \frac{\partial u}{\partial\nu^{o}}(t)=G^{o}(t,u(t)) \ \ \forall t\in\partial {\mathbb{I}}^{o}\,, \] for $\epsilon>0$ small. Under suitable conditions on ${\mathbb{I}}^{o}$, ${\mathbb{I}}^{i}$, $G^{o}$, we show that for $\epsilon>0$ sufficiently small, such a boundary value problem admits locally around $\tilde{u}$ a unique solution $u(\epsilon,\cdot)$. Then we show that (suitable restrictions of) $u(\epsilon,\cdot)$ and the energy integral of $u(\epsilon,\cdot)$ can be continued real analytically in the parameter $\epsilon$ around $\epsilon=0$