Asymptotic behaviour of the solutions of a non-linear transmission problem for the Laplace operator in a domain with a small hole. A functional analytic approach

We consider a bounded open subset ${\mathbb{I}}^{o}$ of ${\mathbb{R}}^{n}$ with $0\in{\mathbb{I}}^{o}$, and a function $f^{o}$ of $\partial{\mathbb{I}}^{o}$ to ${\mathbb{R}}$. Under reasonable assumptions, the Dirichlet problem $ \Delta u=0$ in $ {\mathbb{I}}^{o}$, $ u=f^{o} $ on $\partial {\mathbb{I}}^{o}$, has one and only one solution $\tilde{u}^{o}$. Then we consider another bounded open subset ${\mathbb{I}}^{i}$ of ${\mathbb{R}}^{n}$ with $0\in{\mathbb{I}}^{i}$, and an increasing diffeomorphism $F$ of ${\mathbb{R}}$ onto itself, and a constant $\gamma\in ]0,+\infty[$, and a function $g$ of $\partial{\mathbb{I}}^{i}$ to ${\mathbb{R}}$, and we consider the nonlinear transmission boundary value problem \[ \begin{array}{c} \Delta u^{i}=0 \quad {\mathrm{in}}\ \epsilon {\mathbb{I}}^{i}\, \qquad \Delta u^{o}=0 \quad {\mathrm{in}}\ {\mathbb{I}}^{o}\setminus\epsilon{\mathrm{cl}}{\mathbb{I}}^{i}\,, \\ u^{o}=F(u^{i}) \quad {\mathrm{on}}\ \epsilon\partial{\mathbb{I}}^{i}\,, \qquad \frac{\partial u^{o}}{\partial\nu_{\epsilon {\mathbb{I}}^{i}}} (x)= \gamma \frac{\partial u^{i}}{\partial\nu_{\epsilon {\mathbb{I}}^{i}}} (x) +g(x/\epsilon) \quad \forall x\in \epsilon\partial{\mathbb{I}}^{i}\,, \\ u^{o}=f^{o} \ {\mathrm{on}}\quad \partial {\mathbb{I}}^{o}\,, \end{array} \] for $\epsilon > 0 $ small, where $\nu_{\epsilon {\mathbb{I}}^{i}}$ is the outward unit normal to $\epsilon\partial{\mathbb{I}}^{i}$. Under suitable conditions on the data, we show that for $\tilde{\epsilon}>0$ sufficiently small, such a boundary value problem admits locally around $(F^{(-1)}(\tilde{u}^{o}(0)),\tilde{u}^{o}) $ a family of solutions $\{ (u^{i}(\epsilon,\cdot),u^{o}(\epsilon,\cdot)) \}_{\epsilon\in ]0,\tilde{\epsilon}[}$. Then we show that $u^{i}(\epsilon,\epsilon\cdot)$ and (suitable restrictions of) $u^{o}(\epsilon,\cdot)$ and $u^{o}(\epsilon,\epsilon\cdot)$ can be continued real analytically in the parameter $\epsilon$ around $\epsilon=0$ for $n\geq 3$, and can be represented in terms of real analytic functions of $\epsilon$, $\log^{-1}\epsilon$, $\epsilon\log^{2}\epsilon$ for $n=2$.