An existence result for a nonlinear transmission problems

Abstract: Let $\Omega^o$ and $\Omega^i$ be open bounded subsets of $\mathbb{R}^n$ of class $C^{1,\alpha}$ such that the closure of $\Omega^i$ is contained in $\Omega^o$. Let $f^o$ be a function in $C^{1,\alpha}(\partial\Omega^o)$ and let $F$ and $G$ be continuous functions from $\partial\Omega^i\times\mathbb{R}$ to $\mathbb{R}$. By exploiting an argument based on potential theory and on the Leray-Schauder principle we show that under suitable and completely explicit conditions on $F$ and $G$ there exists at least one pair of continuous functions $(u^o, u^i)$ such that [ \left{ \begin{array}{ll} \Delta u^o=0&\text{in }\Omega^o\setminus\mathrm{cl}\Omega^i\,,\ \Delta u^i=0&\text{in }\Omega^i\,,\ u^o(x)=f^o(x)&\text{for all }x\in\partial\Omega^o\,,\ u^o(x)=F(x,u^i(x))&\text{for all }x\in\partial\Omega^i\,,\ \nu_{\Omega^i}\cdot\nabla u^o(x)-\nu_{\Omega^i}\cdot\nabla u^i(x)=G(x,u^i(x))&\text{for all }x\in\partial\Omega^i\,, \end{array} \right. ] where the last equality is attained in certain weak sense. In a simple example we show that such a pair of functions $(u^o, u^i)$ is in general neither unique nor local unique. If instead the fourth condition of the problem is obtained by a small nonlinear perturbation of a homogeneous linear condition, then we can prove the existence of at least one classical solution which is in addition locally unique.