An Overdetermined Problem for an Elliptic Equation

Abstract

We consider the following overdetermined boundary value problem: $\Delta u+\lambda u+\mu =0$ in $\Omega$, $u=0$ on $\partial \Omega$ and $\partial u/\partial n=c$ on $\partial \Omega$, where $c\ne 0$, $\lambda$ and $\mu$ are real constants and $\Omega \subset R^2$ is a smooth bounded convex open set. We first show that it may happen that the problem has no solution. Then we study the existence of solutions for a wide class of domains.