Some Results Related to Schiffer's Problem

Abstract

We consider the following semilinear overdetermined problem on a two dimensional bounded or unbounded domain $\Omega$ with analytic boundary $\partial\Omega$ having at least one bounded connected component \begin{eqnarray*} \left\{ \begin{array}{l} - \Delta u = g(u) \quad \hbox{in } \Omega,\\ \frac{\partial u}{\partial \nu} =0 \, \hbox{ and } \, u = c \hbox{ on } \partial \Omega, \end{array} \right. \end{eqnarray*} where $c$ is a constant. When $g(c) =0$ the constant solution $u \equiv c$ is the unique solution. For $g(c) \not =0$, we show that the boundary is a circle if and only if the problem admits a solution that has constant third or fourth normal derivative along the boundary. A similar result involving the fifth normal derivative is proved.