An Inverse Problem for an Elliptic Equation

Abstract

We consider the following overdetermined boundary value problem: $∆u=\lambda u-\mu$ in $\Omega$, $u=0$ on $\partial \Omega$ and $\frac{\partial u}{\partial n}=\psi$ on $\partial \Omega$, where $\lambda$ and $\mu$ are real constants and $\Omega$ is a smooth bounded planar domain. A very interesting problem is to examine whether ∂_u_ one can identify the constants $\lambda$ and $\mu$ from knowledge of the normal flux $\frac{\partial u}{\partial n}$ on $\partial \Omega$ corresponding to some nontrivial solution. It is well known that if $\Omega$ is a disk then such identification of $(\lambda ,\mu )$ is completely impossible. Some partial results have already been obtained. The purpose of this paper is to extend and to improve these results. Moreover we also examine the interesting case where $\psi$ is constant.