# An Inverse Problem for an Elliptic Equation

### Robert Dalmasso

Equipe EDP, Grenoble, France

## Abstract

We consider the following overdetermined boundary value problem: $∆u=λu−µ$ in $Ω$, $u=0$ on $∂Ω$ and $∂n∂u =ψ$ on $∂Ω$, where $λ$ and $µ$ are real constants and $Ω$ is a smooth bounded planar domain. A very interesting problem is to examine whether ∂_u_ one can identify the constants $λ$ and $µ$ from knowledge of the normal ﬂux $∂n∂u $ on $∂Ω$ corresponding to some nontrivial solution. It is well known that if $Ω$ is a disk then such identiﬁcation of $(λ,µ)$ 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 $ψ$ is constant.

## Cite this article

Robert Dalmasso, An Inverse Problem for an Elliptic Equation. Publ. Res. Inst. Math. Sci. 40 (2004), no. 1, pp. 91–123

DOI 10.2977/PRIMS/1145475967