An Inverse Problem for an Elliptic Equation

Abstract

We consider the following overdetermined boundary value problem: ∆_u_ = −λ_u_−µ in Ω, u = 0 on ∂Ω and ∂_u_/∂_n_ = 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_ 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.