An Inverse Problem for an Elliptic Equation

Abstract

We consider the following overdetermined boundary value problem: ∆u = −λu− 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 one can identify the constants λ and µ from knowledge of the normal flux ∂u ∂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.

A Globally Convergent Algorithm for a PDE-Constrained Optimization Problem Arising in Electrical Impedance Tomography

“…The discretization can be done by finite differences or finite elements. Despite their differences, both methods approximate the solution of the discretized problem ( [4], [5], [15], [16], [18], [30]). …”