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Recovery of an inhomogeneity in an elliptic equation. (English) Zbl 0974.35134

Summary: We consider the inverse problem to identify an unknown domain \(D\) entering an elliptic equation \(\Delta u- \chi(D)u= 0\) in \(\Omega\). We show that solutions of the differential equation can be represented as solutions of an integral equation using the volume potential. We then prove the global uniqueness of the inverse problem within the class of two- or three-dimensional balls. Based on the representation we propose a numerical algorithm to detect the unknown domain \(D\) and show some results of numerical experiments.

MSC:

35R30 Inverse problems for PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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