Recovery of an inhomogeneity in an elliptic equation

We consider the inverse problem to identify an unknown domain D entering an elliptic equation Δu - χ (D)u = 0 in Ω. 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.