An inverse problem in electrostatics

Abstract

We reconstruct the shape of a conductor from knowledge of its Green function on a large sphere. To this end, we consider an integral operator F having as kernel the difference between the Green function of the conductor and the Green function of free space. We use the representation F = (1/2)GSG* with the single-layer potential S and a suitable operator G to show that the range of F1/2 and the range of GS1/2 coincide. This allows us to characterize the range of GS1/2 with the help of the singular system of F via Picard's theorem. The characterization, in turn, provides a possibility of deciding whether a point is an element of D by computing a series involving the singular system of F and the free space Green function.