An inverse problem in electrostatics

We reconstruct the shape of a conductor from knowledge of its Green functio n 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 t he Green function of free space. We use the representation F = (1/2)GSG* wi th the single-layer potential S and a suitable operator G to show that the range of F-1/2 and the range of GS(1/2) coincide. This allows us to charact erize the range of GS(1/2) with the help of the singular system of F via Pi card's theorem. The characterization, in rum, provides a possibility of dec iding whether a point is an element of D by computing a series involving th e singular system of F and the free space Green function.