The layer potential technique for the inverse conductivity problem

We consider the inverse conductivity problem for the equation determining the unknown object D contained in a domain with one measurement on . The method in this paper is the layer potential technique. We find a representation formula for the solution to the equation using single layer potentials on D and . Using this representation formula, we prove that the location and size of a disk D contained in a simply connected bounded Lipschitz domain can be determined with one measurement corresponding to arbitrary non-zero Neumann data on . (Previously, it was known that a disk can be determined with one measurement if is assumed to be the half space.) We also prove a weaker version of the uniqueness for balls in with one measurement corresponding to a certain Neumann data.