The layer potential technique for the inverse conductivity problem

We consider the inverse conductivity problem for the equation div((1 + (k - 1)chi(D))Delta u) = 0 determining the unknown object D contained in a domain Omega with one measurement on an. 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 Omega. Using this representation formula, we prove that the location and size of a disk D contained in a simply connected bounded Lipschitz domain Omega can be determined with one measurement corresponding to arbitrary non-zero Neumann data on partial derivative Omega. (Previously, it was known that a disk can be determined with one measurement if Omega is assumed to be the half space.) We also prove a weaker version of the uniqueness for balls in R(n) (n greater than or equal to 3) with one measurement corresponding to a certain Neumann data.