The layer potential technique for the inverse conductivity problem

We consider the inverse conductivity problem for the equaiton div((1 + (k - 1)XD)▽α = 0 determining the unknown object D contained in a domian Ω with one measurement on ∂Ω 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 Ω. Using this representation formula, we prove that the location and size of a disk D contained in a simply connected bounded Lipschitz domain Ω can be determined with one measurement corresponding to arbitreary non-zero Neumann data on ∂Ω (Previously, it was known that a disk can be determined with one measurement if Ω is assumed to be the half space.) We also prove a weaker version of the uniqueness for balls in Rⁿ (n ≥ 3) with one measurement corresponding to a certain Neumann data.