The determination of a discontinuity in a conductivity from a single boundary measurement

We consider the determination of the interior domain D subset of Omega where D is characterized by a different conductivity from the surrounding medium. This amounts to solving the inverse problem of recovering the piecewise constant conductivity a = I + chi(D) in div(a del u) = 0 from boundary data consisting of Cauchy data on the boundary of the exterior domain Omega. We will compute the derivative of the map from the domain D to this data and use this to obtain both qualitative and quantitative measures of the solution of the inverse problem.