On reconstruction in the inverse conductivity problem, with one measurement

We consider an inverse problem for electrically conductive material occupying a domain Omega in R-2. Let gamma be the conductivity of Omega, and D a subdomain of Omega. We assume that gamma is a positive constant k on D, k not equal 1 and is 1 on Omega \ D; both D and k are unknown. The problem is to find a reconstruction formula of D from the Cauchy data on partial derivative Omega of a non-constant solution u of the equation del . gamma del u = 0 in Omega. We prove that if D is known to be a convex polygon such that diam D < dist (D, partial derivative Omega), there are two formulae for calculating the support function of D from the Cauchy data.