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Recovering an obstacle and a nonlinear conductivity from Cauchy data

We consider the inverse problem of recovering the shape and location of an object where the surrounding medium is both conductive and homogeneous but the conductivity depends on the solution u. The physical situation is then modelled by the equation ∇ ⋅ k(u)∇u = 0. We measure the pair of values , that is, Cauchy data, on an accessible part of the exterior boundary. Given sets of Cauchy data pairs we wish to recover both the shape and location of the unknown obstacle together with the unknown conductivity k(u) of the background medium. We prove a uniqueness result and propose several algorithms designed to provide numerical reconstructions.