Calderón’s inverse conductivity problem in the plane

Abstract

We show that the Dirichlet to Neumann map for the equation $ \nabla\cdot\sigma\nabla u = 0$ in a two-dimensional domain uniquely determines the bounded measurable conductivity $ \sigma$. This gives a positive answer to a question of A. P. Calderón from 1980. Earlier the result has been shown only for conductivities that are sufficiently smooth. In higher dimensions the problem remains open.