MSRI - Partial Data for General Second Order Elliptic Operators in Two Dimensions

Abstract: We consider the inverse problem of determining the coefficients of a general second-order elliptic operator in two dimensions by measuring the corresponding Cauchy data on an arbitrary open subset of the boundary. We show that one can determine the coefficients of the operator up to natural obstruction such as the conformal invariance, gauge transformation and diffeomorphism invariance. Next we apply the main result to prove that the curl of the magnetic field and the electric potential are uniquely determined by measuring partial Cauchy data associated to the magnetic Schroedinger equation measured on an arbitrary open subset of the boundary. We also show that any two of the three coefficients of a generalized second order elliptic operator whose principal part is the Laplacian, are uniquely determined by their partial Cauchy data. This is joint work with G. Uhlmann and M. Yamamoto.