Inverse Problems for Nonsmooth First Order Perturbations of the Laplacian

We consider inverse boundary value problems in Rn, n ≥ 3, for operators which may be written as first order perturbations of the Laplacian. The purpose is to obtain global uniqueness theorems for such problems when the coefficients are nonsmooth. We use complex geometrical optics solutions of Sylvester-Uhlmann type to achieve this. A main tool is an extension of the Nakamura-Uhlmann intertwining method to operators which have continuous coefficients. For the inverse conductivity problem for a C1+ε conductivity, we construct complex geometrical optics solutions whose properties depend explicitly on ε. This implies the uniqueness result of Päivärinta-PanchenkoUhlmann for C3/2 conductivities. For the magnetic Schrödinger equation, the result is that the Dirichlet-to-Neumann map uniquely determines the magnetic field corresponding to a Dini continuous magnetic potential in C1,1 domains. For the steady state heat equation with a convection term, we obtain global uniqueness of Lipschitz continuous convection terms in Lipschitz domains.