Some Questions in Inverse Boundary Value Problems and the Mixed Problems for Laplace's Equation in Lipschitz Domains.

This project will study several questions in partial differential equations which lead to interesting questions in harmonic analysis. The first set of questions are related to the inverse conductivity problem as studied by Sylvester and Uhlmann. My main interest is considering the a priori regularity assumption which seem to be necessary to study this problem. I propose a technique to establish a uniqueness theorem in dimensions 3 and larger for coefficients which only have one derivative. In addition, I will consider the two-dimensional problem, where such a theorem is known. In two dimensions, I propose to extend the result from equations to systems. The second set of questions are related to the mixed problem for Laplace's equation. The goal here is to obtain optimal regularity results for solutions to these problems. Examples indicate that the positive result depend strongly on the geometry of the domain and the sets where Dirichlet and Neumann data are posed. The inverse conductivity problem is a mathematical formulation of the problem of determining the interior physical properties of an object by making electrical measurements at the boundary. This and related problems are of practical importance in medical imaging and in the nondestructive evaluation of materials. The theoretical investigations proposed in this project may shed some light on how to improve practical implementation of these problems. The mixed problem for Laplace's equation models the problem of determining the temperature in the interior of a solid where part of the boundary is insulated. My research is focused on understanding how the geometry of the region of the region affects our ability to solve this problem.