Some Questions in Inverse Problems and the Mixed Problem for Laplace's Equation in Lipschitz Domains

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This project will study several questions in partial differentialequations which lead to interesting questions in harmonicanalysis. The first set of questions are related to the inverseconductivity problem as studied by Sylvester and Uhlmann. My maininterest is considering the a priori regularity assumption which seemto be necessary to study this problem. I propose a technique toestablish a uniqueness theorem in dimensions 3 and larger forcoefficients which only have one derivative. In addition, I willconsider the two-dimensional problem, where such a theorem isknown. In two dimensions, I propose to extend the result fromequations to systems. The second set of questions are related to themixed problem for Laplace's equation. The goal here is to obtainoptimal regularity results for solutions to these problems. Examplesindicate that the positive result depend strongly on the geometry ofthe domain and the sets where Dirichlet and Neumann data areposed.The inverse conductivity problem is a mathematical formulation of theproblem of determining the interior physical properties of an objectby making electrical measurements at the boundary. This and relatedproblems are of practical importance in medical imaging and in thenondestructive evaluation of materials. The theoretical investigationsproposed in this project may shed some light on how to improvepractical implementation of these problems. The mixed problem forLaplace's equation models the problem of determining the temperaturein the interior of a solid where part of the boundary is insulated.My research is focused on understanding how the geometry of theregion of the region affects our ability to solve this problem.