Minimal Smoothness Questions for Inverse Problems and Boundary Value Problems

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Russell Brown will pursue research in two areas: inverse problems for partial differential equations and regularity for boundary value problems in nonsmooth domains. Many of the fundamental laws of nature are expressed as partial differential equations. In an inverse problem, one attempts to recover a coefficient in a partial differential equation from information about solutions of the equation. This provides a mathematical model for determining physical properties of an object (such as its conductivity) from measurements (such as voltage and current at the boundary). In the problem I will study, the inverse conductivity problem, we consider exactly the example described above: determine a spatially inhomogeneous conductivity from measurements of current and voltage made at the boundary. The innovation in my research is to attempt to determine the least restrictive hypotheses under which this determination can be made. The second area of investigation is related to boundary value problems for various partial differential equations in nonsmooth domains. In contrast to the inverse problem considered above, here we are considering the more straightforward problem of obtaining solutions to a partial differential equation given information at the boundary. This is a mathematical model for the problem of (for example) of finding the voltage potential in the interior from knowledge of the potential at the boundary. The innovation in my research is to establish minimal (and more realistic) hypotheses under which the solution can be shown to exist.