The aim of this project is to investigate further some problems originating from my work on (a) parabolic and elliptic measure, (b) quasiconformal mappings, and (c) regularity of PDE's. Under (a) I would like to know when the Dirichlet Problem has a solution for certain parabolic and elliptic PDE's with drag term. Given that the Dirichlet problem for these PDE's has a solution, I would like to know when the corresponding measures possess basic properties such as a doubling property. As for (b), I would like to construct more examples of domains which are quasiconformal to a sphere and for which harmonic measure and n - 1 dimensional Hausdorff measure on the boundary are equal. Finally under (c) I would like to know if the techniques used to prove reverse Holder inequalities for solutions to systems modeled on for the parabolic p Laplacian can be used on other PDE's such as the Navier Stokes equation.
Many physical problems can be described in the language of partial differential equations (PDE's). Well known examples of such equations arising in the 19 th century are Laplace's equation, the heat equation, the wave equation, Maxwell's equations, and the Navier- Stokes' equation. Without question knowledge derived from a theoretical study of these equations led to many fundamental technological advances during the 19 th and 20 th centuries. Three questions often asked by those who study PDE's is (a) does there exist a solution, (b) is it unique and (c) does it possess nice properites or is it regular? As for (a) and (b) one is often concerned with so called boundary values or boundary conditions for a solution in the domain of existence. So called overdetermined boundary value problems have no solution whereas such classical problems as the Dirichlet and Neumann problems have solutions if the boundary of the given domain and the boundary conditions are sufficiently nice (smooth). My work is concerned with how much one can relax these assumptions and still get meaningful theorems. For example, during the last quarter century, many classical theorems for the Laplacian in smooth domains have been shown to hold in a class of rough domains called Lipschitz or sawtooth domains. My coauthors and I have obtained the analogue of Lipschitz domains for the heat equation. Another avenue of investigation has been to consider questions (a)-(c) in a half space for rough parabolic PDE's. My work provides a model for certain free boundary problems such as ice melting (the Stefan problem).
