Questions Concerning Parabolk Measure Uniform Rectifiability and the Kato Square Root Conjecture

The aim of this project is to investigate further some problems originating from my work on (a) caloric measure in parabolic flat domains, (b) inverse problems and (c) Kato type problems. Under (a) I would like to know to what extent such concepts as uniform rectifiability, Reifenberg flatness, and asymptotic optimal doubling which have been extensively studied in regard to Laplace's equation, can be generalized to the heat equation. As regards (b), I would like to investigate whether very weak overdetermined boundary conditions for solutions to certain p-Laplacian type equations in a domain D imply that the boundary of D satisfies a regularity condition similar to uniform rectifiability. Finally under (c) I would like to know if an extrapolation technique, used by the author and co-authors on some parabolic measure and Kato problems, could be applied to other Kato type problems. 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 are (a) does there exist a solution, (b) is it unique and (c) does it possess nice properties or is it regular? As concerns (a) and (b) one is often concerned with so called boundary values or boundary conditions for a solution in its 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, my co-authors and I have obtained nearly optimal results, which show that certain boundary value problems for Laplace's equation can only be solved if the given domain is a ball. As another example of my work, classical theorems for the Laplacian in smooth domains have been shown to hold in a class of rough domains called Lipschitz or sawtooth domains. More recent work has generalized these results to nongraph domains satisfying `uniform rectifiability' assumptions. My co-authors and I have obtained the analogue of Lipschitz and uniformly rectifiable domains for the heat equation. Our work provides a model for certain free boundary problems such as ice melting (the Stefan problem).