Parabolic and elliptic boundary value and free boundary problems

This project is concerned with the theory of boundary value problems and free boundary problems for elliptic and parabolic partial differential equations. Such equations arise, for example, in the mathematical theory of heat conduction: an equation of elliptic type describes steady state (equilibrium) temperature distributions and a related parabolic equation governs heat conduction in the time-evolutive case. In a boundary value problem, one uses mathematical knowledge of either 1) the temperature distribution on the boundary (i.e., perimeter) of some region in space (or of some evolving region in space-time) or 2) the heat flux (the rate at which heat flows across the boundary), to deduce information about the internal temperature distribution inside the region. In free boundary problems, one uses simultaneous knowledge of both the boundary temperature distribution and the heat flux to deduce information about the geometry of the region and its boundary. A central goal of this project is to understand the interplay between analytic information and geometry. This project provides research training opportunities for graduate students. The project has three main areas of focus: 1) to find a geometric characterization of the space-time domains for which the Dirichlet (or initial-Dirichlet) problem is solvable for the heat equation with singular (p-integrable) data, and to study related free boundary problems. The PI and coauthors have previously treated such problems in the steady state (elliptic) setting; in the present project, the PI seeks to treat the more difficult time-evolutive case. 2) to solve the Kato square root problem for elliptic equations in non-divergence form. The solution of the Kato problem for divergence form elliptic operators has led to significant progress in the theory of boundary value problems for divergence form equations. As a first step towards opening up the analogous theory in the nondivergence setting, the PI plans to treat the Kato problem for non-divergence elliptic operators. 3) to solve the Dirichlet problem in Lipschitz domains for non-symmetric divergence from elliptic equations with periodic coefficients. A primary motivation for the study of operators with periodic coefficients is their applicability to the theory of homogenization, which in turn provides a mathematical model for materials with periodic microstructure. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.