This project is primarily concerned with the mathematical theory of heat conduction. The subject matter lies at the intersection of geometric measure theory, partial differential equations, and harmonic analysis. In geometric measure theory, one studies geometric properties of sets via the behavior of some measure on that set, where the concept of "measure" generalizes the notions of length, area and volume. In this project, the kind of set that we consider is typically the boundary of some region in space, or of some evolving region in space-time. Partial differential equations describe mathematically the conduction of heat, the propagation of waves, and many other physical phenomena. Harmonic analysis is a mathematical tool with which one extracts information by decomposing mathematical functions into fundamental constituent pieces. A principal goal of this project is to understand, in a quantitative way, how the geometry of a region influences the conduction of heat through the region, and across its boundary. This project will contribute to the development of the US workforce through the training of graduate students.
The project has three main areas of focus: 1) the Neumann Problem. The PI plans to solve the Neumann problem for Laplace's equation, with p-integrable data, in domains satisfying a quantitative, scale invariant version of a measure theoretic condition which is equivalent to the existence of a measure-theoretic outer unit normal at almost every boundary point; thus, the condition is natural for the Neumann problem, and may be sharp. Eventually, the PI seeks to characterize geometrically the domains in which such solvability is possible. 2) Parabolic quantitative rectifiability. At present, the theory of quantitative rectifiability in the time-evolutive parabolic setting is quite rudimentary in comparison to the rich theory available in the steady-state elliptic setting. In recent work of the PI and co-authors, elliptic quantitative rectifiability has played a central role in the characterization of those domains for which the Dirichlet problem for Laplace's equation is solvable. The PI expects that the proposed work would be a first step towards an analogous characterization in the parabolic case. 3) The Kato square root problem for non-divergence form elliptic operators. The solution of the square root problem for divergence form elliptic operators has enabled 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 non-divergence setting, the PI proposes to treat the Kato problem for non-divergence elliptic operators.
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.