Quantitative Parabolic Rectifiability, Parabolic Partial Differential Equations and Boundary Value Problems

This project studies evolution equations, particularly those that govern heat flow, diffusion, and transport processes in complicated regions. These processes are described mathematically by parabolic partial differential equations. A central question addressed by the project is how the shape and roughness of the boundary of a region influence the behavior of solutions to these equations inside the region. A particular focus is placed on boundaries that may vary in time. The project develops mathematical tools relevant to the study of heat flow, diffusion, and transport in complex or heterogeneous environments. The project also supports education and workforce development through outreach at local K-12 schools and through mentoring of graduate and undergraduate students. The project studies parabolic uniform rectifiability and boundary value problems for parabolic partial differential equations in rough domains. The main goal is to identify sharp analytic and potential-theoretic criteria that characterize the appropriate geometric regularity of boundaries in the time-dependent setting. The first part of the project seeks characterizations of parabolic uniform rectifiability through caloric measure, estimates for bounded caloric functions, and boundedness of certain parabolic singular integral operators. Related qualitative questions concerning when caloric measure and surface measure are mutually absolutely continuous will also be investigated. The second part of the project studies boundary value problems for variable-coefficient parabolic operators in rough domains, with an emphasis on Green function estimates, quantitative absolute continuity of caloric measure, and solvability of Dirichlet and regularity problems under scale-invariant coefficient assumptions. 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.