[Analysis and PDE Seminar] Geometry and the Solvability of the Parabolic Dirichlet Problem

Speaker: James Warta, University of Missouri
 

Abstract: The Dirichlet problem asks if there exists a function that satisfies a differential equation on a given domain and assumes given values on the boundary of that domain. Parabolic differential equations are parallel abstractions to the heat equation as elliptic equations are to Laplace's equation. Existence of solutions to elliptic equations, and the property of such solutions in the interior of a given problem domain, are intimately dependent upon geometric regularity of the boundary and the regularity of the values specified there. The theory for parabolic equations arose largely from elliptic theory and inherited some direct analogs from it, but oftentimes requires nuance given the non-Euclidean geometry of spacetime. This talk will give a brief survey of current research and introduce the contributions of the author to it.