Nonlinear PDE with gradient constraints: Regularity, Eigenvalue problems, and Infinity ground states

University Of Pennsylvania, Philadelphia PA

Investigators

Abstract

Abstract (Hynd, 1301628) The aim of this project is to analyze properties of solutions of partial differential equations (PDE) with gradient constraints. The principle investigator (PI) proposes to establish the best known regularity result for solutions of Hamilton-Jacobi-Bellman equations arising in stochastic control theory, where the underlying control processes have sample paths that are singular. This will allow control theorists to better study a wider class of models as the design of optimal controls in stochastic singular control depends on regularity properties of solutions. In mathematical finance models involving transaction costs, quantities of interest in the limit of small transaction costs may be characterized via a solution of an eigenvalue problem involving a PDE with gradient constraint. The PI proposes to deduce various properties of these solutions including providing new minmax formulae for eigenvalues. The PI will also perform a study of infinity ground states, which are solutions of a PDE involving the infinity Laplacian and a gradient constraint. This equation arises as a limit of Euler-Lagrange equations associated with a sequence of constrained variational problems. The PI will develop techniques for resolving a fundamental unresolved uniqueness question: on exactly which domains are infinity ground states unique up to a multiplicative factor? The proposed research seeks to address a variety of problems at the intersection of diverse fields of mathematics with applications to industries such as telecommunications and finance. Consequently, this project has the potential to serve a wide audience and to attract students and other researchers looking to forge research connections. In attempts to increase the participation of underrepresented groups doing research in these areas of science, the PI will attend meetings such as the Conference for African American Researchers in the Mathematical Sciences and the Blackwell-Tapia conference. The PI will also visit universities that serve large numbers of students from underrepresented groups to give presentations on this research and to build relationships that may ultimately lead to research collaborations.