Sharp estimates for Dirichlet heat kernels of the Laplacian, fractional Laplacian and applications

Pedro Jose Mendez Hernandez, Purdue University

Abstract

In Chapter 2, we study a conjecture concerning a geometrical characterization in terms of the areas of Whitney cubes along quasihyperbolic geodesics, for Intrinsic Ultracontractivity (IU) of the semigroup associated to the Dirichlet Laplacian in domains which have the “wide access” property. In particular, we prove that any such domain which is IU satisfies this geometric condition. We also prove that this condition characterizes IU for tubes along geodesics, and that it implies one-half intrinsic ultracontractivity for domains with the “wide access” property. In Chapter 3, we study the moments of integrability of [special characters omitted], the first exit time of an n-dimensional symmetric α-stable process, α ∈ (0, 2), from a circular cone of angle [special characters omitted]. We show that there exists a constant [special characters omitted] such that for all x in the cone, [special characters omitted], if [special characters omitted] and [special characters omitted], if [special characters omitted]. We characterize [special characters omitted] in terms of the principle eigenvalue of a degenerate differential operator. We also give some explicit upper and lower bounds for [special characters omitted].

Degree

Ph.D.

Advisors

Banuelos, Purdue University.

Subject Area

Mathematics

COinS