A multi-point maximum principle to prove global parabolic Harnack inequalities

Abstract

In this work, we introduce a novel methodology for proving global pointwise Harnack inequalities for parabolic partial differential equations on a Riemannian manifold. The main idea of our approach is to apply a multi-point maximum principle. We demonstrate our techniques by studying the Harnack inequalities satisfied by positive solutions of the linear Schrödinger equation and the doubly nonlinear heat equation.

In Chapter 1, we recount the history of parabolic Harnack inequalities, before reviewing the existence of solutions to our aforementioned equations of interest in Chapter 2. In Chapter 3, we present the first proofs of the Harnack inequality using our multi-point maximum principle approach, focusing on classical solutions. In Section 3.1, we analyse the Schrödinger equation, first in Euclidean space and then on a Riemannian manifold with nonnegative Ricci curvature. This section contains applications to Schrödinger equations with a gradient drift term, including the heat equation governed by the Ornstein-Uhlenbeck operator. In addition, we use our Harnack inequality to recover a differential Harnack inequality comparable to the famous result of Li and Yau.

In Section 3.2, we demonstrate how our techniques can be extended to prove the Harnack inequality for positive classical solutions of the doubly nonlinear heat equation. However, since solutions of this equation do not in general possess sufficient regularity to be treated as classical solutions, we dedicate Chapter 4 to adapting our proof techniques to viscosity solutions. After reviewing the basic notions associated with viscosity solutions, we develop a modified version of the parabolic theorem on sums by Crandall and Ishii, which is crucial in our methodology. Finally, we present a new proof of the Harnack inequality satisfied by positive viscosity solutions of the doubly nonlinear heat equation.