Harnack-type inequalities for nonlinear evolution equations

TL;DR

This paper develops Harnack inequalities for nonlinear evolution equations, extending classical results to broader classes of PDEs, and demonstrates their applications in regularity and continuity properties of solutions.

Contribution

It introduces new Harnack inequalities for nonlinear PDEs, building on methods by Li, Yau, Auchmuty, and Bao, and applies these to porous medium and p-Laplace equations.

Findings

Established general Harnack inequalities for nonlinear evolution equations.

Applied inequalities to prove regularity of solutions to porous medium and p-Laplace equations.

Demonstrated local Hölder continuity of solutions using Harnack inequalities.

Abstract

Harnack inequalities are useful qualitative tools for understanding the properties of partial differential equations. Originally discovered as a property of harmonic functions, Harnack inequalities have since been studied for solutions of wider classes of elliptic and parabolic problems. In this monograph, we take particular interest in deriving Harnack inequalities for solutions of nonlinear evolution equations. We focus on exploring the methods introduced by Li and Yau in the case of the linear heat equation and later extended to nonlinear problems by Auchmuty and Bao. Prior to presenting these results, we study a minimisation problem, which appears naturally in the proofs. After establishing a family of three general Harnack inequality results by Auchmuty and Bao, we investigate applications to deriving Harnack inequalities satisfied by solutions of the porous medium equation and weak solutions of the parabolic problem associated with the $p$-Laplace operator, which we refer to here as the $p$-diffusion equation. Finally, we demonstrate a common application of Harnack inequalities by proving the local space-time H\"older continuity of solutions to a class of linear evolution problems. The proof is based on methods introduced by Moser during his seminal work on Harnack inequalities during the 1960s. We conclude by suggesting potential opportunities for future work following on from the topics discussed here.