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Harnack-type inequalities for nonlinear evolution equations

Jessica Slegers

TL;DR

The work addresses deriving Harnack-type inequalities for nonlinear evolution equations by extending Li–Yau type gradient methods to nonlinear settings through the Auchmuty–Bao framework. It combines a variational minimisation in weighted Sobolev spaces with an Aronson–Benilan type gradient estimate to derive three general Harnack inequalities and then specializes to the heat, PME, and p-diffusion equations, including explicit space-time bounds. It also establishes local Hölder continuity via Moser's method and discusses weak solutions and potential future work on boundary-value problems and nonlocal operators. The results advance qualitative understanding of nonlinear diffusion and provide tools for regularity theory in nonlinear parabolic PDEs.

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 Hölder 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.

Harnack-type inequalities for nonlinear evolution equations

TL;DR

The work addresses deriving Harnack-type inequalities for nonlinear evolution equations by extending Li–Yau type gradient methods to nonlinear settings through the Auchmuty–Bao framework. It combines a variational minimisation in weighted Sobolev spaces with an Aronson–Benilan type gradient estimate to derive three general Harnack inequalities and then specializes to the heat, PME, and p-diffusion equations, including explicit space-time bounds. It also establishes local Hölder continuity via Moser's method and discusses weak solutions and potential future work on boundary-value problems and nonlocal operators. The results advance qualitative understanding of nonlinear diffusion and provide tools for regularity theory in nonlinear parabolic PDEs.

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 -Laplace operator, which we refer to here as the -diffusion equation. Finally, we demonstrate a common application of Harnack inequalities by proving the local space-time Hölder 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.
Paper Structure (19 sections, 36 theorems, 285 equations)

This paper contains 19 sections, 36 theorems, 285 equations.

Table of Contents

  1. Introduction
  2. Modelling
  3. Main results
  4. Preliminaries
  5. Minimisation of a Convex Functional
  6. General Harnack-type Inequalities
  7. Applications to Nonlinear Evolution Equations
  8. The Heat Equation
  9. The Porous Medium Equation
  10. The p-diffusion Equation
  11. Harnack Inequalities and Hölder Continuity
  12. Final Remarks
  13. Future Work
  14. The Sobolev Spaces $W^{1,p}(a,b)$ and $W^{1,p}(\Omega)$
  15. The Sobolev Space $W^{1,p}(a,b)$
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $\Omega \subseteq \mathbb{R}^d$ be an open convex set and let $f$ be a positive continuously differentiable function on $\Omega_T:= \Omega \times (0,T)$, for which there exist constants $C>0$, $p>1$, $r \in \mathbb{R}$, and a function $a \in L_{\mathop{\mathrm{loc}}\nolimits}^1(0,T)$ such that Then for all $x_1,x_2 \in \Omega$ and $0 < t_1 < t_2 <T$: where $q:= \frac{p}{p-1}$, $m:=\frac{r}{p

Theorems & Definitions (53)

  • Theorem 1.1: General Harnack inequalities, Auchmuty-Bao-1994
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • ...and 43 more