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Harnack Inequality for Nonlinear Equations Driven by the Normalized Infinity-Laplacian

Ahmed Mohammed, Carson Pocock

TL;DR

This work addresses the Harnack inequality for nonnegative viscosity solutions of the nonlinear, degenerate equation $\Delta_\infty^N u = f(u) + g(u)|Du|^q$ with $0\le q\le 1$. The authors first prove a Harnack-type bound for supersolutions of a linearized model $\Delta_\infty^N u = A(x)u + B(x)|Du|^q|u|^{1-q}$, and then develop a Keller–Osserman type growth framework (KO$_q$) together with a structural function $h(s)$ to obtain uniform a priori bounds and a global oscillation control. Key innovations include the integration of the KO$_q$ condition, the barrier/radial-ODE analysis for the local comparison, and the construction of a domain-wise Harnack via a chain of balls, all in the context of the normalized infinity-Laplacian with gradient terms. The results yield local and global Harnack inequalities with constants depending only on the data and the domain's distance to the boundary, advancing the understanding of positivity and oscillation for infinity-type equations with nonlinear absorption and gradient terms.

Abstract

This paper aims to investigate a Harnack inequality for non-negative solutions of the normalized infinity Laplacian with nonlinear absorption and gradient terms. More specifically, we establish a Harnack inequality for non-negative viscosity solutions of the PDE $Δ_\infty^Nu=f(u)+g(u)|Du|^q$, where $0\le q\le 1$, and for a large class of non-decreasing continuous functions $f$ and $g$ that meet suitable growth conditions at infinity.

Harnack Inequality for Nonlinear Equations Driven by the Normalized Infinity-Laplacian

TL;DR

This work addresses the Harnack inequality for nonnegative viscosity solutions of the nonlinear, degenerate equation with . The authors first prove a Harnack-type bound for supersolutions of a linearized model , and then develop a Keller–Osserman type growth framework (KO) together with a structural function to obtain uniform a priori bounds and a global oscillation control. Key innovations include the integration of the KO condition, the barrier/radial-ODE analysis for the local comparison, and the construction of a domain-wise Harnack via a chain of balls, all in the context of the normalized infinity-Laplacian with gradient terms. The results yield local and global Harnack inequalities with constants depending only on the data and the domain's distance to the boundary, advancing the understanding of positivity and oscillation for infinity-type equations with nonlinear absorption and gradient terms.

Abstract

This paper aims to investigate a Harnack inequality for non-negative solutions of the normalized infinity Laplacian with nonlinear absorption and gradient terms. More specifically, we establish a Harnack inequality for non-negative viscosity solutions of the PDE , where , and for a large class of non-decreasing continuous functions and that meet suitable growth conditions at infinity.
Paper Structure (5 sections, 8 theorems, 119 equations)

This paper contains 5 sections, 8 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Preliminaries
  4. Proof of the Harnack Inequalities
  5. Appendix

Key Result

Theorem 2.1

Let $\Omega\subseteq\mathbb{R}^n$ be an open set, and $0\le q\le 1$. There are positive constants $r_0$ and $C$, that depend on $q$, $A_0$ and $B_0$, such that for any given ball $B(x_0,2r)\subset\Omega$ with $0<r<r_0$ and any non-negative viscosity supersolution $u\in\textup{LSC}(\Omega)$ of (e-1)

Theorems & Definitions (13)

  • Theorem 2.1
  • Theorem 2.2
  • Definition 3.1
  • Proposition 3.2: Comparison Principle
  • Lemma 3.3
  • Proposition 3.4: Global $L^\infty$ Estimate
  • Remark 3.5
  • Lemma A.1
  • Remark A.2
  • Remark A.3
  • ...and 3 more