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

TL;DR

This paper proves a Harnack inequality for non-negative solutions of a nonlinear PDE involving the normalized infinity Laplacian, with applications to equations with absorption and gradient terms.

Contribution

It establishes a new Harnack inequality for viscosity solutions of a class of nonlinear PDEs with the normalized infinity Laplacian, covering a range of growth conditions.

Findings

Harnack inequality holds for solutions with gradient terms up to order 1.

Results apply to a broad class of nonlinear functions f and g.

Provides tools for regularity analysis of infinity Laplacian equations.

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 $\Delta_\infty^Nu=f(u)+g(u)|Du|^q$, 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.