A Strong Maximum Principle and a Compact Support Principle for infinity Laplacian

TL;DR

This paper establishes necessary and sufficient conditions for the strong maximum principle and compact support principle for solutions to certain quasilinear elliptic inequalities involving the infinity Laplacian, expanding understanding of their qualitative behavior.

Contribution

It provides a complete characterization of when the strong maximum principle and compact support principle hold for these inequalities, which was previously not fully understood.

Findings

Characterization of conditions for the maximum principle

Conditions for the compact support principle

Extension of principles to a broad class of inequalities

Abstract

In this article we find necessary and sufficient conditions for the strong maximum principle and compact support principle for non-negative solutions to the quasilinear elliptic inequalities $$\Delta_\infty u + G(|Du|) - f(u)\,\leq 0\quad \text{in}\; \mathcal{O},$$ and $$\Delta_\infty u + G(|Du|) - f(u)\,\geq 0\quad \text{in}\; \mathcal{O},$$ where $\mathcal{O}$ denotes the infinity Laplacian, $G$ is an appropriate continuous function and $f$ is a nondecreasing, continuous function with $f(0)=0$.