Positivity sets of supersolutions of degenerate elliptic equations and the strong maximum principle

TL;DR

This paper studies the geometric properties of positivity sets of supersolutions to fully nonlinear degenerate elliptic equations and establishes conditions under which the strong maximum principle holds.

Contribution

It provides geometric characterizations of positivity sets and introduces conditions for the validity of the strong maximum principle in degenerate elliptic equations.

Findings

Geometric description of positivity sets for supersolutions

Conditions for the strong maximum principle to hold

Validation of the principle under specific geometric assumptions

Abstract

We investigate positivity sets of nonnegative supersolutions of the fully nonlinear elliptic equations $F(x,u,Du,D^2u)=0$ in $\Omega$, where $\Omega$ is an open subset of ${\mathbb R}^N$, and the validity of the strong maximum principle for $F(x,u,Du,D^2u)=f$ in $\Omega$, with $f\in\text{C}(\Omega)$ being nonpositive. We obtain geometric characterizations of positivity sets $\left\{x\in\Omega\,:\, u(x)>0\right\}$ of nonnegative supersolutions $u$ and establish the strong maximum principle under some geometric assumption on the set $\left\{x\in\Omega\,:\, f(x)=0\right\}$.