Comparison principles and Lipschitz regularity for some nonlinear degenerate elliptic equations

TL;DR

This paper proves interior Lipschitz regularity for viscosity solutions of certain degenerate elliptic equations and introduces a weak comparison principle called the propagation of touching points.

Contribution

It establishes Lipschitz regularity for fully nonlinear degenerate elliptic equations and introduces a new weak comparison principle for specific operators.

Findings

Proved interior Lipschitz regularity for viscosity solutions.

Established a weak form of the strong comparison principle.

Developed methods applicable to conformally invariant degenerate elliptic equations.

Abstract

We establish interior Lipschitz regularity for continuous viscosity solutions of fully nonlinear, conformally invariant, degenerate elliptic equations. As a by-product of our method, we also prove a weak form of the strong comparison principle, which we refer to as the principle of propagation of touching points, for operators of the form $\nabla^2 \psi + L(x,\psi,\nabla \psi)$ which are non-decreasing in $\psi$.