Gradient bounds for viscosity solutions to certain elliptic equations

TL;DR

This paper investigates the gradient bounds of viscosity solutions to certain degenerate elliptic equations, establishing a one-dimensional equation for the modulus of continuity that helps derive bounds and properties of solutions.

Contribution

It introduces a one-dimensional equation for the modulus of continuity of solutions, enabling new gradient bounds for a class of degenerate elliptic equations.

Findings

Identified a one-dimensional equation for the modulus of continuity.

Derived gradient bounds for solutions under certain conditions.

Extended previous parabolic results to elliptic equations.

Abstract

Our principal object of study is the modulus of continuity of a periodic or uniformly vanishing function \( u: \mathbb{R} ^{n} \rightarrow \mathbb{R} \) which satisfies a degenerate elliptic equation \( F(x, u, \nabla u, D^{2} u) = 0 \) in the viscosity sense. The equations under consideration here have second-order terms of the form \( -{\rm Trace} \, (\mathcal{A} (\|\nabla u \|) \cdot D^{2} u) , \) where \( \mathcal{A} \) is an \( n\times n\) matrix which is symmetric and positive semi-definite. Following earlier work, \cite{Li21}, of the second author, which addressed the parabolic case, we identify a one-dimensional equation for which the modulus of continuity is a subsolution. In favorable cases, this one-dimensional operator can be used to derive a gradient bound on $u$ or to draw other conclusions about the nature of the solution.