Moduli of Continuity for Viscosity Solutions

TL;DR

This paper studies how the smoothness of solutions to certain nonlinear evolution equations, including the level set mean curvature flow, can be characterized by a one-dimensional subsolution equation, extending previous results.

Contribution

It extends the understanding of moduli of continuity to nonsingular and singular degenerate equations, including the level set mean curvature flow, as viscosity solutions.

Findings

Moduli of continuity are viscosity subsolutions of a 1D equation.

Extension of Andrews' results to nonsingular and degenerate equations.

Provides a unified framework for analyzing regularity of viscosity solutions.

Abstract

In this paper, we investigate the moduli of continuity for viscosity solutions of a wide class of nonsingular quasilinear evolution equations and also for the level set mean curvature flow, which is an example of singular degenerate equations. We prove that the modulus of continuity is a viscosity subsolution of some one dimensional equation. This work extends B. Andrews' recent result on moduli of continuity for smooth spatially periodic solutions.