Modulus of Continuity Estimates for Fully Nonlinear Parabolic Equations

TL;DR

This paper establishes that the moduli of continuity for viscosity solutions of fully nonlinear parabolic PDEs are subsolutions of related one-dimensional equations, enabling sharp gradient bounds and extending previous quasilinear results.

Contribution

It introduces a novel approach linking moduli of continuity to subsolutions of one-dimensional equations for fully nonlinear parabolic PDEs, extending prior quasilinear findings.

Findings

Sharp Lipschitz bounds for solutions

Gradient estimates derived from comparison principles

Extension of results from quasilinear to fully nonlinear equations

Abstract

We prove that the moduli of continuity of viscosity solutions to fully nonlinear parabolic partial differential equations are viscosity subsolutions of suitable parabolic equations of one space variable. As applications, we obtain sharp Lipschitz bounds and gradient estimates for fully nonlinear parabolic equations with bounded initial data, via comparison with one-dimensional solutions. This work extends multiple results of Andrews and Clutterbuck for quasilinear equations to fully nonlinear equations.