Estimates on parabolic equations that hold only where the space-time gradient is large

TL;DR

This paper investigates H"older estimates and Harnack inequalities for viscosity solutions of parabolic equations, specifically focusing on regions where the space-time gradient exceeds a certain threshold, advancing understanding of solutions under gradient constraints.

Contribution

It introduces new estimates and inequalities applicable to parabolic equations only in regions with large space-time gradients, a novel focus in the analysis of viscosity solutions.

Findings

Established H"older estimates in regions with large gradients.

Proved Harnack inequalities under gradient constraints.

Extended classical results to gradient-restricted domains.

Abstract

We study the Krylov-Safonov type H\"older estimate and Harnack inequality for viscosity solutions satisfying a uniformly parabolic equation only where the gradient with respect to the space-time variables is large.