Estimates on elliptic equations that hold only where the gradient is large

TL;DR

This paper demonstrates that classical regularity results like Hölder estimates and Harnack inequality apply to viscosity solutions of elliptic equations, even when these solutions are only valid at points with large gradients.

Contribution

It extends the applicability of Krylov and Safonov's regularity theory to functions satisfying elliptic equations only where the gradient is large.

Findings

Hölder estimates hold under the new conditions

Harnack inequality applies to these solutions

Regularity theory extends to gradient-restricted solutions

Abstract

We consider a function which is a viscosity solution of a uniformly elliptic equation only at those points where the gradient is large. We prove that the H{\"o}lder estimates and the Harnack inequality, as in the theory of Krylov and Safonov, apply to these functions.