A new proof of H\"older estimates for the gradient of quasilinear elliptic equations

TL;DR

This paper introduces a novel proof technique for establishing H"older continuity of gradients in quasilinear and fully nonlinear elliptic equations, enhancing understanding of their regularity properties.

Contribution

The paper presents a new covering method-based proof for H"older estimates, applicable to both quasilinear and fully nonlinear elliptic equations, inspired by Evans-Krylov theorem.

Findings

H"older estimates for gradients of quasilinear elliptic equations established

Extension of H"older estimates to fully nonlinear elliptic equations

New proof technique simplifies regularity analysis

Abstract

In this paper, we give a new proof of H\"older estimates for the gradient of quasilinear elliptic equations, using a covering method inspired by the proof of Evans-Krylov theorem for fully nonlinear elliptic equations. Moreover, H\"older estimates for the gradient of fully nonlinear elliptic equations are also obtained by the same method.