Borderline regularity for fully nonlinear equations in Dini domains

TL;DR

This paper establishes borderline boundary gradient regularity for solutions to fully nonlinear elliptic equations in domains with Dini continuous boundaries, refining previous interior estimates with a geometric compactness approach.

Contribution

It provides a new boundary gradient regularity result for fully nonlinear elliptic equations in Dini domains, extending interior regularity techniques to the boundary context.

Findings

Proves boundary gradient continuity in Dini domains.

Sharpens previous boundary gradient estimates.

Introduces a geometric, compactness-based proof technique.

Abstract

In this paper, we prove borderline gradient continuity of viscosity solutions to Fully nonlinear elliptic equations at the boundary of a $C^{1,\dini}$-domain. Our main result Theorem 3.1 is a sharpening of the boundary gradient estimate proved by Ma-Wang following the borderline interior gradient regularity estimates established Daskalopoulos-Kuusi-Mingione. We however mention that, differently from the approach in the interior case which depends on $W^{1,q}$ estimates, our proof is slightly more geometric and is based on compactness arguments inspired by the techniques in the fundamental works of Caffarelli.