Boundary Lipschitz Regularity and the Hopf Lemma on Reifenberg Domains for Fully Nonlinear Elliptic Equations

TL;DR

This paper establishes boundary Lipschitz regularity and the Hopf lemma for fully nonlinear elliptic equations on Reifenberg domains, extending classical results under less restrictive geometric conditions.

Contribution

It introduces a unified approach to prove boundary regularity and the Hopf lemma on Reifenberg domains with $C^{1, ext{Dini}}$ conditions, broadening applicability.

Findings

Lipschitz continuity of solutions at boundary points under exterior Reifenberg conditions

Hopf lemma validity at boundary points under interior Reifenberg conditions

Extension of classical regularity results to less smooth Reifenberg domains

Abstract

In this paper, we prove the boundary Lipschitz regularity and the Hopf Lemma by a unified method on Reifenberg domains for fully nonlinear elliptic equations. Precisely, if the domain $\Omega$ satisfies the exterior Reifenberg $C^{1,\mathrm{Dini}}$ condition at $x_0\in \partial \Omega$ (see Definition 1.3), the solution is Lipschitz continuous at $x_0$; if $\Omega$ satisfies the interior Reifenberg $C^{1,\mathrm{Dini}}$ condition at $x_0$ (see Definition 1.4), the Hopf lemma holds at $x_0$. Our paper extends the results under the usual $C^{1,\mathrm{Dini}}$ condition.