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

Abstract

In this paper, we study the boundary regularity for viscosity solutions of fully nonlinear elliptic equations. We use a unified, simple method to prove that if the domain $\Omega$ satisfies the exterior $C^{1,\mathrm{Dini}}$ condition at $x_0\in \partial \Omega$ (see Definition 1.2), the solution is Lipschitz continuous at $x_0$; if $\Omega$ satisfies the interior $C^{1,\mathrm{Dini}}$ condition at $x_0$ (see Definition 1.3), the Hopf lemma holds at $x_0$. The key idea is that the curved boundaries are regarded as perturbations of a hyperplane. Moreover, we show that the $C^{1,\mathrm{Dini}}$ conditions are optimal.