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

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.