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 O satisfies the exterior C(1,Dini )condition at x(0) ? ?O, the solution is Lipschitz continuous at x(0); if O satisfies the interior C-1,C-Dini condition at x(0), 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,C-Dini conditions are optimal.