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 \(C^{1,\textrm{Dini}}\) condition at \(x_0\in \partial \Omega \), the solution is Lipschitz continuous at \(x_0\); if \(\Omega \) satisfies the interior \(C^{1,\textrm{Dini}}\) 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 \(C^{1,\textrm{Dini}}\) conditions are optimal.

在本文中，我们研究完全非线性椭圆方程粘度解的边界正则性。我们用统一、简单的方法来证明，如果域\(\Omega \)满足\(x_0\in \partial \Omega \)处的外部\(C^{1,\textrm{Dini}}\)条件，解在\(x_0\)处是 Lipschitz 连续的；如果\(\Omega \)满足\(x_0\)处的内部\(C^{1,\textrm{Dini}}\)条件，则 Hopf 引理在\(x_0\)处成立。关键思想是弯曲边界被视为超平面的扰动。此外，我们表明\(C^{1,\textrm{Dini}}\)条件是最优的。