Boundary Pointwise $C^{1,\alpha }$ and $C^{2,\alpha }$ Regularity for Fully Nonlinear Elliptic Equations

Abstract

In this paper, we obtain the boundary pointwise $C^{1,\alpha }$ and $C^{2,\alpha }$ regularity for viscosity solutions of fully nonlinear elliptic equations. I.e., If $\partial \Omega$ is $C^{1,\alpha }$ (or $C^{2,\alpha }$) at $x_0\in \partial \Omega$, the solution is $C^{1,\alpha }$ (or $C^{2,\alpha }$) at $x_0$. Our results are new even for the Laplace equation. Moreover, our proofs are simple.