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

## TL;DR

This paper establishes boundary pointwise $C^{1,eta}$ and $C^{2,eta}$ regularity for viscosity solutions of fully nonlinear elliptic equations, extending known results even for classical Laplace equations with simpler proofs.

## Contribution

It provides new boundary regularity results for fully nonlinear elliptic equations, including the Laplace equation, with simplified proof techniques.

## Key findings

- Boundary regularity results for fully nonlinear elliptic equations.
- Extension of regularity results to classical Laplace equation.
- Simplified proofs for boundary regularity.

## 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.