# Regularity for fully nonlinear elliptic equations with oblique boundary conditions

## TL;DR

This paper establishes regularity results for viscosity solutions of fully nonlinear elliptic equations with oblique boundary conditions, including pointwise $C^{eta}$, $C^{1,eta}$, and $C^{2,eta}$ regularity, along with fundamental principles like maximum principle and Harnack inequality.

## Contribution

It provides new regularity results and fundamental principles for solutions of fully nonlinear elliptic equations with oblique boundary conditions.

## Key findings

- Proved pointwise $C^{eta}$, $C^{1,eta}$, and $C^{2,eta}$ regularity.
- Established the A-B-P maximum principle and Harnack inequality.
- Demonstrated uniqueness and solvability of the equations.

## Abstract

In this paper, we obtain a series of regularity results for viscosity solutions of fully nonlinear elliptic equations with oblique derivative boundary conditions. In particular, we derive the pointwise $C^{\alpha}$, $C^{1,\alpha}$ and $C^{2,\alpha}$ regularity. As byproducts, we also prove the A-B-P maximum principle, Harnack inequality, uniqueness and solvability of the equations.