# $C^{1, \alpha}$ Regularity for degenerate fully nonlinear elliptic equations with Neumann boundary conditions

## TL;DR

This paper proves boundary regularity results for a class of degenerate fully nonlinear elliptic equations with Neumann boundary conditions, extending interior regularity to the boundary using compactness and new boundary estimates.

## Contribution

It establishes $C^{1, eta}$ regularity up to the boundary for degenerate fully nonlinear elliptic equations with Neumann conditions, extending previous interior results.

## Key findings

- Boundary $C^{1, eta}$ regularity is achieved for degenerate equations.
- New boundary Hölder estimates are developed for different gradient regimes.
- The proof combines compactness arguments with boundary estimates.

## Abstract

In this paper, we establish $C^{1, \alpha}$ regularity upto the boundary for a class of degenerate fully nonlinear elliptic equations with Neumann boundary conditions. Our main result Theorem 2.1 constitutes the boundary analogue of the interior $C^{1, \alpha}$ regularity result established in [21] for equations with similar structural assumptions. The proof of our main result is achieved via compactness arguments combined with new boundary H\"{o}lder estimates for equations which are uniformly elliptic when the gradient is either small or large.