Local $C^{0,\alpha}$ Estimates for Viscosity Solutions of Neumann-type Boundary Value Problems

TL;DR

This paper establishes local Hölder continuity estimates for viscosity solutions of fully nonlinear elliptic equations with Neumann boundary conditions, including degenerate cases, broadening regularity understanding in boundary value problems.

Contribution

It provides the first regularity results and Hölder estimates for viscosity solutions of nonlinear elliptic equations with Neumann boundary conditions, even in degenerate cases.

Findings

Proved local $C^{0,eta}$ regularity for viscosity solutions.

Established estimates applicable to degenerate elliptic equations.

Handled general linear and nonlinear Neumann boundary conditions.

Abstract

In this article, we prove the local $C^{0,\alpha}$ regularity and provide $C^{0,\alpha}$ estimates for viscosity solutions of fully nonlinear, possibly degenerate, elliptic equations associated to linear or nonlinear Neumann type boundary conditions. The interest of these results comes from the fact that they are indeed regularity results (and not only a priori estimates), from the generality of the equations and boundary conditions we are able to handle and the possible degeneracy of the equations we are able to take in account in the case of linear boundary conditions.