$C^{1,\alpha}$ regularity for degenerate fully nonlinear elliptic equations with oblique boundary conditions on $C^1$ domains

TL;DR

This paper establishes sharp boundary regularity estimates for viscosity solutions of degenerate fully nonlinear elliptic equations with oblique boundary conditions on $C^1$ domains, advancing understanding of boundary behavior in degenerate elliptic PDEs.

Contribution

It provides the first sharp $C^{1,eta}$ boundary regularity results for degenerate fully nonlinear elliptic equations with oblique boundary conditions on $C^1$ domains.

Findings

Established uniform boundary Hölder estimates in almost $C^1$-flat domains.

Proved $C^{1,eta}$ regularity up to the boundary for solutions.

Extended regularity results to degenerate elliptic equations with oblique boundary conditions.

Abstract

We provide a sharp $C^{1,\alpha}$ estimate up to the boundary for a viscosity solution of a degenerate fully nonlinear elliptic equation with the oblique boundary condition on a $C^1$ domain. To this end, we first obtain a uniform boundary H{\"o}lder estimate with the oblique boundary condition in an "almost $C^1$-flat" domain for the equations which is uniformly elliptic only where the gradient is far from some point, and then we establish a desired $C^{1,\alpha}$ regularity based on perturbation and compactness arguments.