$C^{1,\alpha}$ regularity of solutions of degenerate fully non-linear elliptic equations

TL;DR

This paper establishes $C^{1,eta}$ regularity for solutions of a class of degenerate fully nonlinear elliptic equations, combining existing Hölder estimates with new Lipschitz bounds.

Contribution

It introduces a novel approach by integrating Hölder and Lipschitz estimates to prove $C^{1,eta}$ regularity for degenerate elliptic equations.

Findings

Proves $C^{1,eta}$ regularity for solutions

Combines Hölder and Lipschitz estimates effectively

Extends regularity results to degenerate equations

Abstract

In the present paper, a class of fully non-linear elliptic equations are considered, which are degenerate as the gradient becomes small. H\"older estimates obtained by the first author (2011) are combined with new Lipschitz estimates obtained through the Ishii-Lions method in order to get $C^{1,\alpha}$ estimates for solutions of these equations.