Boundary $ C^{1}$ regularity for degenerate fully nonlinear elliptic equations on $ C^{2} $ domain

TL;DR

This paper proves boundary regularity results for degenerate fully nonlinear elliptic equations on smooth domains, extending interior regularity findings and establishing sharp boundary regularity conditions.

Contribution

It provides the first global boundary regularity results for degenerate fully nonlinear elliptic equations on $C^2$ domains, including sharp boundary regularity conditions.

Findings

Established global $C^{0,eta}$, $C^{0,1}$, and $C^{1}$ estimates for degenerate equations.

Showed that $C^{1,eta}$ regularity of boundary data is optimal within Hölder spaces.

Derived global $C^{1,eta}$ regularity for certain singular fully nonlinear equations.

Abstract

In this article, we establish global regularity results ($ C^{0,\gamma}$, $ C^{0,1} $ and $ C^{1}$ estimates) for a class of degenerate fully nonlinear equation on $ C^{2} $-domain. This corresponds to the boundary counterpart of the interior $ C^{1}$ regularity results by \cite{APPT22} and \cite{AN25}. By example we show that $ C^{1,\alpha} $ regularity of boundary datum is sharp within the scale of H\"{o}lder spaces. As a byproduct, we also provide global $ C^{1,\beta} $ regularity for a class singular fully nonlinear equation.