Sharp boundary and global regularity for degenerate fully nonlinear elliptic equations

TL;DR

This paper establishes optimal boundary and global regularity estimates for solutions of degenerate fully nonlinear elliptic equations, showing solutions are $C^{1,eta}$ up to the boundary with explicit regularity depending on boundary data and degeneracy.

Contribution

It provides the first sharp global regularity results for degenerate fully nonlinear elliptic equations, including explicit regularity exponents based on boundary data and degeneracy degree.

Findings

Solutions are $C^{1,eta}$ up to the boundary with explicit $eta$.

Global regularity estimates are sharp and optimal.

Results apply even to simple degenerate Laplacian models.

Abstract

We obtain optimal boundary and global regularity estimates for viscosity solutions of fully nonlinear elliptic equations whose ellipticity degenerates at the critical points of a given solution. We show that any solution is $C^{1,\alpha}$ on the boundary of the domain, for an optimal and explicit $\alpha$ given only in terms of the regularity of the boundary datum and the elliptic degeneracy degree, no matter how possibly low is the interior regularity for that class of equations. We also obtain sharp global estimates. Our findings are new even for model equations, involving only a degenerate Laplacian; all previous results of global nature give $C^{1,\alpha}$ regularity only for some small $\alpha>0$.