Sharp boundary and global regularity for degenerate fully nonlinear elliptic equations

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 $\alpha>0$.