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 C1,α on the boundary of the domain, for an optimal and explicit α given only in terms of the regularity of the boundary datum and the elliptic degeneracy degree, and independent of the elliptic operator. We also obtain sharp global estimates. We use a different method from all previous results of global nature, which give C1,α regularity only for some small α>0. Our findings are new even for the model equations, involving a degenerate Laplacian.