Second-order estimates and regularity for fully nonlinear elliptic equations on Riemannian manifolds

Abstract

We derive a priori second-order estimates for solutions of a class of fully nonlinear elliptic equations on Riemannian manifolds under structure conditions which are close to optimal. We treat both equations on closed manifolds and the Dirichlet problem on manifolds with boundary without any geometric restrictions to the boundary. These estimates yield regularity and existence results, some of which are new even for equations in Euclidean space.