Second Order Estimates and Regularity for Fully Nonlinear Elliptic Equations on Riemannian Manifolds

TL;DR

This paper establishes second order estimates for fully nonlinear elliptic equations on Riemannian manifolds, leading to regularity and existence results without restrictive geometric assumptions.

Contribution

It provides general a priori second order estimates for solutions of nonlinear elliptic equations on Riemannian manifolds, applicable to both closed and boundary cases.

Findings

Derived second order estimates under general conditions

Achieved regularity results for solutions

Proved existence of solutions using these estimates

Abstract

We derive a priori second order estimates for solutions of a class of fully nonlinear elliptic equations on Riemannian manifolds under some very general structure conditions. We treat both equations on closed manifolds, and the Dirichlet problem on manifolds with boundary without any geometric restrictions to the boundary except being smooth and compact. As applications of these estimates we obtain results on regularity and existence.