A priori estimates for the obstacle problem of Hessian type equations on Riemannian manifolds

TL;DR

This paper develops new techniques to establish a priori second order estimates for obstacle problems involving fully nonlinear Hessian equations on Riemannian manifolds, leading to existence results for viscosity solutions.

Contribution

It introduces novel methods for second order estimates in obstacle problems of Hessian type equations on Riemannian manifolds, expanding the theoretical framework.

Findings

Established new second order a priori estimates

Proved existence of C^{1,1} viscosity solutions

Extended techniques to singular perturbation problems

Abstract

We are concerned with a priori estimates for the obstacle problem of a wide class of fully nonlinear equations on Riemannian manifolds. We use new techniques introduced by Bo Guan and derive new results for a priori second order estimates of its singular perturbation problem under fairly general conditions. By approximation, the existence of a C^{1,1} viscosity solution is proved.