The Dirichlet problem for a class of Hessian type equations

TL;DR

This paper addresses the Dirichlet problem for Hessian type equations, establishing $C^2$ estimates and proving the existence of classical solutions using new methods under optimal conditions.

Contribution

Introduces new techniques to obtain $C^2$ estimates for Hessian equations and proves classical solutions exist under optimal structure conditions.

Findings

Established $C^2$ estimates for an approximating problem

Proved the existence of classical solutions

Developed new methods for Hessian type equations

Abstract

We are concerned with the Dirichlet problem for a class of Hessian type equations. Applying some new methods we are able to establish the $C^2$ estimates for an approximating problem under essentially optimal structure conditions. Based on these estimates, the existence of classical solutions is proved.