The Dirichlet Problem for mixed Hessian type equations on Riemannian manifolds

TL;DR

This paper establishes $C^2$ estimates and proves the existence of solutions for mixed Hessian type equations with Dirichlet boundary conditions on Riemannian manifolds, advancing the understanding of these complex nonlinear PDEs.

Contribution

It provides new a priori estimates and existence results for mixed Hessian equations on Riemannian manifolds, extending previous work to more general settings.

Findings

Derived $C^2$ estimates for mixed Hessian equations

Proved existence of admissible solutions for Dirichlet problems

Extended theory to Riemannian manifolds

Abstract

In this paper, we derive $C^2$ estimates for a class of mixed Hessian type equations with Dirichlet boundary condition, and obtain the existence theorem of admissible solutions for the classical Dirichlet problem of these mixed Hessian type equations.