The Dirichlet problem for mixed Hessian equations on Hermitian manifolds

TL;DR

This paper investigates the Dirichlet problem for a class of mixed Hessian equations on Hermitian manifolds, establishing a priori estimates and solvability under certain conditions.

Contribution

It introduces new methods for solving complex mixed Hessian equations on Hermitian manifolds with specific a priori estimates.

Findings

Established a priori estimates for the complex mixed Hessian equation.

Proved the existence of solutions to the Dirichlet problem under given conditions.

Extended the theory of Hessian equations to Hermitian manifolds.

Abstract

In this paper we study the Dirichlet problem for a class of Hessian type equation with its structure as a combination of elementary symmetric functions on Hermitian manifolds. Under some conditions with the initial data on manifolds and admissible subsolutions, we derive a priori estimates for this complex mixed Hessian equation and solvability of the corresponding Dirichlet problem.