The Dirichlet problem for a class of Hessian quotient equations on Riemannian manifolds

TL;DR

This paper establishes existence and uniqueness results for the Dirichlet problem involving Hessian quotient equations on Riemannian manifolds, using a priori estimates and admissible subsolutions without boundary geometric restrictions.

Contribution

It introduces a method to solve the Dirichlet problem for Hessian quotient equations on Riemannian manifolds under minimal boundary assumptions.

Findings

Existence and uniqueness of solutions are proven.

A priori estimates are developed for Hessian quotient equations.

The results hold without boundary geometric restrictions.

Abstract

In this paper, we consider the Dirichlet problem for a class of Hessian quotient equations on Riemannian manifolds. Under the assumption of an admissible subsolution, we solve the existence and the uniquness for the Dirichlet problem in a domain without any geometric restrictions on the boundary, based on the a priori estimates for the solutions to the Hessian quotient type equations.