The Dirichlet problem for Monge-Amp\`ere type equations on Riemannian manifolds

TL;DR

This paper addresses the Dirichlet problem for Monge-Ampère type equations on Riemannian manifolds, establishing a priori estimates and proving existence of solutions using the continuity method.

Contribution

It provides new a priori second-order derivative estimates and proves existence results for Monge-Ampère type equations on Riemannian manifolds.

Findings

Established second-order a priori estimates for solutions.

Proved existence of solutions via the continuity method.

Extended analysis to p-plurisubharmonic functions on Riemannian manifolds.

Abstract

In this paper, we study the Dirichlet problem for Monge-Amp\`ere type equations for $p$-plurisubharmonic functions on Riemannian manifolds. The $a$ $priori$ estimates up to the second order derivatives of solutions are established. The existence of a solution then follows by the continuity method.