A Dirichlet type problem for non-pluripolar complex Monge-Amp\`ere equations

TL;DR

This paper investigates a Dirichlet problem for non-pluripolar complex Monge-Ampère equations in bounded domains, establishing existence and uniqueness results and extending previous work in the field.

Contribution

It extends existing theorems on complex Monge-Ampère equations to include prescribed singularities and provides new local existence and uniqueness results.

Findings

Proved local existence and uniqueness for the Dirichlet problem.

Extended previous results to broader classes of singularities.

Enhanced understanding of complex Monge-Ampère equations in bounded domains.

Abstract

In this paper, we study a Dirichlet type problem for the non-pluripolar complex Monge - Amp\`ere equation with prescribed singularity on a bounded domain of $\mathbb{C}^n$. We provide a local version for an existence and uniqueness theorem proved by Darvas, Di Nezza and Lu. Our work also extends a result of Ahag, Cegrell, Czyz and Pham.