On Dirichlet's principle and problem

TL;DR

This paper provides a new variational proof for characterizing measures that admit solutions to the Dirichlet problem involving the complex Monge-Ampère operator within the class of finite energy plurisubharmonic functions.

Contribution

It introduces a novel variational approach to fully characterize measures allowing solutions to the Dirichlet problem for the complex Monge-Ampère operator.

Findings

Complete measure characterization for Dirichlet problem solutions

New proof technique using variational methods

Enhanced understanding of complex Monge-Ampère equations

Abstract

The aim of this paper is to give a new proof of the complete characterization of measures for which there exist a solution of the Dirichlet problem for the complex Monge-Ampere operator in the set of plurisubharmonic functions with finite pluricomplex energy. The proof uses variational methods.