The Dirichlet problem for the Monge-Amp\`ere equation on Hermitian manifolds with boundary

TL;DR

This paper investigates the Dirichlet problem for the complex Monge-Ampère equation on Hermitian manifolds with boundary, establishing existence, continuity, and optimal subsolution theorems for weak solutions under various measure conditions.

Contribution

It provides new existence and regularity results for weak solutions to the Monge-Ampère equation on Hermitian manifolds with boundary, including optimal subsolution theorems for bounded and Hölder continuous functions.

Findings

Proved existence of weak solutions under measure domination conditions.

Established continuity of solutions for measures with $L^p$, $p>1$ densities.

Developed optimal subsolution theorems for bounded and Hölder continuous functions.

Abstract

We study weak quasi-plurisubharmonic solutions to the Dirichlet problem for the complex Monge-Am\`ere equation on a general Hermitian manifold with non-empty boundary. We prove optimal subsolution theorems: for bounded and H\"older continuous quasi-plurisubharmonic functions. The continuity of the solution is proved for measures that well dominated by capacity, for example measures with $L^p$, $p>1$ densities, or moderate measures in the sense of Dinh-Nguyen-Sibony.