Complex Monge-Amp\`ere equation in strictly pseudoconvex domains

TL;DR

This paper investigates the complex Monge-Ampère equation in strictly pseudoconvex domains, establishing conditions under which solutions are continuous outside small sets, advancing understanding of boundary value problems in complex analysis.

Contribution

It introduces a new sufficient condition ensuring the continuity of solutions to the complex Monge-Ampère equation outside small sets in pseudoconvex domains.

Findings

Established a sufficient condition for solution continuity

Extended regularity results for boundary value problems

Analyzed solutions in the context of complex analysis

Abstract

We study the complex Monge-Amp\`ere equation $(dd^c u)^n=\mu$ in a strictly pseudoconvex domain $\Omega$ with the boundary condition $u=\varphi$, where $\varphi\in C(\partial\Omega)$. We provide a non-trivial sufficient condition for continuity of the solution $u$ outside "small sets".