On the existence and regularity of the Dirichlet problem for complex Monge-Ampère equations on weakly pseudoconvex domains
Let \(\Omega\) be a bounded pseudoconvex domain of plurisubharmonic type \(m\) in \(\mathbb{C}^n\) with \(C^2\) boundary and \(m\geq 2\). For any \(0<\alpha \leq \frac 2m\), \(\phi\in C^{m\alpha}(\partial \Omega)\), and \(f\geq 0\) in \(\Omega\) with \(f^{\frac 1m}\in C^\alpha(\overline \Omega)\). The main result of the paper under review is that there exists a unique plurisubharmonic function \(u\in C^\alpha(\overline \Omega )\) such that \(\det (\frac {\partial^2 u}{\partial z_i \overline \partial z_j})=f\) in \(\Omega\) in the weak sense and \(u=\phi \) on \(\partial \Omega\). This is proved by using arguments of \textit{E. Bedford} and \textit{B. A.Taylor} [Invent. Math. 37, 1--44 (1976; Zbl 0315.31007)]. For the complex ellipsoid in \(\mathbb{C}^2\), this result is sharp.
scientific article; zbMATH DE number 2123234