The Dirichlet problem for complex Monge-Ampère equations and regularity of the pluri-complex Green function (Q1281358)

Regularity property are studied for solutions of the Dirichlet problem \[ (dd^c u)^n= \psi(z,u,\nabla u) \quad\text{in }\Omega, \qquad u=\varphi\quad\text{on }\partial\Omega, \tag{1} \] \(\Omega\) being a bounded domain in \(\mathbb{C}^n\) with \(C^\infty\) boundary. The main results obtained are as follows. Theorem 1.1. If \(\varphi,\psi\) are smooth functions, \(\psi>0\), then there exists a strictly plurisubharmonic solution \(u\in C^\infty (\overline{\Omega})\) of (1), provided there is a strictly plurisubharmonic subsolution \(v\in C^2 (\overline{\Omega})\) of (1) (i.e. \((dd^c v)^n\geq \psi\), \(v=\varphi\) on \(\partial \Omega\)). Theorem 1.2. If \(\Omega\) is strongly pseudoconvex then the pluricomplex Green function \(g_s\) with the pole at \(s\in \Omega\) belongs to \(C^{1,\alpha} (\overline{\Omega}\setminus \{s\})\) for any \(0< \alpha< 1\). Note that in Theorem 1.1, \(\Omega\) is not assumed to be pseudoconvex.