The Dirichlet problem for the complex Monge-Ampère operator: Perron classes and rotation-invariant measures
The Dirichlet problem corresponding to an open, bounded and strictly pseudoconvex subset \(\Omega\) of \(\mathbb{C}^ n\) and the Monge-Ampère operator \((dd^ c)^ n\) can be stated as follows: Given \(\varphi\in C(\partial \Omega)\) and a positive measure \(\mu\) on \(\Omega\), find \(u\in \text{PSH}\cap L^ \infty (\Omega)\) such that \((dd^ c u)^ n= \mu\) on \(\Omega\) and \(\varlimsup_{z\to\xi} u(z)= \varphi(\xi)\) for every \(\xi\) on \(\partial \Omega\). The solution to this problem exists, for example, if \(\mu\) is the Lebesgue measure with an \(L^ 2\)-density function. In this paper, it is proved that if \(\Omega= B\) is the unit ball and if \(\nu\) is a positive rotation invariant measure in \(B\) for which there is a \(u\in \text{PSH}\cap L^ \infty (B)\) with \((dd^ c u)^ n\geq \nu\) and \(\varlimsup_{z\to\xi} u(z)\leq 0\) on \(\partial B\) then for every \(0\leq f\in L^ \infty (B, \nu)\) and for every \(\varphi\in C(\partial B)\) there is a unique solution to the Dirichlet problem with \(\mu= f d\nu\); moreover, if \(f\) is also continuous on \(B\) then the solution is in \(C(\overline {B})\).
scientific article; zbMATH DE number 731836