The global Dirichlet problem for the complex Monge-Ampère equation
Let \({\mathcal L}_+\) denote the collection of all plurisubharmonic functions \(u\) on \(\mathbb{C}^n\) such that \(u(z)={1\over 2}\log (1+ |z|^2) +O(1)\). The following Dirichlet problem is considered: given a function \(f\geq 0\) on \(\mathbb{C}^n\), find \(u\in {\mathcal L}_+\) satisfying \((dd^lu)^n= fdV\). A necessary condition is that \(\int_{\mathbb{C}^n} fdV=(2\pi)^n\). The authors show that the condition is not sufficient even when \(f\in C^\infty (\mathbb{C}^n)\). On the other hand, they prove that the problem has a solution for any \(f\in C^3 (\mathbb{C}^n)\) satisfying the additional condition \(f(z)\leq C(1+|z|^2)^{-n-1}\).
scientific article; zbMATH DE number 1542088