The global Dirichlet problem for the complex Monge-Ampère equation
type: Item
instance of: scholarly article
title: The global Dirichlet problem for the complex Monge-Ampère equation
zbMATH Open document ID: 0990.32008
DOI: 10.1007/BF02923087
author: Urban Cegrell
author: Slawomir Kołodziej
published in: The Journal of Geometric Analysis
publication date: 2000-12-11T00:00:00Z
review text: 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 \(f(z)\leq C(1+|z|^2)^{-n-1}\).
Mathematics Subject Classification ID: 32W20
zbMATH DE Number: 1542088
zbMATH Keywords: logarithmic growth
zbMATH Keywords: complex Monge-Ampère equation
zbMATH Keywords: plurisubharmonic functions
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02923087
OpenAlex ID: W1985293675
description: scientific article; zbMATH DE number 1542088