Levi-equation in higher dimensions
type: Item
instance of: scholarly article
title: Levi-equation in higher dimensions
zbMATH Open document ID: 0765.32011
author: Zbigniew Slodkowski
author: Giuseppe Tomassini
published in: Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Serie IX. Rendiconti Lincei. Matematica e Applicazioni
publication date: 1992-09-27T00:00:00Z
full work available at URL: https://eudml.org/doc/244335
review text: The authors announce two results about the Dirichlet problem for the Levi equation in \(\mathbb C^ 3\). Let \(\Omega\) be a bounded domain in \(\mathbb R^ 5\) with connected boundary, and let \(u\) be a continuous real-valued function on \(\Omega\). The graph of \(u\) is then a surface in \(\mathbb C^ 3\), and this surface has a Levi form. The problem is to find a function \(u\), with given continuous boundary values on \(\partial\Omega\), such that the determinant of the Levi form equals \(k(\cdot,u)(1+| Du|^ 2)^ 2\) in \(\Omega\), where \(k\) is a given continuous function on \(\Omega\times\mathbb R\). The authors' first proposition states that if \(u\) is a weak solution of this problem with \(k=0\), then the graph of \(u\) is neither strictly pseudoconvex nor strictly pseudoconcave at any point. The second proposition states that the problem can be reduced to a certain ``Bellman problem'' for a family of quasilinear degenerate elliptic operators.
reviewed by: Harold P. Boas
Mathematics Subject Classification ID: 32W20
Mathematics Subject Classification ID: 35J96
zbMATH DE Number: 61694
zbMATH Keywords: Monge-Ampère equation
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: Levi form
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 61694