Geometric properties of solutions of the Levi curvature equation in \(\mathbb C^ 2\)
type: Item
instance of: scholarly article
title: Geometric properties of solutions of the Levi curvature equation in \(\mathbb C^ 2\)
zbMATH Open document ID: 0874.47023
author: Zbigniew Slodkowski
author: Giuseppe Tomassini
published in: Journal of Functional Analysis
publication date: 1996-07-15T00:00:00Z
review text: The authors study for \(L(u)=-\text{det}\left(\begin{smallmatrix} 0 & u_{z_1} & u_{z_2}\\ u_{z_1} & u_{z_1z_1} & u_{z_1z_2}\\ u_{z_2} & u_{z_2z_1} & u_{z_2z_2}\end{smallmatrix}\right)\) the Dirichlet problem: \[ L(u)= k|\partial u|^3\text{ in }\Omega\subset\mathbb C^2,\;u=g\text{ on }b\Omega,\tag{\(*\)} \] where \(\Omega\) is bounded and strictly pseudoconvex, \(k=k(z,t)\) is a real function on \(\Omega\times\mathbb R\) and \(g:b\Omega\to\mathbb R\). Introducing the notion of weak solution of \(L(u)=k|\partial u|^3\), they prove an existence theorem for the Dirichlet problem \((*)\). Furthermore, they study weak solutions of the homogeneous problem \(L(u)=0\) in terms of geometric properties of their level and sublevel sets.
Mathematics Subject Classification ID: 47F05
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 47N20
Mathematics Subject Classification ID: 37D30
zbMATH DE Number: 889773
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: strictly pseudoconvex
zbMATH Keywords: weak solution
zbMATH Keywords: homogeneous problem
zbMATH Keywords: geometric properties
zbMATH Keywords: level and sublevel sets
reviewed by: S. K. Chatterjea
MaRDI profile type: Publication
full work available at URL: jfan.1996.0061
OpenAlex ID: W1992283637
DOI: 10.1006/JFAN.1996.0061
description: scientific article; zbMATH DE number 889773