Spacelike graphs with prescribed mean curvature
type: Item
instance of: scholarly article
title: Spacelike graphs with prescribed mean curvature
zbMATH Open document ID: 0892.53025
published in: Differential and Integral Equations
publication date: 1998-05-17T00:00:00Z
review text: The authors study rotationally symmetric spacelike hypersurfaces with prescribed mean curvature \(f(u)\) for the Minkowski metric \(dx^2_1 + \ldots + dx^2_N - dt^2\), \(N>1\). The problem is reduced to the equations \[ \frac {u''}{(1-u^{'2})^{3/2}} + \frac {N-1}{r} \frac {u'}{(1-u^{'2})^{1/2}} = f(u), \qquad u({| }x{| }) \rightarrow u_0 \quad\text{as } {| }x{| }\rightarrow 0, \] which are thoroughly analyzed by using quite elementary but rather notrivial methods. Any solution \(u=u({| }x{| })\), \(0<x<r_0\), admits a unique extension to \(0<x<\infty \) and either \(u'({| }x{| })\rightarrow 0\) (the regular case) or \(u'({| }x{| })\rightarrow \pm 1\) (the singular case) as \({| }x{| }\rightarrow 0\). For any \(u_0\), there exists a unique regular solution and an infinity of singular ones. The main part of the paper is devoted to the investigation of the global behaviour under various assumptions on the function \(f\).
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53C50
zbMATH DE Number: 1153444
zbMATH Keywords: mean curvature
zbMATH Keywords: Minkowski space
author: Marie-Françoise Bidaut-Véron
author: Andrea Ratto
MaRDI profile type: Publication
Recommended article: Spacelike graphs with parallel mean curvature
Recommended article: Graphs with prescribed mean curvature on hyperbolic spaces
Recommended article: Spacelike graphs of prescribed mean curvature in the steady state space
Recommended article: Spacelike graphs with prescribed mean curvature on exterior domains in the Minkowski spacetime
Recommended article: Graphs with prescribed mean curvature in the sphere
Recommended article: Q3398878
Recommended article: Mean curvature flow of spacelike graphs
Recommended article: On a class of graphs with prescribed mean curvature
Recommended article: Graphs with Prescribed Mean Curvature on Poincaré Disk
Recommended article: Spacelike radial graphs of prescribed mean curvature in the Lorentz-Minkowski space
description: scientific article; zbMATH DE number 1153444