Dirichlet problem for space-like hypersurfaces with prescribed scalar curvature in \(\mathbb R^{n,1}\)
type: Item
instance of: scholarly article
title: Dirichlet problem for space-like hypersurfaces with prescribed scalar curvature in \(\mathbb R^{n,1}\)
zbMATH Open document ID: 1043.53027
author: Pierre Bayard
published in: Calculus of Variations and Partial Differential Equations
publication date: 2003-10-16T00:00:00Z
review text: In this paper under review, the author proves a Dirichlet problem for space-like hypersurfaces with prescribed scalar curvature in Minkowski space. Let \({\mathbb R}^{n,1}\) be the Minkowski space \(\displaystyle{{\mathbb R}^{n,1} = \bigl({\mathbb R}^{n+1}, \sum_{i=1}^n dx_i^2 - dx_{n+1}^2 \bigr)}\) with the canonical coordinates \((x_1, \dots, x_{n+1})\). Let \(\Omega\) be a smooth bounded domain in \({\mathbb R}^{n}= \{x_{n+1} = 0\}\) and let \(u\) be a smooth function defined on \(\overline{\Omega}\) satisfying \(\displaystyle{\sup_{\overline{\Omega}} | Du| < 1}\). Such a function \(u\) is called a spacelike function. Let \(M\) be the graph of the function \(u\), i.e., \(M=\{(x, u(x)): x \in \overline{\Omega}\}\) which is a spacelike hypersurface of \({\mathbb R}^{n,1}\). Setting \(\hat e_i= e_i + u_i e_{n+1}\), where \(e_i\) denotes the canonical basis of \({\mathbb R}^{n+1}\) and \(u_i\) denotes the partial derivative, the matrix, \(A(Du)\), of the induced scalar product with respect to the basis \(\{\hat e_i\}\) is given by \(A(Du)_{ij} = \delta_{ij} - u_iu_j\) and the inverse \(\displaystyle{A^{-1}(Du)_{ij} = \delta_{ij} + \frac{u_iu_j}{\omega^2}}\), where \(\omega = \sqrt{1-| Du| ^2}\). Then the matrix of the curvature endomorphism of \(M\) in the basis \(\{\hat e_i\}\) is \(\displaystyle{\frac{1}{\omega} A^{-1}(Du) D^2u}\) and the \(m\)-th order curvature of \(M\) at each point \(p\in M\) is defined by \[ {\mathcal H}_m (p) = \frac{1}{_nC_m} \sigma_m(\lambda_1, \dots, \lambda_n), \] where \(\lambda_1, \dots, \lambda_n\) are the principal curvatures of \(M\) at the point \(p\) and \(\sigma_m\) the \(m\)-th elementary symmetric function. Thus the first order curvature is just the mean curvature and the second order curvature is the scalar curvature of \(M\). One can thus define the \(m\)-th order curvature operator acting on the spacelike function over \(\Omega\) by \[ {\mathcal H}_m[u] = \frac{1}{_nC_m \omega^m} \sum_{I, J} A^{IJ}(Du) (D^2u)_{IJ}, \] where t…
reviewed by: Gabjin Yun
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 53C50
zbMATH DE Number: 1993489
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: Mikowski space
zbMATH Keywords: scalar curvature
zbMATH Keywords: spacelike hypersurface
MaRDI profile type: Publication
full work available at URL: s00526-002-0178-5
OpenAlex ID: W2050566971
DOI: 10.1007/S00526-002-0178-5
Recommended article: Dirichlet problem for the scalar curvature in \(\mathbb{R}^{3,1}\)
Recommended article: The Dirichlet problem for the equation of prescribed scalar curvature in Minkowski space
Recommended article: The Dirichlet problem for the prescribed curvature equations
Recommended article: The Dirichlet problem for a class of prescribed curvature equations
Recommended article: The Dirichlet problem for the equation of prescribed mean curvature
Recommended article: The Dirichlet problem for a prescribed mean curvature equation
Recommended article: The Dirichlet problem for prescribed principal curvature equations
Recommended article: The Dirichlet problem for curvature equations in Riemannian manifolds
Recommended article: The Dirichlet problem for the prescribed curvature quotient equations
Recommended article: On the Dirichlet problem for the prescribed mean curvature equation over general domains
description: scientific article; zbMATH DE number 1993489