A nonlinear inequality arising in geometry and Calabi-Bernstein type problems
type: Item
instance of: scholarly article
title: A nonlinear inequality arising in geometry and Calabi-Bernstein type problems
zbMATH Open document ID: 1223.26035
DOI: 10.1155/2010/950380
author: Alfonso Romero
author: Rafael Rubio
published in: Journal of Inequalities and Applications
publication date: 2010-12-06T00:00:00Z
full work available at URL: https://eudml.org/doc/224034
review text: The authors study the following nonlinear differential inequality \[ H(u)^ 2 \leq \frac{f'(u)^ 2}{f(u)^ 2}, \] under the assumption \[ | Du| < \lambda f(u) \quad 0<\lambda <1, \] where \(f\) is a positive smooth function, \(u \in C^{\infty}(\Omega)\), \(| Du| \) is the length of the gradient \(Du\) of \(u\) and \[ H(u)=-\text{div}\Big(\frac{Du}{2f(u)\sqrt{f(u)^ 2 -| Du|^ 2}}\Big)-\frac{f'(u)}{2\sqrt{f(u)^ 2 -| Du|^ 2}}\Big(2+\frac{| Du|^ 2}{f(u)^ 2}\Big). \] A characterization of the entire solutions of the above nonlinear inequality is presented, which has a natural interpretation in terms of certain nonflat Robertson-Walker spacetimes. Furthermore, new Calabi-Bernstein-type problems are raised and solved.
reviewed by: James A. Oguntuase
Mathematics Subject Classification ID: 26D07
zbMATH DE Number: 5823208
zbMATH Keywords: Calabi-Bernstein-type problems
zbMATH Keywords: warping function
zbMATH Keywords: Riemannian manifold
Wikidata QID: Q59253173
MaRDI profile type: Publication
OpenAlex ID: W2125783917
cites work: Q3312869
cites work: Entire spacelike hypersurfaces of constant mean curvature in Minkowski space
cites work: Q4820357
cites work: Q3029855
cites work: On the mean curvature of spacelike surfaces in certain three-dimensional Robertson-Walker spacetimes and Calabi-Bernstein's type problems
cites work: Uniqueness of noncompact spacelike hypersurfaces of constant mean curvature in generalized Robertson-Walker spacetimes
cites work: New examples of Calabi-Bernstein problems for some nonlinear equations.
cites work: General Relativity for Mathematicians
cites work: Causally symmetric spacetimes
cites work: Uniqueness of complete spacelike hypersurfaces of constant mean curvature in generalized Robertson-Walker spacetimes
cites work: Q4522320
cites work: New proof of the Calabi-Bernstein theorem
cites work: Zero mean curvature surfaces with non–negative curvature in flat Lorentzian 4–spaces
cites work: Constant mean curvature spacelike surfaces in three-dimensional generalized Robertson-Walker spacetimes
description: scientific article; zbMATH DE number 5823208