Remarks on energy inequalities for wave and Maxwell equations on a curved background
type: Item
instance of: scholarly article
title: Remarks on energy inequalities for wave and Maxwell equations on a curved background
zbMATH Open document ID: 1065.35075
author: S. Alinhac
published in: Mathematische Annalen
publication date: 2004-12-29T00:00:00Z
review text: In this note the author discusses energy inequalities for perturbed wave equations and Maxwell's equations and presents an illustration of what he calls {}``ghost weight'' method. The paper starts by presenting energy estimates for the perturbed wave equation in \(\mathbb{R}^{1+n}\), \(n\) a positive integer, in terms of derivatives tangential to the light cone utilizing Cartesian coordinates. Here a typical amplification factor occurs. In a second part a completely geometrical approach is taken in which \(\mathbb{R}^{1+n}\) is considered with an arbitrary Lorentz metric tensor and its corresponding d'Alembert equation. Due to utilizing geometric methods an energy inequality with no amplification factor is obtained. A comparable result is derived for the 2-form solution of Maxwell's equations in \(\mathbb{R}^{1+3}\) with a general Lorentz metric tensor. In the last part, a particular case for the d'Alembert equation is considered, which the author calls {}``quasiradial'' and allows a discussion in more simple geometric terms.
reviewed by: Rainer Picard
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35L05
Mathematics Subject Classification ID: 35Q60
zbMATH DE Number: 2123989
zbMATH Keywords: curved space-time
zbMATH Keywords: geometric methods
zbMATH Keywords: d'Alembert equation
MaRDI profile type: Publication
full work available at URL: s00208-004-0534-1
OpenAlex ID: W2063523617
DOI: 10.1007/S00208-004-0534-1
description: scientific article; zbMATH DE number 2123989