Remarks on energy inequalities for wave and Maxwell equations on a curved background
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.
scientific article; zbMATH DE number 2123989