Asymptotic behavior at infinity of the Green function of a class of systems including wave propagation in crystals
type: Item
instance of: scholarly article
title: Asymptotic behavior at infinity of the Green function of a class of systems including wave propagation in crystals
zbMATH Open document ID: 0587.35058
author: Koji Kikuchi
published in: Osaka Journal of Mathematics
publication date: 1985-01-01T00:00:00Z
review text: The first order symmetric hyperbolic system can be written in the form \[ (1)\quad E(x)D_ tu-\sum A_ jD_ ju=\tilde f(t,x), \] where \(iD_ t=\partial /\partial t\), \(iD_ j=\partial /\partial x_ j\), E(x) is a positive definite Hermitian matrix, \(A_ j\) are \(m\times m\) Hermitian matrices. Multiplying (1) by \(E^{-1}\) reduces it to the form \(D_ tu- \Lambda^ 0u=f(t,x).\) If f(x,t) is given by \[ -f(t,x)=e^{i\lambda t}f(x),\quad \lambda \in {\mathbb{R}}\setminus \{0\}, \] then the solution u(t,x) also must be of the form \(u(x,t)=e^{i\lambda t}v(x,\lambda),\) and v(x,\(\lambda)\) solves the eigenvalue problem \((2)\quad \Lambda^ 0v- \lambda v=f(x).\) The Green function G(x,\(\xi)\) is defined by the inverse Fourier transform \(G(x,\xi)={\mathcal F}^{-1}[(\Lambda^ 0-\xi I)^{- 1}],\) but it is useful to think of it as the elementary solution of (2), i.e. \((\Lambda^ 0-\lambda I)G(x,\lambda)=\delta (x)\cdot I.\) The author develops representation of Green's function in terms of the slowness integral in the style of C. Wilcox and J. R. Schulenberger. (The slowness surface is the polar inverse with respect to the unit sphere of the wave surface.) The author proceeds to develop a quite impressive list of results concerning estimates for Green's function, properties of the slowness surface, which for electromagnetic waves in crystals is called the Fresnel surface, existence of stationary points, and asymptotic properties of the Green function. Some of the results are quite lengthy and technically difficult because of consideration of special cases, odd or even spatial dimensions and a selection of different assumptions concerning the conditions on the slowness surface S, such as: \((S_ i)\) stating that the set of all algebraic singularities forms an (n-d) smooth manifold, with \(d\geq (n+1)/2\). Small misprints mar the presentation. On page 1 the Hermitian matrices are not constant, \(\Delta\) should be \(\Lambda\), etc., but these are minor points hardly detracting from…
Mathematics Subject Classification ID: 35L10
Mathematics Subject Classification ID: 35A22
Mathematics Subject Classification ID: 35C15
Mathematics Subject Classification ID: 42B10
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 3942068
zbMATH Keywords: symmetric hyperbolic system
zbMATH Keywords: Green function
zbMATH Keywords: Fourier transform
zbMATH Keywords: slowness integral
zbMATH Keywords: Fresnel surface
zbMATH Keywords: asymptotic properties
reviewed by: Vadim Komkov
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3942068