Distribution of resonances and local energy decay in the transmission problem. II
type: Item
instance of: scholarly article
title: Distribution of resonances and local energy decay in the transmission problem. II
zbMATH Open document ID: 0968.35035
author: Fernando Cardoso
author: Georgi Vodev
author: Georgi Popov
published in: Mathematical Research Letters
publication date: 2000-08-13T00:00:00Z
review text: For part I see \textit{G. Popov} and \textit{G. Vodev} [Asymptotic Anal. 19, No. 3-4, 253-265 (1999; Zbl 0931.35115)]. From the introduction: This paper is concerned with the resonances of the transmission problem for a transparent bounded strictly convex obstacle \({\mathcal O}\) with a smooth boundary (which may contain an impenetrable body). If the speed of propagation inside \({\mathcal O}\) is bigger than that outside \({\mathcal O}\), we prove under some natural conditions, that there exists a strip in the upper half plane containing the real axis, which is free of resonances. We also obtain an uniform decay of the local energy for the corresponding mixed problem with an exponential rate of decay when the dimension is odd, and polynomial otherwise. It is well known that such a decay of the local energy holds for the wave equation with Dirichlet (Neumann) boundary conditions for any nontrapping obstacle. In our case, however, \({\mathcal O}\) is a trapping obstacle for the corresponding classical system.
Mathematics Subject Classification ID: 35J15
Mathematics Subject Classification ID: 35B34
Mathematics Subject Classification ID: 35P20
zbMATH DE Number: 1489509
zbMATH Keywords: a priori estimates
zbMATH Keywords: cutoff resolvent
zbMATH Keywords: uniform decay
zbMATH Keywords: exponential rate of decay
zbMATH Keywords: trapping obstacle
MaRDI profile type: Publication
full work available at URL: mrl.1999.v6.n4.a2
OpenAlex ID: W2030473241
DOI: 10.4310/MRL.1999.V6.N4.A2
description: scientific article; zbMATH DE number 1489509