Transmission eigenvalues and a problem of Hans Lewy
type: Item
instance of: scholarly article
title: Transmission eigenvalues and a problem of Hans Lewy
zbMATH Open document ID: 0957.65093
DOI: 10.1016/S0377-0427(99)00334-9
author: David Colton
author: Lassi Päivärinta
published in: Journal of Computational and Applied Mathematics
publication date: 2000-07-09T00:00:00Z
review text: Under certain assumptions on the matrix \(N= N(x)\), the authors consider the transmission eigenvalue problem \[ \nabla\cdot N\nabla u+ k^2u= 0\quad\text{and}\quad \Delta v+ k^2v= 0\quad\text{in}\quad D \] \[ \text{together with}\quad u= v\quad\text{and}\quad \partial u/\partial\nu= \partial v/\partial\nu\quad\text{on}\quad \partial D. \] Here, \(D\subset\mathbb{R}^n\) denotes a bounded and simply connected region with smooth boundary \(\partial D\) and unit normal \(\nu\). A main difficulty is due to the fact that \(N(x)- I\) vanishes at \(\partial D\). Establishing in particular an associated weigthed Poincaré inequality, it is shown that the problem exhibits at most a countable set of transmission eigenvalues \(k\geq 0\). The result has important applications to the inverse scattering problem for anisotropic media.
reviewed by: Waldemar Velte
Mathematics Subject Classification ID: 65N25
Mathematics Subject Classification ID: 35P15
Mathematics Subject Classification ID: 35P25
Mathematics Subject Classification ID: 35R30
zbMATH DE Number: 1472300
zbMATH Keywords: transmission eigenvalue problem
zbMATH Keywords: Poincaré inequality
zbMATH Keywords: inverse scattering
MaRDI profile type: Publication
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full work available at URL: s0377-0427(99)00334-9
OpenAlex ID: W1988116167
description: scientific article; zbMATH DE number 1472300