A perturbation problem for transmission eigenvalues
type: Item
instance of: scholarly article
title: A perturbation problem for transmission eigenvalues
zbMATH Open document ID: 1491.35302
author: David M. Ambrose
author: Fioralba Cakoni
author: Shari Moskow
published in: Research in the Mathematical Sciences
publication date: 2022-01-25T00:00:00Z
review text: In this paper, the authors study the following scattering problem. Consider the Helmholtz equation for the incident wave \(\nu\) of monochromatic radiation with frequency \(\omega\) \[ \triangle\nu+k^{2}\nu=0\text{ \ in \ }\mathbb{R}^{d}, \] with \(d=2,3\), in a bounded region \(D\) with the refractive index \(n\) being a bounded real-valued function such that \(\sup\left( n-1\right) =\bar{D}\), and \(k\) is the wave number proportional to \(\omega\). The total field \(u\) is decomposed as \(u=u^{s}+\nu\), where the scattered field \(u^{s}\in H_{loc} ^{2}\left( \mathbb{R}^{d}\right) \) satisfies \[ \triangle u^{s}+k^{2}nu^{s}=-k^{2}\left( n-1\right) \nu,\text{ \ in \ }\mathbb{R}^{d}, \] with the outgoing Sommerfeld radiation condition \[ \lim_{r\rightarrow\infty}r^{\frac{d-1}{2}}\left( \frac{\partial u^{s} }{\partial r}-iku^{s}\right) =0, \] uniformly with respect to \(\hat{x}=x/\left\vert x\right\vert \), \(r=\left\vert x\right\vert \). \(k\) is a non-scattering wave number if \(u^{s}\) is zero outside \(D\). Non-scattering wave numbers are a subset of real transmission eigenvalues. From the previous results in the literature, if the media \(D\) is Lipschitz and \(n\) a bounded function, an infinite discrete set of real transmission eigenvalues is shown to exist under the assumption that either \(n-1\geq\alpha>0\), or else, \(1-n\geq\alpha>0\) a.e. in \(D\). Moreover, when \(D\) is a ball of radius \(a\) centered at the origin and \(n(r)\) is a radial function, the existence of real transmission eigenvalues is known for \(n\in C^{2}[0,a]\) under the additional assumption \(\frac{1}{a}\int_{0}^{a} \sqrt{n\left( \rho\right) }d\rho\neq1\). The aim of the present paper is to extend the above results and to provide examples of existence of real transmission eigenvalues for classes of refractive index that do not satisfy the above conditions. The authors use a perturbation method based on the implicit function theorem, meaning that the ''irregular'' refractive index is a…
Mathematics Subject Classification ID: 35P25
Mathematics Subject Classification ID: 35P05
Mathematics Subject Classification ID: 35J57
Mathematics Subject Classification ID: 35R30
zbMATH DE Number: 7463878
zbMATH Keywords: perturbation theory
zbMATH Keywords: spectral problems
zbMATH Keywords: transmission eigenvalues
zbMATH Keywords: scattering theory for inhomogeneous media
zbMATH Keywords: non-scattering waves
zbMATH Keywords: Sommerfeld radiation condition
zbMATH Keywords: implicit function theorem
reviewed by: I. P. Naumkin
MaRDI profile type: Publication
full work available at URL: s40687-021-00308-w
OpenAlex ID: W4206227776
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DOI: 10.1007/S40687-021-00308-W
description: scientific article; zbMATH DE number 7463878