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Classification of spectra of the Neumann–Poincaré operator on planar domains with corners by resonance

Helsing, Johan LU ; Kang, Hyeonbae and Lim, Mikyoung (2017) In Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis 34(4). p.991-1011
Abstract

We study spectral properties of the Neumann–Poincaré operator on planar domains with corners with particular emphasis on existence of continuous spectrum and pure point spectrum. We show that the rate of resonance at continuous spectrum is different from that at eigenvalues, and then derive a method to distinguish continuous spectrum from eigenvalues. We perform computational experiments using the method to see whether continuous spectrum and pure point spectrum appear on domains with corners. For the computations we use a modification of the Nyström method which makes it possible to construct high-order convergent discretizations of the Neumann–Poincaré operator on domains with corners. The results of experiments show that all three... (More)

We study spectral properties of the Neumann–Poincaré operator on planar domains with corners with particular emphasis on existence of continuous spectrum and pure point spectrum. We show that the rate of resonance at continuous spectrum is different from that at eigenvalues, and then derive a method to distinguish continuous spectrum from eigenvalues. We perform computational experiments using the method to see whether continuous spectrum and pure point spectrum appear on domains with corners. For the computations we use a modification of the Nyström method which makes it possible to construct high-order convergent discretizations of the Neumann–Poincaré operator on domains with corners. The results of experiments show that all three possible spectra, absolutely continuous spectrum, singularly continuous spectrum, and pure point spectrum, may appear depending on domains. We also prove rigorously two properties of spectrum which are suggested by numerical experiments: symmetry of spectrum (including continuous spectrum), and existence of eigenvalues on rectangles of high aspect ratio.

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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Lipschitz domain, Neumann–Poincaré operator, RCIP method, Resonance, Spectrum
in
Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis
volume
34
issue
4
pages
21 pages
publisher
Elsevier
external identifiers
  • scopus:85011102783
  • wos:000404306900008
ISSN
0294-1449
DOI
10.1016/j.anihpc.2016.07.004
language
English
LU publication?
yes
id
6d947317-934e-492c-84bf-a4132004ec08
date added to LUP
2017-07-25 13:16:26
date last changed
2017-09-18 11:39:18
@article{6d947317-934e-492c-84bf-a4132004ec08,
  abstract     = {<p>We study spectral properties of the Neumann–Poincaré operator on planar domains with corners with particular emphasis on existence of continuous spectrum and pure point spectrum. We show that the rate of resonance at continuous spectrum is different from that at eigenvalues, and then derive a method to distinguish continuous spectrum from eigenvalues. We perform computational experiments using the method to see whether continuous spectrum and pure point spectrum appear on domains with corners. For the computations we use a modification of the Nyström method which makes it possible to construct high-order convergent discretizations of the Neumann–Poincaré operator on domains with corners. The results of experiments show that all three possible spectra, absolutely continuous spectrum, singularly continuous spectrum, and pure point spectrum, may appear depending on domains. We also prove rigorously two properties of spectrum which are suggested by numerical experiments: symmetry of spectrum (including continuous spectrum), and existence of eigenvalues on rectangles of high aspect ratio.</p>},
  author       = {Helsing, Johan and Kang, Hyeonbae and Lim, Mikyoung},
  issn         = {0294-1449},
  keyword      = {Lipschitz domain,Neumann–Poincaré operator,RCIP method,Resonance,Spectrum},
  language     = {eng},
  month        = {07},
  number       = {4},
  pages        = {991--1011},
  publisher    = {Elsevier},
  series       = {Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis},
  title        = {Classification of spectra of the Neumann–Poincaré operator on planar domains with corners by resonance},
  url          = {http://dx.doi.org/10.1016/j.anihpc.2016.07.004},
  volume       = {34},
  year         = {2017},
}