Classification of spectra of the Neumann–Poincaré operator on planar domains with corners by resonance
(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.
(Less)
- author
- Helsing, Johan LU ; Kang, Hyeonbae and Lim, Mikyoung
- organization
- publishing date
- 2017-07-01
- 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
- 2025-01-07 17:43:06
@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}},
keywords = {{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}},
doi = {{10.1016/j.anihpc.2016.07.004}},
volume = {{34}},
year = {{2017}},
}