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Spectral properties of the Neumann–Poincaré operator and cloaking by anomalous localized resonance for the elasto-static system

Published online by Cambridge University Press:  12 April 2017

KAZUNORI ANDO ,
YONG-GWAN JI ,
HYEONBAE KANG ,
KYOUNGSUN KIM  and
SANGHYEON YU
KAZUNORI ANDO
Affiliation:
Department of Electrical and Electronic Engineering and Computer Science, Ehime University, Ehime 790-8577, Japan email: ando@cs.ehime-u.ac.jp
YONG-GWAN JI
Affiliation:
Department of Mathematics, Inha University, Incheon 22212, South Korea emails: 22151063@inha.edu, hbkang@inha.ac.kr
HYEONBAE KANG
Affiliation:
Department of Mathematics, Inha University, Incheon 22212, South Korea emails: 22151063@inha.edu, hbkang@inha.ac.kr
KYOUNGSUN KIM
Affiliation:
Department of Mathematical Sciences, Seoul National University, Seoul 08826, South Korea email: kqsunsis@snu.ac.kr
SANGHYEON YU
Affiliation:
Seminar for Applied Mathematics, ETH Zürich, Rämistrasse 101, CH-8092 Zürich, Switzerland email: sanghyeon.yu@sam.math.ethz.ch
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Abstract

We first investigate spectral properties of the Neumann–Poincaré (NP) operator for the Lamé system of elasto-statics. We show that the elasto-static NP operator can be symmetrized in the same way as that for the Laplace operator. We then show that even if elasto-static NP operator is not compact even on smooth domains, it is polynomially compact and its spectrum on two-dimensional smooth domains consists of eigenvalues that accumulate to two different points determined by the Lamé constants. We then derive explicitly eigenvalues and eigenfunctions on discs and ellipses. Using these resonances occurring at eigenvalues is considered. We also show on ellipses that cloaking by anomalous localized resonance takes place at accumulation points of eigenvalues.

Keywords

Neumann-Poincaré operatorLamé systemlinear elasticityspectrumresonancecloaking by anomalous localized resonance
Type
Papers
Information
European Journal of Applied Mathematics , Volume 29 , Issue 2 , April 2018 , pp. 189 - 225
Copyright
Copyright © Cambridge University Press 2017 

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