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

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 eve...

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Bibliographic Details
Published inEuropean journal of applied mathematics Vol. 29; no. 2; pp. 189 - 225
Main Authors ANDO, KAZUNORI, JI, YONG-GWAN, KANG, HYEONBAE, KIM, KYOUNGSUN, YU, SANGHYEON
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 01.04.2018
Subjects
Applied mathematics
Eigenvalues
Eigenvectors
Ellipses
linear elasticity
cloaking by anomalous localized resonance
Neumann-Poincaré operator
resonance
spectrum
Lamé system
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Summary: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.
ISSN:0956-7925
1469-4425
DOI:10.1017/S0956792517000080