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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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References

[1] Ammari, H., Ciraolo, G., Kang, H., Lee, H. & Milton, G. W. (2013) Spectral theory of a Neumann–Poincaré-type operator and analysis of cloaking due to anomalous localized resonance. Arch. Ration. Mech. An. 208, 667–692.Google Scholar
[2] Ammari, H. & Kang, H. (2007) Polarization and Moment Tensors with Applications to Inverse Problems and Effective Medium Theory, Applied Mathematical Sciences, Vol. 162, Springer-Verlag, New York.Google Scholar
[3] Ando, K. & Kang, H. (2016) Analysis of plasmon resonance on smooth domains using spectral properties of the Neumann-Poincaré operators. J. Math. Anal. Appl., 435, 162–178.Google Scholar
[4] Ando, K., Kang, H., Kim, K. & Yu, S. Cloaking by anomalous localized resonance for the Lamé system on a coated structure, arXiv:1612.08384.Google Scholar
[5] Chang, T. K. & Choe, H. J. (2007) Spectral properties of the layer potentials associated with elasticity equations and transmission problems on Lipschitz domains. J. Math. Anal. Appl. 326, 179–191.Google Scholar
[6] Dahlberg, B. E. J., Kenig, C. E. & Verchota, G. C. (1988) Boundary value problems for the systems of elastostatics in Lipschitz domains. Duke Math. J. 57 (3), 795–818.Google Scholar
[7] Dassios, G. (2012) Ellipsoidal Harmonics: Theory and Applications, Cambridge University Press, Cambridge.Google Scholar
[8] Folland, G. B. (1995) Introduction to Partial Differential Equations, 2nd ed., Princeton Univ., Princeton.Google Scholar
[9] Helsing, J., Kang, H. & Lim, M. Classification of spectra of the Neumann–Poincaré operator on planar domains with corners by resonance, Ann. I. H. Poincaré-AN, to appear, arXiv:1603.03522.Google Scholar
[10] Kang, H. (2015) Layer potential approaches to interface problems. Inverse Problems and Imaging, Vol. 44, Panoramas et Syntheses, Societe Mathematique de France, Paris.Google Scholar
[11] Kang, H., Lim, M. & Yu, S. Spectral resolution of the Neumann-Poincaré operator on intersecting disks and analysis of plamson resonance, arXiv:1501.02952.Google Scholar
[12] Khavinson, D., Putinar, M. & Shapiro, H. S. (2007) Poincaré's variational problem in potential theory. Arch. Ration. Mech. An. 185, 143–184.Google Scholar
[13] Kochmann, D. M. & Milton, G. W. (2014) Rigorous bounds on the effective moduli of composites and inhomogeneous bodies with negative-stiffness phases. J. Mech. Phys. Solids 71, 46–63.Google Scholar
[14] Kohn, R. V., Lu, J., Schweizer, B. & Weinstein, M. I. (2014) A variational perspective on cloaking by anomalous localized resonance. Comm. Math. Phys. 328, 1–27.CrossRefGoogle Scholar
[15] Kupradze, V. D. (1965) Potential Methods in the Theory of Elasticity, Daniel Davey & Co., New York.Google Scholar
[16] Lakes, R. S., Lee, T., Bersie, A. & Wang, Y. (2001) Extreme damping in composite materials with negative-stiffness inclusions. Nature 410, 565–567.Google Scholar
[17] Mayergoyz, I. D., Fredkin, D. R. & Zhang, Z. (2005) Electrostatic (plasmon) resonances in nanoparticles. Phys. Rev. B 72, 155412.Google Scholar
[18] Milton, G. W. & Nicorovici, N.-A.P. (2006) On the cloaking effects associated with anomalous localized resonance. Proc. R. Soc. A 462, 3027–3059.Google Scholar
[19] Mitrea, I. (1999) Spectral radius properties for layer potentials associated with the elastostatics and hydrostatics equations in nonsmooth domains. J. Fourier Anal. Appl. 5 (4), 385–408.Google Scholar
[20] Muskhelishvili, N. I. (1977) Some Basic Problems of the Mathematical Theory of Elasticity. English ed., Noordhoff International Publishing, Leiden, the Netherlands.Google Scholar
[21] Perfekt, K. & Putinar, M. (2014) Spectral bounds for the Neumann–Poincaré operator on planar domains with corners. J. Anal. Math. 124, 39–57.Google Scholar
[22] Perfekt, K. & Putinar, M. The essential spectrum of the Neumann–Poincaré operator on a domain with corners. Arch. Rational Mech. Anal. 223 (2017), 1019–1033.Google Scholar
[23] Verchota, G. C. (1984) Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains. J. Funct. Anal. 59, 572–611.Google Scholar