Skip to main content
×
Home
    • Aa
    • Aa

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

  • KAZUNORI ANDO (a1), YONG-GWAN JI (a2), HYEONBAE KANG (a2), KYOUNGSUN KIM (a3) and SANGHYEON YU (a4)...
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.

Copyright
Footnotes
Hide All

This work is supported by the Korean Ministry of Education, Sciences and Technology through NRF grants Nos. 2010-0017532 (to H.K) and 2012003224 (to S.Y).

Footnotes
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[3] K. Ando & H. Kang (2016) Analysis of plasmon resonance on smooth domains using spectral properties of the Neumann-Poincaré operators. J. Math. Anal. Appl., 435, 162178.

[6] B. E. J. Dahlberg , C. E. Kenig & G. C. Verchota (1988) Boundary value problems for the systems of elastostatics in Lipschitz domains. Duke Math. J. 57 (3), 795818.

[7] G. Dassios (2012) Ellipsoidal Harmonics: Theory and Applications, Cambridge University Press, Cambridge.

[14] R. V. Kohn , J. Lu , B. Schweizer & M. I. Weinstein (2014) A variational perspective on cloaking by anomalous localized resonance. Comm. Math. Phys. 328, 127.

[19] I. Mitrea (1999) Spectral radius properties for layer potentials associated with the elastostatics and hydrostatics equations in nonsmooth domains. J. Fourier Anal. Appl. 5 (4), 385408.

[22] K. Perfekt & M. Putinar The essential spectrum of the Neumann–Poincaré operator on a domain with corners. Arch. Rational Mech. Anal. 223 (2017), 10191033.

[23] G. C. Verchota (1984) Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains. J. Funct. Anal. 59, 572611.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

European Journal of Applied Mathematics
  • ISSN: 0956-7925
  • EISSN: 1469-4425
  • URL: /core/journals/european-journal-of-applied-mathematics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 10 *
Loading metrics...

Abstract views

Total abstract views: 99 *
Loading metrics...

* Views captured on Cambridge Core between 12th April 2017 - 27th July 2017. This data will be updated every 24 hours.