Spectrum of Neumann-Poincaré operator on annuli and cloaking by anomalous localized resonance for linear elasticity

Kazunori Ando, Hyeonbae Kang, Kyoungsun Kim, Sanghyeon Yu

Research output: Contribution to journalArticlepeer-review

5 Citations (Scopus)

Abstract

We investigate anomalous localized resonance on the circular coated structure and cloaking due to it in the context of elastostatic systems. The structure consists of the circular core with constant Lamé parameters and the circular shell with negative Lamé parameters proportional to those of the core. We show that there is a nonzero number k0 determined by Lamé parameters such that two nonempty eigenvalue sequences of the Neumann{Poincaré operator associated with the structure converge to k0 and -k0, respectively, and derive precise asymptotics of the convergence. We then show by qualitative estimates based on asymptotics of eigenvalues that cloaking by anomalous localized resonance takes place if and only if the dipole-type source lies inside critical radii determined by radii of the core and the shell. The critical radii corresponding to k0 and -k0 are different.

Original languageEnglish
Pages (from-to)4232-4250
Number of pages19
JournalSIAM Journal on Mathematical Analysis
Volume49
Issue number5
DOIs
Publication statusPublished - 2017
Externally publishedYes

Bibliographical note

Funding Information:
∗Received by the editors January 30, 2017; accepted for publication (in revised form) June 13, 2017; published electronically October 30, 2017. http://www.siam.org/journals/sima/49-5/M111408.html Funding: The work of the authors was supported by the A3 Foresight Program among China (NSFC), Japan (JSPS), and Korea (NRF 2014K2A2A6000567). The work of the second author was supported by NRF 2016R1A2B4011304. †Department of Electrical and Electronic Engineering and Computer Science, Ehime University, Ehime 790-8577, Japan (ando@cs.ehime-u.ac.jp). ‡Department of Mathematics, Inha University, Incheon 22212, S. Korea (hbkang@inha.ac.kr). §Department of Mathematical Sciences, Seoul National University, Seoul 08826, S. Korea (kgsunsis@snu.ac.kr). ¶Seminar for Applied Mathematics, ETH Zürich, CH-8092 Zürich, Switzerland (sanghyeon.yu@ sam.math.ethz.ch).

Publisher Copyright:
© 2017 Society for Industrial and Applied Mathematics.

Keywords

  • Annulus
  • Cloaking by anomalous localized resonance
  • Lamé system
  • Linear elasticity
  • Neumann-Poincaré operator
  • Resonance
  • Spectrum

ASJC Scopus subject areas

  • Analysis
  • Computational Mathematics
  • Applied Mathematics

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