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Spectral Resolution of the Neumann–Poincaré Operator on Intersecting Disks and Analysis of Plasmon Resonance

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Abstract

The purpose of this paper is to investigate the spectral nature of the Neumann–Poincaré operator on the intersecting disks, which is a domain with the Lipschitz boundary. The complete spectral resolution of the operator is derived, which shows, in particular, that it admits only the absolutely continuous spectrum; no singularly continuous spectrum and no pure point spectrum. We then quantitatively analyze using the spectral resolution of the plasmon resonance at the absolutely continuous spectrum.

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References

  1. Ammari H., Ciraolo G., Kang H., Lee H., Milton G.W.: Spectral theory of a Neumann–Poincaré-type operator and analysis of cloaking due to anomalous localized resonance. Arch. Ration. Mech. Anal. 208, 667–692 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ammari H., Ciraolo G., Kang H., Lee H., Yun K.: Spectral analysis of the Neumann–Poincaré operator and characterization of the stress concentration in anti-plane elasticity. Arch. Ration. Mech. Anal. 208, 275–304 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ammari, H., Kang, H.: Polarization and moment tensors. Applied Mathematical Sciences, vol. 162, Springer, New York (2007)

  4. Ando, K., Kang, H.: Analysis of plasmonic resonance on smooth domains using spectral properties of the Neumann–Poincaré operator. J. Math. Anal. Appl. 435, 162–178, 2016.

  5. Bonnetier E., Triki F.: Pointwise bounds on the gradient and the spectrum of the Neumann–Poincaré operator: the case of 2 discs. Contemp. Math. 577, 79–90 (2012)

    MATH  Google Scholar 

  6. Bonnetier E., Triki F.: On the spectrum of Poincaré variational problem for two close-to-touching inclusions in 2D. Arch. Ration. Mech. Anal. 209, 541–567 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Carleman, T.: Über das Neumann–Poincarésche Problem für ein Gebiet mit Ecken. Almquist and Wiksells, Uppsala, 1916

  8. Folland, G.B.: Introduction to Partial Differential Equations, 2nd Ed. Princeton University Press, Princeton, 1995

  9. Grieser D.: Plasmonic eigenvalue problem. Rev. Math. Phys. 26, 1450005 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Helsing, J., Kang, H., Lim, M.: Classification of spectra of the Neumann–Poincaré operator on planar domains with corners by resonance. Ann. I. H. Poincare-AN, to appear

  11. Kang, H.: Layer potential approaches to interface problems. In: Inverse Problems and Imaging. Panoramas et Syntheses 44, Societe Mathematique de France, Paris (2015)

  12. Kang, H., Kim, K., Lee, H., Shin, J., Yu, S.: Spectral properties of the Neumann–Poincaré operator and uniformity of estimates for the conductivity equation with complex coefficients. J. London Math. Soc. 93, 519–546, 2016

  13. Kang, H., Seo, J-K.: Recent progress in the inverse conductivity problem with single measurement. In: Nakamura, G., Saitoh, S., Seo, JK., Yamamota, M. (eds.) Inverse Problems and Related Fields, pp. 69–80. CRC Press, Boca Raton (2000)

  14. Kellogg, O.D.: Foundations of Potential Theory. Dover, New York, 1953

  15. Kenig, C.: Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems. CBMS Series 83, American Mathematics Society, Providence, 1991

  16. Khavinson D., Putinar M., Shapiro H.S.: Poincaré’s variational problem in potential theory. Arch. Ration. Mech. Anal. 185, 143–184 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kohn R.V., Lu J., Schweizer B., Weinstein M.I.: A variational perspective on cloaking by anomalous localized resonance. Commun. Math. Phys. 328, 1–27 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. Lei D.Y., Aubry A., Luo Y., Maier S.A., Pendry J.B.: Plasmonic interaction between overlapping nanowires. ACS Nano 5(1), 597–607 (2011)

    Article  Google Scholar 

  19. Lim M.: Symmetry of a boundary integral operator and a characterization of a ball. Ill. J. Math. 45, 537–543 (2001)

    MathSciNet  MATH  Google Scholar 

  20. Lim M., Yu S.: Asymptotics of the solution to the conductivity equation in the presence of adjacent circular inclusions with finite conductivities. J. Math. Anal. Appl. 421, 131–156 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Neumann, C.: Über die Methode des arithmetischen Mittels, Erste and zweite Abhandlung. Leipzig 1887/88, in Abh. d. Kgl. Sächs Ges. d. Wiss., IX and XIII.

  22. Nguyen H-M., Nguyen H.L.: Complete resonance and localized resonance in plasmonic structures. ESAIM Math. Model Numer. Anal. 49, 741–754 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  23. Perfekt K.-M., Putinar M.: Spectral bounds for the Neumann–Poincaré operator on planar domains with corners. J. Anal. Math. 124, 39–57 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  24. Perfekt K., Putinar M.: The essential spectrum of the Neumann–Poincaré operator on a domain with corners. Arch. Ration. Mech. Anal. 223, 1019–1033 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  25. Poincaré H.: La méthode de Neumann et le problème de Dirichlet. Acta Math. 20, 59–152 (1897)

    Article  MathSciNet  MATH  Google Scholar 

  26. Verchota G.C.: Layer potentials and boundary value problems for Laplace’s equation in Lipschitz domains. J. Funct. Anal. 59, 572–611 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  27. Yosida, K.: Functional Analysis, 4th Ed. Springer, Berlin, 1974

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Author notes

  1. Sanghyeon Yu

    Present address: Seminar for Applied Mathematics, ETH Zürich, Rämistrasse 101, 8092, Zürich, Switzerland

Authors and Affiliations

  1. Department of Mathematics, Inha University, Incheon, 22212, South Korea

    Hyeonbae Kang

  2. Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, Daejeon, 34141, South Korea

    Mikyoung Lim & Sanghyeon Yu

Authors
  1. Hyeonbae Kang
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  2. Mikyoung Lim
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  3. Sanghyeon Yu
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Correspondence to Hyeonbae Kang.

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Communicated by F. Otto

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Kang, H., Lim, M. & Yu, S. Spectral Resolution of the Neumann–Poincaré Operator on Intersecting Disks and Analysis of Plasmon Resonance. Arch Rational Mech Anal 226, 83–115 (2017). https://doi.org/10.1007/s00205-017-1129-9

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  • DOI: https://doi.org/10.1007/s00205-017-1129-9

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