The Essential Spectrum of the Neumann–Poincaré Operator on a Domain with Corners


Exploiting the homogeneous structure of a wedge in the complex plane, we compute the spectrum of the anti-linear Ahlfors–Beurling transform acting on the associated Bergman space. Consequently, the similarity equivalence between the Ahlfors–Beurling transform and the Neumann–Poincaré operator provides the spectrum of the latter integral operator on a wedge. A localization technique and conformal mapping lead to the first complete description of the essential spectrum of the Neumann–Poincaré operator on a planar domain with corners, with respect to the energy norm of the associated harmonic field.

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  1. 1.

    Abrahamse, M.B., Kriete, T.L.: The spectral multiplicity of a multiplication operator. Indiana Univ. Math. J. 22, 845–857 (1972/73)

  2. 2.

    Ahlfors L.V.: Remarks on the Neumann–Poincaré integral equation. Pacific J. Math. 2, 271–280 (1952)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    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(2), 667–692 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    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(1), 275–304 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Ammari H., Deng Y., Kang H., Lee H.: Reconstruction of inhomogeneous conductivities via the concept of generalized polarization tensors. Ann. Inst. H. Poincaré Anal. Non Linéaire 31(5), 877–897 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Ammari, H., Kang, H.: Reconstruction of small inhomogeneities from boundary measurements, Lecture Notes in Mathematics, Vol. 1846. Springer, Berlin, 2004

  7. 7.

    Bergman S., Schiffer M.: Kernel functions and conformal mapping. Compositio Math. 8, 205–249 (1951)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Carleman, T.: Über das Neumann–Poincarésche problem für ein gebiet mit ecken. Almquist & Wiksells, Uppsala, 1916

  9. 9.

    Dunford, N., Schwartz, J.T.: Linear operators. Part III: Spectral operators. Interscience Publishers [John Wiley & Sons, Inc.], New York, 1971. (With the assistance of William G. Bade and Robert G. Bartle, Pure and Applied Mathematics, Vol. VII)

  10. 10.

    Garcia, S.R., Prodan, E., Putinar, M.: Mathematical and physical aspects of complex symmetric operators. J. Phys. A 47(35), 353001, 54 (2014)

  11. 11.

    Helsing, J., Kang, H., Lim, M.: Classification of spectra of the Neumann–Poincaré operator on planar domains with corners by resonance. Ann. Inst. H. Poincaré Anal. Non Linéaire (2016). doi:10.1016/j.anihpc.2016.07.004

  12. 12.

    Kang, H., Lim, M., Yu, S.: Spectral resolution of the Neumann–Poincaré operator on intersecting disks and analysis of plasmon resonance. arXiv:1501.02952 [math.AP] (2015)

  13. 13.

    Kress, R.: Linear integral equations, Applied Mathematical Sciences, Vol. 82, 3rd edn. Springer, New York, 2014

  14. 14.

    Krushkal, S.: Fredholm eigenvalues of Jordan curves: geometric, variational and computational aspects, Analysis and mathematical physics, Trends Math., pp. 349–368. Birkhäuser, Basel, 2009

  15. 15.

    Maue A.-W.: Zur Formulierung eines allgemeinen Beugungsproblems durch eine Integralgleichung. Z. Physik 126, 601–618 (1949)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Mitrea I.: On the spectra of elastostatic and hydrostatic layer potentials on curvilinear polygons. J. Fourier Anal. Appl. 8(5), 443–487 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Miyanishi, Y., Suzuki, T.: Eigenvalues and eigenfunctions of double layer potentials. Trans. Am. Math. Soc. (to appear)

  18. 18.

    Nelson, E.: Topics in dynamics. I: Flows, Mathematical Notes. Princeton University Press, Princeton; University of Tokyo Press, Tokyo, 1969

  19. 19.

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

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Putinar, M., Shapiro, H.S.: The Friedrichs operator of a planar domain, Complex analysis, operators, and related topics, Oper. Theory Adv. Appl., Vol. 113, pp. 303–330. Birkhäuser, Basel, 2000

  21. 21.

    Radon, J.: Gesammelte Abhandlungen. Band 1. Verlag der Österreichischen Akademie der Wissenschaften, Vienna; Birkhäuser Verlag, Basel, 1987. (With a foreword by Otto Hittmair, Edited and with a preface by Peter Manfred Gruber, Edmund Hlawka, Wilfried Nöbauer and Leopold Schmetterer)

  22. 22.

    Tucsnak, M., Weiss, G.: Observation and control for operator semigroups, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]. Birkhäuser, Basel, 2009

  23. 23.

    Wendland, W.L.: On the double layer potential, Analysis, partial differential equations and applications, Oper. Theory Adv. Appl., Vol. 193., pp. 319–334. Birkhäuser, Basel, 2009

  24. 24.

    Werner S.: Spiegelungskoeffizient und Fredholmscher Eigenwert für gewisse Polygone. Ann. Acad. Sci. Fenn. Math. 22(1), 165–186 (1997)

    MathSciNet  Google Scholar 

  25. 25.

    Zaremba S.: Les fonctions fondamentales de M. Poincaré et la méthode de Neumann pour une frontière composée de polygones curvilignes. Journal de Mathématiques Pures et Appliquées 10, 395–444 (1904)

    MATH  Google Scholar 

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Correspondence to Karl-Mikael Perfekt.

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Communicated by F. Lin

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Perfekt, K., Putinar, M. The Essential Spectrum of the Neumann–Poincaré Operator on a Domain with Corners. Arch Rational Mech Anal 223, 1019–1033 (2017).

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