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Spectral gaps in the dirichlet and neumann problems on the plane perforated by a doubleperiodic family of circular holes

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We show that the spectrum of the Dirichlet and Neumann problems for the Laplace operator in the plane perforated by a double–periodic family of circular holes contains gaps (even any a priori given number of gaps) of certain radii of holes. The result is obtained by asymptotic analysis of the cell spectral problem, interpreted as a problem in a domain with thin bridges. Some open questions are stated.

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Authors and Affiliations

  1. Institute for Problems in Mechanical Engineering RAS, 61, Bolshoi pr. V.O., St. Petersburg, 199178, Russia

    S. A. Nazarov

  2. University of Oulu, P.O. Box 4500, 90014, Oulu, Finland

    K. Ruotsalainen

  3. University of Helsinki, P.O.Box 68 FIN-00014, Helsinki, Finland

    J. Taskinen

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  1. S. A. Nazarov
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Correspondence to S. A. Nazarov.

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Translated from Problems in Mathematical Analysis 62, December 2011, pp. 51–100

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Nazarov, S.A., Ruotsalainen, K. & Taskinen, J. Spectral gaps in the dirichlet and neumann problems on the plane perforated by a doubleperiodic family of circular holes. J Math Sci 181, 164–222 (2012). https://doi.org/10.1007/s10958-012-0681-y

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