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Integral Equations and Operator Theory

, Volume 88, Issue 3, pp 373–386 | Cite as

Effects of Rayleigh Waves on the Essential Spectrum in Perturbed Doubly Periodic Elliptic Problems

  • F. L. Bakharev
  • G. CardoneEmail author
  • S. A. Nazarov
  • J. Taskinen
Article

Abstract

We give an example of a scalar second order differential operator in the plane with double periodic coefficients and describe its modification, which causes an additional spectral band in the essential spectrum. The modified operator is obtained by applying to the coefficients a mirror reflection with respect to a vertical or horizontal line. This change gives rise to Rayleigh type waves localized near the line. The results are proven using asymptotic analysis, and they are based on high contrast of the coefficient functions.

Keywords

Periodic media Open waveguides High contrast of coefficients Asymptotics Spectral bands 

Mathematics Subject Classification

Primary 35P05 Secondary 47A75 

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Notes

Acknowledgements

F.B. was supported by Grants 0.38.237.2014 and 6.57.61.2016 of St.Petersburg University, and by Grant 15-01-02175 of RFBR; S.N. was supported by Grant 15-01-02175 of RFBR and by the Academy of Finland Project 268973; G.C. is a member of GNAMPA of INDAM; J.T. was partially supported by the Väisälä Foundation of the Finnish Academy of Sciences and Letters.

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Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  1. 1.Mathematics and Mechanics FacultySt. Petersburg State UniversitySt. PetersburgRussia
  2. 2.Department of EngineeringUniversità del SannioBeneventoItaly
  3. 3.St. Petersburg State UniversityStary PeterhofRussia
  4. 4.Peter the Great St. Petersburg State Polytechnical UniversitySt. PetersburgRussia
  5. 5.Institute of Problems of Mechanical Engineering RASSt. PetersburgRussia
  6. 6.Department of Mathematics and StatisticsUniversity of HelsinkiHelsinkiFinland

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