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Waveguide with non-periodically alternating Dirichlet and Robin conditions: homogenization and asymptotics

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Abstract

We consider a magnetic Schrödinger operator in a planar infinite strip with frequently and non-periodically alternating Dirichlet and Robin boundary conditions. Assuming that the homogenized boundary condition is the Dirichlet or the Robin one, we establish the uniform resolvent convergence in various operator norms and we prove the estimates for the rates of convergence. It is shown that these estimates can be improved by using special boundary correctors. In the case of periodic alternation, pure Laplacian, and the homogenized Robin boundary condition, we construct two-terms asymptotics for the first band functions, as well as the complete asymptotics expansion (up to an exponentially small term) for the bottom of the band spectrum.

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Author information

Authors and Affiliations

  1. Institute of Mathematics of Ufa Scientific Center of RAS, Chernyshevskogo 112, 450008, Ufa, Russia

    Denis Borisov

  2. Bashkir State Pedagogical University, October rev. st. 3a, 450000, Ufa, Russia

    Denis Borisov

  3. LMAM, UMR 7122, Université de Lorraine et CNRS, Ile du Saulcy, 57045, Metz Cedex 1, France

    Renata Bunoiu

  4. Department of Engineering, Giuseppe Cardone University of Sannio, Corso Garibaldi 107, 82100, Benevento, Italy

    Giuseppe Cardone

Authors
  1. Denis Borisov
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  2. Renata Bunoiu
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  3. Giuseppe Cardone
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Corresponding author

Correspondence to Denis Borisov.

Additional information

D. Borisov was partially supported by RFBR and Federal Task Program “Scientific and Pedagogical Staff of Innovative Russia for 2009-2013” (contract No. 02.640.11.0612).

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Borisov, D., Bunoiu, R. & Cardone, G. Waveguide with non-periodically alternating Dirichlet and Robin conditions: homogenization and asymptotics. Z. Angew. Math. Phys. 64, 439–472 (2013). https://doi.org/10.1007/s00033-012-0264-2

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  • DOI: https://doi.org/10.1007/s00033-012-0264-2

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