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

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 vario...

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Published inZeitschrift für angewandte Mathematik und Physik Vol. 64; no. 3; pp. 439 - 472
Main Authors Borisov, Denis, Bunoiu, Renata, Cardone, Giuseppe
Format Journal Article
LanguageEnglish
Published Basel SP Birkhäuser Verlag Basel 01.06.2013
Springer Verlag
Subjects
Asymptotic properties
Engineering
Mathematical Methods in Physics
Mathematics
Theoretical and Applied Mechanics
35B27
Uniform resolvent convergence
Band spectrum
35P05
Alternating boundary conditions
35J15
Asymptotics
Waveguide
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ISSN0044-2275
1420-9039
DOI10.1007/s00033-012-0264-2

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Summary: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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ISSN:0044-2275
1420-9039
DOI:10.1007/s00033-012-0264-2