Remark on Justification of Asymptotics of Spectra of Cylindrical Waveguides with Periodic Singular Perturbations of Boundary and Coefficients

To perform an asymptotic analysis of spectra of singularly perturbed periodic waveguides, it is required to estimate remainders of asymptotic expansions of eigenvalues of a model problem on the periodicity cell uniformly with respect to the Floquet parameter. We propose two approaches to this proble...

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Bibliographic Details
Published in:Journal of mathematical sciences (New York, N.Y.) N.Y.), 2021-09, Vol.257 (5), p.597-623
Main Authors: Gómez, D., Nazarov, S. A., Orive-Illera, R., Pérez-Martínez, M.-E.
Format: Article
Language:English
Subjects:
Analysis
Asymptotic series
Boundary value problems
Eigenvalues
Mathematical models
Mathematics
Mathematics and Statistics
Parameters
Periodic variations
Perturbation
Spectra
Waveguides
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Summary:To perform an asymptotic analysis of spectra of singularly perturbed periodic waveguides, it is required to estimate remainders of asymptotic expansions of eigenvalues of a model problem on the periodicity cell uniformly with respect to the Floquet parameter. We propose two approaches to this problem. The first is based on the max–min principle and is sufficiently easily realized, but has a restricted application area. The second is more universal, but technically complex since it is required to prove the unique solvability of the problem on the cell for some value of the spectral parameter and the Floquet parameter in a nonempty closed segment, which is verified by constructing an almost inverse operator of the operator of an inhomogeneous model problem in variational setting. We consider boundary value problems on the simplest periodicity cell: a rectangle with a row of fine holes.
ISSN:1072-3374
1573-8795
DOI:10.1007/s10958-021-05506-z