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Asymptotic expansions for eigenvalues of the Steklov problem in singularly perturbed domains


Author: S. A. Nazarov
Translated by: A. Plotkin
Original publication: Algebra i Analiz, tom 26 (2014), nomer 2.
Journal: St. Petersburg Math. J. 26 (2015), 273-318
MSC (2010): Primary 35P20
DOI: https://doi.org/10.1090/S1061-0022-2015-01339-3
Published electronically: February 3, 2015
MathSciNet review: 3242037
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Abstract: Full asymptotic expansions are constructed and justified for two series of eigenvalues and the corresponding eigenfunctions of the spectral Steklov problem in a domain with a singular boundary perturbation having the form of a small cavity. The terms of those series are of type $ \lambda _k+o(1)$ and $ \varepsilon ^{-1}(\mu _m+o(1))$, where $ \lambda _k$ and $ \mu _m$ are the eigenvalues of the Steklov problem in a bounded domain without cavity and the exterior Steklov problem for a cavity of unit size. A similar problem of the surface wave is also treated. The smoothness requirements on the boundary are discussed and unsolved problems are stated.


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Additional Information

S. A. Nazarov
Affiliation: Institute of Engineering Problems, Russian Academy of Sciences, V.O., Bol′shoĭ pr. 61, St. Petersburg 199178, Russia; Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskiĭ pr. 28, Petergof, St. Petersburg 198505, Russia
Email: srgnazarov@yahoo.co.uk

DOI: https://doi.org/10.1090/S1061-0022-2015-01339-3
Keywords: Spectral Steklov problem, singular boundary perturbation, small cavity, full asymptotic expansions of eigenvalues and eigenfunctions, surface waves
Received by editor(s): December 1, 2012
Published electronically: February 3, 2015
Additional Notes: Supported by RFBR (grant no. 12-01-00348)
Dedicated: Dedicated to Vladimir Andreevich Steklov
Article copyright: © Copyright 2015 American Mathematical Society