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

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Abstract

© 2015 American Mathematical Society.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 λk+o(1) and ε-1(μm+o(1)), where λk and μm 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.
Original languageEnglish
Pages (from-to)273-318
JournalSt. Petersburg Mathematical Journal
Issue number2
Publication statusPublished - 2015

Cite this

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title = "Asymptotic expansions for eigenvalues of the Steklov problem in singularly perturbed domains",
abstract = "{\circledC} 2015 American Mathematical Society.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 λk+o(1) and ε-1(μm+o(1)), where λk and μm 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.",
author = "S.A. Nazarov",
year = "2015",
language = "English",
pages = "273--318",
journal = "St. Petersburg Mathematical Journal",
issn = "1061-0022",
publisher = "American Mathematical Society",
number = "2",

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T1 - Asymptotic expansions for eigenvalues of the Steklov problem in singularly perturbed domains

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N2 - © 2015 American Mathematical Society.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 λk+o(1) and ε-1(μm+o(1)), where λk and μm 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.

AB - © 2015 American Mathematical Society.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 λk+o(1) and ε-1(μm+o(1)), where λk and μm 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.

M3 - Article

SP - 273

EP - 318

JO - St. Petersburg Mathematical Journal

JF - St. Petersburg Mathematical Journal

SN - 1061-0022

IS - 2

ER -