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Asymptotics of the eigenvalues of boundary value problems for the Laplace operator in a three-dimensional domain with a thin closed tube


Author: S. A. Nazarov
Translated by: V. E. Nazaikinskii
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 76 (2015), vypusk 1.
Journal: Trans. Moscow Math. Soc. 2015, 1-53
MSC (2010): Primary 35J25; Secondary 35B25, 35B40, 35B45, 35P20, 35S05
DOI: https://doi.org/10.1090/mosc/243
Published electronically: November 17, 2015
MathSciNet review: 3467259
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Abstract: We construct and justify asymptotic representations for the eigenvalues and eigenfunctions of boundary value problems for the Laplace operator in a three-dimensional domain $ \Omega (\varepsilon )=\Omega \setminus \Bar {\Gamma }_\varepsilon $ with a thin singular set $ \Gamma _\varepsilon $ lying in the  $ c\varepsilon $-neighborhood of a simple smooth closed contour $ \Gamma $. We consider the Dirichlet problem, a mixed boundary value problem with the Neumann conditions on  $ \partial \Gamma _\varepsilon $, and also a spectral problem with lumped masses on  $ \Gamma _\varepsilon $. The asymptotic representations are of diverse character: we find an asymptotic series in powers of the parameter $ \vert{\ln \varepsilon }\vert^{-1}$ or  $ \varepsilon $. The most comprehensive and complicated analysis is presented for the lumped mass problem; namely, we sum the series in powers of $ \vert{\ln \varepsilon }\vert^{-1}$ and obtain an asymptotic expansion with the leading term holomorphically depending on  $ \vert{\ln \varepsilon }\vert^{-1}$ and with the remainder $ O(\varepsilon ^\delta )$, $ \delta \in (0,1)$. The main role in asymptotic formulas is played by solutions of the Dirichlet problem in $ \Omega \setminus \Gamma $ with logarithmic singularities distributed along the contour $ \Gamma $.


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Additional Information

S. A. Nazarov
Affiliation: Laboratory of Nanomanufacturing, Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russia; Mathematics and Mechanics Faculty, St. Petersburg State University, St. Petersburg, Russia; Laboratory of Mathematical Methods in Mechanics of Materials, Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, St. Petersburg, Russia
Email: srgnazarov@yahoo.co.uk

DOI: https://doi.org/10.1090/mosc/243
Keywords: Eigenvalue and eigenfunction asymptotics, convergence theorem, singular perturbation of a domain, thin toroidal cavity, Dirichlet and Neumann problems, lumped mass.
Published electronically: November 17, 2015
Additional Notes: Supported by grant no. 6.37.671.2013 from St. Petersburg State University.
Article copyright: © Copyright 2015 S. A. Nazarov

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