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 Tr. Mosk. Mat. Obs., 2015, Volume 76, Issue 1, Pages 1–66 (Mi mmo570)

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

S. A. Nazarovabc

a Laboratory of Mathematical Methods in Mechanics of Materials, Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, St. Petersburg, Russia
b Laboratory of Nanomanufacturing, Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russia
c Mathematics and Mechanics Faculty, St. Petersburg State University, St. Petersburg, Russia

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 $\Omega (\varepsilon )=\Omega \setminus \overline {\Gamma }_\varepsilon$ 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 $\Gamma$.

Key words and phrases: Eigenvalue and eigenfunction asymptotics, convergence theorem, singular perturbation of a domain, thin toroidal cavity, Dirichlet and Neumann problems, lumped mass.

 Funding Agency Grant Number Saint Petersburg State University 6.37.671.2013

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English version:
Transactions of the Moscow Mathematical Society, 2015, 76:1, 1–53

UDC: 517.957:517.956.227
MSC: 35J25, 35B25, 35B40, 35B45, 35P20, 35S05
Revised: 02.06.2014

Citation: S. A. Nazarov, “Asymptotics of the eigenvalues of boundary value problems for the Laplace operator in a three-dimensional domain with a thin closed tube”, Tr. Mosk. Mat. Obs., 76, no. 1, MCCME, M., 2015, 1–66; Trans. Moscow Math. Soc., 76:1 (2015), 1–53

Citation in format AMSBIB
\Bibitem{Naz15} \by S.~A.~Nazarov \paper Asymptotics of the eigenvalues of boundary value problems for the Laplace operator in a three-dimensional domain with a thin closed tube \serial Tr. Mosk. Mat. Obs. \yr 2015 \vol 76 \issue 1 \pages 1--66 \publ MCCME \publaddr M. \mathnet{http://mi.mathnet.ru/mmo570} \elib{https://elibrary.ru/item.asp?id=24850128} \transl \jour Trans. Moscow Math. Soc. \yr 2015 \vol 76 \issue 1 \pages 1--53 \crossref{https://doi.org/10.1090/mosc/243} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84960078236} 

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This publication is cited in the following articles:
1. S. A. Nazarov, M. Eugenia Perez, “On multi-scale asymptotic structure of eigenfunctions in a boundary value problem with concentrated masses near the boundary”, Rev. Mat. Complut., 31:1 (2018), 1–62
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