RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
General information
Latest issue
Archive
Impact factor
Subscription
Journal history

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Tr. Mosk. Mat. Obs.:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


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

This article is cited in 1 scientific paper (total in 1 paper)

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


Full text: PDF file (830 kB)
References: PDF file   HTML file

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
Received: 30.10.2012
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{http://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{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84960078236}


Linking options:
  • http://mi.mathnet.ru/eng/mmo570
  • http://mi.mathnet.ru/eng/mmo/v76/i1/p1

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    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  crossref  mathscinet  zmath  isi  scopus
  • Trudy Moskovskogo Matematicheskogo Obshchestva
    Number of views:
    This page:205
    Full text:77
    References:23

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2019