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 Sibirsk. Mat. Zh., 2013, Volume 54, Number 3, Pages 655–672 (Mi smj2449)

The localization for eigenfunctions of the Dirichlet problem in thin polyhedra near the vertices

S. A. Nazarovab

a Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, St. Petersburg, Russia
b St. Petersburg State University, Department of Mathematics and Mechanics, St. Petersburg, Russia

Abstract: Under some geometric assumptions, we show that eigenfunctions of the Dirichlet problem for the Laplace operator in an $n$-dimensional thin polyhedron localize near one of its vertices. We construct and justify asymptotics for the eigenvalues and eigenfunctions. For waveguides, which are thin layers between periodic polyhedral surfaces, we establish the presence of gaps and find asymptotics for their geometric characteristics.

Keywords: Dirichlet problem, asymptotics of spectrum, localization of eigenfunctions, spectral gaps.

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English version:
Siberian Mathematical Journal, 2013, 54:3, 517–532

Bibliographic databases:

Document Type: Article
UDC: 517.956.227

Citation: S. A. Nazarov, “The localization for eigenfunctions of the Dirichlet problem in thin polyhedra near the vertices”, Sibirsk. Mat. Zh., 54:3 (2013), 655–672; Siberian Math. J., 54:3 (2013), 517–532

Citation in format AMSBIB
\Bibitem{Naz13} \by S.~A.~Nazarov \paper The localization for eigenfunctions of the Dirichlet problem in thin polyhedra near the vertices \jour Sibirsk. Mat. Zh. \yr 2013 \vol 54 \issue 3 \pages 655--672 \mathnet{http://mi.mathnet.ru/smj2449} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3112622} \transl \jour Siberian Math. J. \yr 2013 \vol 54 \issue 3 \pages 517--532 \crossref{https://doi.org/10.1134/S0037446613030166} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000322243600016} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84881066808} 

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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, “Asymptotics of eigenvalues of the Dirichlet problem in a skewed $\mathcal{T}$-shaped waveguide”, Comput. Math. Math. Phys., 54:5 (2014), 793–814
2. 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”, Trans. Moscow Math. Soc., 76:1 (2015), 1–53
3. S. A. Nazarov, “Eigenmodes of a thin elastic layer between periodic rigid profiles”, Comput. Math. Math. Phys., 55:10 (2015), 1684–1697
4. S. A. Nazarov, “Localization of longitudinal and transverse oscillations in a thin curved elastic gasket”, Dokl. Phys., 60:10 (2015), 446–450
5. S. A. Nazarov, “Asymptotics of the natural oscillations of a thin elastic gasket between absolutely rigid profiles”, J. Appl. Math. Mech., 79:6 (2015), 577–586
6. S. A. Nazarov, “Discrete spectrum of cranked quantum and elastic waveguides”, Comput. Math. Math. Phys., 56:5 (2016), 864–880
7. S. A. Nazarov, E. Perez, J. Taskinen, “Localization effect for Dirichlet eigenfunctions in thin non-smooth domains”, Trans. Am. Math. Soc., 368:7 (2016), 4787–4829
8. F. L. Bakharev, S. G. Matveenko, S. A. Nazarov, “Examples of plentiful discrete spectra in infinite spatial cruciform quantum waveguides”, Z. Anal. Anwend., 36:3 (2017), 329–341
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