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

This article is cited in 8 scientific papers (total in 8 papers)

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:

UDC: 517.956.227
Received: 25.02.2012

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
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\paper The localization for eigenfunctions of the Dirichlet problem in thin polyhedra near the vertices
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\pages 655--672
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\pages 517--532
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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  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    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  mathnet  crossref  elib
    3. S. A. Nazarov, “Eigenmodes of a thin elastic layer between periodic rigid profiles”, Comput. Math. Math. Phys., 55:10 (2015), 1684–1697  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    4. S. A. Nazarov, “Localization of longitudinal and transverse oscillations in a thin curved elastic gasket”, Dokl. Phys., 60:10 (2015), 446–450  crossref  crossref  mathscinet  isi  elib  elib  scopus
    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  crossref  mathscinet  isi  elib  elib  scopus
    6. S. A. Nazarov, “Discrete spectrum of cranked quantum and elastic waveguides”, Comput. Math. Math. Phys., 56:5 (2016), 864–880  mathnet  crossref  crossref  isi  elib
    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  crossref  mathscinet  zmath  isi  scopus
    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  crossref  mathscinet  zmath  isi  scopus
  • Сибирский математический журнал Siberian Mathematical Journal
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