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Mat. Zametki, 2009, Volume 86, Issue 4, Pages 571–587 (Mi mz4157)  

This article is cited in 6 scientific papers (total in 6 papers)

Asymptotics of the Solution of the Steklov Spectral Problem in a Domain with a Blunted Peak

S. A. Nazarov

Institute of Problems of Mechanical Engineering, Russian Academy of Sciences

Abstract: We construct and justify the asymptotics of the eigenvalues and eigenfunctions of the Laplace equation with Steklov boundary conditions in a domain with an acute peak whose end of size $O(\varepsilon)$ is broken off. In particular, we establish that any positive eigenvalue with a fixed number turns out to be infinitesimal as $\varepsilon\to+0$ and the corresponding eigenfunction is localized in the $c\varepsilon$-neighborhood of the vertex of the peak.

Keywords: Steklov spectral problem, Laplace operator, domain with a blunted peak, Sobolev space, elliptic boundary-value problem, Neumann problem, Hardy inequality, Poincaré inequality, Green's formula, Hilbert space

DOI: https://doi.org/10.4213/mzm4157

Full text: PDF file (590 kB)
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English version:
Mathematical Notes, 2009, 86:4, 542–555

Bibliographic databases:

UDC: 517.956.8:517.956.227
Received: 14.03.2007

Citation: S. A. Nazarov, “Asymptotics of the Solution of the Steklov Spectral Problem in a Domain with a Blunted Peak”, Mat. Zametki, 86:4 (2009), 571–587; Math. Notes, 86:4 (2009), 542–555

Citation in format AMSBIB
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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. F. L. Bakharev, S. A. Nazarov, “On the structure of the spectrum for the elasticity problem in a body with a supersharp spike”, Siberian Math. J., 50:4 (2009), 587–595  mathnet  crossref  mathscinet  isi  elib  elib
    2. Nazarov S.A., Ruotsalainen K., Taskinen J., “Essential spectrum of a periodic elastic waveguide may contain arbitrarily many gaps”, Appl. Anal., 89:1 (2010), 109–124  crossref  mathscinet  zmath  isi  elib  scopus
    3. Gryshchuk S., de Cristoforis M.L., “Singular perturbation of simple Steklov eigenvalues”, Numerical Analysis and Applied Mathematics (ICNAAM 2012), v. A, B, AIP Conf. Proc., 1479, eds. Simos T., Psihoyios G., Tsitouras C., Anastassi Z., Amer Inst Physics, 2012, 700–703  crossref  mathscinet  adsnasa  isi  scopus
    4. S. Gryshchuk, M. L. de Cristoforis, “Simple eigenvalues for the Steklov problem in a domain with a small hole. A functional analytic approach”, Math. Methods Appl. Sci., 37:12 (2014), 1755–1771  crossref  mathscinet  zmath  isi  scopus
    5. Bakharev F.L., Taskinen J., “Bands in the Spectrum of a Periodic Elastic Waveguide”, Z. Angew. Math. Phys., 68:5 (2017), 102  crossref  mathscinet  zmath  isi  scopus
    6. Nazarov S.A., ““Wandering” Natural Frequencies of An Elastic Cuspidal Plate With the Clamped Peak”, Mater. Phys. Mech., 40:1 (2018), 47–55  crossref  isi  scopus
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