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Sibirsk. Mat. Zh., 2008, Volume 49, Number 5, Pages 1105–1127 (Mi smj1907)  

This article is cited in 16 scientific papers (total in 16 papers)

The spectrum of the elasticity problem for a spiked body

S. A. Nazarov

Institute of Problems of Mechanical Engineering, Russian Academy of Sciences

Abstract: We establish the existence of continuous spectrum for the operator of the linear elasticity problem in a three-dimensional domain with a sufficiently sharp spiked singularity of the boundary. We obtain some information about the structure of the spectrum and verify the weighted Korn inequality, which enables us to prove that the spectrum is discrete for insufficiently sharp spikes. We state some open questions.

Keywords: elasticity equations, zero cusp, spike, discrete spectrum, continuous spectrum.

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English version:
Siberian Mathematical Journal, 2008, 49:5, 874–893

Bibliographic databases:

UDC: 517.946
Received: 12.03.2007

Citation: S. A. Nazarov, “The spectrum of the elasticity problem for a spiked body”, Sibirsk. Mat. Zh., 49:5 (2008), 1105–1127; Siberian Math. J., 49:5 (2008), 874–893

Citation in format AMSBIB
\by S.~A.~Nazarov
\paper The spectrum of the elasticity problem for a~spiked body
\jour Sibirsk. Mat. Zh.
\yr 2008
\vol 49
\issue 5
\pages 1105--1127
\jour Siberian Math. J.
\yr 2008
\vol 49
\issue 5
\pages 874--893

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    This publication is cited in the following articles:
    1. S. A. Nazarov, “The Essential Spectrum of Boundary Value Problems for Systems of Differential Equations in a Bounded Domain with a Cusp”, Funct. Anal. Appl., 43:1 (2009), 44–54  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. 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
    3. Cardone G., Nazarov S.A., Taskinen J., “A criterion for the existence of the essential spectrum for beak-shaped elastic bodies”, J. Math. Pures Appl. (9), 92:6 (2009), 628–650  crossref  mathscinet  zmath  isi  scopus
    4. Cardone G., Nazarov S.A., Taskinen J., ““Absorption” effect for elastic waves by the beak-shaped boundary irregularity”, Dokl. Phys., 54:3 (2009), 146–150  mathnet  crossref  mathscinet  zmath  adsnasa  isi  elib  elib  scopus
    5. Campbell A., Nazarov S.A., Sweers G.H., “Spectra of Two-Dimensional Models for Thin Plates with Sharp Edges”, SIAM J. Math. Anal., 42:6 (2010), 3020–3044  crossref  mathscinet  zmath  isi  elib  scopus
    6. Nazarov S.A., “Localized elastic fields in periodic waveguides with defects”, Journal of Applied Mechanics and Technical Physics, 52:2 (2011), 311–320  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    7. Nazarov S.A., Taskinen J., “Radiation conditions at the top of a rotational cusp in the theory of water-waves”, ESAIM Math. Model. Numer. Anal., 45:5 (2011), 947–979  crossref  mathscinet  zmath  isi  scopus
    8. Nazarov S.A., Ruotsalainen K., Taskinen J., “Gaps in the Spectrum of the Neumann Problem on a Perforated Plane”, Dokl. Math., 86:1 (2012), 574–578  crossref  mathscinet  zmath  isi  elib  elib  scopus
    9. Kamotski I.V. Maz'ya V.G., “On the Linear Water Wave Problem in the Presence of a Critically Submerged Body”, SIAM J. Math. Anal., 44:6 (2012), 4222–4249  crossref  mathscinet  zmath  isi  elib  scopus
    10. Nazarov S.A., Slutskij A.S., Taskinen J., “Korn Inequality For a Thin Rod With Rounded Ends”, Math. Meth. Appl. Sci., 37:16 (2014), 2463–2483  crossref  mathscinet  zmath  isi  elib  scopus
    11. Kozlov V. Nazarov S.A., “on the Spectrum of An Elastic Solid With Cusps”, Adv. Differ. Equat., 21:9-10 (2016), 887–944  mathscinet  zmath  isi
    12. Nazarov S.A., Ruotsalainen K.M., Silvola M., “Trapped Modes in Piezoelectric and Elastic Waveguides”, J. Elast., 124:2 (2016), 193–223  crossref  mathscinet  zmath  isi  scopus
    13. Nazarov S.A., ““Wandering” Eigenfrequencies of a Two-Dimensional Elastic Body With a Blunted Cusp”, Dokl. Phys., 62:11 (2017), 512–516  crossref  mathscinet  isi  scopus
    14. Kozlov V.A. Nazarov S.A., “Waves and Radiation Conditions in a Cuspidal Sharpening of Elastic Bodies”, J. Elast., 132:1 (2018), 103–140  crossref  mathscinet  zmath  isi  scopus
    15. 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
    16. S. A. Nazarov, “‘Blinking’ and ‘gliding’ eigenfrequencies of oscillations of elastic bodies with blunted cuspidal sharpenings”, Sb. Math., 210:11 (2019), 1633–1662  mathnet  crossref  crossref  adsnasa  isi  elib
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