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

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

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
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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, “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
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
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
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
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
6. Nazarov S.A., “Localized elastic fields in periodic waveguides with defects”, Journal of Applied Mechanics and Technical Physics, 52:2 (2011), 311–320
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
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
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
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
11. Kozlov V. Nazarov S.A., “on the Spectrum of An Elastic Solid With Cusps”, Adv. Differ. Equat., 21:9-10 (2016), 887–944
12. Nazarov S.A., Ruotsalainen K.M., Silvola M., “Trapped Modes in Piezoelectric and Elastic Waveguides”, J. Elast., 124:2 (2016), 193–223
13. Nazarov S.A., ““Wandering” Eigenfrequencies of a Two-Dimensional Elastic Body With a Blunted Cusp”, Dokl. Phys., 62:11 (2017), 512–516
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
15. Nazarov S.A., ““Wandering” Natural Frequencies of An Elastic Cuspidal Plate With the Clamped Peak”, Mater. Phys. Mech., 40:1 (2018), 47–55
16. S. A. Nazarov, “‘Blinking’ and ‘gliding’ eigenfrequencies of oscillations of elastic bodies with blunted cuspidal sharpenings”, Sb. Math., 210:11 (2019), 1633–1662
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