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 Sibirsk. Mat. Zh., 2009, Volume 50, Number 4, Pages 746–756 (Mi smj1996)

On the structure of the spectrum for the elasticity problem in a body with a supersharp spike

F. L. Bakhareva, S. A. Nazarovb

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

Abstract: We establish that the continuous spectrum of the Neumann problem for the system of elasticity equations occupies the entire closed positive real semiaxis in the case that a three-dimensional body with a sharp-spiked cusp whose cross-section contracts to a point with the velocity $O(r^{1+\gamma})$, where $r$ is the distance to the vertex of the spike and $\gamma>1$ is the sharpness exponent.

Keywords: system of elasticity equations, spike, cusp, peak, continuous spectrum.

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English version:
Siberian Mathematical Journal, 2009, 50:4, 587–595

Bibliographic databases:

UDC: 517.984.5:517.958:539(4)

Citation: F. L. Bakharev, S. A. Nazarov, “On the structure of the spectrum for the elasticity problem in a body with a supersharp spike”, Sibirsk. Mat. Zh., 50:4 (2009), 746–756; Siberian Math. J., 50:4 (2009), 587–595

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. 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
2. 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
3. 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
4. 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
5. Chesnel L., Claeys X., Nazarov S.A., “a Curious Instability Phenomenon For a Rounded Corner in Presence of a Negative Material”, Asymptotic Anal., 88:1-2 (2014), 43–74
6. Martin J., Nazarov S.A., Taskinen J., “Spectrum of the Linear Water Model For a Two-Layer Liquid With Cuspidal Geometries At the Interface”, ZAMM-Z. Angew. Math. Mech., 95:8 (2015), 859–876
7. Kozlov V., Nazarov S.A., “on the Spectrum of An Elastic Solid With Cusps”, Adv. Differ. Equat., 21:9-10 (2016), 887–944
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