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 Mat. Sb., 2019, Volume 210, Number 11, Pages 129–158 (Mi msb9160)

‘Blinking’ and ‘gliding’ eigenfrequencies of oscillations of elastic bodies with blunted cuspidal sharpenings

S. A. Nazarov

Faculty of Mathematics and Mechanics, St Petersburg State University, St Petersburg, Russia

Abstract: The spectrum of a two-dimensional problem in elasticity theory is investigated for a body $\Omega^h$ with a cuspidal sharpening with a short tip of length $h>0$ that is broken off. It is known that when the tip is in place, the spectrum of the problem for $\Omega^0$ has a continuous component $[\Lambda_\dagger,+\infty)$ with positive cut-off point $\Lambda_\dagger>0$. We show that each point $\Lambda>\Lambda_\dagger$ is a ‘blinking’ eigenvalue, that is, it is an actual eigenvalue of the problem in $\Omega^h$ ‘almost periodically’ in the scale of $|\ln h|$. Among families of eigenvalues $\Lambda^h_{m(h)}$, which continuously depend on $h$, we discover ‘gliding’ eigenvalues, which fall down along the real axis at a great rate, $O((\Lambda^h_{m(h)}-\Lambda_\dagger)h^{-1}|\ln h|^{-1})$, but then land softly on the threshold $\Lambda_\dagger$. This reveals a new way of forming the continuous spectrum of the problem for a cuspidal body $\Omega^0$ from the system of discrete spectra of the problems in the $\Omega^h$, $h>0$. In addition, there may be ‘hardly movable’ eigenvalues, which remain in a small neighbourhood of a fixed point for all small $h$, in contrast to ‘gliding’ eigenvalues.
Bibliography: 30 titles.

Keywords: blunted cuspidal sharpening, two-dimensional elastic isotropic body, discrete and continuous spectrum, asymptotic behaviour, ‘blinking’ and ‘gliding’ eigenfrequencies.

 Funding Agency Grant Number Russian Science Foundation 17-11-01003 This research was supported by the Russian Science Foundation under grant no. 17-11-01003.

DOI: https://doi.org/10.4213/sm9160

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English version:
Sbornik: Mathematics, 2019, 210:11, 1633–1662

Bibliographic databases:

UDC: 517.958:539.3(2)+517.956.227+517.956.8
MSC: Primary 35Q74; Secondary 35P05

Citation: S. A. Nazarov, “‘Blinking’ and ‘gliding’ eigenfrequencies of oscillations of elastic bodies with blunted cuspidal sharpenings”, Mat. Sb., 210:11 (2019), 129–158; Sb. Math., 210:11 (2019), 1633–1662

Citation in format AMSBIB
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