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Two-term asymptotics of the spectrum of natural frequencies of a thin elastic shell. (English. Russian original) Zbl 0707.73041
Sov. Math., Dokl. 41, No. 1, 108-112 (1990); translation from Dokl. Akad. Nauk SSSR 310, No. 4, 777-780 (1990).
In this very interesting paper, the author considers the asymptotic distribution of the eigenvalues of the free oscillations of a thin elastic shell (with half thickness h) with connected compact mean surface G and boundary $$\Gamma$$. $${\mathcal G}$$ is assumed to be $$C^{\infty}$$ and $$\Gamma$$ piecewise smooth. Associated with the problem is a functional $${\mathcal F}(u,v)=(h^ 3/3)g(u,v)+f(u,v)$$ (modulo a multiplicative constant the potential energy) and the related operators $${\mathcal L}_{pq}=(h^ 3/3)h_{pq}+\ell_{pq}$$, $$\ell_{pq}$$ the moment free operators and $$n_{pq}$$ the moment operators.
The author observes that the validity of the two dimensional theory of shells is valid only for $$\lambda <<h^{-2}$$ while the usual single term asymptotics for the eigenvalue distribution function $$N_ h(\lambda)$$ functions for $$\lambda >>h^{-2}$$. Accordingly, he considers $$\lambda \sim R^{-2}$$ fixed (R a characteristic linear dimension of G) and seeks the asymptotics of $$N_ h(\lambda)$$ as h/R$$\to 0$$. An auxiliary problem is introduced, namely $$\ell_{pq}u_ q=\lambda u_ p$$, $$p,q=1,2,3$$; $$d_{jq}u_ q|_{\Gamma}=0$$, $$j=1,2$$, $$q=1,2,3.$$
This problem has a continuous spectrum which may be classified in terms of the normal curvature K(x,$$\xi$$) over the cotangent bundle and some other characteristics. If $$\lambda$$ does not belong to the continuous spectrum of this auxiliary problem and $$\lambda >\max K^ 2$$ then $$N_ h(\lambda)=a(\lambda)h^{-1}+O(h^{-1/2})$$. The proof of this result is fairly intricate and involves the wave front operator theory of certain pseudo-differential operators.
Reviewer: M.Thompson

##### MSC:
 74H45 Vibrations in dynamical problems in solid mechanics 74K15 Membranes 35P20 Asymptotic distributions of eigenvalues in context of PDEs 58J47 Propagation of singularities; initial value problems on manifolds