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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

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