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Uniform Estimates of Remainders in Asymptotic Expansions of Solutions to the Problem on Eigenoscillations of a Piezoelectric Plate

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Abstract

A method for obtaining estimates of asymptotic remainders is presented. The constants in estimates are independent of the number of the eigenvalue, as well as of the small parameter h, the thickness of the plate. Owing to an information about connections between frequencies of eigenoscillations of the three-dimensional plates and its two-dimensional model obtained under various restrictions to h, it is possible to divide the asymptotics in collective and individual ones. Only in the case of the individual asymptotics, i.e., under rigid restrictions on h, it is possible to construct asymptotic expansions for the corresponding eigenvectors. We consider arbitrarily anizotropic composed cylindrical plates in whcih piezoeffects can dominate along longitudinal directions, as well as along transverse directions. The connectedness of elastic and electric fields Implies the appearance of a nontrivial dissipative components of the operator of the problem under consideration, but its spectrum remains real and positive. Bibliography: 43 titles.

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Nazarov, S.A. Uniform Estimates of Remainders in Asymptotic Expansions of Solutions to the Problem on Eigenoscillations of a Piezoelectric Plate. Journal of Mathematical Sciences 114, 1657–1725 (2003). https://doi.org/10.1023/A:1022364812273

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