We introduce the operator of transverse displacements in L 2-space on a periodic singular grid F = F 0 with zero thickness and a natural measure. In the case of a model grid, we find the spectrum of this operator: a countable sequence of eigenvalues of infinite multiplicity. This result is useful for describing the asymptotic behavior of the spectrum in the problem of elasticity theory on the corresponding thin grid F h if the thickness parameter h tends to 0 and, in particular, for justifying the existence of gaps in the spectrum of the problem on the thin grid F h. Bibliography: 4 titles. Illustrations: 5 figures.
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V. V. Zhikov and S. E. Pastukhova, “Derivation of the limit equations of elasticity theory on thin nets” [in Russian], Tr. Semin. im. I. G. Petrovskogo 25, 55–97 (2006); English transl.: J. Math. Sci., New York 135, No. 1, 2637–2665 (2006).
V. V. Zhikov and S. E. Pastukhova, “Averaging of problems in the theory of elasticity on periodic grids of critical thickness” [in Russian], Mat. Sb. 194, No. 5, 61–96 (2003); English transl.: Sb. Math. 194, No. 5, 697–732 (2003).
M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press, New York etc. (1978).
E. Kamke, Manuel of Ordinary Differential Equations [Russian translation], Nauka, Moscow (1976).
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Translated from Problemy Matematicheskogo Analiza 68, January 2013, pp. 113–123.
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Zhikov, V.V., Pastukhova, S.E. Spectrum of the operator of transverse displacements on a periodic model grid. J Math Sci 189, 459–471 (2013). https://doi.org/10.1007/s10958-013-1199-7
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DOI: https://doi.org/10.1007/s10958-013-1199-7