We study homogenization of linear equations with coefficients oscillating relative to different groups of variables with periods of different smallness order. In this case, the constant effective coefficient matrix is found by successive application of classical homogenization. We construct a two-scale matrix whose effective matrix depends on the order of reiterated homogenizations. Such a dependence is not observed in some simplest cases, for example, if the spatial variable is one-dimensional or the medium has layer structure. Bibliography: 2 titles.
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S. E. Pastukhova and R. N. Tikhomirov, “Operator estimates in reiterated and locally periodic homogenization” [in Russian], Dokl. Akad. Nauk, Ross. Akad. Nauk 415, No. 3, 304–309 (2007); English transl.: Dokl. Math. 76, No. 1, 548–553 (2007).
V. V. Zhikov, S. M. Kozlov, and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals [in Russian], Nauka, Moscow (1993); English transl.: Springer, Berlin (1994).
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Translated from Problemy Matematicheskogo Analiza 77, December 2015, pp. 153-158.
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Pastukhova, S.E., Tikhomirov, R.N. On Reiterated Homogenization Identities. J Math Sci 205, 297–303 (2015). https://doi.org/10.1007/s10958-015-2248-1
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DOI: https://doi.org/10.1007/s10958-015-2248-1