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Operator error estimates for homogenization of fourth order elliptic equations


Author: S. E. Pastukhova
Translated by: Yu. Meshkova
Original publication: Algebra i Analiz, tom 28 (2016), nomer 2.
Journal: St. Petersburg Math. J. 28 (2017), 273-289
MSC (2010): Primary 35B27
DOI: https://doi.org/10.1090/spmj/1450
Published electronically: February 15, 2017
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Abstract: Homogenization of elliptic divergence-type fourth-order operators with $ \varepsilon $-periodic coefficients is studied. Here $ \varepsilon $ is a small parameter. Approximations for the resolvent are obtained in the $ (L^2\to L^2)$- and $ (L^2\to H^2)$-operator norms with an error of order $ \varepsilon $. A particular focus is on operators with bi-Laplacian, which, as compared with the general case, have their own special features that result in simplification of proofs. Operators of the type considered in the paper appear in the study of the elastic properties of thin plates. The operator estimates are proved with the help of the so-called shift method suggested by V. V. Zhikov in 2005.


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Additional Information

S. E. Pastukhova
Affiliation: Moscow Technical University (MIREA), pr. Vernadskogo 78, Moscow 119454, Russia
Email: pas-se@yandex.ru

DOI: https://doi.org/10.1090/spmj/1450
Keywords: Homogenization, operator estimates for homogenization, fourth-order equations, Steklov average, corrector
Received by editor(s): August 4, 2015
Published electronically: February 15, 2017
Additional Notes: The author was supported by RFBR (grant no. 14-01-00192) and by RSF (project no. 14-11-00398)
Article copyright: © Copyright 2017 American Mathematical Society