Operator error estimates for homogenization of fourth order elliptic equations

Pastukhova, S.

doi:10.1090/spmj/1450

Operator error estimates for homogenization of fourth order elliptic equations
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by S. E. Pastukhova
Translated by: Yu. Meshkova

St. Petersburg Math. J. 28 (2017), 273-289

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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