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Operator error estimates for homogenization of fourth order elliptic equations. (English. Russian original) Zbl 1361.35023

St. Petersbg. Math. J. 28, No. 2, 273-289 (2017); translation from Algebra Anal. 28, No. 2, 204-226 (2016).
Summary: Homogenization of elliptic divergence-type fourth-order operators with \( \epsilon \)-periodic coefficients is studied. Here \( \epsilon \) is a small parameter. Approximations for the resolvent are obtained in the \( (L^2\rightarrow L^2)\)- and \( (L^2\rightarrow H^2)\)-operator norms with an error of order \( \epsilon \). 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 [Dokl. Math. 72, No. 1, 534–538 (2005; Zbl 1130.35312); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 403, No. 3, 305–308 (2005)].

MSC:

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35J40 Boundary value problems for higher-order elliptic equations
74K20 Plates

Citations:

Zbl 1130.35312
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Full Text: DOI

References:

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