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Homogenization with corrector for periodic differential operators. Approximation of solutions in the Sobolev class $ H^1({\mathbb{R}}^d)$


Authors: M. Sh. Birman and T. A. Suslina
Translated by: T. A. Suslina
Original publication: Algebra i Analiz, tom 18 (2006), nomer 6.
Journal: St. Petersburg Math. J. 18 (2007), 857-955
MSC (2000): Primary 35P99, 35Q99
DOI: https://doi.org/10.1090/S1061-0022-07-00977-6
Published electronically: October 5, 2007
MathSciNet review: 2307356
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Abstract | References | Similar Articles | Additional Information

Abstract: Investigation of a class of matrix periodic elliptic second-order differential operators $ {{\mathcal{A}}_\varepsilon}$ in $ {\mathbb{R}}{^d}$ with rapidly oscillating coefficients (depending on $ {{\mathbf{x}}/\varepsilon}$) is continued. The homogenization problem in the small period limit is studied. Approximation for the resolvent $ {({\mathcal{A}}_\varepsilon + I)^{-1}}$ in the operator norm from $ {L_2({\mathbb{R}}^d)}$ to $ H^1({\mathbb{R}}^d)$ is obtained with an error of order $ {\varepsilon}$. In this approximation, a corrector is taken into account. Moreover, the ( $ {L_2} \to {L_2}$)-approximations of the so-called fluxes are obtained.


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Additional Information

M. Sh. Birman
Affiliation: Department of Physics, St. Petersburg State University, Petrodvorets, Ul′yanovskaya 3, 198504 St. Petersburg, Russia
Email: mbirman@list.ru

T. A. Suslina
Affiliation: Department of Physics, St. Petersburg State University, Petrodvorets, Ul′yanovskaya 3, 198504 St. Petersburg, Russia
Email: suslina@list.ru

DOI: https://doi.org/10.1090/S1061-0022-07-00977-6
Keywords: Periodic operators, threshold approximations, homogenization, corrector, energy estimates
Received by editor(s): September 20, 2006
Published electronically: October 5, 2007
Additional Notes: Supported by RFBR (grants no. 05-01-01076-a, 05-01-02944-YaF-a).
Article copyright: © Copyright 2007 American Mathematical Society

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