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Homogenization with corrector term for periodic elliptic differential operators


Authors: M. Sh. Birman and T. A. Suslina
Translated by: T. A. Suslina
Original publication: Algebra i Analiz, tom 17 (2005), nomer 6.
Journal: St. Petersburg Math. J. 17 (2006), 897-973
MSC (2000): Primary 35J99
DOI: https://doi.org/10.1090/S1061-0022-06-00935-6
Published electronically: September 21, 2006
MathSciNet review: 2202045
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Abstract | References | Similar Articles | Additional Information

Abstract: We continue to study the class of matrix periodic elliptic differential operators $ {{\mathcal{A}}_\varepsilon}$ in $ {\mathbb{R}^d}$ with coefficients oscillating rapidly (i.e., depending on $ {{\mathbf{x}}/\varepsilon}$). This class was introduced in the authors' earlier work of 2001 and 2003. The problem of homogenization in the small period limit is considered. Approximation for the resolvent $ {({\mathcal{A}}_\varepsilon + I)^{-1}}$ is obtained in the operator norm in $ {L_2(\mathbb{R}^d)}$ with error term of order $ {\varepsilon^2}$. The so-called corrector is taken into account. We develop the approach of our paper of 2003, where approximation with no corrector term but with remainder term of order $ {\varepsilon}$ was found. The paper is based on the operator-theoretic material obtained in our paper in the previous issue of this journal. Though the present paper is a continuation of the earlier work, it can be read independently.


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Additional Information

M. Sh. Birman
Affiliation: Department of Physics, St. Petersburg State University, Ulyanovskaya 1, Petrodvorets, St. Petersburg 198504, Russia
Email: mbirman@list.ru

T. A. Suslina
Affiliation: Department of Physics, St. Petersburg State University, Ulyanovskaya 1, Petrodvorets, St. Petersburg 198504, Russia

DOI: https://doi.org/10.1090/S1061-0022-06-00935-6
Keywords: Periodic operators, threshold approximations, homogenization, corrector
Received by editor(s): October 17, 2005
Published electronically: September 21, 2006
Additional Notes: Supported by RFBR (grant nos. 05-01-01076-a and 05-01-02944-YaF-a)
Article copyright: © Copyright 2006 American Mathematical Society

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