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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 

 

Homogenization in the Sobolev class $ H^1(\mathbb{R}^d)$ for second order periodic elliptic operators with the inclusion of first order terms


Author: T. A. Suslina
Translated by: the author
Original publication: Algebra i Analiz, tom 22 (2010), nomer 1.
Journal: St. Petersburg Math. J. 22 (2011), 81-162
MSC (2010): Primary 35B27
DOI: https://doi.org/10.1090/S1061-0022-2010-01135-X
Published electronically: November 17, 2010
MathSciNet review: 2641084
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Abstract: Matrix periodic elliptic second order differential operators $ {\mathcal B}_{\varepsilon}$ in $ \mathbb{R}^d$ with rapidly oscillating coefficients (depending on $ \mathbf{x}/\varepsilon$) are studied. The principal part of the operator is given in a factorized form $ b(\mathbf{D})^* g(\varepsilon^{-1}\mathbf{x})b(\mathbf{D})$, where $ g$ is a periodic, bounded and positive definite matrix-valued function and $ b(\mathbf{D})$ is a matrix first order operator whose symbol is a matrix of maximal rank. The operator also has zero and first order terms with unbounded coefficients. The problem of homogenization in the small period limit is considered. Approximation for the generalized resolvent of the operator $ {\mathcal B}_\varepsilon$ is obtained in the operator norm in $ L_2(\mathbb{R}^d;\mathbb{C}^n)$ with error term $ O(\varepsilon)$. Also, approximation for this resolvent is obtained in the norm of operators acting from $ L_2(\mathbb{R}^d;\mathbb{C}^n)$ to $ H^1(\mathbb{R}^d;\mathbb{C}^n)$ with error term of order $ \varepsilon$ and with the corrector taken into account. The general results are applied to homogenization problems for the Schrödinger operator and the two-dimensional Pauli operator with potentials involving singular terms.


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Additional Information

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

DOI: https://doi.org/10.1090/S1061-0022-2010-01135-X
Keywords: Periodic differential operators, homogenization, effective operator, corrector
Received by editor(s): July 20, 2009
Published electronically: November 17, 2010
Additional Notes: Supported by RFBR (grant no. 08-01-00209-a), by a “Scientific schools” grant no. 816.2008.1, and by a “Development of scientific potential of high school” grant no. 2.1.1/2501
Article copyright: © Copyright 2010 American Mathematical Society