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Homogenization of parabolic and elliptic periodic operators in $ L_2(\mathbb{R}^d)$ with the first and second correctors taken into account


Authors: E. S. Vasilevskaya and T. A. Suslina
Translated by: T. A. Suslina
Original publication: Algebra i Analiz, tom 24 (2012), nomer 2.
Journal: St. Petersburg Math. J. 24 (2013), 185-261
MSC (2010): Primary 35B27
DOI: https://doi.org/10.1090/S1061-0022-2013-01236-2
Published electronically: January 22, 2013
MathSciNet review: 3013323
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Abstract: In the space $ L_2(\mathbb{R}^d;{\mathbb{C}}^n)$, a wide class of matrix elliptic second order differential operators (DO's) $ \mathcal {A}_\varepsilon $ is studied; the $ \mathcal {A}_\varepsilon $ are assumed to admit a factorization of the form $ \mathcal {A}_\varepsilon = \mathcal {X}_\varepsilon ^* \mathcal {X}_\varepsilon $, where $ \mathcal {X}_\varepsilon $ is a homogeneous first order DO. The coefficients of these operators are periodic and depend on $ \mathbf {x}/\varepsilon $, $ \varepsilon >0$. The behavior of the operator exponential $ e^{-\mathcal {A}_\varepsilon \tau }$, $ \tau >0$, and of the resolvent $ ({\mathcal {A}}_\varepsilon +I)^{-1}$ for small $ \varepsilon $ is investigated. An approximation for the exponential $ e^{-\mathcal {A}_\varepsilon \tau }$ in the operator norm in $ L_2(\mathbb{R}^d; \mathbb{C}^n)$ with an error term of order $ \tau ^{-3/2}\varepsilon ^3$ is obtained. For the resolvent $ ({\mathcal {A}}_\varepsilon +I)^{-1}$, approximation in the norm of operators acting from $ H^1(\mathbb{R}^d; \mathbb{C}^n)$ to $ L_2( \mathbb{R}^d; \mathbb{C}^n)$ is found with an error term of order $ \varepsilon ^3$. In these approximations, the first and second order correctors are taken into account.


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Additional Information

E. S. Vasilevskaya
Affiliation: Department of Physics, St. Petersburg State University, Ul′yanovskaya 3, Petrodvorets, St. Petersburg 198504, Russia
Email: vasilevskaya-e@yandex.ru

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

DOI: https://doi.org/10.1090/S1061-0022-2013-01236-2
Keywords: Periodic differential operators, homogenization, effective operator, corrector
Received by editor(s): November 1, 2011
Published electronically: January 22, 2013
Additional Notes: The authors were supported by RFBR (grant no. 11-01-00458-a) and the Program of Support of Leading Scientific Schools (grant NSh-357.2012.1)
Article copyright: © Copyright 2013 American Mathematical Society