Homogenization of parabolic and elliptic periodic operators in $L_2(\mathbb {R}^d)$ with the first and second correctors taken into account
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E. S. Vasilevskaya and T. A. Suslina
Translated by: T. A. Suslina - St. Petersburg Math. J. 24 (2013), 185-261
- DOI: https://doi.org/10.1090/S1061-0022-2013-01236-2
- Published electronically: January 22, 2013
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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.References
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Bibliographic 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
- 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)
- © Copyright 2013 American Mathematical Society
- Journal: St. Petersburg Math. J. 24 (2013), 185-261
- MSC (2010): Primary 35B27
- DOI: https://doi.org/10.1090/S1061-0022-2013-01236-2
- MathSciNet review: 3013323