On homogenization for a periodic elliptic operator in a strip
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T. A. Suslina
Translated by: the author - St. Petersburg Math. J. 16 (2005), 237-257
- DOI: https://doi.org/10.1090/S1061-0022-04-00849-0
- Published electronically: December 17, 2004
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Abstract:
In a strip $\Pi = \mathbb {R}\times (0,a)$, the operator \begin{equation*} A_\varepsilon = D_1 g_1(x_1/\varepsilon ,x_2) D_1 + D_2 g_2(x_1/\varepsilon ,x_2) D_2 \end{equation*} is considered, where $g_1$, $g_2$ are periodic with respect to the first variable. Periodic boundary conditions are put on the boundary of the strip. The behavior of the operator $A_\varepsilon$ in the limit $\varepsilon \to 0$ is studied. It is proved that, with respect to the operator norm in $L_2(\Pi )$, the resolvent $(A_\varepsilon +I)^{-1}$ tends to the resolvent of the effective operator $A^0$. A sharp order estimate for the norm of the difference of the resolvents is obtained. The operator $A^0$ is of the same type, but its coefficients depend only on $x_2$.References
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Bibliographic Information
- T. A. Suslina
- Affiliation: St. Petersburg State University, Faculty of Physics, Petrodvorets, Ul’yanovskaya 1, St. Petersburg 198504, Russia
- Email: tanya@petrov.stoic.spb.su
- Received by editor(s): September 1, 2003
- Published electronically: December 17, 2004
- Additional Notes: Supported by RFBR (grant no. 02-01-00798).
- © Copyright 2004 American Mathematical Society
- Journal: St. Petersburg Math. J. 16 (2005), 237-257
- MSC (2000): Primary 35B27
- DOI: https://doi.org/10.1090/S1061-0022-04-00849-0
- MathSciNet review: 2068354
Dedicated: Dedicated to my dear teacher Mikhail Shlemovich Birman with love and gratitude