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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

On homogenization for a periodic elliptic operator in a strip


Author: T. A. Suslina
Translated by: the author
Original publication: Algebra i Analiz, tom 16 (2004), nomer 1.
Journal: St. Petersburg Math. J. 16 (2005), 237-257
MSC (2000): Primary 35B27
Published electronically: December 17, 2004
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Abstract: In a strip $\Pi = \mathbb{R}\times (0,a)$, the operator

\begin{displaymath}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{displaymath}

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 $x_2$.


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Additional 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

DOI: http://dx.doi.org/10.1090/S1061-0022-04-00849-0
PII: S 1061-0022(04)00849-0
Keywords: Periodic operator, homogenization, effective operator
Received by editor(s): September 1, 2003
Published electronically: December 17, 2004
Additional Notes: Supported by RFBR (grant no. 02-01-00798).
Dedicated: Dedicated to my dear teacher Mikhail Shlemovich Birman with love and gratitude
Article copyright: © Copyright 2004 American Mathematical Society