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

 

Homogenization of elliptic systems with periodic coefficients: Weighted $ L^p$ and $ L^{\infty}$ estimates for asymptotic remainders


Author: S. A. Nazarov
Translated by: A. Plotkin
Original publication: Algebra i Analiz, tom 18 (2006), nomer 2.
Journal: St. Petersburg Math. J. 18 (2007), 269-304
MSC (2000): Primary 35J45
Published electronically: March 19, 2007
MathSciNet review: 2244938
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Abstract: The difference between the fundamental matrix for a second order selfadjoint elliptic system with sufficiently smooth periodic coefficients and the fundamental matrix for the corresponding homogenized system in $ \mathbb{R}^n$ is shown to decay as $ O(1+\vert x\vert^{1-n}$) at infinity, $ n\ge 2$. As a consequence, weighted $ L^p$ and $ L^{\infty}$ estimates are obtained for the difference $ u^{\varepsilon}-u^0$ of the solutions of a system with rapidly oscillating periodic coefficients and the homogenized system in $ \mathbb{R}^n$ with right-hand side belonging to an appropriate weighted $ L^p$-class in $ \mathbb{R}^n$.


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Additional Information

S. A. Nazarov
Affiliation: Institute of Mechanical Engineering Problems, Bol′shoĭ pr. V.O. 61, 199178 St. Petersburg, Russia
Email: serna@snark.ipme.ru, srgnazarov@yahoo.co.uk

DOI: http://dx.doi.org/10.1090/S1061-0022-07-00951-X
PII: S 1061-0022(07)00951-X
Keywords: Homogenization, periodic differential operators, weighted estimates
Received by editor(s): October 1, 2005
Published electronically: March 19, 2007
Additional Notes: Supported by RFBR (grant no. 04-01-00567)
Article copyright: © Copyright 2007 American Mathematical Society