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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Homogenization of periodic differential operators of high order

Author: N. Veniaminov
Translated by: the author
Original publication: Algebra i Analiz, tom 22 (2010), nomer 5.
Journal: St. Petersburg Math. J. 22 (2011), 751-775
MSC (2010): Primary 35B27
Published electronically: June 27, 2011
MathSciNet review: 2828827
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Abstract: A periodic differential operator of the form $ A_\varepsilon = (\mathbf{D}^p)^\ast g(\mathbf{x} / \varepsilon) \mathbf{D}^p$ is considered on $ L_2(\mathbb{R}^d)$; here $ g(x)$ is a positive definite symmetric tensor of order $ 2 p$ periodic with respect to a lattice $ \Gamma$. The behavior of the resolvent of the operator $ A_\varepsilon$ as $ \varepsilon \to 0$ is studied. It is shown that the resolvent $ (A_\varepsilon + I)^{-1}$ converges in the operator norm to the resolvent of the effective operator $ A^0$ with constant coefficients. For the norm of the difference of resolvents, an estimate of order $ \varepsilon$ is obtained.

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Additional Information

N. Veniaminov
Affiliation: Department of Physics, St. Petersburg State University, Peterhoff, St. Petersburg 198504, Russia; Laboratory of Analysis, Geometry, and Applications, University Paris 13, Paris, France

Keywords: Periodic differential operators, averaging, homogenization, threshold effect, operators of high order
Received by editor(s): January 28, 2010
Published electronically: June 27, 2011
Article copyright: © Copyright 2011 American Mathematical Society

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