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Homogenization with corrector for a stationary periodic Maxwell system

Author: T. A. Suslina
Translated by: the author
Original publication: Algebra i Analiz, tom 19 (2007), nomer 3.
Journal: St. Petersburg Math. J. 19 (2008), 455-494
MSC (2000): Primary 35P20, 35Q60
Published electronically: March 21, 2008
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Abstract: The homogenization problem in the small period limit for a stationary periodic Maxwell system in $ {\mathbb{R}^3}$ is studied. It is assumed that the dielectric permittivity and the magnetic permeability are rapidly oscillating (depending on $ \mathbf{x}/\varepsilon$), positive definite, and bounded matrix-valued functions. For all four physical fields (the strength of the electric field, the strength of the magnetic field, the electric displacement vector, and the magnetic displacement vector), uniform approximations in the $ {L_2(\mathbb{R}^3)}$-norm are obtained with the (order-sharp) error term of order $ \varepsilon$. Besides solutions of the homogenized Maxwell system, the approximations contain rapidly oscillating terms of zero order that weakly tend to zero. These terms can be interpreted as correctors of zero order.

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Additional Information

T. A. Suslina
Affiliation: Department of Physics, St. Petersburg State University, Ul’yanovskaya 3, Petrodvorets, 198504 St. Petersburg, Russia

Keywords: Periodic Maxwell operator, homogenization, effective medium, corrector
Received by editor(s): February 8, 2007
Published electronically: March 21, 2008
Additional Notes: Supported by RFBR (grant no. 05-01-01076-a) and the President grant “Scientific Schools” (grant no. 5403.2006.1).
Article copyright: © Copyright 2008 American Mathematical Society

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