Homogenization with corrector for a stationary periodic Maxwell system

Suslina, T.

doi:10.1090/S1061-0022-08-01006-6

Homogenization with corrector for a stationary periodic Maxwell system
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by T. A. Suslina
Translated by: the author

St. Petersburg Math. J. 19 (2008), 455-494

DOI: https://doi.org/10.1090/S1061-0022-08-01006-6

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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