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 Mat. Zametki, 2017, Volume 102, Issue 5, Pages 700–720 (Mi mz11594)

Homogenization of a Nonstationary Model Equation of Electrodynamics

M. Dorodnyi, T. A. Suslina

Saint Petersburg State University

Abstract: In $L_2(\mathbb R^3;\mathbb C^3)$, we consider a self-adjoint operator $\mathscr L_\varepsilon$, $\varepsilon >0$, generated by the differential expression $\operatorname{curl}\eta(\mathbf x /\varepsilon)^{-1}\operatorname{curl} -\nabla\nu(\mathbf x/\varepsilon)\operatorname{div}$. Here the matrix function $\eta(\mathbf x)$ with real entries and the real function $\nu(\mathbf x)$ are periodic with respect to some lattice, are positive definite, and are bounded. We study the behavior of the operators $\cos(\tau\mathscr L_\varepsilon^{1/2})$ and $\mathscr L_\varepsilon^{-1/2} \sin(\tau\mathscr L_\varepsilon^{1/2})$ for $\tau\in\mathbb R$ and small $\varepsilon$. It is shown that these operators converge to $\cos(\tau(\mathscr L^0)^{1/2})$ and $(\mathscr L^0)^{-1/2}\sin(\tau(\mathscr L^0)^{1/2})$, respectively, in the norm of the operators acting from the Sobolev space $H^s$ (with a suitable $s$) to $L_2$. Here $\mathscr L^0$ is an effective operator with constant coefficients. Error estimates are obtained and the sharpness of the result with respect to the type of operator norm is studied. The results are used for homogenizing the Cauchy problem for the model hyperbolic equation $\partial^2_\tau\mathbf v_\varepsilon =-\mathscr L_\varepsilon\mathbf v_\varepsilon$, $\operatorname{div}\mathbf v_\varepsilon=0$, appearing in electrodynamics. We study the application to a nonstationary Maxwell system for the case in which the magnetic permeability is equal to $1$ and the dielectric permittivity is given by the matrix $\eta(\mathbf x/\varepsilon)$.

Keywords: periodic differential operator, homogenization, operator error estimate, nonstationary Maxwell system.

 Funding Agency Grant Number Russian Science Foundation 17-11-01069 This work was supported by the Russian Science Foundation under grant 17-11-01069.

DOI: https://doi.org/10.4213/mzm11594

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English version:
Mathematical Notes, 2017, 102:5, 645–663

Bibliographic databases:

Document Type: Article
UDC: 517.956.2

Citation: M. Dorodnyi, T. A. Suslina, “Homogenization of a Nonstationary Model Equation of Electrodynamics”, Mat. Zametki, 102:5 (2017), 700–720; Math. Notes, 102:5 (2017), 645–663

Citation in format AMSBIB
\Bibitem{DorSus17} \by M.~Dorodnyi, T.~A.~Suslina \paper Homogenization of a Nonstationary Model Equation of Electrodynamics \jour Mat. Zametki \yr 2017 \vol 102 \issue 5 \pages 700--720 \mathnet{http://mi.mathnet.ru/mz11594} \crossref{https://doi.org/10.4213/mzm11594} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3716505} \elib{http://elibrary.ru/item.asp?id=30512312} \transl \jour Math. Notes \yr 2017 \vol 102 \issue 5 \pages 645--663 \crossref{https://doi.org/10.1134/S0001434617110050} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000418838500005} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85039450816} 

• http://mi.mathnet.ru/eng/mz11594
• https://doi.org/10.4213/mzm11594
• http://mi.mathnet.ru/eng/mz/v102/i5/p700

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