RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PERSONAL OFFICE
General information
Latest issue
Archive
Impact factor
Subscription
Guidelines for authors
License agreement
Submit a manuscript

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Mat. Zametki:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


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

Full text: PDF file (685 kB)
First page: PDF file
References: PDF file   HTML file

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

Bibliographic databases:

Document Type: Article
UDC: 517.956.2
Received: 10.04.2017

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}


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

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles
  • Математические заметки Mathematical Notes
    Number of views:
    This page:188
    References:13
    First page:15

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2019