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 Funktsional. Anal. i Prilozhen.: Year: Volume: Issue: Page: Find

 Funktsional. Anal. i Prilozhen., 2016, Volume 50, Issue 4, Pages 91–96 (Mi faa3257)

Brief communications

Homogenization of Hyperbolic Equations

M. Dorodnyia, T. A. Suslina

a St. Petersburg State University, St. Petersburg, Russia

Abstract: We consider a self-adjoint matrix elliptic operator $A_\varepsilon$, $\varepsilon >0$, on $L_2({\mathbb R}^d;{\mathbb C}^n)$ given by the differential expression $b({\mathbf D})^* g({\mathbf x}/\varepsilon)b({\mathbf D})$. The matrix-valued function $g({\mathbf x})$ is bounded, positive definite, and periodic with respect to some lattice; $b({\mathbf D})$ is an $(m\times n)$-matrix first order differential operator such that $m \ge n$ and the symbol $b(\boldsymbol{\xi})$ has maximal rank. We study the operator cosine $\cos (\tau A^{1/2}_\varepsilon)$, where $\tau \in {\mathbb R}$. It is shown that, as $\varepsilon \to 0$, the operator $\cos (\tau A^{1/2}_\varepsilon)$ converges to $\cos(\tau (A^0)^{1/2})$ in the norm of operators acting from the Sobolev space $H^s({\mathbb R}^d;{\mathbb C}^n)$ (with a suitable $s$) to $L_2({\mathbb R}^d;{\mathbb C}^n)$. Here $A^0$ is the effective operator with constant coefficients. Sharp-order error estimates are obtained. The question about the sharpness of the result with respect to the type of the operator norm is studied. Similar results are obtained for more general operators. The results are applied to study the behavior of the solution of the Cauchy problem for the hyperbolic equation $\partial^2_\tau {\mathbf u}_\varepsilon ({\mathbf x}, \tau) =- A_\varepsilon {\mathbf u}_\varepsilon({\mathbf x}, \tau)$.

Keywords: periodic differential operators, hyperbolic equations, homogenization, operator error estimates

 Funding Agency Grant Number Russian Foundation for Basic Research 16-01-00087 Supported by the Russian Foundation for Basic Research (project no. 16-01-00087).

DOI: https://doi.org/10.4213/faa3257

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English version:
Functional Analysis and Its Applications, 2016, 50:4, 319–324

Bibliographic databases:

UDC: 517.956.2

Citation: M. Dorodnyi, T. A. Suslina, “Homogenization of Hyperbolic Equations”, Funktsional. Anal. i Prilozhen., 50:4 (2016), 91–96; Funct. Anal. Appl., 50:4 (2016), 319–324

Citation in format AMSBIB
\Bibitem{DorSus16} \by M.~Dorodnyi, T.~A.~Suslina \paper Homogenization of Hyperbolic Equations \jour Funktsional. Anal. i Prilozhen. \yr 2016 \vol 50 \issue 4 \pages 91--96 \mathnet{http://mi.mathnet.ru/faa3257} \crossref{https://doi.org/10.4213/faa3257} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3646712} \elib{http://elibrary.ru/item.asp?id=28119107} \transl \jour Funct. Anal. Appl. \yr 2016 \vol 50 \issue 4 \pages 319--324 \crossref{https://doi.org/10.1007/s10688-016-0162-z} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000390093200007} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85006427492} 

• http://mi.mathnet.ru/eng/faa3257
• https://doi.org/10.4213/faa3257
• http://mi.mathnet.ru/eng/faa/v50/i4/p91

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This publication is cited in the following articles:
1. M. Dorodnyi, T. A. Suslina, “Homogenization of a Nonstationary Model Equation of Electrodynamics”, Math. Notes, 102:5 (2017), 645–663
2. M. A. Dorodnyi, T. A. Suslina, “Spectral approach to homogenization of hyperbolic equations with periodic coefficients”, J. Differential Equations, 264:12 (2018), 7463–7522
3. Yu. M. Meshkova, “Ob usrednenii periodicheskikh giperbolicheskikh sistem”, Matem. zametki, 105:6 (2019), 937–942
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