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

 Funktsional. Anal. i Prilozhen., 2017, Volume 51, Issue 3, Pages 87–93 (Mi faa3492)

Brief communications

Homogenization of the Dirichlet problem for elliptic and parabolic systems with periodic coefficients

Yu. M. Meshkovaa, T. A. Suslinab

a Chebyshev Laboratory, St. Petersburg State University, St. Petersburg, Russia
b Department of Physics, St. Petersburg State University, St. Petersburg, Russia

Abstract: Let $\mathcal{O}\subset\mathbb{R}^d$ be a bounded domain of class $C^{1,1}$. Let $0<\varepsilon\leqslant 1$. In $L_2(\mathcal{O};\mathbb{C}^n)$ we consider a positive definite strongly elliptic second-order operator $B_{D,\varepsilon}$ with Dirichlet boundary condition. Its coefficients are periodic and depend on $\mathbf{x}\varepsilon$. The principal part of the operator is given in factorized form, and the operator has lower order terms. We study the behavior of the generalized resolvent $(B_{D,\varepsilon}-\zeta Q_0(\cdot/\varepsilon))^{-1}$ as $\varepsilon \to 0$. Here the matrix-valued function $Q_0$ is periodic, bounded, and positive definite; $\zeta$ is a complex-valued parameter. We find approximations of the generalized resolvent in the $L_2(\mathcal{O};\mathbb{C}^n)$-operator norm and in the norm of operators acting from $L_2(\mathcal{O};\mathbb{C}^n)$ to the Sobolev space $H^1(\mathcal{O};\mathbb{C}^n)$ with two-parameter error estimates (depending on $\varepsilon$ and $\zeta$). Approximations of the generalized resolvent are applied to the homogenization of the solution of the first initial-boundary value problem for the parabolic equation $Q_0({\mathbf x}/\varepsilon)\partial_t {\mathbf v}_\varepsilon({\mathbf x},t)=- ( B_{D,\varepsilon} {\mathbf v}_\varepsilon)({\mathbf x},t)$.

Keywords: periodic differential operators, elliptic systems, parabolic systems, homogenization, operator error estimates.

 Funding Agency Grant Number Russian Foundation for Basic Research 16-01-00087 «Ðîäíûå ãîðîäà» ÏÀÎ «Ãàçïðîì íåôòü» ôîíä Äìèòðèÿ Çèìèíà «Äèíàñòèÿ» Rokhlin Scholarship Supported by RFBR (project no. 16-01-00087). The first author is supported by “Native Towns,” a social investment program of PJSC “Gazprom Neft,” by the “Dynasty” foundation, and by the Rokhlin grant.

DOI: https://doi.org/10.4213/faa3492

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English version:
Functional Analysis and Its Applications, 2017, 51:3, 230–235

Bibliographic databases:

UDC: 517.956.2+517.956.4
Accepted:26.05.2017

Citation: Yu. M. Meshkova, T. A. Suslina, “Homogenization of the Dirichlet problem for elliptic and parabolic systems with periodic coefficients”, Funktsional. Anal. i Prilozhen., 51:3 (2017), 87–93; Funct. Anal. Appl., 51:3 (2017), 230–235

Citation in format AMSBIB
\Bibitem{MesSus17} \by Yu.~M.~Meshkova, T.~A.~Suslina \paper Homogenization of the Dirichlet problem for elliptic and parabolic systems with periodic coefficients \jour Funktsional. Anal. i Prilozhen. \yr 2017 \vol 51 \issue 3 \pages 87--93 \mathnet{http://mi.mathnet.ru/faa3492} \crossref{https://doi.org/10.4213/faa3492} \elib{http://elibrary.ru/item.asp?id=29106594} \transl \jour Funct. Anal. Appl. \yr 2017 \vol 51 \issue 3 \pages 230--235 \crossref{https://doi.org/10.1007/s10688-017-0187-y} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000411338100007} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85029762772} 

• http://mi.mathnet.ru/eng/faa3492
• https://doi.org/10.4213/faa3492
• http://mi.mathnet.ru/eng/faa/v51/i3/p87

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This publication is cited in the following articles:
1. Yu. M. Meshkova, T. A. Suslina, “Homogenization of the first initial boundary value problem for parabolic systems: Operator error estimates”, St. Petersburg Math. J., 29:6 (2018), 935–978
2. N. N. Senik, “Ob usrednenii lokalno periodicheskikh ellipticheskikh i parabolicheskikh operatorov”, Funkts. analiz i ego pril., 54:1 (2020), 87–92
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