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

This article is cited in 2 scientific papers (total in 2 papers)

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
«Родные города» ПАО «Газпром нефть»
фонд Дмитрия Зимина «Династия»
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
Received: 25.05.2017
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
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\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
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\elib{http://elibrary.ru/item.asp?id=29106594}
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\jour Funct. Anal. Appl.
\yr 2017
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\issue 3
\pages 230--235
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    Citing articles on Google Scholar: Russian citations, English citations
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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  mathnet  crossref  mathscinet  isi  elib
    2. N. N. Senik, “Ob usrednenii lokalno periodicheskikh ellipticheskikh i parabolicheskikh operatorov”, Funkts. analiz i ego pril., 54:1 (2020), 87–92  mathnet  crossref
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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