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Mathematics of the USSR-Sbornik, 1991, Volume 68, Issue 2, Pages 503–553
DOI: https://doi.org/10.1070/SM1991v068n02ABEH002111
(Mi sm1679)
 

This article is cited in 15 scientific papers (total in 15 papers)

Averaging principles for eguations with rapidly oscillatory coefficients

G. V. Sandrakov
References:
Abstract: Elliptic equations of arbitrary order with smooth, rapidly oscillating coefficients are considered. An algorithm is set forth for constructing a formal asymptotic expansion of solutions of such equations. The algorithm consists in the successive solution of a number of periodic problems. The solvability conditions for these problems lead to an averaged equation (system) with constant coefficients. It is proved that if the solution of the equation is bounded and converges to some limit in a suitable sense, then the limit function (vector) satisfies the averaged equation (system).
An asymptotic expansion of solutions of an equation of divergence form of arbitrary order is constructed. This makes it possible to obtain for such equations estimates of the form
$$ \|u_\varepsilon-u_s^0\|_s\leqslant C\sqrt\varepsilon, $$
where $2s$ is the order of the equation, $u_\varepsilon$ is a solution of the equation, and $u_s^0$ comprises $s$ terms of the asymptotic expansion.
Bibliography: 22 titles.
Received: 16.12.1988
Russian version:
Matematicheskii Sbornik, 1989, Volume 180, Number 12, Pages 1634–1679
Bibliographic databases:
UDC: 517.955.8
MSC: Primary 35J30, 35A35, 35C10; Secondary 35E99
Language: English
Original paper language: Russian
Citation: G. V. Sandrakov, “Averaging principles for eguations with rapidly oscillatory coefficients”, Mat. Sb., 180:12 (1989), 1634–1679; Math. USSR-Sb., 68:2 (1991), 503–553
Citation in format AMSBIB
\Bibitem{San89}
\by G.~V.~Sandrakov
\paper Averaging principles for eguations with rapidly oscillatory coefficients
\jour Mat. Sb.
\yr 1989
\vol 180
\issue 12
\pages 1634--1679
\mathnet{http://mi.mathnet.ru/sm1679}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1038221}
\zmath{https://zbmath.org/?q=an:0717.35021}
\transl
\jour Math. USSR-Sb.
\yr 1991
\vol 68
\issue 2
\pages 503--553
\crossref{https://doi.org/10.1070/SM1991v068n02ABEH002111}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=A1991FE73700010}
Linking options:
  • https://www.mathnet.ru/eng/sm1679
  • https://doi.org/10.1070/SM1991v068n02ABEH002111
  • https://www.mathnet.ru/eng/sm/v180/i12/p1634
  • This publication is cited in the following 15 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Математический сборник - 1989–1990 Sbornik: Mathematics
    Statistics & downloads:
    Abstract page:481
    Russian version PDF:177
    English version PDF:27
    References:71
    First page:1
     
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