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Sbornik: Mathematics, 2000, Volume 191, Issue 7, Pages 973–1014
DOI: https://doi.org/10.1070/sm2000v191n07ABEH000491
(Mi sm491)
 

This article is cited in 261 scientific papers (total in 261 papers)

On an extension of the method of two-scale convergence and its applications

V. V. Zhikov

Vladimir State Pedagogical University
References:
Abstract: The concept of two-scale convergence associated with a fixed periodic Borel measure $\mu$ is introduced. In the case when $d\mu=dx$ is Lebesgue measure on the torus convergence in the sense of Nguetseng–Allaire is obtained. The main properties of two-scale convergence are revealed by the simultaneous consideration of a sequence of functions and a sequence of their gradients. An application of two-scale convergence to the homogenization of some problems in the theory of porous media (the double-porosity model) is presented. A mathematical notion of “softly or weakly coupled parallel flows” is worked out. A homogenized operator is constructed and the convergence result itself is interpreted as a “strong two-scale resolvent convergence”. Problems concerning the behaviour of the spectrum under homogenization are touched upon in this connection.
Received: 19.04.1999 and 17.02.2000
Russian version:
Matematicheskii Sbornik, 2000, Volume 191, Number 7, Pages 31–72
DOI: https://doi.org/10.4213/sm491
Bibliographic databases:
UDC: 517.9
MSC: Primary 35B27; Secondary 74E05
Language: English
Original paper language: Russian
Citation: V. V. Zhikov, “On an extension of the method of two-scale convergence and its applications”, Mat. Sb., 191:7 (2000), 31–72; Sb. Math., 191:7 (2000), 973–1014
Citation in format AMSBIB
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  • This publication is cited in the following 261 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Математический сборник - 1992–2005 Sbornik: Mathematics
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    Abstract page:1661
    Russian version PDF:497
    English version PDF:41
    References:110
    First page:1
     
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