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Mat. Sb., 2000, Volume 191, Number 7, Pages 31–72 (Mi msb491)  

This article is cited in 176 scientific papers (total in 176 papers)

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

V. V. Zhikov

Vladimir State Pedagogical University

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.


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English version:
Sbornik: Mathematics, 2000, 191:7, 973–1014

Bibliographic databases:

UDC: 517.9
MSC: Primary 35B27; Secondary 74E05
Received: 19.04.1999 and 17.02.2000

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
\by V.~V.~Zhikov
\paper On an extension of the~method of two-scale convergence and its applications
\jour Mat. Sb.
\yr 2000
\vol 191
\issue 7
\pages 31--72
\jour Sb. Math.
\yr 2000
\vol 191
\issue 7
\pages 973--1014

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