On an extension of the~method of two-scale convergence and its applications
Sbornik. Mathematics, Tome 191 (2000) no. 7, pp. 973-1014
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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.
@article{SM_2000_191_7_a2,
author = {V. V. Zhikov},
title = {On an extension of the~method of two-scale convergence and its applications},
journal = {Sbornik. Mathematics},
pages = {973--1014},
publisher = {mathdoc},
volume = {191},
number = {7},
year = {2000},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SM_2000_191_7_a2/}
}
V. V. Zhikov. On an extension of the~method of two-scale convergence and its applications. Sbornik. Mathematics, Tome 191 (2000) no. 7, pp. 973-1014. http://geodesic.mathdoc.fr/item/SM_2000_191_7_a2/