Pure and Applied Mathematics Quarterly

Volume 9 (2013)

Number 3

Two-scale convergence: Some remarks and extensions

Pages: 461 – 486

DOI: http://dx.doi.org/10.4310/PAMQ.2013.v9.n3.a4

Authors

L. Flodén (Department of Engineering and Sustainable Development, Mid Sweden University, Östersund, Sweden)

A. Holmbom (Department of Engineering and Sustainable Development, Mid Sweden University, Östersund, Sweden)

M. Olsson Lindberg (Department of Engineering and Sustainable Development, Mid Sweden University, Östersund, Sweden)

J. Persson (Department of Engineering and Sustainable Development, Mid Sweden University, Östersund, Sweden)

Abstract

We first study the fundamental ideas behind two-scale convergence to enhance an intuitive understanding of this notion. The classical definitions and ideas are motivated with geometrical arguments illustrated by illuminating figures. Then a version of this concept, very weak two-scale convergence, is discussed both independently and briefly in the context of homogenization. The main features of this variant are that it works also for certain sequences of functions which are not bounded in $L^2 (\Omega)$ and at the same time is suited to detect rapid oscillations in some sequences which are strongly convergent in $L^2 (\Omega)$. In particular, we show how very weak two-scale convergence explains in a more transparent way how the oscillations of the governing coefficient of the PDE to be homogenized causes the deviation of the $G$-limit from the weak $L^2 (\Omega)^{N \times N}$-limit for the sequence of coefficients. Finally, we investigate very weak multiscale convergence and prove a compactness result for separated scales which extends a previous result which required well-separated scales.

Keywords

two-scale convergence, multiscale convergence, very weak multiscale convergence, homogenization

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