Two-scale convergence: Some remarks and extensions

Abstract

We first study the fundamental ideas behind two-scale conver-gence to enhance an intuitive understanding of this notion. The classicaldefinitions and ideas are motivated with geometrical arguments illustratedby illuminating figures. Then a version of this concept, very weak two-scaleconvergence, is discussed both independently and brie°y in the context ofhomogenization. The main features of this variant are that it works alsofor certain sequences of functions which are not bounded inL2 and atthe same time is suited to detect rapid oscillations in some sequences whichare strongly convergent inL2 . In particular, we show how very weaktwo-scale convergence explains in a more transparent way how the oscilla-tions of the governing coe±cient of the PDE to be homogenized causes thedeviation of theG-limit from the weak L2 NxN-limit for the sequence ofcoe±cients. Finally, we investigate very weak multiscale convergence andprove a compactness result for separated scales which extends a previousresult which required well-separated scales.