Two-scale convergence: Some remarks and extensions

We first study the fundamental ideas behind two-scale conver- gence 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 brie°y 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 and at the same time is suited to detect rapid oscillations in some sequences which are strongly convergent in L 2 . In particular, we show how very weak two-scale convergence explains in a more transparent way how the oscilla- tions of the governing coe±cient of the PDE to be homogenized causes the deviation of the G -limit from the weak L 2 NxN -limit for the sequence of coe±cients. 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.