Two-scale convergence: Some remarks and extensions

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 (Ω) and at the same time is suited to detect rapid oscillations in some sequences which are strongly convergent in L 2 (Ω). 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 (Ω) N ×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.

Space-time homogenization for nonlinear diffusion

“…We emphasize that the periods in x and t of a( x ε , t ε r ) are ε and ε r , respectively. Then the form of the corresponding cell-problem changes significantly at a critical value of r, and moreover, in contrast to standard homogenization, it is not always elliptic, but parabolic at the critical scale (see also [15,17,18,19,20]).…”

“…Corollary 2.13 (Very weak two-scale convergence, cf. [17,18,19,20]). In addition to the same assumptions as in Theorem 2.9.…”