A note on two scale compactness for $p=1$

Abstract

In this paper the notion of two-scale convergence introduced by G. Nguetseng and G. Allaire is extended to the case of bounded sequences in $L^1(\Omega )$, where $\Omega$ is any open subset of ${\mathbb{R}}^N$. Three different approaches will be considered: an adaptation of the method used in $L^p(\Omega )$ with $p>1$, a measure-theoretic argument, and the periodic unfolding technique.