\Sigma-convergence

Abstract

We discuss two new concepts of convergence in $L^{p}$-spaces, the so-called weak $\Sigma $-convergence and strong $\Sigma $-convergence, which are intermediate between classical weak convergence and strong convergence. We also introduce the concept of $\Sigma $-convergence for Radon measures. Our basic tool is the classical Gelfand representation theory. Apart from being a natural generalization of well-known two-scale convergence theory, the present study lays the foundation of the mathematical framework that is needed to undertake a systematic study of deterministic homogenization problems beyond the usual periodic setting. A few homogenization problems are worked out by way of illustration.