Σ-Convergence

Abstract

Abstract. We discuss two new concepts of convergence in Lp-spaces, the socalled weak Σ -convergence and strong Σ-convergence, which are intermediate between classical weak convergence and strong convergence. We also introduce the concept of Σ-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.