Sequential convergence for functions and operators

Abstract: The mathematical discipline homogenization theory is closely related to convergence issues. In this thesis different types of convergence are studied and put in relation to each other. We consider the classical concepts of G- and H-convergence and compensated compactness. The main focus, however, is on two-scale convergence which was originally adjusted to periodic homogenization but was later generalized to non-periodic cases. We point out some properties of the general version, where we use sequences of linear operators to obtain a two-scale limit, and identify conditions that make these operators compatible with two-scale convergence. A particular type of G-convergence is investigated using a specific choice of these two-scale compatible operators. Some considerations on how to introduce a defect measure for general two-scale convergence are put forward. The relationship between general two-scale convergence and the recent concept of unfolding is also studied. Finally, we illustrate H-convergence by means of periodic homogenization.