Further Investigations of Convergence Results for Homogenization Problems with Various Combinations of Scales

Abstract: This thesis is based on six papers. We study the homogenization of selected parabolic problems with one or more microscopic scales in space and time, respectively. The approaches are prepared by means of certain methods, like two-scale convergence, multiscale convergence and also the evolution setting of multiscale convergence and very weak multiscale convergence. Paper I treats a linear parabolic homogenization problem with rapid spatial and temporal oscillations in perforated domains. Suitable results of two-scale convergence type are established. Paper II deals with further development of compactness results which can be used in the homogenization procedure engaging a certain limit condition. The homogenization procedure deals with a parabolic problem with a certain matching between a fast spatial and a fast temporal scale and a coefficient passing to zero that the time derivative is multiplied with. Papers III and IV are further generalizations of Paper II and investigate homogenization problems with different types of matching between the microscopic scales. Papers III and IV deal with one and two rapid scales in both space and time respectively. Paper V treats the nonlinearity of monotone parabolic problems with an arbitrary number of spatial and temporal scales by applying the perturbed test functions method together with multiscale convergence and very weak multiscale convergence adapted to the evolution setting. In Paper VI we discuss the relation between two-scale convergence and the unfolding method and potential extensions of existing results. The papers above are summarized in Chapter 4. Chapter 1 gives a brief introduction to the topic and Chapters 2 and 3 are surveys over some important previous results.