Homogenization of some partial differential operators and integral functionals

Abstract: This thesis is devoted to some problems connected to the theory of homogenization of partial differential operators and integral functionals. The thesis consists of an introduction and three different parts, A-C. In the introduction we give an elementary presentation of the basic ideas in the homogenization theory. Moreover, the introduction also serves as an overview of the field and points out where the results contained in this thesis fit in. The first part, Part A, consists of one paper and a complementary appendix and it deals with the limit behavior of the solutions of a sequence of quasi-linear equations. Part B consists of three papers in which bounds on the homogenized integrand for non-linear problems are developed. Moreover, examples where the bounds are optimal are given. The third part, Part C, consists of four papers which consider some further mathematical and engineering aspects on the homogenization method. Several numerical results are also presented and compared with theoretical results.