Some problems in Homogenization

Abstract

Some problems in the homogenization of partial differential equations are considered for this study. The process of obtaining the macroscopic or effective properties of materials having heterogeneities on a scale much smaller compared to the material dimensions are dealt with in this study. The problems in homogenization are of the nature of obtaining the global behaviour of solutions of problems in partial differential equations having rapidly oscillating coefficients. The aim is to identify a suitable homogenized problem whose solution approximated the solution of the original problem, for small oscillations. This thesis consists of two parts, the first concerns the homogenization of a class of optimal control problems under several situations; the second deals with the justification of the second term in the asymptotic expansion for a flow in a partially fissured medium. In the second and third chapters, study of some problems considered from a new points of view, and some generalizations of the existing results are obtained. In the fourth chapter the homogenization of optimal control problems governed by elliptic systems in perforated domains is studied. Some results are obtained for the homogenization of the optimal control problem governed by Dirichlet boundary value problems in perforated domains, and discussed in the fifth chapter.