Homogenization

Abstract

The theory of homogenization is a theory which was developed in the last forty years. Its success lies in the fact that practically every partial differential equation can be homogenized. Also during the last decades composite materials have invaded our world. To explain the principle of this theory the composite materials are indeed very well adapted. Suppose that we built a “composite*”i.e., a material made of different materials by juxtaposing small identical cells containing the different type of materials for instance a three material composite — see Figure 8.1 below. Assuming that we make the cells smaller and smaller at the limit we get a new material — a composite — which inherits some properties which can be very different from the ones of the materials composing it. For instance mixing three materials with different heat conductivity in the way above leads at the limit to a new material for which we can study the conductivity by cutting a piece of it — say Ω. This is such an issue which is addressed by homogenization techniques (see [14],[40],[41],[72],[46],[47],[61],[87],[92]).