Homogenization of Multiple Integrals

Abstract

Abstract The object of homogenization theory is the description of the macroscopic properties of structures with fine microstructure, covering a wide range of applications that run from the study of properties of composites to optimal design. The structures under consideration may model cellular elastic materials, fibred materials, stratified or porous media, or materials with many holes or cracks. In mathematical terms, this study can be translated in the asymptotic analysis of fast-oscillating differential equations or integral functionals. The book presents an introduction to the mathematical theory of homogenization of nonlinear integral functionals, with particular regard to those general results that do not rely on smoothness or convexity assumptions. Homogenization results and appropriate descriptive formulas are given for periodic and almost- periodic functionals. The applications include the asymptotic behaviour of oscillating energies describing cellular hyperelastic materials, porous media, materials with stiff and soft inclusions, fibered media, homogenization of HamiltonJacobi equations and Riemannian metrics, materials with multiple scales of microstructure and with multi-dimensional structure. The book includes a specifically designed, self-contained and up-to-date introduction to the relevant results of the direct methods of Gamma-convergence and of the theory of weak lower semicontinuous integral functionals depending on vector-valued functions. The book is based on various courses taught at the advanced graduate level. Prerequisites are a basic knowledge of Sobolev spaces, standard functional analysis and measure theory. The presentation is completed by several examples and exercises.