This book provides an introduction to the mathematical theory of homogenization, which describes the replacement of a real composite material by a fictitious homogeneous one. The aim of the theory is to describe the macroscopic properties of the composite by taking into account the properties of the microscopic structure. The first four chapters cover variational methods for partial differential equations, which is the natural framework of homogenization theory. The text then discusses the homogenization of several kinds of second order boundary value problems. Particular attention is given to the classical examples of the steady and non-steady heat equations, the wave equation and the linearized system of elasticity. All topics are illustrated by figures and numerous examples.
Doina Cioranescu (Professor, University of Paris VI and CNRS, France) Patrizia Donato (Professor, University of Rouen, France)
1. Weak and weak - convergence in Banach spaces ; 2. Rapidly oscillating periodic functions ; 3. Some classes of Sobolev spaces ; 4. Some variational elliptic problems ; 5. Examples of periodic composite materials ; 6. Homogenization of elliptic equations: the convergence result ; 7. The multiple-scale method ; 8. Tartar's method of oscillating test functions ; 9. The two-scale convergence method ; 10. Homogenization in linearized elasticity ; 11. Homogenization of the heat equation ; 12. Homogenization of the wave equation ; 13. General Approaches to the non-periodic case ; References