An introduction to homogenization.

*(English)*Zbl 0939.35001
Oxford Lecture Series in Mathematics and its Applications. 17. Oxford: Oxford University Press. ix, 262 p. (1999).

The authors write the present book by an excellent clearness and, at the same time, by mathematical accuracy. The purpose of the book is to provide detailed proofs and tools adapted to graduate courses. It gives a background of homogenization theory to students and also to researchers – in mathematics as well as in engineering, mechanics, or physics – who are interested in a mathematical introduction to the field.

Homogenization theory established the macroscopic behaviour of a system which is “microscopically” heterogeneous, in order to describe some characteristics of the heterogeneous medium. The heterogeneous material, by a limit process, is replaced by a homogeneous fictitious one (the homogenized “material”), whose global (or overall) characteristics are a good approximation of the initial ones. From the mathematical point of view, this means mainly that the solutions of a boundary value problem, depending of a small parameter, converge to the solution of a limit boundary value problem which is explicitly described.

The book starts with a presentation of the variational approach of partial differential equations (PDEs), which is the natural framework for homogenization theory. The authors focus their attention on the periodic homogenization of linear partial differential equations, which is a realistic assumption for a large class of applications. The multiple-scale method is described, and the two-scale convergence method is introduced. The method of oscillating test functions is also considered, and some important related results, as for instance the convergence of energies and the existence of correctors, are presented. Some chapters are devoted to the homogenization of the linearized system of elasticity, the heat equation and the wave equations. The book is concluded with an overview of some general approach to the study of the nonperiodic case.

Homogenization theory established the macroscopic behaviour of a system which is “microscopically” heterogeneous, in order to describe some characteristics of the heterogeneous medium. The heterogeneous material, by a limit process, is replaced by a homogeneous fictitious one (the homogenized “material”), whose global (or overall) characteristics are a good approximation of the initial ones. From the mathematical point of view, this means mainly that the solutions of a boundary value problem, depending of a small parameter, converge to the solution of a limit boundary value problem which is explicitly described.

The book starts with a presentation of the variational approach of partial differential equations (PDEs), which is the natural framework for homogenization theory. The authors focus their attention on the periodic homogenization of linear partial differential equations, which is a realistic assumption for a large class of applications. The multiple-scale method is described, and the two-scale convergence method is introduced. The method of oscillating test functions is also considered, and some important related results, as for instance the convergence of energies and the existence of correctors, are presented. Some chapters are devoted to the homogenization of the linearized system of elasticity, the heat equation and the wave equations. The book is concluded with an overview of some general approach to the study of the nonperiodic case.

Reviewer: M.Codegone (Torino)

##### MSC:

35-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations |

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

74Q05 | Homogenization in equilibrium problems of solid mechanics |

74Q10 | Homogenization and oscillations in dynamical problems of solid mechanics |

35B37 | PDE in connection with control problems (MSC2000) |