##
**Homogenization of multiple integrals.**
*(English)*
Zbl 0911.49010

Oxford Lecture Series in Mathematics and its Applications. 12. Oxford: Clarendon Press. xiv, 298 p. (1998).

The homogenization (averaging) theory is devoted to the description of macroscopic properties of composite materials and is based on the study of asymptotic behaviour of solutions to partial differential equations or corresponding energies as a small-scale parameter tends to zero. The major task of homogenization theory is the proof of the convergence, the description of the limit problem, and the study of its properties. From the mathematical point of view homogenization is a part of the theory of G-convergence of differential operators and \(\Gamma\)-convergence of functionals. Nowadays the \(\Gamma\)-convergence theory provides the rigorous mathematical framework to formulate correctly many problems coming mainly from physics. It is the main tool for studying asymptotic problems in the calculus of variations.

The authors, who are active contributors to the field, have designed the book as an advanced text which presents a systematical exposition of the homogenization theory of nonlinear multiple integral functionals, with particular regard to those results that do not rely on smoothness or convexity hypotheses.

The book consists of 24 chapters, devoted to basic results in the mathematical theory of homogenization of multiple integrals, an appendix with basic definitions, theorems on almost-periodic functions, extension operators, and some regularity results, references with notes, and an index.

The chapters are grouped in four parts.

Part one “Lower semicontinuity of integral functionals” deals with recent developments on weak convergence. Here one can find the detailed description of minimum problems in Sobolev spaces and both necessary and sufficient conditions for weak lower semicontinuity. Moreover, some results on polyconvex, quasiconvex, and rank-1-convex functions with examples as well as other results used in the further presentation are also given in this part.

The second part “\(\Gamma\)-convergence” contains an introduction to the \(\Gamma\)-convergence theory which originated in the famous paper of De Giorgi and Franzoni from 1975, and which is the important tool for studying the asymptotic behaviour of variational functionals defined on topological spaces. In particular the direct and indirect methods of \(\Gamma\)-convergence are introduced and investigated. Following the book of G. Buttazzo [“Semicontinuity, relaxation and integral representation in the calculus of variations” (1989; Zbl 0669.49005)], the authors present some results on integral representation of functionals on Sobolev spaces. The version of the fundamental estimate and results obtained in a context suited to integrals with degenerate growth conditions generalize similar theorems that can be found in the monograph by G. Dal Maso [“An introduction to \(\Gamma\)-convergence” (1993; Zbl 0816.49001)].

The next two parts of the book are devoted to the homogenization. The “Basic homogenization” problems are discussed in part three. The authors give here results on periodic homogenization of integral functionals in the scalar and the vector-valued case, providing the main convergence theorems. Various types of generalizations including the almost-periodic homogenization techniques are illustrated. Subsequently, a closure theorem for the homogenization is presented and the structure of the homogenized integrands is studied. The chapter on the loss of polyconvexity by homogenization improves the earlier results of the authors. A few applications as homogenization of elliptic operators in divergence form, homogenization of Riemannian metrics and Hamilton-Jacobi equations, and homogenization of Besicovitch almost-periodic functionals and related issues are also considered.

The fourth part presents the finer homogenization results. One can find here the results on the homogenization of functionals describing the properties of porous media with a periodic microstructure (connected media) and the media with stiff and soft inclusions. The last chapters of the book are devoted to homogenization of functionals with non-standard growth conditions and various types of degeneracy, iterated homogenization and correctors, and homogenization of materials with multiple scales of microstructure and multidimensional structures.

Most of the chapters end with exercises which help the reader to find the relations among the chapter, the other parts of the book, and the cited literature. The list of references contains more than one hundred of items.

The book is written in a clear and lucid style and its layout is nice. It contains numerous interesting and important examples illustrating the abstract theory of homogenization. The book is almost self-contained, requiring as prerequisites only a basic knowledge of Sobolev spaces, measure theory and standard functional analysis. I think this book will be a valuable resource for workers in many fields. It can serve as a postgraduate text and as a starting point for advanced research.

The authors, who are active contributors to the field, have designed the book as an advanced text which presents a systematical exposition of the homogenization theory of nonlinear multiple integral functionals, with particular regard to those results that do not rely on smoothness or convexity hypotheses.

The book consists of 24 chapters, devoted to basic results in the mathematical theory of homogenization of multiple integrals, an appendix with basic definitions, theorems on almost-periodic functions, extension operators, and some regularity results, references with notes, and an index.

The chapters are grouped in four parts.

Part one “Lower semicontinuity of integral functionals” deals with recent developments on weak convergence. Here one can find the detailed description of minimum problems in Sobolev spaces and both necessary and sufficient conditions for weak lower semicontinuity. Moreover, some results on polyconvex, quasiconvex, and rank-1-convex functions with examples as well as other results used in the further presentation are also given in this part.

The second part “\(\Gamma\)-convergence” contains an introduction to the \(\Gamma\)-convergence theory which originated in the famous paper of De Giorgi and Franzoni from 1975, and which is the important tool for studying the asymptotic behaviour of variational functionals defined on topological spaces. In particular the direct and indirect methods of \(\Gamma\)-convergence are introduced and investigated. Following the book of G. Buttazzo [“Semicontinuity, relaxation and integral representation in the calculus of variations” (1989; Zbl 0669.49005)], the authors present some results on integral representation of functionals on Sobolev spaces. The version of the fundamental estimate and results obtained in a context suited to integrals with degenerate growth conditions generalize similar theorems that can be found in the monograph by G. Dal Maso [“An introduction to \(\Gamma\)-convergence” (1993; Zbl 0816.49001)].

The next two parts of the book are devoted to the homogenization. The “Basic homogenization” problems are discussed in part three. The authors give here results on periodic homogenization of integral functionals in the scalar and the vector-valued case, providing the main convergence theorems. Various types of generalizations including the almost-periodic homogenization techniques are illustrated. Subsequently, a closure theorem for the homogenization is presented and the structure of the homogenized integrands is studied. The chapter on the loss of polyconvexity by homogenization improves the earlier results of the authors. A few applications as homogenization of elliptic operators in divergence form, homogenization of Riemannian metrics and Hamilton-Jacobi equations, and homogenization of Besicovitch almost-periodic functionals and related issues are also considered.

The fourth part presents the finer homogenization results. One can find here the results on the homogenization of functionals describing the properties of porous media with a periodic microstructure (connected media) and the media with stiff and soft inclusions. The last chapters of the book are devoted to homogenization of functionals with non-standard growth conditions and various types of degeneracy, iterated homogenization and correctors, and homogenization of materials with multiple scales of microstructure and multidimensional structures.

Most of the chapters end with exercises which help the reader to find the relations among the chapter, the other parts of the book, and the cited literature. The list of references contains more than one hundred of items.

The book is written in a clear and lucid style and its layout is nice. It contains numerous interesting and important examples illustrating the abstract theory of homogenization. The book is almost self-contained, requiring as prerequisites only a basic knowledge of Sobolev spaces, measure theory and standard functional analysis. I think this book will be a valuable resource for workers in many fields. It can serve as a postgraduate text and as a starting point for advanced research.

Reviewer: S.Migorski (Krakow)

### MSC:

49J45 | Methods involving semicontinuity and convergence; relaxation |

49-02 | Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control |

49J40 | Variational inequalities |

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

35B27 | Homogenization in context of PDEs; PDEs in media with periodic structure |

74E05 | Inhomogeneity in solid mechanics |