Table of Contents

The past 20 years have witnessed a renaissance of theoretical work on the macroscopic behavior of microscopically heterogeneous materials. This activity brings together a number of related themes, including: (1) the use of weak convergence as a rigorous yet general language for the discussion of macroscopic behavior; (2) interest in new types of questions, particularly the ``G-closure problem,'' motivated in large part by applications of optimal control theory to structural optimization; (3) the introduction of new methods for bounding effective moduli, including one based on ``compensated compactness''; and (4) the identification of deep links between the analysis of microstructures and the multidimensional calculus of variations. This work has implications for many physical problems involving optimal design, composite materials, and coherent phase transitions. As a result it has received attention and support from numerous scientific communities -- including engineering, materials science, and physics as well as mathematics.

There is by now an extensive literature in this area. But for various reasons certain fundamental papers were never properly published, circulating instead as mimeographed notes or preprints. Other work appeared in poorly distributed conference proceedings volumes. Still other work was published in standard books or journals, but written in Russian or French. The net effect is a sort of ``gap'' in the literature, which has made the subject unnecessarily difficult for newcomers to penetrate.

The present book aims to help fill this gap by assembling a coherent selection of this work in a single, readily accessible volume, in English translation. We do not claim that these articles represent the last word -- or the first word -- on their respective topics. But we do believe they represent fundamental work, well worth reading and studying today. They form the foundation upon which subsequent progress has been built.

The decision what to include in a volume such as this is difficult and necessarily somewhat arbitrary. We have restricted ourselves to work originally written in Russian or French, by a handful of authors with different but related viewpoints. It would have been easy to add other fundamental work. We believe, however, that our choice has a certain coherence. This book will interest scientists working in the area, and those who wish to enter it. The book contains papers we want our Ph.D. students to study, to which they have not until now had ready access.

We now list the chapters in this book, and comment briefly on each one. They are presented, here and in the book, in chronological order.

1. On the control of coefficients in partial differential equations by F. Murat and L. Tartar. The article represents some of the earliest work recognizing the ill-posedness of optimal control problems when the ``control'' is the coefficient of a PDE. Other early work of a similar type is described in the review article by Lurie and Cherkaev (see chapter 7 of the present book).
2. Estimation of homogenized coefficients by L. Tartar. This is one of the earliest applications of weak convergence as a tool for bounding the effective moduli of composite materials.
3. H-Convergence by F. Murat and L. Tartar. The theory of H-convergence provides a mathematical framework for analysis of composites in complete generality, without any need for geometrical hypotheses such as periodicity or randomness. When specialized to the self-adjoint case it becomes equivalent to G-convergence. Treatments of G-convergence can be found elsewhere, including the books of Jikov, Kozlov, and Oleinik [1] and Dal Maso [2]. However the treatment by Murat and Tartar has the advantage of being self-contained, elegant, compact, and quite general. As a result it remains, in our opinion, the best exposition of this basic material.
4. A strange term coming from nowhere by F. Murat and D.Cioranescu. The focus of this work is somewhat different from the other chapters of this book. Attention is still on the macroscopic consequences of microstructures, and weak convergence still plays a fundamental role, however in this work the fine-scale boundary condition is of Dirichlet rather than Neumann or transmission type. There has been a lot of work on problems with similar boundary conditions but more general geometry, e.g. Dal Maso, G. and Garroni [3], and to problems involving Stokes flow, e.g. Allaire [4] and Hornung [5]. For work on structural optimization in problems of this type see Butazzo, G. and Dal Maso [6] and Sverak [7].
5. Design of composite plates of extremal rigidity by L. Gibiansky and A. Cherkaev. This work provides an early application of homogenization to a problem of optimal design. Most prior work dealt with second-order scalar problems such as thermal conduction; this article deals instead with plate theory (and, by isomorphism, 2D elasticity). For subsequent related work see Kohn and Strang [8] Allaire and Kohn [9,10] and especially the book of Bendsoe [11] and the review paper by Rozvany, Bendsoe, and Kirsch [12] which have extensive bibliographies.
6. Calculus of variations and homogenization by F. Murat and L. Tartar. This work presents a very complete treatment of optimal design problems in the setting of scalar second-order problems, and structures made from two isotropic materials. Such a treatment was made possible by the solution of the associated ``G-closure problem'' a few years before. The exposition of Murat and Tartar emphasizes the role of optimality conditions. For related work we refer once again to the book of Bendsoe [11], and also the article of Kohn and Strang [8].
7. Effective characteristics of composite materials and the optimal design of structural elements by K.A. Lurie and A. V. Cherkaev. The paper presents a comprehensive review of work by Russian community on homogenization methods applied to structural optimization and can be viewed as a theoretical introduction to optimal design problems illustrated by a number of examples. The approach developed here is strongly influenced by advances in control theory (see the book by Lurie [13]) as well as by practical optimization problems. The paper is supplemented by an Appendix describing early (1972) progress by Lurie and Simkina. That work in Russia was approximately contemporary with work by Murat and Tartar on similar issues in France in the early 1970s, including the first chapter of this book.
8. Microstructures of composites of extremal rigidity and exact bounds on the associated energy density by L.V. Gibiansky and A.V. Cherkaev. This work is a straight continuation of the problem which is discussed above, in chapter 5 of this book. The bounds considered in this chapter by Gibiansky and Cherkaev concern the rigidity or compliance of a two-component elastic composite in three space dimensions; however, the paper reflects a subtle shift of emphasis. The mathematical community gradually realized during the 1980s that bounds on effective moduli are of broad interest in mechanics, beyond their value for relaxing problems of structural optimization. Here the translation method is applied for proving such bounds -- based on the use of lower semicontinuous quadratic forms.