J.-L. Lions - Böcker

This is a reprinting of a book originally published in 1978. At that time it was the first book on the subject of homogenization, which is the asymptotic analysis of partial differential equations with rapidly oscillating coefficients, and as such it sets the stage for what problems to consider and what methods to use, including probabilistic methods. At the time the book was written the use of asymptotic expansions with multiple scales was new, especially their use as a theoretical tool, combined with energy methods and the construction of test functions for analysis with weak convergence methods. Before this book, multiple scale methods were primarily used for non-linear oscillation problems in the applied mathematics community, not for analyzing spatial oscillations as in homogenization. In the current printing a number of minor corrections have been made, and the bibliography was significantly expanded to include some of the most important recent references. This book gives systematic introduction of multiple scale methods for partial differential equations, including their original use for rigorous mathematical analysis in elliptic, parabolic, and hyperbolic problems, and with the use of probabilistic methods when appropriate. The book continues to be interesting and useful to readers of different backgrounds, both from pure and applied mathematics, because of its informal style of introducing the multiple scale methodology and the detailed proofs.

This IMA Volume in Mathematics and its Applications Homogenization and Effective Moduli of Materials and Media represents the proceedings of a workshop which was an integral part of the 19R4-R5 IMA program on CONTINUUM PHYSICS AND PARTIAL DIFFERENTIAL EQUATIONS. We are grateful to the Scientific Committee: J . L. Ericksen D. Kinderlehrer H. Brezis C. Dafermos for their dedication and hard work in rleveloping an imaginative, stimulating, and productive year-long program. George R. Sell Hans Weinherger PREFACE The papers in this volume were presented at a workshop on homogenization of differential equations and the determination of effective moduli of materials and media, primarily in the context of continuum theory. These areas are closely linked to a variety of phenomena, such as the elastic and dielectric responses of composites, and the effective properties of shales and soils. For instance, the ability to predict the effective stiffness response of a composite across a broad range of frequencies allows its performance under given circumstances to be assessed by means of nondestructive testing. A fundamental mathematical tool is homogenization, the study of partial differential equations with rapidly varying coefficients or boundary conditions. The recent alliance of homogenization with optimal design has stimulated the development of both fields. The presentations at the workshop emphasized recent advances and open questions.

IRIA-LABORIA + has organized, this year, an International Conference on Control Theory, Numerical Methods and Computer Systems Modelling. This meeting which was sponsored by the International Federation for Information Proce s sing (IFIP) and by the European Institute for Advanced Studies in Management, took place in June (17-21) with the participation of more than 200 specialists among which 55 participants were repre senting 12 different countrie s. This volume of the Springer-Verlag Series "Lecture Notes" contains the lectures presented during the meeting and demonstrates the high interest of the research which is actually carried out in these fields. We specially wish to thank Monsieur DANZIN, Director of IRIA, for the interest he has shown for this Symposium, Professor BALAKRISHNAN who has arranged for IFIP to sponsor our meeting and Professor GRAVES, Director of the European Institute for Advanced Studies in Management for his support. The IRIA Public Relations has been of a great assistance to the Organization Committee and we wish to thank Mademoiselle BRICHETEAU and her staff for their contribution.At last we expre s s our gratitude to the Se s sions Chairmen and all the speakers for the very interesting discussions they have directed. A. BENSOUSSAN and J.L. LIONS +Institut de Recherche d'Informatique et d'Automatique Laboratoire de Recherche de l'IRIA PREFACE L'IRIA-LABORIA + a organise cette annee une Conference Internationale sur la Theorie du Contrale, les Methodes Nurneriques et la Modelisation des Systernes Informatiques.

IRIA LABORIA, Institut de Recherche d'Informatique et d'Automatique

INRIA, Institut National de Recherche en Informatique et en Automatique

INRIA, Institut National de Recherche en Informatique et en Automatique

These lecture notes by very authoritative scientists survey recent advances of mathematics driven by industrial application showing not only how mathematics is applied to industry but also how mathematics has drawn benefit from interaction with real-word problems.The famous David Report underlines that innovative high technology depends crucially for its development on innovation in mathematics. The speakers include three recent presidents of ECMI, one of ECCOMAS (in Europe) and the president of SIAM.

1. We begin by giving a simple example of a partial differential inequality that occurs in an elementary physics problem. We consider a fluid with pressure u(x, t) at the point x at the instant t that 3 occupies a region Q oflR bounded by a membrane r of negligible thickness that, however, is semi-permeable, i. e., a membrane that permits the fluid to enter Q freely but that prevents all outflow of fluid. One can prove then (cf. the details in Chapter 1, Section 2.2.1) that au (aZu azu aZu) (1) in Q, t>o, -a - du = g du = -a z + -a z + -a z t Xl X X3 z l g a given function, with boundary conditions in the form of inequalities u(X,t»o => au(x,t)/an=O, XEr, (2) u(x,t)=o => au(x,t)/an?:O, XEr, to which is added the initial condition (3) u(x,O)=uo(x). We note that conditions (2) are non linear; they imply that, at each fixed instant t, there exist on r two regions r~ and n where u(x, t) =0 and au (x, t)/an = 0, respectively. These regions are not prescribed; thus we deal with a "free boundary" problem.