This book contains the written versions of lectures delivered since 1997 in the well-known weekly seminar on Applied Mathematics at the College de France in Paris, directed by Jacques-Louis Lions. It is the 14th and last of the series, due to the recent and untimely death of Professor Lions.
The texts in this volume deal mostly with various aspects of the theory of nonlinear partial differential equations. They present both theoretical and applied results in many fields of growing importance such as Calculus of variations and optimal control, optimization, system theory and control, operations research, fluids and continuum mechanics, nonlinear dynamics, meteorology and climate, homogenization and material science, numerical analysis and scientific computations
The book is of interest to everyone from postgraduate, who wishes to follow the most recent progress in these fields.

"18.07"

Pierre Grisvard, one of the most distinguished French mathematicians, died on April 22, 1994. A Conference was held in November 1994 out of which grew the invited articles contained in this volume. All of the papers are related to functional analysis applied to partial differential equations, which was Grisvard's specialty. Indeed his knowledge of this area was extremely broad. He began his career as one of the very first students of Jacques Louis Lions, and in 1965, he presented his "State Thesis" on interpolation spaces, using in particular, spectral theory for linear operators in Banach spaces. After 1970, he became a specialist in the study of optimal regularity for par tial differential equations with boundary conditions. He studied singulari ties coming from coefficients, boundary conditions, and mainly non-smooth domains, and left a legacy of precise results which have been published in journals and books. Pierre Grisvard spent most of his career as a full professor at the University of Nice, where he started in 1967. For shorter or longer periods, he visited several foreign countries, and collaborated with some of the most famous mathematicians in his field. He was also an excellent organizer and directed a large number of Ph.D. students. Finally, this volume contains a bibliography of Grisvard's works as well as one paper which he wrote and which has not been published before."

INRIA, Institut National de Recherche en Informatique et en Automatique

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.

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, to => 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.

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.

From the foreword: "This volume contains most of the 113 papers presented during the Eighth International Conference on Analysis and Optimization of Systems organized by the Institut National de Recherche en Informatique et en Automatique. Papers were presented by speakers coming from 21 different countries. These papers deal with both theoretical and practical aspects of Analysis and Optimization of Systems. Most of the topics of System Theory have been covered and five invited speakers of international reputation have presented the new trends of the field."

INRIA, Institut National de Recherche en Informatique et en Automatique

INRIA, Institut National de Recherche en Informatique et en Automatique

From the foreword: "This volume contains most of the 113 papers presented during the Eighth International Conference on Analysis and Optimization of Systems organized by the Institut National de Recherche en Informatique et en Automatique. Papers were presented by speakers coming from 21 different countries. These papers deal with both theoretical and practical aspects of Analysis and Optimization of Systems. Most of the topics of System Theory have been covered and five invited speakers of international reputation have presented the new trends of the field."

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

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

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