Progress in Nonlinear Differential Equations and Their Applications Volume 68

Transcription

1 Progress in Nonlinear Differential Equations and Their Applications Volume 68 Editor Haim Brezis Université Pierre et Marie Curie Paris and Rutgers University New Brunswick, N.J. Editorial Board Antonio Ambrosetti, Scuola Internazionale Superiore di Studi Avanzati, Trieste A. Bahri, Rutgers University, New Brunswick Felix Browder, Rutgers University, New Brunswick Luis Cafarelli, Institute for Advanced Study, Princeton Lawrence C. Evans, University of California, Berkeley Mariano Giaquinta, University of Pisa David Kinderlehrer, Carnegie-Mellon University, Pittsburgh Sergiu Klainerman, Princeton University Robert Kohn, New York University P.L. Lions, University of Paris IX Jean Mahwin, Université Catholique de Louvain Louis Nirenberg, New York University Lambertus Peletier, University of Leiden Paul Rabinowitz, University of Wisconsin, Madison John Toland, University of Bath

2 Variational Problems in Materials Science Gianni dal Maso Antonio DeSimone Franco Tomarelli Editors Birkhäuser Basel Boston Berlin

3 Editors: Gianni dal Maso Antonio DeSimone Scuola Internazionale Superiore di Studi Avanzati (SISSA) Via Beirut Trieste Italy dalmaso@sissa.it desimone@sissa.it Franco Tomarelli Dipartimento di Matematica Francesco Brioschi Politecnico di Milano Piazza Leonardo da Vinci Milano Italy fratom@mate.polimi.it 2000 Mathematics Subject Classification 35-06, 47J40, 49-06, 74-06, 76A15 A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at < ISBN Birkhäuser Verlag, Basel Boston Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use whatsoever, permission from the copyright owner must be obtained Birkhäuser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced of chlorine-free pulp. TCF Printed in Germany ISBN 10: e-isbn: ISBN 13:

4 Contents Preface... Invited Lectures... vii ix K. Barmak, D. Kinderlehrer, I. Livshits and S. Ta asan Remarks on a Multiscale Approach to Grain Growth in Polycrystals... 1 A. Braides and V. Chiadò Piat Another Brick in the Wall G. Buttazzo and M.O. Rieger Variational Problems for Functionals Involving the Value Distribution G. Del Piero Bi-modal Cohesive Energies G. De Matteis and E.G. Virga Criterion for Tricritical Points in Liquid Crystal Phases N. Desenzani and I. Fragalà Asymptotics of Boundary Value Problems for Supercritical Ginzburg-Landau Energies G.A. Francfort An Introduction to H-measures and Their Applications A. Garroni and G. Palatucci A Singular Perturbation Result with a Fractional Norm D. Percivale and F. Tomarelli Smooth and Creased Equilibria for Elastic-plastic Plates and Beams P. Podio-Guidugli On Concentrated Contact Interactions A. Visintin Discontinuous Hysteresis and P.D.E.s List of Registered Participants

5 Preface This volume contains the proceedings of the international workshop Variational Problems in Materials Science, which was jointly organized by the International School for Advanced Studies (SISSA) of Trieste and by the Dipartimento di Matematica Francesco Brioschi of the Politecnico di Milano. The conference took place at SISSA from September 6 to 10, The study of variational problems in materials science has a long history, and it has contributed a lot in shaping our understanding on how materials work and perform. There is, however, a recent renewed interest in this subject as a consequence of the fruitful interaction between mathematical analysis and the modelling of new, technologically advanced materials. On one hand, a sizable group of analysts has found in materials science a valuable source of inspiration for new variational theories and interesting problems. On the other hand, workers in the fields of theoretical, applied, and computational mechanics are increasingly using innovative variational techniques. The workshop intended to review some of the recent advances stemming from the successful interaction between the two communities, and to identify promising areas for further cooperation. Talks were devoted to a wide spectrum of analytical techniques and of physical systems and phenomena. They included the study of BV vector fields, path functionals over Wasserstein spaces, variational approaches to quasi-static evolution, free-discontinuity problems with applications to fracture and plasticity, systems with hysteresis or with interfacial energies, evolution of interfaces, multiscale analysis in ferromagnetism and ferroelectricity, variational techniques for the study of crystal plasticity, of dislocations, and of concentrations in Ginzburg Landau functionals, concentrated contact interactions, and phase transitions in biaxial liquid crystals. This volume collects contributions authored or co-authored by 11 out of the 20 speakers invited to deliver lectures at the workshop. They all contain original results in fields which are at the forefront of current research, and in rapid evolution. More than sixty researchers with quite different disciplinary expertise (calculus of variations, computational mechanics, continuum mechanics, geometric measure theory, materials science) attended the workshop. The list of participants appears at the end of the volume. We thank them all for their lively and friendly contributions to the scientific discussions, and to the pleasant atmosphere of the event.

6 viii Preface The meeting enjoyed the financial support of SISSA, Politecnico di Milano, MIUR (through the National Research Group Calculus of Variations ), and of the Istituto Nazionale di Alta Matematica Francesco Severi through the groups GNAMPA and GNFM. We are grateful to all these Institutions for their contributions. Most of the burden of organizing the logistics fell on Andrea Brunetta. We wish to warmly thank him for his excellent job. Finally, it is our pleasure to thank once again all the lecturers, and we are especially grateful to those who contributed to this volume. Milano and Trieste, November 15, 2005 Gianni Dal Maso Antonio DeSimone Franco Tomarelli

7 Invited Lectures Luigi Ambrosio (Scuola Normale Superiore, Pisa) On the Cauchy problem for BV vector fields Kaushik Bhattacharya (Caltech) Ferroelectric ceramics Andrea Braides (Università di Roma Tor Vergata) Surface energies in discrete systems Giuseppe Buttazzo (Università dipisa) Path functionals over Wasserstein spaces Antonin Chambolle (École Polytechnique) On the initiation of cracks in Griffith s model Gianpietro Del Piero (Università di Ferrara) Bi-modal cohesive energies Irene Fonseca (Carnegie Mellon University) Variational methods in materials science: the study of foams and quantum dots Ilaria Fragalà (Politecnico di Milano) Concentration of Ginzburg-Landau energies with supercritical growth Gilles Francfort (Université Paris XIII) A variational approach to brittle damage evolution Adriana Garroni (Università diromalasapienza) Gamma-convergence of a phase field model for dislocations Richard D. James (University of Minnesota) Hysteresis and geometry: a way to search for new materials with unlikely physical properties David Kinderlehrer (Carnegie Mellon University) Approaches to understanding interface evolution in polycrystals

8 x Invited Lectures Stefan Müller (Max Planck Institute for Mathematics in the Sciences, Leipzig) A hierarchy of plate and shell theories derived from three-dimensional nonlinear elasticity by Gamma-convergence Michael Ortiz (Caltech) Discrete crystal plasticity Felix Otto (University of Bonn) Multiscale analysis in micromagnetics Danilo Percivale (Università digenova) Regular and non regular solutions for elastic plastic beams and plates Paolo Podio-Guidugli (Università di Roma Tor Vergata) Phenomenology of concentrated contact interactions in simple continuous bodies Lev Truskinovsky (École Polytechnique) Quasi-static deformation of a system with nonconvex energy from a perspective of dynamics Epifanio Virga (Università dipavia) Phase transitions in biaxial nematic liquid crystals Augusto Visintin (Università ditrento) Quasilinear hyperbolic equations with hysteresis