Variational Methods: Proceedings of a Conference Paris, June 1988 PDF
Preview Variational Methods: Proceedings of a Conference Paris, June 1988
Progress in Nonlinear Differential Equations and Their Applications Volume 4 Editor Haim Brezis Universite Pierre et Marie Curie Paris and Rutgers University New Brunswick Editorial Board A. Bahri, Rutgers University, New Brunswick John Ball, Heriot-Watt University, Edinburgh Luis Cafarelli, Institute for Advanced Study, Princeton Michael Crandall, University of California, Santa Barbara Mariano Giaquinta, University of Aorence David Kinderlehrer, Carnegie-Mellon University, Pittsburgh Robert Kohn, New York University P.L. Lions, University of Paris IX Louis Nirenberg, New York University Lambertus Peletier, University of Leiden Paul Rabinowitz, University of Wisconsin, Madison Variational Methods Proceedings of a Conference Paris, June 1988 Edited by Henri Berestycki Jean-Michel Coron 1v ar Ekeland With 13 Illustrations 1990 Springer Science+Business Media, LLC Henri Berestycki Ivar Ekeland Math~matiques CEREMADE Universit~ Pierre et Marie Curie Universit~ de Paris IX-Dauphine 75252 Paris Cedex 05 75775 Paris Cedex 16 France France Jean-Michel Coron ~partement de Mathematiques, Bâtiment 425 Universit~ de Paris-Sud Centre d'Orsay 91405 Orsay Cedex France Printed on acid-free paper. © Springer Science+Business Media New York 1990 Originally published by Birkhlluser Boston in 1990 Softcover reprint of the hardcover 1s t edition 1990 AH rights reserved. No part of this publication may be reproduced, stored in a retrieva! system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or other wise, without prior permission of the copyright owner. Permission to photocopy for interna! or personal use, or the internal or personal use of specific clients, is granted by Birkhiiuser Boston, loc., for libraries and other users registered with the Copy right Clearance Center (CCC), provided that the base fee of $0.00 per copy, plus $0.20 per page is paid directly to CCC, 21 Congress Street, Salem, MA 01970, V.S.A. Special requests should be addressed directly to Springer Science+Business Media, LLC. 3452-5/90 $0.00 + .20 ISBN 978-1-4757-1082-3 ISBN 978-1-4757-1080-9 (eBook) DOI 10.1007/978-1-4757-1080-9 Camera-ready copy prepared by the authors. 9 8 7 6 5 432 1 PREFACE In the framework of the "Annee non lineaire" (the special nonlinear year) sponsored by the C.N.R.S. (the French National Center for Scien tific Research), a meeting was held in Paris in June 1988. It took place in the Conference Hall of the Ministere de la Recherche and had as an organizing theme the topic of "Variational Problems." Nonlinear analysis has been one of the leading themes in mathemat ical research for the past decade. The use of direct variational methods has been particularly successful in understanding problems arising from physics and geometry. The growth of nonlinear analysis is largely due to the wealth of ap plications from various domains of sciences and industrial applica tions. Most of the papers gathered in this volume have their origin in applications: from mechanics, the study of Hamiltonian systems, from physics, from the recent mathematical theory of liquid crystals, from geometry, relativity, etc. Clearly, no single volume could pretend to cover the whole scope of nonlinear variational problems. We have chosen to concentrate on three main aspects of these problems, organizing them roughly around the following topics: 1. Variational methods in partial differential equations in mathemat ical physics 2. Variational problems in geometry 3. Hamiltonian systems and related topics. The conference on "Variational Problems" was sponsored by the fol- lowing institutions: -CNRS (Centre National de la Recherche Scientifique) -DRET (Direction de la Recherche et des Etudes Techniques) -CEA (Commissariat a l'Energie Atomique) -CNES (Centre National d'Etudes Spatiales) -SMAI (Societe de MatMmatiques Appliquees et Industrielles) It has also benefitted from the support of Universities Paris IX (Dau phine), Paris XI (Orsay), and Paris XIII (Villetaneuse). We thank the Steering Committee of the "Annee non lineaire," Pro fessor C. Bardos, O. Pironneau, and J.-P. Puel, for their moral and material help. We are very thankful to Michel Broue and the Departe ment de MatMmatiques et Informatique at the Ecole Normale Super ieure for its warm support throughout the organization. We wish to v vi Preface express our appreciation to Mrs. Ann Kostant for her assistance in the preparation of the manuscripts for the volume and to the Birkhauser staff for their constant support and patience. And lastly, we wish to express gratitude to Mrs. A. Iapteff and Mrs. C. Pastadjian for their kind and competent secretarial assistance. Paris, November 1989 Henri Berestycki Jean-Michel Coron Ivar Ekeland CONTENTS Preface............................................................................................... v Part I: Partial Differential Equations and Mathematical Physics 1. The (Non)continuity of Symmetric Decreasing Rearrangement .......................................................................... 3 Frederick J. Almgren, Jr., and Elliott H. Lieb 2. Counting Singularities in Liquid Crystals .............................. 17 Frederick J. Almgren, Jr., and Elliott H. Lieb 3. Relaxed Energies for Harmonic Maps ..................................... 37 F. Bethuel, H. Brezis, and J.M. Coron 4. Topological Results on Fredholm Maps and Application to a Superlinear Differential Equation ........................................... 53 Vittorio Cafagna 5. Existence Results for Some Quasilinear Elliptic Equations .. 61 Henrik Egnell 6. A New Setting for Skyrme's Problem ...................................... 77 Maria J. Esteban 7. Relative Category and the Calculus of Variations ................. 95 G. Fournier and M. Willem 8. Point and Line Singularities in Liquid Crystals .................... 105 Robert M. Hardt 9. The Variety of Configurations of Static Liquid Crystals ....... 115 Robert Hardt, David Kinderlehrer, and Fang Hua Lin 10. Existence of Multiple Solutions of Semilinear Elliptic Equations in RN......................................................................... 133 Yanyan Li vii viii Contents 11. Lagrange Multipliers, Morses Indices and Compactness ....... 161 P.L. Lions 12. Elliptic Equations with Critical Growth and Moser's Inequality........ .......... ........ ............ ............................................. 185 J.B. McLeod and L.A. Peletier 13. Evolution Equations with Discontinuous Nonlinearities and Non-Convex Constraints........................................................... 197 A. Marino and C. Saccon 14. Nonlinear Variational Two-Point Boundary Value Problems. ................... '" ........................... , ..... .......... .......... ... ...... 209 J. Mawhin 15. Some Relative Isoperimetric Inequalities and Applications to Nonlinear Problems .................................................................. 219 Filomena Pacella Part II: Partial Differential Equations and Problems in Geometry 16. Approximation in Sobolev Spaces Between Two Manifolds and Homotopy Groups............................................................... 239 Fabrice Bethuel 17. The "Magic" of Weitzenbock Formulas ................................... 251 Jean-Pierre Bourguignon 18. A Remark on Minimal Surfaces with Comers ...... ....... .......... 273 Michael Griiter 19. Extremal Surfaces of Mixed Type in Minkowski Space 283 Rn+l Gu Chaohao (C.H. Gu) 20. Convergence of Minimal Submanifolds to a Singular Variety 297 Robert Gulliver 21. Harmonic Diffeomorphisms Between Riemannian Manifolds 309 Frederic Helein 22. Surfaces of Minimal Area Supported by a Given Body in 1R3 ............................................................................................ 319 G. Mancini and R. Musina Contents ix 23. Calibrations and New Singularities in Area-Minimizing Surfaces: A Survey........ ........ .... ............ ...... ....... ....... ........ ........ 329 Frank Morgan 24. Harmonic Maps with Free Boundaries.................................... 343 Klaus Steffen 25. Global Existence of Partial Regularity Results for the Evolution of Harmonic Maps.... .......... ........ ..... ....... ........ ... ....... 359 Michael Struwe Part III: Hamiltonian Systems 26. Multiple Periodic Trajectories in a Relativistic Gravitational Field.................................................................... 373 A. Ambrosetti and U. Bessi 27. Periodic Solutions of Some Problems of 3-Body Type ............ 383 Abbas Bahri and Paul H. Rabinowitz 28. Periodic Solutions of Dissipative Dynamical Systems ........... 395 Vieri Benci and Marco Degiovanni 29. Periodic Trajectories for the Lorentz-Metric of a Static Gravitational Field.................................................................... 413 Vieri Benci and Donato Fortunato 30. Morse Theory for Harmonic Maps ........................................... 431 Kung-Ching Chang 31. Morse Theory and Existence of Periodic Solutions of Elliptic Type .......... .......... ..... ..... ............. ....... .......... .......... ... ..... 447 B. D'Onofrio and I. Ekeland 32. Periodic Solutions of a Nonlinear Second Order System ....... 455 L. Lassoued 33. Existence of Multiple Brake Orbits for a Hamiltonian System ........................................................................................ 469 Andrzej Szulkin PART I Partial Differential Equations and Mathematical Physics
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