Ambrosio Dancer Calculus of Variations and Partial Differential Equations
Springer-Verlag Berlin Heidelberg GmbH
L. Ambrosio N. Dancer Calculus of Variations and Partial Differential Equations Topics on Geometrical Evolution Problems and Degree Theory Edited by G. Buttazzo,A. Marino, M. K. V. Murthy Springer
Authors and Editors Luigi Ambrosio Scuola Normale Superiore Piazza Cavalieri 7 56100 Pisa, Italy Norman Dancer University of Sydney School of Mathematics NSW 2006 Sydney, Australia Giuseppe Buttazzo Antonio Marino M. K. V. Murthy Universita di Pisa Via Buonarroti 2 56127 Pisa, Italy Mathematics Subject Classification (1991): 60J15, 30005, 35Q15, 30F10, 60K25 Library of Congress Cataloging-in-Publication Data Calculus of variatious and partial differential equations : topics on geometrical evolution problems and degree theory I edited by G. Buttazzo, A. Marino, M.K. V. Murthy. p.cm. Includes bibliographical references and index. ISBN 978-3-540-64803-1 ISBN 978-3-642-57186-2 (ebook) DOI 10.1007/978-3-642-57186-2 1. Calculus ofvariatious. 2. Topological degree. 3. Differential equatious, Partial. 4. Mathematical physics. 1. Buttazzo, Giuseppe. II. Marino, A. ill. Murthy, M. K. V. (M. K. Venkatesha) IV. Summer School on "Calculus ofvariatious and Partial Differential Equatious" (1996 : Pisa, Italy) QC20.7.C3 C35 1999 SIS'.64--dc21 98-054672 This work is subject to copyright. AII rights are reserved, whether the whole or part of the material is concerned, specifically th. rights of translation, reprinting, reuse of iilustrations, recitation, broad-casting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its CUITent version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg 2000 Cover design: design.& production GmhH, Heidelberg SPIN: 10688575 41/3143-5 4 3 2 1 o - Printed on acid-free paper
Preface The project of editing this book originated in a two-week Summer School on "Calculus of Variations and Partial Differential Equations" which was held in Pisa in September 1996. The School was attended by more than 150 participants, mostly Ph. D. students, but also by several more experienced scientists. The success achieved encouraged us to collect into a book the course material of the two main lecturers and the complementary articles of the eight other speakers on related subjects. Special importance was attached to the self-contained presentation of these rather advanced research topics, and we hope that our work will contribute to filling the gap between the level of basic courses and more advanced research. Pisa, March 3, 1998 G. Buttazzo A. Marino M.K.V. Murthy
Table of Contents Preface... V Table of Contents... VII List of Authors... IX I Geometric Evolution Problems Introduction................................................... 3 Geometric evolution problems, distance function and viscosity solutions........................................................ 5 L. Ambrosio Variational models for phase transitions, an approach via r -convergence.. 95 G. Alberti Some aspects of De Giorgi's barriers for geometric evolutions... 115 G. Bellettini, M. NotJaga Partial Regularity for Minimizers of Free Discontinuity Problems with p-th Growth... 153 A. Leaci Free discontinuity problems and their non-local approximation... 171 A. Braides II Degree Theory on Convex Sets and Applications to Bifurcation Introduction... 183 Degree theory on convex sets and applications to bifurcation... 185 E. N. Dancer Nonlinear elliptic equations involving critical Sobolev exponents... 227 D. Passaseo On the existence and multiplicity of positive solutions for semilinear mixed and Neumann elliptic problems... 243 G. Cerami
VIII Table of Contents Solitons and Relativistic Dynamics... 259 V. Benci, D. Fortunato An algebraic approach to nonstandard analysis... 285 V. Benci References... 327 Index... 345
List of Authors Giovanni Alberti Via Buonarroti 2 Universita. di Pisa 56127 Pisa (Italy) e-mail: alberti@dm.uniplit Luigi Ambrosio Scuola Normale Superiore Classe di Scienze Piazza dei Cavalieri, 7 56126 Pisa (Italy) e-mail: ambrosio@sns.it Giovanni Bellettini Universita. di Rama "Tor Vergata" Via della Ricerca Scientmca 00133 Rama (Italy) e-mail: bellettini@axp.mat.uniroma2.it Vieri Benci Applicata "U. Dini" Universita. di Pisa, Via Bonanno 25/b 56126 Pisa (Italy) e-mail: benci@dma.uniplit Andrea Braides SISSA Via Beirut 4 34014 Trieste (Italy) e-mail: braides@sissa.it E. N. Dancer School of Mathematics and Statistics University of Sydney NSW 2006 (Australia) e-mail: dancer@maths.su.oz.au Antonio Leaci Universita. di Leece Via Arnesano 73100 Leece (Italy) e-mail: leaci@ingleol.unile.it Giovanna Cerami e Applicazioni Universita. di Palermo Via Archirafi, 34 90123 Palermo (Italy) e-mail: cerami@ipamat.math.unipa.it Donato Fortunato Universita. di Bari, Via Orabona 4 70125 Bari (Italy) e-mail: fortunat@pascal.dm.uniba.it Matteo Novaga Universita. di Pisa Via Buonarroti, 2 56127 Pisa (Italy) e-mail: novaga@dm.uniplit Donato Passaseo Universita. di Leece Via Arnesano 73100 Leece (Italy) e-mail: passaseo@ilenic.unile.it