Progress in Nonlinear Differential Equations and Their Applications Volume 54

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2 Progress in Nonlinear Differential Equations and Their Applications Volume 54 Editor Haim Brezis Universite 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 PL. Lions, University of Paris IX Jean Mahwin, Universite Catholique de Louvain Louis Nirenberg, New York University Lambertus Peletier, University of Leiden Paul Rabinowitz, University of Wisconsin, Madison John Toland, University of Bath

3 Nonlinear Equations: Methods, Models and Applications Daniela Lupo Carlo D. Pagani Bernhard Ruf Editors Springer Basel AG

4 Editors' addresses: Daniela Lupo Dipartimento di Matematica Politecnico di Milano Piazza Leonardo da Vinci, Milano Italy Carlo D. Pagani Dipartimento di Matematica Politecnico di Milano Piazza Leonardo da Vinci, Milano Italy Bernhard Ruf Dipartimento di Matematica Universitä degli Studi di Milano Via Saldini, Milano Italy Mathematics Subject Classification: 35-xx, 46-xx, 49-xx, 92D20 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 This work is subject to copyright. Allrightsare 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 Springer Basel AG Originally published by Birkhäuser Verlag, Basel in 2003 Softcover reprint of the hardcover 1st edition 2003 Printed on acid-free paper produced of chlorine-free pulp. TCF ISBN ISBN (ebook) DOI / science. com

5 Contents Preface Mohameden Quld Ahmedou A Riemann Mapping Type Theorem in Higher Dimensions, Part I: The Conformally Flat Case with Umbilic Boundary Vll 1 Vieri Benci and Giulia Menconi Computable Information Content and a Simple Application to the Study of DNA Lucio Boccardo and Benedetta Pellacci Bounded Positive Critical Points of Some Multiple Integrals of the Calculus of Variations Marta Calanchi and Bernhard Ruf Hilbert Type Numbers for Polynomial ODE's Paolo Caldiroli and Roberta Musina 2-type Parametric Surfaces with Prescribed Mean Curvature and Minimal Energy Italo Capuzzo Dolcetta Representations of Solutions of Hamilton-Jacobi Equations Gabriella Caristi Nonexistence of Global Solutions of Higher Order Evolution Inequalities in ]RN 91 s. Cingolani and G. Vannella Morse Index Computations for a Class of Functionals Defined in Banach Spaces Monica Clapp A Global Compactness Result for Elliptic Problems with Critical Nonlinearity on Symmetric Domains Marco Degiovanni Variational Methods for Functionals with Lack of Strict Convexity Vieri Benci and Donato Fortunato Some Remarks on the Semilinear Wave Equation 141

6 Vi Contents Rafael Jose Iorio, Jr. Unique Continuation Principles for Some Equations of Benjamin-Ono Type H.A. Biagioni and F. Linares Well-posedness Results for the Modified Zakharov-Kuznetsov Equation Orlando Lopes A Class of Isoinertial One Parameter Families of Selfadjoint Operators J. Horak and P. J. McKenna Traveling Waves in Nonlinearly Supported Beams and Plates A.M. Micheletti and D. Visetti Solitary Waves Solutions of a Nonlinear Schrodinger Equation C. O. Alves, P. C. Carriiio and O.H. Miyagaki Nontrivial Solutions of a Class of Quasilinear Elliptic Problems Involving Critical Exponents Filomena Pacella and P.N. Srikanth Solutions of Semilinear Problems in Symmetric Planar Domains - ODE Behavior and Uniqueness of Branches Paul H. Rabinowitz and Ed Stredulinsky Solutions of an Allen-Cahn Model Equation Giorgio Talenti Some Equations of Non-geometrical Optics Speakers

7 Preface This volume contains research articles from the field of Nonlinear Differential Equations which result from the "Workshop on Nonlinear Analysis and Applications" held in Bergamo on July 9 to 13, 200l. This workshop was the third edition of a meeting which first took place in Campinas in 1996 and was founded in part upon scientific cooperation, already well initiated, between some participants, on specific problems in Nonlinear Analysis, and in part upon the whish to extend such cooperation to other researchers and to other topics. The scientific collaboration between Italy and Brazil is not new; it dates back at least to the thirties, and includes, among others, the name of Luigi Fantappie, just to mention only one of the earliest Italians that developed part of their scientific activity in Brazil. If the first workshop had mainly an informal character, the second, which took place in 1998 again in Campinas, already had the structure and the breath of a true international congress. At this point it was the Italians turn to organize the third meeting. The main purpose of the conference was to provide a forum for the discussion of recent work and modern trends in various fields of Nonlinear Analysis. About 130 researchers coming from 17 countries attended the conference. Their participation was remarkable both for the number and for the quality; we take here the opportunity to thank them all for their contribution to the scientific activities and to the friendly atmosphere of the meeting, which was further enhanced by the beautiful surroundings of the Citta Alta of Bergamo. The classical topics of Nonlinear Analysis such as Calculus of Variations, Variational Inequalities, Critical Point Theory and their use in various aspects of the study of elliptic differential equations and systems, equations of Hamilton Jacobi, Schrodinger and Navier-Stokes, and free boundary problems were considered. Moreover various models were focused upon: travelling waves in supported beams and plates, vortex condensation in electroweak theory, information theory, non-geometrical optics, Dirac-Fock models for heavy atoms... This volume contains the contributions of 17 of the 50 invited speakers. A complete list of speakers is included. The meeting took place in Bergamo at the Palazzo dell'universita and was organized by Daniela E. Lupo and Carlo D. Pagani from the Dipartimento di Matematica "F. Brioschi", Politecnico di Milano, Bernhard Ruf from the Dipartimento di Matematica "F. Enriques", U niversita di Milano, Djairo de Figueiredo from UNICAMP, University of Campinas and Rafael Iorio from IMPA, Rio de Janeiro.

8 viii Preface The congress would not have been possible without the financial support of several institutions. We wish here to thank: - Dipartimento di Matematica "F. Brioschi", Politecnico di Milano, - Dipartimento di Matematica "F. Enriques", Universita di Milano, - Fondi MURST: Metodi variazionali ed equazioni differenziali nonlineari, Metodi variazionali e topologici nello studio di fenomeni nonlineari, - Gruppo INDAM per l' Analisi Matematica, la Probabilita e le loro Applicazioni. We also wish to thank the Universita di Bergamo for their hospitality, the Azienda Promozione Turistica Bergamo for their collaboration and Miss Pamela Palazzini who efficiently provided secretarial assistance during the conference. Daniela Lupo Carlo Pagani Bernhard Ruf