Elliptic and Parabolic Problems: A Special Tribute to the Work of Haim Brezis PDF
Preview Elliptic and Parabolic Problems: A Special Tribute to the Work of Haim Brezis
Progress in Nonlinear Differential Equations and Their Applications Volume 63 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 Elliptic and Parabolic Problems A Special Tribute to the Work of Haim Brezis Catherine Bandle Henri Berestycki Bernhard Brighi Alain Brillard Michel Chipot Jean-Michel Coron Carlo Sbordone Itai Shafrir Vanda Valente Giorgio Vergara Caffarelli Editors Birkhäuser Basel Boston Berlin ● ● Editors: Catherine Bandle Jean-Michel Coron Mathematisches Institut Département de mathématiques Universität Basel Bâtiment 425 Rheinsprung 21 Université de Paris-Sud 4051 Basel, Switzerland 91405 Orsay, France [email protected] [email protected] Henri Berestycki Carlo Sbordone Ecole des hautes études Dipartimento di Matematica e Applicazioni en sciences sociales (EHESS) Università di Napoli “Federico II” CAMS Via Cintia 54, Boulevard Raspail 80126 Napoli, Italy 75006 Paris, France [email protected] [email protected] Bernard Brighi Itai Shafrir Université de Haute-Alsace Department of Mathematics Faculté des Sciences et Techniques Technion – Israel Institute of Technology 4 rue des frères Lumières 32000 Haifa, Israel 68093 Mulhouse Cedex, France [email protected] [email protected] Alain Brillard Vanda Valente Université de Haute-Alsace CNR-IAC Laboratoire de Gestion Viale del Policlinico, 137 des Risques et Environnement 00161 Roma, Italy 25, rue de Chemnitz [email protected] 68200 Mulhouse, France [email protected] Giorgio Vergara Caffarelli Michel Chipot Dipartimento di metodi e modelli Universität Zürich matematici per le Scienze Aplicate Angewandte Mathematik Università di Roma “La Sapienza” Winterthurerstr. 190 Via A. Scarpa 16 8057 Zürich, Switzerland 00161 Roma, Italy [email protected] [email protected] 2000 Mathematics Subject Classification 35Bxx, 35Jxx, 35Kxx 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 <http://dnb.ddb.de>. ISBN 3-7643-7249-4 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. © 2005 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: 3-7643-7249-4 ISBN 13: 978-3-7643-7249-1 9 8 7 6 5 4 3 2 1 www.birkhauser.ch Contents Preface .................................................................. ix Andrew Acker One-Layer Free Boundary Problems with Two Free Boundaries ...... 1 Catherine Bandle and Simon Stingelin New Numerical Solutions for the Brezis-Nirenberg Problem on Sn ...................................................... 13 H. Beira˜o da Veiga On some Boundary Value Problems for Incompressible Viscous Flows with Shear Dependent Viscosity ...................... 23 A. Bermu´dez, R. Leira, M.C. Mun˜iz and F. Pena Radiative Heat Transfer in Silicon Purification ...................... 33 Said Berrimi and Salim A. Messaoudi A Decay Result for a Quasilinear Parabolic System ................. 43 S.I. Betelu´, M.A. Fontelos and U. Kindela´n The Shape of Charged Drops: Symmetry-breaking Bifurcations and Numerical Results ................................. 51 A. Blanchet, J. Dolbeault and R. Monneau On the One-dimensional Parabolic Obstacle Problem with Variable Coefficients ........................................... 59 Lucio Boccardo Hardy Potentials and Quasi-linear Elliptic Problems Having Natural Growth Terms ...................................... 67 Bernard Brighi and Jean-David Hoernel Recent Advances on Similarity Solutions Arising During Free Convection ..................................... 83 R. Brossard, J.-P. Loh´eac and M. Moussaoui Rellich Relations for Mixed Boundary Elliptic Problems ............. 93 A. Can˜ada, J.A. Montero and S. Villegas Lyapunov-type Inequalities and Applications to PDE ............... 103 vi Contents A. Can˜ada and D. Ruiz Gaeta 2004. Elliptic Resonant Problems with a Periodic Nonlinearity ........................................ 111 Raffaela Capitanelli Harnack Inequality for p-Laplacians on Metric Fractals .............. 119 Ana Carpio Wave Propagationin Discrete Media ................................ 127 Thierry Cazenave, Fla´vio Dickstein and Fred B. Weissler A Solution of the Heat Equation with a Continuum of Decay Rates ........................................ 135 Claire Chainais-Hillairet and Yue-Jun Peng Finite Volume Scheme for Semiconductor Energy-transportModel ............................................ 139 Michel Chipot and Yitian Xie Asymptotic Behavior of Nonlinear Parabolic Problems with Periodic Data .................................................. 147 Gabriel Peyr´e and Laurent Cohen Geodesic Computations for Fast and Accurate Surface Remeshing and Parameterization ........................... 157 M. Comte On the Newton Body Type Problems ............................... 173 Jean-Michel Coron Some Open Problems on Water Tank Control Systems .............. 179 Juan D´avila and Marcelo Montenegro H¨older Estimates for Solutions to a Singular Nonlinear Neumann Problem ....................................... 189 U. De Maio and T.A. Mel’nyk Asymptotic Analysis of the Neumann Problem for the Ukawa Equation in a Thick Multi-structure of Type 3:2:2 ......... 207 J.I. D´ıaz On the Ha¨ım Brezis Pioneering Contributions on the Location of Free Boundaries .................................................. 217 J´eroˆme Droniou Fractal Conservation Laws: Global Smooth Solutions and Vanishing Regularization ............................................ 235 M. Escobedo Stationary and Self-similar Solutions for Coagulation and Fragmentation Equations ....................................... 243 Contents vii Alberto Fiorenza Orlicz Capacities and Applications to PDEs and Sobolev Mappings .............................................. 259 Uta Renata Freiberg and Maria Rosaria Lancia Energy Forms on Non Self-similar Fractals .......................... 267 Thierry Gallou¨et Measure Data and Numerical Schemes for Elliptic Problems ......... 279 Yuxin Ge Brezis-Nirenberg Problem and Coron Problem for Polyharmonic Operators ......................................... 291 Mohammed Guedda Local and Global Properties of Solutions of a Nonlinear Boundary Layer Equation ............................ 299 M.A. Herrero Mathematical Models of Aggregation: The Role of Explicit Solutions ...................................... 309 S. Kamin, H. Berestycki, L. Kagan and G. Sivashinsky Metastable Behavior of Premixed Gas Flames ....................... 319 Satyanad Kichenassamy Recent Progresson Boundary Blow-up .............................. 329 Jean Mawhin Maximum Principle for Bounded Solutions of the Telegraph Equation: The Case of High Dimensions .................. 343 Andrea Pascucci KolmogorovEquations in Physics and in Finance ................... 353 Sergio Polidoro Harnack Inequalities and Gaussian Estimates for a Class of Hypoelliptic Operators ................................... 365 Augusto C. Ponce How to Construct Good Measures ................................... 375 Vicen¸tiu Ra˘dulescu Bifurcation and Asymptotics for Elliptic Problems with Singular Nonlinearity .......................................... 389 Marc Oliver Rieger A Model for Hysteresis in Mechanics Using Local Minimizers of Young Measures .................................................. 403 Carlo Sbordone The Precise Lp-theory of Elliptic Equations in the Plane ............ 415 viii Contents V. Valente Essential Spectrum and Noncontrollability of Membrane Shells ...... 423 J.L. V´azquez The Porous Medium Equation. New Contractivity Results .......... 433 Laurent V´eron Large Solutions of Elliptic Equations with Strong Absorption ....... 453 Elvira Zappale Relaxation in Presence of Pointwise Gradient Constraints ........... 465 Preface The goal of these proceedings and of the meeting of Gaeta was to celebrate and honorthe mathematicalachievementsofHaimBrezis.The prodigiousinfluence of histalentandhispersonalityinthedomainofnonlinearanalysisisunanimouslyac- claimed!Thisimpactisvisibleinthehugenumberofhisformerstudents(dozens), students of former students (hundreds) and collaborators (hundreds). Thus the Gaeta meeting was, to some extent, the family reunion of part of this large com- munity sharing a joint interest in the field of elliptic and parabolic equations and pushing it to a very high standard. Italyhasalongtraditionandtasteforanalysisandwecouldnotfindabetter placeneitheramorecompletesupportfortherealisationofourproject.Wehaveto thankheretheuniversityofCassino,Napoli,Roma“laSapienza”,theGNAMPA- Istituto di Alta Matematica, CNR-IAC, MEMOMAT, RTN Fronts-Singularities, the commune of Gaeta. Additional founding came from the universities of Mul- house and Zu¨rich. Finally, we are grateful to Birkha¨user and Dr. Hempfling who allowed us to record the talks of this conference in a prestigious volume. The organizers ProgressinNonlinearDifferentialEquations andTheirApplications,Vol.63,1–12 (cid:1)c 2005Birkh¨auserVerlagBasel/Switzerland One-Layer Free Boundary Problems with Two Free Boundaries Andrew Acker Abstract. Westudytheuniquenessandsuccessiveapproximationofsolutions of a class of two-dimensional steady-state fluid problems involving infinite periodic flows between two periodic free boundaries, each characterized by a flow-speed condition related to Bernoulli’s law. 1. Introduction We study a class of generic double-free-boundary problems involving ideal fluid-flowsintwo-dimensional,periodic,strip-likeflow-domains.The periodicflow boundaries are both free, and each is characterized by a condition expressing the boundary flow-speed as a given function of position. Such conditions often arise from application of Bernoulli’s law. We study the questions of existence and uniquenessofsolutionsoftheabovedouble-free-boundaryproblems,aswellasthe successiveapproximationoftheirsolutionsbya trialfreeboundarymethodcalled the Operator Method. Theexistence,uniqueness,andconvergencequestionswereallfirststudiedin the contextofthecorrespondingone-free-boundaryproblem,inwhichonebound- ary component is specified. The one-free-boundary existence results essentially follow (in 2 dimensions) from a theorem of Beurling [9], and the corresponding uniqueness results follow from the Lavrentiev Principle (called the Lindelof Prin- ciple by M.A. Lavrentiev [12]). The analytical trial-free boundary convergence proof for this case was obtained by the author in [3, 4]. The author has also previously studied some aspects of the double-free-boundary problem, including existence (see [1, 2]). He has more recently generalized the Operator Method to various related free boundary problems (see [5, 6], for example). Other general- izations and modified versions have been studied by Meyer [13], Acker and Meyer [8], Kadakal [11], and Acker, Kadakal and Miller [7]. The operator method has been implemented numerically with some striking results in [7, 11]. The present
The list of books you might like
