Table Of ContentProgress 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
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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
