Free Boundary Problems: Theory and Applications

460 Pages·2007·3.99 MB·English

by  Toyohiko Aiki

ISNM
International Series of Numerical Mathematics
Volume 154
Managing Editors:
Karl-Heinz Hoffmann, Bonn
D. Mittelmann, Tempe
Associate Editors:
R.E. Bank, La Jolla
H. Kawarada, Chiba
R.J. LeVeque, Seattle
C. Verdi, Milano
Honorary Editor:
J. Todd, Pasadena
NFroenel iBnoeaurnPdaarryti aPlroblems
Differential Equations
Theory and Applications
with Applications
Isabel N. Figueiredo
José F. Rodrigues
Lisa Santos
Editors
Birkhäuser Verlag
Basel . Boston . Berlin
Editors:
Isabel Narra Figueiredo José Francisco Rodrigues
Departamento de Matemática Universidade de Lisboa / CMAF
Faculdade de Ciências e Tecnologia Av. Prof. Gama Pinto 2
Universidade de Coimbra 1649-003 Lisboa
Apartado 3008 Portugal
3001-454 Coimbra [email protected]
Portugal
[email protected]
Lisa Santos
Departamento de Matemática
Universidade do Minho
Campus de Gualtar
4710-057 Braga
Portugal
[email protected]
(cid:21)(cid:19)(cid:19)(cid:19)(cid:3)(cid:48)(cid:68)(cid:87)(cid:75)(cid:72)(cid:80)(cid:68)(cid:87)(cid:76)(cid:70)(cid:86)(cid:3)(cid:54)(cid:88)(cid:69)(cid:77)(cid:72)(cid:70)(cid:87)(cid:3)(cid:38)(cid:79)(cid:68)(cid:86)(cid:86)(cid:76)(cid:192)(cid:70)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:3)(cid:22)(cid:24)(cid:16)(cid:19)(cid:25)(cid:15)(cid:3)(cid:22)(cid:24)(cid:53)(cid:22)(cid:24)(cid:15)(cid:3)(cid:27)(cid:19)(cid:36)(cid:21)(cid:21)(cid:15)(cid:3)(cid:25)(cid:24)(cid:48)(cid:91)(cid:91)(cid:15)(cid:3)(cid:25)(cid:24)(cid:49)(cid:91)(cid:91)(cid:15)(cid:3)(cid:25)(cid:24)(cid:61)(cid:19)(cid:24)(cid:15)(cid:3)(cid:26)(cid:23)(cid:49)(cid:21)(cid:19)(cid:15)(cid:3)(cid:26)(cid:23)(cid:49)(cid:22)(cid:19)(cid:15)(cid:3)(cid:26)(cid:23)(cid:49)(cid:28)(cid:28)(cid:15)(cid:3)(cid:26)(cid:25)(cid:39)(cid:21)(cid:26)(cid:17)
Library of Congress Control Number: 2006935948
Bibliographic information published by Die Deutsche Bibliothek
(cid:39)(cid:76)(cid:72)(cid:3)(cid:39)(cid:72)(cid:88)(cid:87)(cid:86)(cid:70)(cid:75)(cid:72)(cid:3)(cid:37)(cid:76)(cid:69)(cid:79)(cid:76)(cid:82)(cid:87)(cid:75)(cid:72)(cid:78)(cid:3)(cid:79)(cid:76)(cid:86)(cid:87)(cid:86)(cid:3)(cid:87)(cid:75)(cid:76)(cid:86)(cid:3)(cid:83)(cid:88)(cid:69)(cid:79)(cid:76)(cid:70)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:3)(cid:76)(cid:81)(cid:3)(cid:87)(cid:75)(cid:72)(cid:3)(cid:39)(cid:72)(cid:88)(cid:87)(cid:86)(cid:70)(cid:75)(cid:72)(cid:3)(cid:49)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:68)(cid:79)(cid:69)(cid:76)(cid:69)(cid:79)(cid:76)(cid:82)(cid:74)(cid:85)(cid:68)(cid:192)(cid:72)(cid:30)(cid:3)(cid:71)(cid:72)(cid:87)(cid:68)(cid:76)(cid:79)(cid:72)(cid:71)(cid:3)(cid:69)(cid:76)(cid:69)(cid:79)(cid:76)(cid:82)(cid:74)(cid:85)(cid:68)(cid:83)(cid:75)(cid:76)(cid:70)(cid:3)(cid:71)(cid:68)(cid:87)(cid:68)(cid:3)(cid:76)(cid:86)(cid:3)(cid:68)(cid:89)(cid:68)(cid:76)(cid:79)(cid:68)(cid:69)(cid:79)(cid:72)(cid:3)
in the Internet at http://dnb.ddb.de.
ISBN 978-3-7643-7718-2 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,
(cid:86)(cid:83)(cid:72)(cid:70)(cid:76)(cid:192)(cid:70)(cid:68)(cid:79)(cid:79)(cid:92)(cid:3)(cid:87)(cid:75)(cid:72)(cid:3)(cid:85)(cid:76)(cid:74)(cid:75)(cid:87)(cid:86)(cid:3)(cid:82)(cid:73)(cid:3)(cid:87)(cid:85)(cid:68)(cid:81)(cid:86)(cid:79)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:15)(cid:3)(cid:85)(cid:72)(cid:83)(cid:85)(cid:76)(cid:81)(cid:87)(cid:76)(cid:81)(cid:74)(cid:15)(cid:3)(cid:85)(cid:72)(cid:16)(cid:88)(cid:86)(cid:72)(cid:3)(cid:82)(cid:73)(cid:3)(cid:76)(cid:79)(cid:79)(cid:88)(cid:86)(cid:87)(cid:85)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:86)(cid:15)(cid:3)(cid:85)(cid:72)(cid:70)(cid:76)(cid:87)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:15)(cid:3)(cid:69)(cid:85)(cid:82)(cid:68)(cid:71)(cid:70)(cid:68)(cid:86)(cid:87)(cid:76)(cid:81)(cid:74)(cid:15)(cid:3)(cid:85)(cid:72)(cid:83)(cid:85)(cid:82)(cid:71)(cid:88)(cid:70)(cid:87)(cid:76)(cid:82)(cid:81)(cid:3)(cid:82)(cid:81)(cid:3)(cid:80)(cid:76)(cid:70)(cid:85)(cid:82)(cid:192)(cid:79)(cid:80)(cid:86)(cid:3)
or in other ways, and storage in data banks. For any kind of use, permission of the copyright owner must be obtained.
(cid:139)(cid:3)(cid:21)(cid:19)(cid:19)(cid:26)(cid:3)(cid:37)(cid:76)(cid:85)(cid:78)(cid:75)(cid:108)(cid:88)(cid:86)(cid:72)(cid:85)(cid:3)(cid:57)(cid:72)(cid:85)(cid:79)(cid:68)(cid:74)(cid:15)(cid:3)(cid:51)(cid:17)(cid:50)(cid:17)(cid:37)(cid:82)(cid:91)(cid:3)(cid:20)(cid:22)(cid:22)(cid:15)(cid:3)(cid:38)(cid:43)(cid:16)(cid:23)(cid:19)(cid:20)(cid:19)(cid:3)(cid:37)(cid:68)(cid:86)(cid:72)(cid:79)(cid:15)(cid:3)(cid:54)(cid:90)(cid:76)(cid:87)(cid:93)(cid:72)(cid:85)(cid:79)(cid:68)(cid:81)(cid:71)
Part of Springer Science+Business Media
Printed on acid-free paper produced from chlorine-free pulp TCF(cid:102)
Printed in Germany
ISBN-10: 3-7643-7718-6 e-ISBN-10: 3-7643-7719-4
ISBN-13: 978-3-7643-7718-2 e-ISBN-13: 978-3-7643-7719-9
9 8 7 6 5 4 3 2 1 www.birkhauser.ch
Contents
Preface ................................................................... ix
T. Aiki and T. Okazaki
One-dimensional Shape Memory Alloy Problem with
Duhem Type of Hysteresis Operator ................................. 1
F. Andreu, N. Igbida, J.M. Mazo´n and J. Toledo
Existence and Uniqueness Results for Quasi-linear Elliptic and
Parabolic Equations with Nonlinear Boundary Conditions ............ 11
S. Antontsev and H.B. de Oliveira
Finite Time Localized Solutions of Fluid Problems with
Anisotropic Dissipation .............................................. 23
S. Antontsev and S. Shmarev
Parabolic Equations with Anisotropic Nonstandard
Growth Conditions .................................................. 33
M. Aso, M. Fr´emond and N. Kenmochi
Parabolic Systems with the Unknown Dependent Constraints
Arising in Phase Transitions ......................................... 45
A. Azevedo, J.F. Rodrigues and L. Santos
The N-membranes Problem with Neumann Type
Boundary Condition ................................................. 55
M. Bause and W. Merz
Modelling, Analysis and Simulation of Bioreactive
Multicomponent Transport .......................................... 65
G. Bellettini and R. March
Asymptotic Properties of the Nitzberg-Mumford Variational Model
for Segmentation with Depth ........................................ 75
vi Contents
M. Belloni
The ∞-Laplacian First Eigenvalue Problem .......................... 85
A. Bermu´dez, M. Rodr´ıguez-Nogueiras and C. Va´zquez
Comparison of Two Algorithms to Solve the Fixed-strike
Amerasian Options Pricing Problem ................................. 95
P. Biler and R. Stan´czy
Nonlinear Diffusion Models for Self-gravitating Particles ............. 107
A.C. Briozzo and D.A. Tarzia
Existence, Uniqueness and an Explicit Solution for a One-Phase
Stefan Problem for a Non-classicalHeat Equation .................... 117
P. Cardaliaguet, F. Da Lio, N. Forcadel and R. Monneau
Dislocation Dynamics: a Non-local Moving Boundary ................ 125
E. Chevalier
Bermudean Approximation of the Free Boundary Associated
with an American Option ............................................ 137
L. Consiglieri and J.F. Rodrigues
Steady-state Bingham Flow with Temperature Dependent
Nonlocal Parameters and Friction .................................... 149
M. Eleuteri
Some P.D.E.s with Hysteresis ........................................ 159
T. Fukao
Embedding Theorem for Phase Field Equation with Convection ...... 169
G. Galiano and J. Velasco
A Dynamic Boundary Value Problem Arising in the Ecology
of Mangroves ........................................................ 179
M. Garzon and J.A. Sethian
Wave Breaking over Sloping Beaches Using a Coupled Boundary
Integral-LevelSet Method ........................................... 189
F. Gibou, C. Min and H. Ceniceros
Finite Difference Schemes for Incompressible Flows
on Fully Adaptive Grids ............................................. 199
Y. Giga, H. Kuroda and N. Yamazaki
Global Solvability of Constrained Singular Diffusion Equation
Associated with Essential Variation .................................. 209
Contents vii
M.E. Glicksman, A. Lupulescu and M.B. Koss
Capillary Mediated Melting of Ellipsoidal Needle Crystals ............ 219
E. Henriques and J.M. Urbano
Boundary Regularity at {t=0} for a Singular Free
Boundary Problem .................................................. 231
D. Hilhorst, R. van der Hout, M. Mimura and I. Ohnishi
Fast Reaction Limits and Liesegang Bands ........................... 241
A.J. James and J. Lowengrub
Numerical Modeling of Surfactant Effects in Interfacial
Fluid Dynamics ..................................................... 251
J. Kampen
The Value of an American Basket Call with Dividends Increases
with the Basket Volatility ........................................... 261
J.R. King and S.J. Franks
Mathematical Modelling of Nutrient-limited Tissue Growth .......... 273
P. Krejˇc´ı
Asymptotic Hysteresis Patterns in a Phase Separation Problem ...... 283
C. Leone
Obstacle Problems for Monotone Operators with Measure Data ...... 291
H. Li and X.-C. Tai
Piecewise Constant Level Set Method for Interface Problems ......... 307
A. Muntean and M. B¨ohm
Dynamics of a Moving Reaction Interface in a Concrete Wall ........ 317
J. Narski and M. Picasso
Adaptive Finite Elements with High Aspect Ratio for
Dendritic Growth of a Binary Alloy
Including Fluid Flow Induced by Shrinkage .......................... 327
C. Nitsch
A Free Boundary Problem for Nonlocal Damage Propagation
in Diatomites ........................................................ 339
A. Pistoia
Concentrating Solutions for a Two-dimensional Elliptic Problem
with Large Exponent in Nonlinearity ................................ 351
viii Contents
M. Ro¨ger
Existence of Weak Solutions for the Mullins-Sekerka Flow ............ 361
R. Rossi
Existence and Approximation Results for General
Rate-independent Problems via
a Variable Time-step Discretization Scheme .......................... 369
A. Segatti
Global Attractors for the Quasistationary Phase Field Model:
a Gradient Flow Approach ........................................... 381
H. Shahgholian and G.S. Weiss
Aleksandrov and Kelvin Reflection and the Regularity
of Free Boundaries ................................................... 391
K. Shirakawa, A. Ito and A. Kadoya
Solvability for a PDE Model of Regional Economic Trend ............ 403
B. Stinner
Surface Energies in Multi-phase Systems with
Diffuse Phase Boundaries ............................................ 413
M. Sussman and M. Ohta
High-order Techniques for Calculating Surface Tension Forces ........ 425
Y. Tao and Q. Guo
Simulation of a Model of Tumors with Virus-therapy ................. 435
X. Zhang and E. Zuazua
Hyperbolic-parabolic Coupled System Arising in
Fluid-structure Interaction ........................................... 445
List of Participants ....................................................... 457
Preface
Thisbookgathersacollectionofrefereedarticlescontainingoriginalresultsreport-
ingtherecentoriginalcontributionsofthelecturesandcommunicationspresented
at the Free Boundary Problems (FBP2005) Conference that took place at the
University of Coimbra, Portugal, from 7 to 12 of June 2005. They deal with the
Mathematics of a broad class of models and problems involving nonlinear partial
differentialequationsarisinginPhysics,Engineering,BiologyandFinance.Among
the main topics, the talks considered free boundary problems in biomedicine, in
porous media, in thermodynamic modeling, in fluid mechanics, in image process-
ing, in financial mathematics or in computations for inter-scale problems.
FBP2005 was the 10th Conference of a Series started in 1981 in Monteca-
tini, Italy, that has had a continuous development in the following conferences
in Maubuisson, France (1984), Irsee, Germany (1987), Montreal, Canada (1990),
Toledo, Spain (1993), Zakopone, Poland (1995), Crete, Greece (1997), Chiba,
Japan (1999), Trento, Italy (2002) and will be followed by the next one foreseen
to be held in Stockholm, Sweden, in 2008.
In fact, the mathematical analysis and fine properties of solutions and in-
terfaces in free boundary problems have been an active subject in the last three
decades and their mathematical understanding continues to be an important in-
terdisciplinary tool for the scientific applications, on one hand, and an intrinsic
aspect of the current development of several important mathematical disciplines.
This wasrecognized,inparticular,by theFree Boundary Problems Scientific Pro-
gramme of the European Science Foundation, that sponsored three conferences in
the nineties in Europe, and is reflected in an electronic newsletter-forum (FBP-
News, http://fbpnews.org), that started in 2003 and continues to have an impor-
tant role to promote the exchange of information and ideas between mathemati-
cians interested in this area.
Over 150 participants have gathered during the FBP2005, to present and
discuss,inmorethan120talks,thelastresultsonthe Mathematicsoffreebound-
aryproblems.ThestructureoftheConference,advisedbyaScientificCommittee,
combined Main Lectures and Focus Sessions by invitation and was complemented
with Focus Discussions and Contribution Talks with selected open proposals by
the worldwide scientific community, that constituted almost half of the commu-
nications. The conference also integrated in its programme, for the first time, an
EuropeanMathematicalSociety(EMS)Lecture.DuringtheFBP2005Conference,
x Preface
newpeopleandnewproblems,withrenewedclassicalsubjects,wereonstage.This
has confirmedthat these conferences continue to be an important catalyst for the
identification and development of this interdisciplinary mathematical field. They
promote, not only in Europe, but all over the world, an interdisciplinary scope
in the broadest possible mathematical sense: from experimental observations to
modeling, from abstract mathematical analysis to numerical computations.
ThecreditofthesuccessoftheFPB2005conferenceismainlyduetothelec-
turers, the organizersof the focus sessions and all the speakers of the invited and
contributed talks, for their valuable contributions. Of course, our acknowledge-
ments also go to the members of the scientific committee, that was constituted
by C. Bandle (University of Basel), H. Berestycki (EHESS, Paris), L. Caffarelli
(University of Austin, Texas, USA), P. Colli (University of Pavia,Italy), C.J. van
Duijn (University of Eindhoven, Netherlands), G. Dziuk (University of Freiburg,
Germany), C. Elliott (University of Sussex, UK), A. Fasano (University of Flo-
rence, Italy), A. Friedman (University of Ohio, USA), B. Kawohl (University of
Koln,Germany),M.Mimura(UniversityofTokyo,Japan),S.Osher(Universityof
Los Angeles, USA), J.F. Rodrigues (University of Lisbon/CMU Coimbra, Portu-
gal),H.Shahgholian(UniversityofStockholm,Sweden),J.Sprekels(WIASBerlin,
Germany) and J.L. Vazquez (University Autonoma of Madrid, Spain), as well as
to our co-organizer L.N. Vicente (University of Coimbra), the reviewers for per-
forming the evaluation of the articles presented in this book of Proceedings and
to K.-H. Hoffmann for accepting it in this Birkha¨user Series. Our thanks also go
to the secretariat of the conference, in particular, we wish to acknowledge Rute
Andrade for her excellent collaboration, and the Department of Mathematics of
the University of Coimbra, for the facilities and active assistance.
Finally,wewishtothankalsotheimportantfinancialsupportfromESF(Eu-
ropean Science Foundation) Scientific Programme (Global) on “Global and Geo-
metricalAspectsofNonlinearPartialDifferentialEquations”,aswellas,thefinan-
cial support from CMUC (Centro de Matem´atica da Universidade de Coimbra),
CMAF(CentrodeMatem´aticaeAplica¸c˜oesFundamentaisdaUniversidadedeLis-
boa),EMS(EuropeanMathematicalSociety),FLAD(Fundac¸˜aoLuso-Americana)
and FCT (Fundac¸˜ao para a Ciˆencia e a Tecnologia).
The Editors
Isabel Narra Figueiredo (Coimbra)
Jos´e Francisco Rodrigues (Lisboa)
Lisa Santos (Braga)