Table Of ContentProgress in Nonlinear Differential Equations
and Their Applications
Volume 51
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
P.L. Lions, University of Paris IX
Jean Mahwin, Universit6 Catholique de Louvain
Louis Nirenberg, New York University
Lambertus Peletier, University of Leiden
Paul Rabinowitz, University of Wisconsin, Madison
John Toland, University of Bath
Variational Methods for
Discontinuous Structures
International Workshop at Villa Erba (Cemobbio),
Italy, July 2001
Gianni daI Maso
Franco Tomarelli
Editors
Springer Base! AG
Editors' addresses:
Gianni daI Maso Franco Tomarelli
ScuoIa Intemazionale Superiore di Studi Dipartimento di Matematica
Avanzati (SISSA) "Francesco Brioschi"
Via Beirut 4 Politecnico di Milano
34014 Trieste Piazza Leonardo da Vinei 32
Italy 20133 Milano
e-mail: [email protected] ltaIy
e-mail: [email protected]
2000 Mathematics Subject Classification 49-06; 65KlO, 68U10, 74B99, 74C99, 74E25,
74N05, 82D40
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 Nationa1bibliografie;
detailed bibliographic data is available in the Internet at <http://dnb.ddb.de>.
ISBN 978-3-0348-9470-8 ISBN 978-3-0348-8193-7 (eBook)
DOI 10.1007/978-3-0348-8193-7
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© 2002 Springer Basel AG
Originally published by Birkhăuser Verlag in 2002
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Contents
Preface ................................................................... Vll
Invited Lectures ix
E. ACERB!
A Model for Mixtures of Micromagnetic Materials
allowing Existence and Regularity .................................... 1
G. BELLETTINI, R. MARCH
Variational Properties of a Model for Image
Segmentation with Overlapping Regions.............................. 9
G. BOUCHITTE, 1. FRAGALA
Variational Theory of Weak Geometric Structures.................... 19
G. BUTTAZZO, E. OUDET & E. STEPANOV
Optimal Transportation Problems with Free Dirichlet Regions 41
M. CARRIERO, A. LEACI & F. TOMARELLI
Local Minimizers for a Free Gradient Discontinuity Problem
in Image Segmentation .............................................. 67
V. CASELLES J.M. MOREL
Irrigation ...... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
A. CIANCHI, N. FUSCO
Symmetrization and Functionals Defined on BV ..................... 91
G. DEL PIERO
Interface Energies and Structured Deformations
in Plasticity ......................................................... 103
1. FONSECA, G. LEONI
Higher Order Variational Problems and Phase
Transitions in Nonlinear Elasticity ................................... 11 7
VI Contents
M.PAOLINI, F. PASQUARELLI
Unstable Crystalline Wulff Shapes in 3D ............................. 141
D. PERCIVALE
Dimension Reduction in Continuum Mechanics...................... 155
J. SHAH
Reaction-Diffusion Equations and Learning 171
Contributors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 183
List of Participants ....................................................... 187
Preface
This volume contains the Proceedings of the International Workshop Variational
Methods For Discontinuous Structures, which was jointly organized by the Dipar
timento di Matematica Francesco Brioschi of Milano Politecnico and the Interna
tional School for Advanced Studies (SISSA) of Trieste. The Conference took place
at Villa Erba Antica (Cernobbio) on the Lago di Como on July 4-6, 2001.
In past years the calculus of variations faced mainly the study of continuous
structures, say particularly problems with smooth solutions. One of the deepest
and more delicate problems was the regularity of weak solutions. More recently,
new sophisticated tools have been introduced in order to study discontinuities: in
many variational problems solutions develop singularities, and sometimes the most
interesting part of a solution is the singularity itself. The conference intended to
focus on recent developments in this direction.
Some of the talks were devoted to differential or variational modelling of
image segmentation, occlusion and textures synthesizing in image analysis, varia
tional description of micro-magnetic materials, dimension reduction and structured
deformations in elasticity and plasticity, phase transitions, irrigation and drainage,
evolution of crystalline shapes; in most cases theoretical and numerical analysis of
these models were provided.
viii Preface
Other talks were dedicated to specific problems of the calculus of variations:
variational theory of weak or lower-dimensional structures, optimal transport prob
lems with free Dirichlet regions, higher order variational problems, symmetrization
in the BV framework.
This volume contains contributions by 12 of the 16 speakers invited to deliver
lectures in the workshop. Most of the contributions present original results in fields
which are rapidly evolving at present.
The meeting was successful in focusing on advanced results and techniques
in pure and applied analysis of variational problems. The scientific level of the
lectures was remarkable.
About seventy researchers in various fields (calculus of variations, geometric
measure theory, materials science and computer science) attended the Conference.
We take here the opportunity to thank them all for their contributions to the
scientific discussion and the friendly atmosphere of the event.
The meeting would not have been possible without the support of several in
stitutions and research projects. Here we wish to mention: national project Calcolo
delle Variazioni co-financed by MIUR (2000), G.N.A.M.P.A. (INDAM), SIEMENS
SpA Automation & Drives, Dipartimento di Matematica Francesco Brioschi (Mi
lano Politecnico) and SIS SA (Trieste).
Moreover we wish to thank the Centro di Cultura Scientifica Alessandro Volta
(Como) for logistic support and for allowing to held the conference sessions in the
beautiful Salone of Villa Erba.
We also thank for the excellent organizing work done by Haria Fragala and
Fabrizio Colombo, the efficient secretarial work done by Nadia Tansini, and for the
hard typographical job done by Pamela Palazzini on the individual manuscripts.
Finally we thank all the lecturers, and we are especially grateful to those who
contributed to this volume.
Levico, February 20, 2002 Gianni Dal Maso
Franco Tomarelli
Invited Lectures
EMILIO ACERBI, (Universita di Parma)
A Model for Mixtures of Micromagnetic Materials allowing
Existence and Regularity
GUY BOUCHITTE, (Universite de Toulon et du Var)
Variational Theory of Weak Geometric Structures: the Measure
Methods and its Applications
GIUSEPPE BUTTAZZO, (Universita di Pisa)
Optimal Transportation Problems with Free Dirichlet Regions
GIANPIETRO DEL PIERO, (Universita di Ferrara)
Interface Energies and Structured Deformations in Plasticity
IRENE FONSECA, (Carnegie-Mellon University of Pittsburgh)
Higher Order Variational Problems and Phase Transitions
in Nonlinear Elasticity
NICOLA FUSCO, (Universita di Firenze)
Symmetrization and Functionals Defined on B V
ROBERT V. KOHN, University of New York)
A Continuum Approach to the Structure of Twist Grain Boundaries
ANTONIO LEACI, (Universita di Leece)
Local Minimizers for a Free Gradient Discontinuity Problem
in Image Segmentation
RICCARDO MARCH, (CNR, Roma)
Variational Properties of a Model for Image Segmentation
with Overlapping Regions
JEAN MICHEL MOREL, (CMLA, France)
Irrigation
x Invited Lectures
UMBERTO MOSCO, (Universita di Roma)
Effective Distances and Fractal Ramifications
DAVID OWEN, (Carnegie-Mellon University of Pittsburgh)
Toward Field Theories for Bodies Undergoing Disarrangements
MAURIZIO PAOLINI, (Universita Cattolica di Brescia)
Unstable Crystalline Wulff Shapes in 3D
DANILO PERCIVALE, (Universita di Genova)
Dimension Reduction in Continuum Mechanics
JAYANT SHAH, (Northeastern University)
Reaction-Diffusion Equations and Learning
LEV TRUSKINOVSKY, (University of Minneapolis)
Multiple Cracking: from Reinforced Concrete
to Skeletal Muscles
