Calculus of Variations and Partial Differential Equations: Topics on Geometrical Evolution Problems and Degree Theory

346 Pages·2000·15.376 MB·English

by  Luigi Ambrosio, Norman Dancer (auth.), Giuseppe Buttazzo, Antonio Marino, M. K. V. Murthy (eds.)

Ambrosio· Dancer
Calculus of Variations
and Partial Differential Equations
Springer-Verlag Berlin Heidelberg GmbH
L. Ambrosio· N. Dancer
Calculus
of Variations and
Partial Differential
Equations
Topics on
Geometrical Evolution Problems
and Degree Theory
Edited by G. Buttazzo,A. Marino,
M. K. V. Murthy
Springer
Authors and Editors
Luigi Ambrosio
Scuola Normale Superiore
Piazza Cavalieri 7
56100 Pisa, Italy
Norman Dancer
University of Sydney
School of Mathematics
NSW 2006 Sydney, Australia
Giuseppe Buttazzo
Antonio Marino
M. K. V. Murthy
Universita di Pisa
Dipartimento di Matematica
Via Buonarroti 2
56127 Pisa, Italy
Mathematics Subject Classification (1991): 60J15, 30005, 35Q15, 30F10, 60K25
Library of Congress Cataloging-in-Publication Data
Calculus of variatious and partial differential equations : topics on geometrical evolution
problems and degree theory I edited by G. Buttazzo, A. Marino, M.K. V. Murthy.
p.cm.
Includes bibliographical references and index.
ISBN 978-3-540-64803-1 ISBN 978-3-642-57186-2 (eBook)
DOI 10.1007/978-3-642-57186-2
1. Calculus ofvariatious. 2. Topological degree. 3. Differential equatious, Partial. 4.
Mathematical physics. 1. Buttazzo, Giuseppe. II. Marino, A. ill. Murthy, M. K. V. (M.
K. Venkatesha) IV. Summer School on "Calculus ofVariatious and Partial Differential
Equatious" (1996 : Pisa, Italy)
QC20.7.C3 C35 1999
SIS' .64--dc21
98-054672
This work is subject to copyright. AII rights are reserved, whether the whole or part of the material
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© Springer-Verlag Berlin Heidelberg 2000
Cover design: design .& production GmhH, Heidelberg
SPIN: 10688575 41/3143 -5 4 3 2 1 o - Printed on acid-free paper
Preface
The project of editing this book originated in a two-week Summer School on
"Calculus of Variations and Partial Differential Equations" which was held in
Pisa in September 1996.
The School was attended by more than 150 participants, mostly Ph. D. stu
dents, but also by several more experienced scientists. The success achieved en
couraged us to collect into a book the course material of the two main lecturers
and the complementary articles of the eight other speakers on related subjects.
Special importance was attached to the self-contained presentation of these
rather advanced research topics, and we hope that our work will contribute to
filling the gap between the level of basic courses and more advanced research.
Pisa, March 3, 1998
G. Buttazzo A. Marino M.K.V. Murthy
Table of Contents
Preface ......................................................... V
Table of Contents ............................................. VII
List of Authors ............................................... IX
I Geometric Evolution Problems
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Geometric evolution problems, distance function and viscosity
solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
L. Ambrosio
r
Variational models for phase transitions, an approach via -convergence .. 95
G. Alberti
Some aspects of De Giorgi's barriers for geometric evolutions ............ 115
G. Bellettini, M. NotJaga
Partial Regularity for Minimizers of Free Discontinuity Problems with p-th
Growth ........................................................... 153
A. Leaci
Free discontinuity problems and their non-local approximation ........... 171
A. Braides
II Degree Theory on Convex Sets and Applications to Bifurcation
Introduction ................................................... 183
Degree theory on convex sets and applications to bifurcation . .. 185
E. N. Dancer
Nonlinear elliptic equations involving critical Sobolev exponents ......... 227
D. Passaseo
On the existence and multiplicity of positive solutions for semilinear mixed
and Neumann elliptic problems ...................................... 243
G. Cerami
VIII Table of Contents
Solitons and Relativistic Dynamics ................................... 259
V. Benci, D. Fortunato
An algebraic approach to nonstandard analysis ........................ 285
V. Benci
References ..................................................... 327
Index ........................................................... 345
List of Authors
Giovanni Alberti Luigi Ambrosio
Dipartimento di Matematica Scuola Normale Superiore
Via Buonarroti 2 Classe di Scienze
Universita. di Pisa Piazza dei Cavalieri, 7
56127 Pisa (Italy) 56126 Pisa (Italy)
e-mail: [email protected] e-mail: [email protected]
Giovanni Bellettini Vieri Benci
Universita. di Rama "Tor Vergata" Dipartimento di Matematica
Via della Ricerca Scientmca Applicata "U. Dini"
00133 Rama (Italy) Universita. di Pisa, Via Bonanno 25/b
e-mail: [email protected] 56126 Pisa (Italy)
e-mail: [email protected]
Andrea Braides Giovanna Cerami
SISSA Dipartimento di Matematica e Appli
Via Beirut 4 cazioni
34014 Trieste (Italy) Universita. di Palermo
e-mail: [email protected] Via Archirafi, 34
90123 Palermo (Italy)
e-mail: [email protected]
E. N. Dancer Donato Fortunato
School of Mathematics and Statistics Dipartimento di Matematica
University of Sydney Universita. di Bari, Via Orabona 4
NSW 2006 (Australia) 70125 Bari (Italy)
e-mail: [email protected] e-mail: [email protected]
Antonio Leaci Matteo Novaga
Dipartimento di Matematica Dipartimento di Matematica
Universita. di Leece Universita. di Pisa
Via Arnesano Via Buonarroti, 2
73100 Leece (Italy) 56127 Pisa (Italy)
e-mail: [email protected] e-mail: [email protected]
Donato Passaseo
Dipartimento di Matematica
Universita. di Leece
Via Arnesano
73100 Leece (Italy)
e-mail: [email protected]
Part I
Geometric Evolution
Problems
Introduction, Part I 3
Introduction to Part I
In many problems arising from Mathematical Physics and from Applied Math
ematics the objects involved are surfaces, or more generally lower dimensional
sets. Some nowadays well-known examples that could be quoted in this respect
as model cases are the phase transitions problem, the Mumford and Shah image
segmentation problem in computer vision, and the geometric evolution problems
such as for instance the motion by mean curvature.
In some phase transitions problems one has to deal with the minimization of
a double well potential with a volume constraint:
{In In
min W(u) dx : udx = constant} . (p)
Since problem (p) above has too many nonphysical solutions, the singular per
turbation In In
min { [clDul2 + W(u)] dx : udx = const}.
was proposed by Cahn and Hilliard in 1958 to select, by taking the limit as
c -+ 0+, physically reasonable solutions.
It was first conjectured by De Giorgi in 1975 and then proved by Modica and
Mortola in 1977 that the solutions selected in this way jump between two zeros
of the potential W across a surface of least area.
The Mumford and Shah image segmentation problem can be described by
the following minimization problem:
+ +,1
{al
min IVul2 dx /31ln-l(K n n) lu - gl2 dx} , (MS)
il\K il\K
where n represents the domain of a digitalized image g, and a, /3, , are three
given positive constants. By minimizing the functional in (M S) among all closed
subsets K of n and all functions u E C1 (n \ K) one looks for a piecewise smooth
approximation u of the input g, filtering noise effects. The feature of this model
is that diffusion across sharp discontinuities of 9 is prevented, thus leading to a
good reconstruction of the edges of the image.
The motion of a surface according to its mean curvature has been widely
studied in the last two decades and is, among geometric evolution problems,
the best known. Since singularities may appear during the evolution, several
weak formulations have been proposed to describe the long time behaviour of
surfaces. One of the possibilities is to represent the evolving surface as the level
set of an auxiliary function u solving (in the weak sense of the theory of viscosity
solutions) the partial differential equation
au .
at = IDul dlV (DIDuul ) . (MC)
The formal interpretation of this equation is that any level set of u(t,·) flows
by mean curvature. In this connection we mention that Huisken and llmanen
L. Ambrosio et al., Calculus of Variations and Partial Differential Equations
© Springer-Verlag Berlin Heidelberg 2000