Table Of ContentProgress in Nonlinear Differential Equations
and Their Applications
Volume 15
Editor
Haim Brezis
Universite Pierre et Marie Curie
Paris
and
Rutgers University
New Brunswick, N.J.
Editorial Board
A. Bahri, Rutgers University, New Brunswick
John Ball, Heriot-Watt University, Edinburgh
Luis Cafarelli, Institute for Advanced Study, Princeton
Michael Crandall, University of California, Santa Barbara
Mariano Giaquinta, University of Florence
David Kinderlehrer, Carnegie-Mellon University, Pittsburgh
Robert Kohn, New York University
P. L. Lions, University of Paris IX
Louis Nirenberg, New York University
Lambertus Peletier, University of Leiden
Paul Rabinowitz, University of Wisconsin, Madison
Topological Nonlinear Analysis
Degree, Singularity, and Variations
Michele Matzeu
Alfonso Vignoli
Editors
Birkhauser
Boston • Basel • Berlin
Editors:
Michele Matzeu Alfonso Vignoli
Department of Mathematics Department of Mathematics
University of Rome, Tor Vergata University of Rome, Tor Vergata
00133 Rome, Italy 00133 Rome, Italy
Library of Congress Cataloging-in-Publication Data
Topological nonlinear analysis : degree, singularity, and variations I
Michele Matzeu, Alfonso Vignoli, editors
p. cm. --(Progress in nonlinear differential equations and
their applications ; v. 15))
Includes bibliographical references.
ISBN ·13:978·1-4612· 7584·8 e· ISBN ·13: 978·1·4612·25711·6
DOl: 111.10117/978·1·4612·2570·6
I. Nonlinear functional analysis. 2. Topological algebras.
I. Matzeu, M. (Michele) II. Vignoli, Alfonso, (1940-
III. Series.
QA321.5.T67 1994 94-5251
515.'355--dc20 CIP
Printed on acid-free paper ~®
Birkhiiuser
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ISBN-13:978-1-4612-7584-8
Reformatted from authors' disks.
987 6 5 4 3 2 I
Contents
Preface vii
Variational Methods and Nonlinear Problems:
Classical Results and Recent Advances
Antonio Ambrosetti ........ . 1
• Introduction. Lusternik-Schnirelman Theory. Applications
to Nonlinear Eigenvalues. Unbounded Functionals • Elliptic
Dirichlet Problems • Singular Potentials • References
Introduction to Morse Theory: A New Approach
Vieri Benci .. 37
• Introduction. Contents. The Abstract Theory. The Morse Index
• The Poincare Polynomial • The Conley Blocks • The Morse
Relations • Morse Theory for Degenerate Critical Points • Some
Existence Theorems • An Application to Riemannian Geometry
• Riemannian Manifolds • Geodesics • The Morse Theory for
Geodesics. The Index Theorem. An Application to Space-Time
Geometry. Introduction. Some Examples of Lorentzian Manifolds
• Morse Theory for Lorentzian Manifolds • Preliminary Lemmas
• Proof of The Morse Relations For Static Space- Time. Some
Application to a Semilinear Elliptic Equation. Introduction
• The Sublinear Case • The Superl inear Case Morse Relations for
Positive Solutions • The Functional Setting • Some Hard Analysis
• The Photography Method • The Topology of The Strip • References
Applications of Singularity Theory to the Solutions
of Nonlinear Equations
James Damon .... 178
• The Full Lyapunov-Schmidt Reduction. Mather's Theory of
Coo-Stability of Mappings - Global Theory. Mather's Local
Theory as Paradigm. Singularity Theory with Special Conditions
• The Structure of Nonlinear Fredholm Operators. Multiplicities
of Solutions to Nonlinear Equations. The Theory for Topological
Equivalence. Bibliography
Fixed Point Index Calculations and Applications
E.N. Dancer . . . . . . . . . . . . . . . . 303
• The Fixed Point Index • Some Remarks on Convex Sets • A Basic
Index Calculation. Index Calculations in Product Cones. Applications
of Index Formulae - I • Applications of Index Formulae - II • Some Global
Branches • Monotone Dynamical Systems • Preliminaries • Connecting
Orbits and Related Results. Generic Convergence. References
vi Contents
Topological Bifurcation
Jorge Ize 341
• Abstract. Introduction. Preliminaries. One Parameter Bifurcation
• Local Bifurcation. Global Bifurcation • Special Nonlinearities
• Multiparameter Bifurcation. Sufficient Conditions for Local Bifurcation
• Necessary Conditions for Linearized Local Bifurcation. Multiparameter
Global Bifurcation. A Summation Formula and A Generalized Degree
• Structure and Dimension of Global Branches. O-EPI Maps. Dimension
• Application to Bifurcation Problems. Equivariant Bifurcation
• Preliminaries. Consequences of the Symmetry. r-EPI Maps. r-Degree
• The Equivariant J-Homomorphism and Sufficient Conditions. Necessary
and Sufficient Conditions for Equivariant Bifurcation. Bibliography
Critical Point Theory
Paul H. Rabinowitz 464
• Introduction • The Mountain Pass Theorem. The Saddle Point
Theorem. Linking and A General Critical Point Theorem. Periodic
Solutions of Hamiltonian Systems • Introduction. The Technical
Framework. Periodic Solutions of Prescribed Energy. Periodic Solutions
of Prescribed Period. Connecting Orbits. Introduction. Homoclinic
Solutions. Heteroclinic Solutions. References
Symplectic Topology: An Introduction
Claude Viterbo 514
• The Classical Uncertainty Principle, Symplectic Rigidity
• Construction of Symplectic Invariants. Generating Functions
• Historical Remarks. Appendix: Rigidity for Finite
Dimensional Lie Groups
Preface
Topological tools in Nonlinear Analysis had a tremendous develop
ment during the last few decades. The three main streams of research in
this field, Topological Degree, Singularity Theory and Variational Meth
ods, have lately become impetuous rivers of scientific investigation. The
process is still going on and the achievements in this area are spectacular.
A most promising and rapidly developing field of research is the study
of the role that symmetries play in nonlinear problems. Symmetries appear
in a quite natural way in many problems in physics and in differential or
symplectic geometry, such as closed orbits for autonomous Hamiltonian
systems, configurations of symmetric elastic plates under pressure, Hopf
Bifurcation, Taylor vortices, convective motions of fluids, oscillations of
chemical reactions, etc ... Some of these problems have been tackled recently
by different techniques using equivariant versions of Degree, Singularity and
Variations.
The main purpose of the present volume is to give a survey of some
of the most significant achievements obtained by topological methods in
Nonlinear Analysis during the last two-three decades. The survey articles
presented here reflect the personal taste and points of view of the authors
(all of them well-known and distinguished specialists in their own fields)
on the subject matter. A common feature of these papers is that of start
ing with an historical introductory background of the different disciplines
under consideration and climbing up to the heights of the most recent re
sults. As a consequence, we obtain a very dynamic picture of the state of
affairs. Actually, we hope to be able in the near future to involve other
distinguished specialists to get their own versions on these topics. Most
probably a never-ending fascinating tale!
Finally let us mention the fact that most of the material of this book
was presented by the authors at the Topological Analysis Workshop on
Degree, Singularity and Variations, held in May 1993 at Villa Campitelli,
Frascati, near Rome.
Contributors
Antonio Ambrosetti, Scuola Normale Superiore, Piazza dei Cavalieri 7,
56100 Pisa, Italy
Vieri Benci, Istituto di Matematiche Applicate "U. Dini", Universita di
Pisa, 56100 Pisa, Italy
James Damon, Department of Mathematics, University of North Car
olina, Chapel Hill, NC 27514, USA
E. N. Dancer, Department of Mathematics, Statistics and Computing
Science, The University of New England, Armidale, NSW 2351, Australia
Jorge Ize, Departamento de Matematicas y Mecanica, ITMAS-UNAM,
Apartado Postal 20-726, D. F. Mexico 20
Michele Matzeu, Dipartimento di Matematica, Universita degli Studi di
Roma, "Tor Vergata", Via della Ricerca Scientifica, 00133 Rome, Italy
Paul H. Rabinowitz, Department of Mathematics, University of
Wisconsin-Madison, Van Vleck Hall, 480 Lincoln Drive, Madison, WI
53706-1388
Alfonso Vignoli, Dipartimento di Matematica, Universita degli Studi di
Roma, "Tor Vergata", Via della Ricerca Scientifica, 00133 Rome, Italy
Claude Viterbo, Departement de MatMmatique, Bat. 425, Universite
Paris-Sud, F-91405 Orsay Cedex, France
Variational Methods and Nonlinear Problems:
Classical Results and Recent Advances
Antonio Ambrosetti
Scuola Normale Superiore
56100 Pisa, Italy
1. Introduction
Around the end of the Twenties two memoires, a first one by Morse [63]
and a second one by Lusternik and Schnirelman [59], marked the birth of
those variational methods known under the name of Calculus of Variation
in the Large. These tools are mainly concerned with the existence of critical
points, distinct from minima, which give rise to solutions of nonlinear dif
ferential equations. The elegance of the abstract tools and the broad range
of applications to problems that had been considered of formidable diffi
culty, such as the existence of closed geodesics on a compact anifold or the
problem of minimal surfaces, have rapidly made the Calculus of Variation
in the Large a very fruitful field of research.
The natural fields of application of these theories are all those nonlinear
variational problems where the search of minima is not satisfactory, or
useless, or else impossible, and thus solutions have to be found by means
of min-max procedures.
In particular, the Lusternik-Schnirelman theory of critical points has
been largely emploied in Nonlinear Analysis because it does not require any
a-priori nondegeneracy assumption. An important progress was marked by
the works of Browder [35], Krasnoselski [55], Palais [65], Schwartz [76] and
Vainberg [87] who extended the theory to infinite dimensional manifolds.
For example, a typical result is that a C1 functional f has infinitely many
critical points on any infinite dimensional Hilbert sphere S, provided: (i)
f is bounded from below on S; (ii) f satisfies the Palais-Smale, (PS) for
2 A. Ambrosetti
short, compactness condition; and (iii) f is even. These critical points give
rise to solutions of nonlinear elliptic eigenvalue problems. A short review
of these classical results is made in Sections 2 and 3.
A further step forward was taken by studying critical points of func
tionals which are possibly unbounded from below. The Mountain-Pass
Theorem and the Linking Theorems establish that critical points can be
found by means of appropriate min-max procedures, provided f satisfies
the (PS) condition and some suitable geometric assumptions. An exten
sion of the Lusternik-Schnirelman theory which covers a class of symmetric
functionals, unbounded from below, has also been obtained. These abstract
tools will be discussed in Section 4, while some applications to semilinear
Elliptic Boundary value Problems are given in Section 5. Here we limit
ourselves to an overview of some more classical results concerning the exis
tence and multiplicity of solutions for sub-linear and super-linear Dirichlet
boundary problems, as well as to give some indications on further topics of
great interest, where the research is still very active.
Finally, in Section 6 we briefly discuss some recent advances concern
ing periodic motions of second order Hamiltonian systems with singular
potentials, like those arising in Celestial Mechanics. This also proved to be
a topic where the power of variational methods apply and, although several
interesting results have been obtained in these last years, many important
problems are still left open and will require a great deal of future researches.
This survey is necessarily incomplete. The interested reader can find
other material on variational methods in the books [44], [62], [82] and in
the survey papers [64], [6].
2. Lusternik-Schnirelman Theory
Let M be a Riemannian Cl,l manifold modelled on a Hilbert space E and
let f E Cl(M, R). We will use the following notation:
fb = {u EMf (u) :::; b}
fa = {u E M feu) 2: a}
f~ = {u E M a:::; f (u ) :::; b}
A critical points of f on M is a u E M such that df(u) =I- O. We set
K={UEM: f'(u)=O}andKc={uEK: f(u)=c}. We say that cis
a critical level of f on M whenever Kc =I- 0.
The following examples show that critical points are naturally related
with differential problems.
Example 2.1. Let V be a compact Riemannian manifold with a Rie
mannian structure < .,. >. The problem of finding closed geodesics on V
