Progress in Nonlinear Diffef ,lad..d :Equ is and Their Applications
Kung-ching Chang
Infinite Dimensional Morse Theory and Multiple Solution Problems
Birkhauser
m
W
Progress in Nonlinear Differential Equations and Their Applications Volume 6
Editor Haim Brezis University 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
Kung-ching Chang
Infinite Dimensional Morse Theory and Multiple Solution Problems
Birkhauser Boston Basel Berlin
Kung-ching Chang Department of Mathematics Peking University Beijing, 100871 People's Republic of China
Library of Congress Cataloging-in-Publication Data Chang, Kung-ching Infinite dimensional Morse theory and multiple solution problems / by Kung-ching Chang p. cm. -- (Progress in nonlinear differential equations and their applications ; v. 6) Includes bibliographical references and index. ISBN 0-8176-3451-7 (acid free) 1. Morse Theory 1. Title. 11. Series. QA331.C445 1991 91-12511 515--dc2O CIP
Printed on acid-free paper.
© Birkhauser Boston 1993. Copyright is not claimed for works of U.S. Government employees. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior permission of the copyright owner. Permission to photocopy for internal or personal use of specific clients is granted by Birkhttuser Boston for libraries and other users registered with the Copyright Clearance Center (CCC), provided that the base fee of $5.00 per copy, plus $0.20 per page is paid directly to CCC, 21 Congress Street, Salem, MA 01970, U.S.A. Special requests should be addressed directly to Birkhauser Boston, 675 Massachusetts Avenue, Cambridge, MA 02139. U.S.A. ISBN 0-8176-3451-7 ISBN 3-7643-3451-7
Typeset in TeX by Ark Publications, Inc., Newton Centre, MA Printed and bound by Quinn-Woodbine, Woodbine, NJ. Printed in the U.S.A.
987654321
TABLE OF CONTENTS
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vii
Introduction
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ix
Preface
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Chapter I: Infinite Dimensional Morse Theory 1. A Review of Algebraic Topology . . . . . . . . . . . . . 2. A Review of the Banach-Finsler Manifold . . . . . . . . . 3. Pseudo Gradient Vector Field and the Deformation Theorems 4. Critical Groups and Morse Type Numbers . . . . . . . . . 5. Gromoll-Meyer Theory . . . . . . . . . . . . . . . . . . . . . . . . . 6. Extensions of Morse Theory . . . . . . . 6.1. Morse Theory Under General Boundary Conditions 6.2. Morse Theory on a Locally Convex Closed Set . . . . 7. Equivariant Morse Theory . . . . . . . . . . . . . . . . 7.1. Preliminaries . . . . . . . . . . . . . . . . . . . 7.2. Equivariant Deformation . . . . . . . . . . . . . 7.3. The Splitting Theorem and the Handle Body Theorem for Critical Manifolds . . . . . . . . . . . . . . . 7.4. G-Cohomology and G-Critical Groups . . . . . . . .
Chapter II: Critical Point Theory. 1. Topological Link . . . . . . . . . . . 2. Morse Indices of Minimax Critical Points 2.1. Link . . . . . . . . . . . . . . . . . . . 2.2. Genus and Cogenus 3. Connections with Other Theories . . . 3.1. Degree theory . . . . . . . . . 3.2. Ljusternik-Schnirelman Theory . . 3.3. Relative Category . . . . . . . 4. Invariant Functionals . . . . . . . . . 5. Some Abstract Critical Point Theorems 6. Perturbation Theory . . . . . . . . . 6.1. Perturbation on Critical Manifolds 6.2. Uhlenbeck's Perturbation Method
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19 32 43 54 55 60 65 66 67
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69 74 83 92 92 96 99 99
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131 131
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136
105 109 111 121
Chapter III: Applications to Semilinear Elliptic Boundary Value Problems.
1. Preliminaries . . . . . . . . 2. Superlinear Problems . . . . 3. Asymptotically Linear Problems
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140 144 153
Infinite Dimensional Morse Theory
vi
3.1. Nonresonance and Resonance with the Landesman-Lazer Condition . . . . . . . . . . . . . . . . . . . . 3.2. Strong Resonance . . . . . . . . . . . . . . . . 3.3. A Bifurcation Problem . . . . . . . . . . . . . 3.4. Jumping Nonlinearities . . . . . . . . . . . . . . 3.5. Other Examples . . . . . . . . . . . . . . . . . 4. Bounded Nonlinearities . . . . . . . . . . . . . . . . 4.1. Functionals Bounded From Below . . . . . . . . . 4.2. Oscillating Nonlinearity . . . . . . . . . . . . . 4.3. Even Functionals . . . . . . . . . . . . . . . . . 4.4. Variational Inequalities . . . . . . . . . . . . . . .
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Chapter IV: Multiple Periodic Solutions of Hamiltonian Systems 1. Asymptotically Linear Systems . . . . . . . . . . . . . . 2. Reductions and Periodic Nonlinearities . . . . . . . . . . . . . . 2.1. Saddle Point Reduction . . . 2.2. A Multiple Solution Theorem . . . . . . . . . . 2.3. Periodic Nonlinearity . . . . . . . . . . . . . . 3. Singular Potentials . . . . . . . . . . . . . . . . . . . . . 4. The Multiple Pendulum Equation . . . . . 5. Some Results on Arnold Conjectures . . . . . . . . . . . . . . . . . 5.1. Conjectures . . . . . . . . . 5.2. The Fixed Point Conjecture on (T2s,wo) . . . . . 5.3. Lagrange Intersections for (CP", RP") . . . . . .
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153 156 161
164 169 172 172 173 176 177
182 188 188 195 198 203
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209 215 215 218 220
Chapter V: Applications to Harmonic Maps and Minimal Surfaces 1. Harmonic Maps and the Heat Flow . . . . . . . . . . . . 2. The Morse Inequalities . . . . . . . . . . . . . . . . . 3. Morse Decomposition . . . . . . . . . . . . . . . . . 4. The Existence and Multiplicity for Harmonic Maps . . . . . 5. The Plateau Problem for Minimal Surfaces . . . . . . . .
229 246 250 257 260
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References
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Index of Notation Index . . . . .
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Appendix: Witten's Proof of the Morse Inequalities 1. A Review of Hodge Theory . . . . . . . . 2. The Witten Complex . . . . . . . . . . . . . 3. Weak Morse Inequalities . . . . . . . . . . . 4. Morse Inequalities . . . .
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274 282 287 295 298 310 311
PREFACE
The book is based on my lecture notes "Infinite dimensional Morse theory and its applications", 1985, Montreal, and one semester of graduate lectures delivered at the University of Wisconsin, Madison, 1987. Since the aim of this monograph is to give a unified account of the topics in critical point theory, a considerable amount of new materials has been added. Some of them have never been published previously. The book is of interest both to researchers following the development of new results, and to people seeking an introduction into this theory. The main results are designed to be as self-contained as possible. And for the reader's convenience, some preliminary background information has been organized. The following people deserve special thanks for their direct roles in helping to prepare this book. Prof. L. Nirenberg, who first introduced me to this field ten years ago, when I visited the Courant Institute of Math Sciences.
Prof. A. Granas, who invited me to give a series of lectures at SMS, 1983, Montreal, and then the above notes, as the primary version of a part of the manuscript, which were published in the SMS collection. Prof. P. Rabinowitz, who provided much needed encouragement during the academic semester, and invited me to teach a semester graduate course after which the lecture notes became the second version of parts of this book.
Professors A. Bahri and H. Brezis who suggested the publication of the book in the Birkhauser series. Professors E. Zehnder and A. Ambrosetti, who provided a favorable environment during the period in which this book was written. Mrs. Ann Kostant, for aiding me in editing and typesetting the manuscript. My teacher Prof. M. T. Cheng for his constant support and influence over the many years. And, of course, I thank my wife and my children for their love, patience and understanding while I was writing this book. Kung-Ching Chang Mathematical Institute, Peking University, Beijing.
INTRODUCTION
This book deals with Morse theory as a way of studying multiple solutions of differential equations which arise in the calculus of variations. The theory consists of two aspects: the global one, in which existence, including the estimate of the number of solutions, is obtained by the relative homology groups of two certain level sets, and the local one, in which a sequence of groups, which we call critical groups, is attached to an isolated critical
point (or orbit) to describe the local behavior of the functional. Morse relations link these two ideas. In comparison with degree theory, which has proved very useful in nonlinear analysis in proving existence and in estimating the number of solutions to an operator equation, Morse theory has a great advantage if the equation is variational. Relative homology groups and critical groups are series of groups that provide both a finer structure and better estimate of the number of solutions than does the degree, which is only an integer. The relationship between the Leray-Schauder index and critical groups is established. The minimax method is another important tool in critical point theory. In this volume it is treated in a unified manner from the Morse theoretic point of view. The mountain pass theorem, the saddle point theorem and multiple solution theorems, discussed in Ljusternik-Schnirelman theory, index theory and pseudo index theory, are studied by observing the relative homology groups for specific level sets. Critical groups for critical points are also estimated. The purpose of this treatment is to provide a unified framework which contains different theories so that various techniques are able to be combined in estimating the number of critical points. Applications to semilinear elliptic boundary value problems, periodic solutions of Hamiltonian systems, and geometric variational problems are also emphasized. These problems are chosen for their own interest as well as for explaining how Morse theory is applied. The book is organized into five chapters and an appendix. Chapter 1 is devoted to Morse theory. Sections 1 and 2 review the basic facts of algebraic topology and infinite dimensional manifolds, respectively. Two deformation theorems, which play a fundamental role in critical point theory, are proved in detail in Section 3. Morse relations and the Morse handle body theorem are studied in Section 4. Section 5 deals with Gromoll-Meyer theory and discusses the main properties of critical groups for isolated critical points, including homotopy invariance and a shifting lemma. The MarinoProdi approximation theorem is also studied in this section. In the rest of the chapter, Morse theory is extended: in Section 6.1, to manifolds with boundaries together with certain boundary value conditions, and, in Section 6.2, to locally convex closed sets. The latter extension is motivated
x
lntnoduction
by variational inequalities. G-equivariant Morse theory is investigated in Section 7, where all the main results of Sections 4 and 5 are completely extended to invariant functions under a compact Lie group action. Chapter 2 views critical point theory with respect to homology groups. Sections 1 through 4 are devoted to this study. The homological link, subordinate homology classes, and Cech-Alexander-Spanier cohomological
rings are used to link up minimax principles with Morse theory. Morse index estimates in Minimax theorems are also presented. In Section 5, we give some abstract critical point theorems which will be applied in subsequent chapters. Two perturbation theories are studied in Section 6, one of which is concerned with the perturbation effect on a critical manifold,
and the other with Uhlenbeck's perturbation theory. Semilinear elliptic BVPs are considered to be models in the applications of critical point theory. The reader will find that there are many different and very interesting results presented in Chapter 3. Although some of them will be familiar, the proofs given here are new and are based on the above unified framework. Problems with superlinear, asymptotically linear and bounded nonlinear terms are studied by example in Sections 2-4. Variational inequalities are also discussed. Chapter 4 deals with some topics on Hamiltonian systems. Since there are special books on this subject, we satisfy ourselves with introducing material that does not overlap. The following problems were selected: asymptotically linear systems, Hamiltonians with periodic nonlinearities, second order systems with singular potentials, the double pendulum equation, Arnold conjectures on symplectic fixed points and on Lagrangian intersections. Our treatment of these is limited to examples. In the final chapter, we analyze two-dimensional harmonic maps and the Plateau problem for minimal surfaces as examples from geometric variational problems. Because of the lack of the Palais-Smale condition, Morse theory for harmonic maps is established by the heat flow. The Plateau problem is considered to be a function defined on a closed convex set in a Banach space. Extended Morse theory is applied to give a proof of the Morse-Toinpkins-Shiffman theorem on unstable coboundary minimal surfaces.
In the appendix, Witten's proof of the Morse inequalities is presented in a self-contained way. Although the material is totally independent of the context of this book, we introduce Witten's idea because the proof is so beautiful and surprising; moreover, it is a good example of the interplay between analysis and topology. This book is not intended to be complete, either as a systematic study
of Morse theory or as the presentation of many applications. We do not deal with Conley theory [Con1], stratified Morse theory, and the beautiful applications in the study of closed geodesics. (For an overview of the literature, the reader is referred to the book by Klingenberg [Kli1J) as well as to the study of gauge theory [AtBiJ.
CHAPTER I
Infinite Dimensional Morse Theory
The basic results in Morse theory are the Morse inequalities and the Morse handle body theorem. They are established on the Banach Finsler manifolds or on the Hilbert Riemannian manifolds in Section 4. The tool in this study is the deformation theorem, which is introduced in Section 3. Some preliminaries on algebraic topology and on infinite dimensional manifolds are reviewed in Sections 1 and 2 respectively. Readers who are familiar with the background material may skip over these two sections. Gromoll-Meyer theory on isolated critical points plays an important role in the applications of Morse theory because the nondegeneracy assumption in the handle body theorem might not hold for concrete problems. Section 5 is devoted to introducing Gromoll-Meyer theory systematically and examines the splitting lemma, the homotopy invariance theorem, the shifting theorem, and the Marino Prodi approximation theorem. The rest of the chapter consists of the extensions of the basic results of Morse theory in different directions: in Section 6.1, to the extension to manifolds with boundaries as well as to the functions satisfying certain boundary value conditions, in Section 6.2, to the extension from manifolds to the locally convex closed subsets; and, in Section 7, to functions with symmetry under a compact Lie group action.
1. A Review of Algebraic Topology
The idea of algebraic topology is to assign algebraic data to topological spaces so that topological problems may be translated into algebraic ones. The singular homology group is an example of algebraic data. It is constructed of the maps of geometric simplexes into arbitrary topological spaces so that it is applicable to infinite dimensional problems. Let X be a topological space, and let
Infinite Dimensional Morse Theory
2
be the standard q-simplex, q = 0, 1, ... where eo _ (0,0.... 0,...)
ei = (1,0,...0,...)
eq = (0,0,...41^,...)
are vectors in R1. A singular q-simplex is defined as a continuous map W : Aq X. Also, let Eq denote the set of all singular q-simplexes. Given an Abelian group C, we define the formal linear combinations:
a = I: gjai, gi E C, of E Eq. These sums are called singular q-chains. The set of all singular q-chains is denoted by Cq(X,G). Suppose that X, X' are two topological spaces, and that
f:X-+X' is continuous, then
f:a=E Mai-`>gif(a:) is a reduced homomorphism: Cq(X, G) ---+ Cq(X', C). For each a E Eq, we define the boundary operator 4
8v = E(-I)iQ(i) i=o
where O(ff) = t'IO,el,...,aj,...,egj,(eo,el,...,ej,...,eq) denotes the q- 1 simplex generated by the vectors eo,e1i...,eq except ej,j = 0,1,...,q. Then we extend the operator 0 linearly onto Cq(X,C), i.e.,
a>giQi = Egia0i. It is not difficult to verify: Cq_1(X, G) is a homomorphism, q = 1, 2, ... . ( 1 ) 8 : C 4 (X, G)
(2) 82c=88c=OVcE Cq(X,G). A different boundary opertor a# can be defined on 0-chains as follows:
a# > giai = E gi The relation
V of E Co(X, G), d i.
a#a=o
1.
A Review of Algebraic Topology
3
also holds.
Suppose that (X, Y) is a pair of topological spaces, with Y C X (being a subspace of X). We call (X, Y) a topological pair.
For two topological pairs (X, Y) and (X', Y'), we say that a map f
(X, Y) -+ (X', Y') is continuous if f : X --+ X' is continuous with f (Y) C Y'. Two maps f, g : (X, Y) -+ (X', Y') are called homotopic if 3 F (0, 11 x X - X', which is continuous and satisfies
F(0, ) = f, F(1, ) = g, and
F: [0, 11 xY-+Y'. Let (X, Y) be a topological pair, since
8: Cq(X,G)
Cq_1(X,G)
implies
8: Cq(Y,G) -+ Cq_I(Y,G). The boundary operator induces a homomorphism a which makes the diagram Cq (X, G) 81
Cq (X, G)/Cq (Y, G) 8j
Cq_I(X,G)
Cq_,(X,C)/Cq(Y,G)
commutative. Clearly 88 = 0. We call CC(X,Y,G) = Cq(X,C)/Cq(Y,G) the singular q-relative chain module. Then we define Zq(X, Y, G) = ker(a), the singular q-relative closed chain module, Bq(X, Y, G) = Im(5), the singular q-relative boundary module, and Hq(X, Y, C) = Zq(X, Y, G)/Bq(X, Y, G), the singular q-relative homology module. The rank of Hq (X, Y, G) is called the singular q-Betti number.
In the case where Y = 0, we write HQ(X, Y, G) = H9 (X, G). For q = 0,
Ho V, G) is defined as the quotient of ker(8#) by Im(8), and for q > 0, let Hq (X, G) = Hq(X, G). We call Hq (X, G) the q-reduced homology module. The 0-reduced relative homology module Ho (X, Y, C) is defined
as Ho(X, Y, C) if Y 54 0 and Ho (X) if Y = 0. The basic properties of singular homology modules are summarized as follows. Their proofs can be found in the book of M. J. Greenberg (Gr 11.
Infinite Dimensional Morse Theory
4
1. Suppose that f : (X, Y) - (X', Y') is continuous, then there is a reduced homomorphism
f. : HQ(X, Y; G) -+ IIq(X', Y'; G) V q.
(a) If f = id, then f. = id; (b) If g : (X',Y') (X",Y") is another continuous map, then the reduced homomorphism g. satisfies
(.qf). = g. f..
(c) 5f. = Mi. 2.
Ilomotopy invariance: If f, ,q : (X, Y) -p (X', Y') are homotopic,
then f. = g.. Two topological pairs (X, Y) and (X', Y') are called homotopically equivalent if there exist continuous maps
(X, Y) - (X', Y'), b : (X" Y') - (X, Y), :
satisfying
V)o m= id(x,y),
O o V) = id(x,,y,).
Thus, if (X, Y) and (X', Y') are homotopically equivalent, then Hq(X, Y, C) °-t 119 (X', Y', G) V q.
We say (X', Y') is a deformation retract of (X, Y) if X' C X, Y' C Y, and if 3 Ti: [0, 11 x X -' X satisfying
77(0, ) = idx, n(I, X) C X', +7(1, Y) C Y, 77(t, Y) C Y and n(t, ) I x' = idx,, `d t E (0,1].
Thus, if (X', Y') is a deformation retract of (X, Y), then
Hq(X',Y',G)
Hq(X,Y,G).
3. Excision: If U C X satisfies U C int(Y), then 119(X\U, Y\U, G) ~ IIq(X,Y,C). 4. Exactness: If Z C Y C X are three topological spaces, and we define the injections i : (Y, Z) - . (X, Z), and j : (X, Z) -. (X, Y), then we have the following exact sequence:
Ilq(Y,Z,G) !+ I1,, (X,Z,G)'-% IIq(X,Y,G)
IIg_i(Y,Z,C) -
A Review of Algebraic Topology
1.
5
In particular, since H. (X, G) = H9 (X, 0, G), we have
ffq(Y,G) 3 Hq(X,G) 2+ Hq(X,Y,C)
e' Hq(Y,G) The same exact sequence also holds for reduced homology modules.
5. If X consists of a family of path-connected components {Xk}, then
®1: Hq(Xk,XknY;C) Vq.
Hq(X,Y;G) 6. H9
0,
dq.
7. Ho(X,C) is a free group on as many generators as there are path components of X. If Y 34 0, Y C X, and X is path-connected, then H0(X, Y; G) °10. 8.
Kiinneth formula: Let X1 and X2 be subspaces of the topological
space X. Denote i : X -. X as the injection, v = 1, 2. (X1,X2) is said to be an excisive couple of subspaces if the inclusion chain map
C9(X1,G)+C9(X2iG) - Cq(X1 UX2,G) induces an isomorphism of homology.
For given topological pairs (X, Y), (X', Y'), we define their product (X, Y) x (X', Y') to be the pair (X x X', X x Y' U Y x X'). If G is a field, and if {X x Y', Y x X') is an excisive couple in X x X', then the cross product is an isomorphism:
H.(X,Y;C)0H.(X',Y';G)
H.((X,Y) x (X',Y');G),
g
H,7(X x X', X x Y' U Y x X'; G) °- ®Hq(X, Y; G)Hq_n(X', Y'; G), D-0
dq=0,1,2,.... In the case where C = a field Q, rank Hq(X,Y;Q) = dim Hq(X,Y;Q), we write
00
X(X, Y; Q) = 1: (-1)q dim H9 (X, Y; Q), g-o
Infinite Dimcn ,onal Morse Theory
6
and call it the Euler characteristic of (X, Y). The following homology groups are often used.
(1)
119(S", G)
n, when q,n> 1
0
q
C
q=n> 1, andq=0,n> 1,
G2 q=n=0. 0
(2)
q
n,
Hq(B",S"-1,G)= {l G q=n,
where B" is the n-ball, and Sn-1 = 8B". Ilq(T", G)
(3)
0 < q < n, q > n,
{0GC
where T" = S' x . . . x S' is the n-torus.
q>n
(4)
119 (f'", Z2) °f
Z2
q < n,
where P" is the real n-projective space. (5)
H9(CP", G) ^_ { 0
q > 2n or q odd,
G q even such that 0 < q < 2n,
where CP" is the complex n-projective space, and C = Q, the rational field, or Z. Now we turn our study to singular cohomology. The singular q-cochain
is defined to be the homomorphism c : Cq(X,G) - G : [al +02,c] _ (a1,c]+[a2,c], `da1,a2 E C9(X,C),
daECq(X,C). The set of all singular q-cochains llom(CG(X, C), G) is denoted by C9(X, G). C9(X, C) is a module: [a,CI + c21 = [a,cl] + [a, c21 d CI, C2 E Cq(X,G), V a E Cq(X,G),
VgEG,VaECq(X,G),VcEC9(X,G). , ] is a bilinear form on Cq(X,G) x C9(X,C). The dual operator of the boundary operator 8 with respect to (,
Thus the duality [
called the coboundary operator and is denoted by 6: [8a, r.] = [a, hc:] d a E C9 (X, G), V e E C9-' (X, G).
]
is
A Review of Algebraic Topology
1.
7
Hence, 6: C9-' (X, G) -' C'(X, G) is a homomorphism, and 02 = 0 implies
62c=OdcECq(X,G). Singular cohomology is defined as follows: For a topological pair (X, Y), let
C (X, Y; G) = Hom(Cq(X, G)/CC(Y, C), G),
and let :
?79-'(X, Y) - Zv4(X,Y)
Cq(X, Y; C) --
be the dual operator of the boundary operator Cq_ I (X, Y; G). Then define
llq(X,Y;G) = ker(b)/Im(3). It is easily seen that ZS9(X, Y; C) is isomorphic to
C9(X,Y;G) = {c E C'(X,G) I (v,cJ = 0
V a E Cq(Y,G)) .
The isomorphism is realized by the dual homomorphism
P' : G' (X,Y;G)
Z`9(X,C)
of the homomorphism
P : CQ(X,G) - Cq(X,Y;C). Therefore Zq(X, Y; G) := ker(b) = {c E Cq(X, G)I(o, cJ = 0 V a E Bq(X, Y; G)),
Bq(X,Y;C) := lm(b) = {c E Cq(X,G)I(a,cJ = 0V a E Zq(X,Y;G)) . In general, we have a canonical homomorphism:
a: HP(X,Y;G) - Hq(X,Y;C)'. In the case where C is a field, a is surjective. The properties of cohomology are very similar to those of homology. The important difference is as follows: Singular homology is a covariant functor of topological pairs, but singular cohomology is a contravariant functor. (1') If f : (X, Y) - (X', Y') is continuous, then
f' : H'(X',Y',G) We have
fl'(X,Y;G)
Infinite Dimensional Morse Theory
8
(a) If f = id, then f = id. (X', Y') -i (X", Y") is continuous, then (gf )' _
(b) If g (2')
:
(c) bf = f'a. If, f,g : (X,Y) -i (X',Y') are homotopic, then f' = g'. If (X, Y) ti (X', Y'), then 11' (X, Y; G)
11'(X', Y'; G).
(3') (Excision) 11' (X \U, Y\U; G) °° 11' (X, Y; G), if U C int(Y).
(4') (Exactness) If Z C Y C X, then the sequence +- H9(Y,Z;G) +- 114(X,Z;G) - H4(X,Y;G) is exact.
(5') H°({P} G) =
a
H9-1(y'
Z;G)
!Gq=0
0 g340. (6') If (X x Y', Y x X') is an excisive couple in X x X', and 11' (X, Y; G) is of finite type, i.e., HO(X, Y; C) is finitely generated for each q,
and C is a field, then
1I'(X,Y;G) ® 11'(X',Y';G)
11'((X,Y) x (X',Y');G).
We can define a product on singular cohomolgy groups such that the singular cohomology groups become graded algebras.
We denote C'(X,G) = ©q 0C4(X,G), and define a cup product as follows: V c E CP(X, C), `d d E C9(X, G), V o E Cp+q(X, G), we consider affine maps
) , : AP - AP+q Pq :
Oq - np+q
to be
Ap = (co, ... , ep),
Pq = (ep, ep+1, ... , e,,4.q).
and then define (o, c U d) = (oAp, c] - (op., d].
The cup product is bilinear, associative, and possesses the unit, i.e., the 0-cochain 1, which is defined by (x, 11 = e `d x E X. We may easily prove that 6(c u d) = be U d + (- 1)"c U 6d
b' c E CP(X, G), V d E C9(X, G).
Hence, Z' (X, G) is a subalgebra of C' (X, G) and 13' (X, G) is an ideal
of Z'(X,G). The cup product U is well defined on H'(X,G), and makes it a graded algebra. Furthermore, if f : X Y is continuous, then
f' :
11' (Y) 11' (X) is a ring homomorphism: f' (cUd) = f' (c) U f' (d), which satisfies f'b = b f'.
1.
A Review of Algebraic Topology
9
The cap product is defined as the dual operator of the cup product, i.e., n : Cp+y(X) X CP(X) Cq(X),
VCECP(X), VdECy(X), VaECp+y(X), (onc,tA = [a,cUd], or, equivalently, a n c = ((A,, copy.
The boundary operators relate the cap product as follows: 8(o n c) =(-1)P((8o) n c- or n bc).
da ECp+g(X), V c E CP(X). If / : X Y is continuous,then we have
/.)o n f'(c)) - /.(o) n c. Since V a E Zp+y(X), V C E ZP(X), we have o n c E Zq(X), and V a E Bp+q(X), d c E ZP(X), we have a n c E By(X), the cap product is well-defined on homology groups:
n : Hp+y(X) x HP(X) - Hq(X). The definition of cup product and cap product can be extended to topological pairs. In fact, we have n : Hp+9(X,Y;G) X HP (X, Y; C) ----+ H9 (X, C)
n : Hp+y(X,Y;G) x HP(X,C) -+ H9(X,Y;C), and
U : HP(X, Y1; G) x HP(X, Y2 : G) -+ HP+y(X, Y1 U Y2; G),
if (Y1iY2) is an excisive couple in X. The cup length of a topological space X is defined as
CL(X) = max{1 E Z+1 3c1,...,c1 E H'(X,G), dim(c;) > 0, i = 1,... ,1, such that c1 U . . . U ci # 0} . This is a topological invariant which is very useful in critical point theory. More generally, we define the cup length for a topological pair (X, Y).
CL(X,Y)=max{1EZ+13coEH'(X,Y), 3c1ic2,...,c1 E11 (X), with dim(c,) > 0, i = 1, 2, ... ,1, such that co U c1 U
U cl 96 0).
10
Infinite Dimensional Morse Theory
In the case where Y = 0, we just take co E H°(X ). These two definitions are the same. We may characterize CL(X, Y) by its dual.
Definition 1.1. Let (X,Y) be a pair of topological spaces and Y C X. For two nontrivial singular homology classes [a]], [02] E H.(X,Y), we say
that [a,] is subordinate to (a2), denoted by [a,] < [0`2), if there exists c E II' (X), with dime > 0 such that fall= [a21 n c,
where fl is the cap product. Let us define L(X, Y) = max {1 E Z.} 13 nontrivial classes [aj) E H. (X, Y), 1 < j:5 1, such that jai] < (a2] < <
Theorem 1.1. L(X, Y) = CL(X, Y) + 1. Proof. For L(X, Y) = I + 1 if and only if 3 nontrivial classes [ao] < [all < < [a,) in fI. (X, Y), i.e., 3 c, E H' (X ), dim c, > 0, 1 < i < 1 , such that jai+,1 = fail flc;, i = 1,2,...,1. However, 3 co E II' (X, Y) such that [ fool, co ] # 0 is equivalent to the nontriviality of [aol. And since [fall, c U c1_, U ... U co) = [[a,_,), c,_, U c,_2 U ... U co)
_
[[aol,co),
L(X, Y) = 1 + 1 if and only if CL(X, Y) = 1.
0
The honiotopy group is another important topological invariant. Let us recall some basic definitions and properties in homotopy theory.
Let X be a topological space and p be a point in X. We call (X, p) a pointed space with base point p. A topological pair (X, Y), in which Y is a subspace of X that contains p, is called a pointed pair (often written (X,Y,p)). A reap f from pointed space (X, p) to a pointed space (X', p'), f : X X', with f (p) = p', is called a pointed map. Similarly, we define a pointed pair map, pointed homotopy, pointed pair homotopy, and so forth. Let I" denote the n-dimensional unit cube, n > 1, 1"-' C I" the bottom
space (t = (t,,t2,... ,tn) E I", if 0 < t, < 1, i = 1,2,... ,n, and t E
In-1
1.
A Review of Algebraic Topology
11
if, further, to = 0). Set Jn-' = 8In\In'', i.e., the union of all the other faces. We denote by On (X, Y, p) the set of all continuous maps
0: (In,BIn-',Jn_1) - (X,Y,P), 0 : In
X, q(8In-1) C Y and O(Jn-1) = p.
(Denote by fln(X, p) the set of all continuous maps
(In, 8In'')
(X,p)) The n-relative homotopy group An (X, Y; p) is defined as the set of all components of iln(X, Y; p) (the n homotopy group lrn(X, p) for those of Dn(X,P)). In fact, irn(X,P) = 7rn(X,P;P) One may define a multiplication on fln(X, Y,p) : For n > 2 (and, on Hn(X,P), f o r n > 1) : b 0, 0 E fln(X,Y,p), (2t1, t2, ... , tn)
(0 V') (t)
0 < t1 < 3
f!J(2ti -1,t2,...,tn)
21 -
`dtEIn.
The homotopy class 10 * 01 clearly depends only on (0] and (>'J. Hence we may define a multiplication on 7rn(X, Y, p) by taking 101 * (01= (m *
PJ.
According to the multiplication, which is obviously associative, the identity element (e] is the class which contains the unique constant map
An inverse element of 101 is the class (0 o 01, where 0 : In the map defined by 9(t) _ (1 - tl,t2,...
In denotes
tn)
for every t E In. With the multiplication structure, Jrn (X, Y, p) is a group for n > 2, and irn(X,p) is a group for n > 1. For n = 0, we define no (X, p) to be the set of path-connected components of X with the path component of p as a distinguished element. So, iro(X, p) is only a set, without group structure, as is 7r1(X, y, p). Moreover, one can show that 1rn (X, p), for n > 3, and 9rn (X, Y, p), for n > 2, are abelian groups. There are alternative definitions of homotopy groups.
Since Jn-' = 8I"\In-' is contractible, let zo = 0 in In, and the inclusion map
(In,8In,zo) _ (I",BIn, in-1)
Infinite Dimensional Morse Theory
12
is a homotopy equivalence. Hence 7rn(X, Y, p) is defined as equivalent to the set of homotopy classes of maps:
0: (1n,8I',zo)
(X,Y,P)
Again, let x0 = ( 1, 0, ... , 0) E R , D" = the unit ball in 1!R", and Sn-1 = 8D", it" (X, Y, p) as the set of homotopy classes of maps
0: (D",S"-1,x0) _ (X,Y,P) The homotopy groups enjoy analogues of all the axioms of homology groups except the excision axiom.
(1) Suppose. f : (X, Y, p) -. (X', Y', p') is a continuous pointed pair map, then there is a reduced homomorphism f. : ak(X,Y,P) (a) If f (b) If q
7rk(X',Y" P')
-
= id, then f. = id.
(X', Y', p') (X", Y", p") is another continuous pointed pair map, then Of ). = g. f.. :
(c) Let 8 : irk (X, Y, p) -. ,rk_ I (Y, p) be defined by restriction, i.e., for a given 0: (In,81",81"\In-') (X,Y,p), we define af4>1= [4>larnl
We have the following commutative diagram:
nk(X,Y,P)
a Irk -I (Y, P)
If. irk (X', Y', P')
I (f I'). a
Irk -I (Y',p').
(2) (Ilomotopy invariance). Suppose that f is pointed pair hornotopic to g, where f, g : (X, Y, p) -. (X', Y', p'). Then
f. = g.. Thus, if (X, Y, p) and (X', Y', p') are pointed pair hornotopic equivalent., then irk(X,Y,P) = Irk (X',Y',P )
(3) (Exactness). Let p E Z C Y C X, then the following sequence is exact: -
irk+t(X,Y,P) a'-. nk(Y,Z,P)
nk(X,Z,P) -' nk(X,Y,P) -i ...
1.
A Review of Algebraic Topology
13
where i : (Y, Z) (X, Z), j : (X, Z) - (X, Y) are inclusions, and define 8' : lrk+l (X, Y, p) -' Irk(Y, Z, p), k > 1, to equal the composition 71k+1(X, Y, P) -0' lrk(Y, P) inclusion.
,rk(Y, Z, p), where j' : (Y, p)
- (Y, Z) is an
(4) If Y is path-connected, then wk (X, Y, p) does not depend on p E Y. In this case, we abbreviate lrk (X, Y). (5) If X = {p}, then irk (X, p) = 0 V k > 1. Thus, if X is contractible, i.e., X is homotopic to a point, then 7rk (X, p) _
0 dk> 1. (6) If (X, p), (Y, q) are pointed spaces, then Irk (X X
ED
(7) If (X, p) is a pointed space, we define its loop space 12(X, p) to be the function space consisting of all continuous maps : (Il, 69P) - (X, p) endowed with compact open topology. Then Irk-1((1,w) = nk(X,P) for k > 2,
where Il = f2(X, p), and w is the constant map w(t) = p d t E I. (8) If 0 < k < n, then Irk(S") = 0, and Irn(S") 54 0,
`dk>2, Irk
(S2"+1) L Irk (CP")
d k > 3.
The following important property establishes the relationship between homology and homotopy.
(9) Theorem 1.2. (Hurewicz Isomorphism Theorem). Let (X, Y, p) be a pointed pair, and assume that Y and X are simply connected. If there is a k > 2 such that H9(X, Y) ?5 0 d q < k, then ir9(X, Y, p) or 0 V q < k, and there is an isomorphism 7rk(X,Y,P)
d Hk(X,Y)
For (absolute) homotopy groups, the same conclusion holds for k > 1 and Y = {p}, where these homology groups are of integral coefficients. (10) Let (X, Y) be a (pointed) pair (with base point p) such that Y is a
retract of X. Then
H.(X) (and
H.(Y) ® H.(X,Y)
Irk(X,P) °-` Irk(Y,P) ®7rk(X,Y,P) d k > 2).
Claim. Let i : Y - X and j : (X, 0) -. (X, Y) (or (X, p) - (X, Y), where p is the base point of the pair (X, Y)), be the inclusions, and r
Infinite Dimensional Morse Theory
14
is the retraction: X -. Y. It follows that r.i, is the identity map on II.(Y) (or lrk(Y,p) V k > 1). This implies that i. is a monomorphism and r. is an epimorphism, i.e., H. (X) H. (Y) ® ker r. (or irk (X, p) 7rk(Y, p) ® ker r., d k > 1). From the exactness of the homology (homotopy) sequence
Hk+I(X,Y)
H, (Y)
Ilk(X)
'-' Ilk(X,Y) -. (-' ak+I(X,Y,P) '9' 'k(Y,P)
\
lrk(X,P)
L. nk(X,Y,P) -) ,
because ker i. = 0, 0. is a trivial map, and therefore j. is an epimorphism. Since kerj. = Im i., j. induces an isomorphism from ker r. onto H. (X, Y) (onto irk(X,},p) if k > 2, resp.). (11) The following theorem is useful in the sequel.
Theorem 1.3. (Palais). Let V be a Banach space and E be a dense subspace (linear) of V with finite topology.
Given 0, an open set in V, let O = 0 fl E be a subspace of E. Then the inclusion map i : O -. 0 is a homotopy equivalence.
Corollary 1.1. Let Vl and V2 be Banach spaces, and let f : V, -i V2 be a linear continuous map of V, onto a dense linear subspace of V2. Given 0
open in V2, let O = f -1(O) and f = f io. Then f : O -. 0 is a homotopy equivalence. (cf. Palais (Pal 2, Theorems 16, 17].) 't'hus, if X is a Banach space embedded continuously into a Hilbert space H as a dense linear subspace, then we have
H.(A,B;G) = H. (AIx,Blx;G), for any pair of open sets (A, B) in II, where (AIx, BIx) is the restriction of (A, B) on X.
2. A Review of the Banach-Finsler Manifold Most manifolds interested in analysis are infinite dimensional. Definition 2.1. Let X be a Banach space, and let M be a connected
Hausdorff space. We say that M is a Banach C' manifold, r > 1 (integer), modeled on X, if (1) 3 a family of open coverings {Ui I i E A}, (2) 3 a family of coordinates {0i I : U; -+ O,(U,) C X, homeomorphism,
iEA),
A Review of the Banach-Finsler Manifold
2.
(3) O, o 0j' `di,i'EA.
15
4,-(U, n U,-) -y O:(Ui fl U,-) is a C'-diffeomorphism,
:
Each pair (.,, U,) is called a chart. The set ((O,, U,) I i E A) is called an atlas.
In a similar way, we define C'(C''0) maps between two C' Banach manifolds, and vector bundles over Banach manifolds (we omit the defini-
tions and basic properties), in particular, the tangent bundle T(M) and cotangent bundle T'(M). Let E = (E, 7r, M) be a vector bundle. : M --y E is called a section, if it o C = idM. A section is called C' (or C'-0) continuous, if it is a C' (or C'-0) map from M to E. A section of the tangent bundle is called a vector field, and a section of the cotangent bundle is called a co-vector field. For a given C'-0 vector field , and a given point p E M, there exists an unique maximal semiflow a : 10, T) - M satisfying
cr(t) =f(a(t)), a(0) = p.
A Riemannian manifold (M, g) is metrizable. The metric d is defined by the arclength of the geodesics, and thus it is defined by the Riemannian metric g: d(x, y) = inf {
/
1
g(o(t), a* (t)) I dt
0
IaEC'(I0,11,M) 0(0)=x,0`(1)=y}. As a metric space (M, d), the topology coincides with (is equivalent to) the topology of the manifold.
Generally speaking, in order to define the distance (or metric d), the inner product g is not necessary. A norm is enough. We may introduce a metric d on a Banach manifold. Since the Riemannian metric is defined on T(M) globally, we shall introduce a Finsler structure on a Banach manifold in a similar manner. Definition 2.2. Let it : E -' M be a Banach vector bundle. R' is called a Finsler structure if (1) II II is continuous, (2) V p E M, II . llp:=II . 111 Ep is an equivalent norm on Ep
II
7t
.
II: E -'
(P),
(3) `d po E M, for any neighborhood U of po which trivializes the vector
bundle E, i.e., E I U = 7r-' (U) : U x Epo; d k> 1, 3 a neighborhood V of M with V D U such that `dpEV.
Infinite Dimensional Morse Theory
16
Example 1. If E is a trivial bundle U x X, then II(p,x)II = UxIIx is a Finsler structure.
Example 2. Suppose that 7f : E M is a Hilbert bundle, with g : C°°(E) x C°D(E) - C°°(M), a bilinear symmetric positive definite C°° map (Riemannian metric). Then 11(p' x)11 =
g(a, a)p
(where a(p) = x)
is a Finsler structure. Claim. V P E M, 3 a trivialized neighborhood U; and the corresponding map r, a-1(U;) -. U. x Ifs, where Ili is a Ililbert space. It follows that 3 U1 L(11;, H,) such that A,(p) is positive definite and satisfies
g(a,a)p = (Aj(p)x,x);. (1) is obvious, (2) follows from the positive definiteness of A;(p). And Aj (p) implies (3). the continuity of p,# A;(p), as well as of p
Lemma 2.1. Suppose that M is a paracompact C'-Banach manifold and that 0 = {(O;, U,)li E Al is an atlas, then there exists a C''0-partition of unity subordinate to 0.
Proof. Since Al is paracompact, there is a locally finite refinement {op I f3 E B} of 0, i.e., V 0 E B, 3 a = a(13) E A such that op C UQlpi. Let Vp = (op) C X, and let
hp(x) = inf {IIx - yII I Y V Vo} ,
then hp E CI-0(X, R+) and Vp = {x E X I hp(x) > 0}. Claim. V X1, x2 E X, V c> 0, 3 y E V0 such that
IIx2-A<-ho (x2)+c, and so hp (xl) < Ilxi - yll <- Ilxl - x211 + IIx2 - yII llxl
-x211+hp(x2)+E,
i.e., hp(x1) - hp(x2) < Ilx, - x211 + c. We may change the positions of x1 and x2i and since c > 0 is arbitrary, we obtain Ihp(xi)-ho (x2)1 511x1 -x211
2.
A Review of the Banach-Finsler Manifold
17
Define fp = hp o 0a(p),
and define 17p(p) =
fp(P)
E fa(P) OE B
Then we have
sup gpCa CU.(#),VQEB, and E no (P) = 1.
IEB
0
Theorem 2.1. Suppose that M is a paracompact Banach manifold and that 7r : E -. M is a Banach vector bundle, then there exists a Finsler structure on the vector bundle.
Proof. Choose an open covering {(r;, U1) I i E A) which trivializes the vector bundle. On each 7r-1(U;), we have a natural Finsler structure II' Since M is paracompact, there is a locally finite refinement {op 10 E B} of the above open covering and the corresponding CI-0-partition of unity
(op, np) 10 E B}, say, V# E B, 3 a= a(#) E A such that ap c U,(#). Let 111
111= E 170 0
Ila(p)
OE B
This is a Finsler structure on E. Claim. The continuity of III III is trivial. V p E M, 3 finitely many 0 E B such that r/p(p) # 0, since IIxIIp,Q(p) = II(P,x)IIo(p) is a norm V /3 E B. Therefore IIIxIIIP = E np 0 ir(x)II (P, x)IIa(p) = E np(P)IlXll P.o(p) $EB
$EB
is also a norm, which is an equivalent norm on Ep. As to (3), V p0 E M, V k > 1, by definition, 3 a neighborhood U of Po such that there are at most finitely many 1,6i I" such that j70(p) = 0, V,6 V p E U. Since II IIi is a Finsler structure, 3 V C U, a smaller neighborhood of po such that k-}IIxIIp.a(ai) < Ilxllpo.a(pi) !5 k4IIxIIp,a(pi) and
k-4np;(p) S'ti (po) <_ k4goi(P)
18
Infinite Dimensional Morse Theory
d p E U, i = 1, 2, ... , n, we obtain k-' IIxIIp S 11x11
<
d p E V.
0
Definition 2.3. A regular C'-Banach manifold M, together with a Finsler structure on its tangent bundle T(M), is called a Finsler manifold. Example 3. Any paracompact Banach manifold possesses a Finsler structure on its tangent bundle, making it to be a Finsler manifold.
Example 4. If Al is a Finsler manifold, with Finsler structure II . II, then we can define II(p,x')II = Slip {(x',x)IIIxIIp S 1 V x E Tp(A1)}.
V x' E TT(M)', it is a Finsler structure on T* (M). Particularly, if f : Al - R', where Al is a paracompact Banach manifold, then 11(p, df (p))II is well defined and p '-+ II(p, df (p))II is a continuous function. In the following, we omit p in (p, df (p)), and denote it by II df (p)II For a Finsler manifold At, we may define a metric d !LS follows:
d(x, y) = inf {L(a)Ia E C'((0,11, A1), a(0) = x, a(1) = y} , where
L(a) =
I1
II(a(t),a'(t))Ildt.
One can prove the following:
Theorem 2.2. Suppose that Al is a Finsler manifold, and that the function d is defined above. Then d is a metric on At, and the reduced topology is equivalent to the topology on the manifold. For a proof we refer to Palais [Pal4]. Finally, we come to the main subject of our theory.
Definition 2.4. Let Al be a C'-Finsler manifold. Let f E C' (AI, a point p E At is called a critical point of f if df (p) = 0. The set K = {p E At I df (p) = B} is called the critical set. A real number c is called a critical value if f -' (c) n K # 0. A real number, if it is not a critical value of f, is called a regular value. The complement set of K, i.e., M' = At \K, is called regular set. A point in A1' is called a regular point.
3.
A Pseudo Gradient Vector Field and Deformation Theorems
19
3. A Pseudo Gradient Vector Field and Deformation Theorems The basic idea in critical point theory is to investigate the variations of topological structures of the level sets of a given function f. Certain flows,
depending on the gradient of the function f, are used to deform these level sets. In finite dimensional manifolds, or more generally, in Hilbert Riemannian manifolds, gradient flow is a natural candidate. However, there are two disadvantages to gradient flow: (1) It needs more smoothness, say
C", and (2) it only works on Hilbert Riemannian manifolds, because, for a Banach Finsler manifold M, f : M - R1, d/ E T* (M) rather than T(M), the gradient flow is not well-defined. We are introduced to Definition 3.1. (Pseudo gradient vector field). Let M be a Finsler manifold, and let f : M -. R' be differentiable at p E M. X E Tp(M) is called a pseudo gradient if (1)
IIXII 5 211df(p)II,
(2)
(df(p), X) ? lld!(p)112,
is the Finsler structure. where (, )p is the duality on Tp(M), and II Let S C M be a subset, and, if f is differentiable on S, X is called a pseudo gradient vector field on S (p.g.v.f. in short), if V p E S, Xp is a pseudo gradient of f at p. 11
Lemma 3.1. Suppose that M is a Finsler manifold, and that f : M
C
R'. Let M' = M\K, where K = (p E M I df (p) = 0) is the critical set; then d p E M', there exists a pseudo gradient of f at p.
Proof. Since Ildf(p)11 & 0 V p E M', by definition, 3 X E Tp(M) such that IIXII = 1 and (df (p), X) > 2 Ildf 3 (p) II Let Y = 2 Il 2 df (p) II X , then 0 IlYll = Ildf(P)II < 2IIdf(P)II, and (df(p),}')p > IIdf(P)IIZ
Theorem 3.1. Suppose that M is a C2-Finsler manifold and that f :
Mc R1. There exists a C1-0 p.g.v.f. off on M'. Proof. According to the lemma, V p0 E M*, 3 Xp0 E Tp(M) such that IIXp0I1 < 211df(po)11,
(df(po), Xpo)po > Ild!(po)112.
The continuity of the Finsler structure and the continuity of df (p) imply that there is a neighborhood Vp0 of prr Vp0 C M', such that IIXp0I1 < 211df(p)11,
20
Infinite Dimensional Morse Theory
and
(df (P), XPO) P > 11df (P)112, d p E V.
Since M' is metrizable, it is paracompact. There is a locally finite C1-0 partition of unity {qp I )3 E B), with supp 1713 C Vim, for some p0 = po(13) E
M' . Let X = X(P) =
rl0(P)XPO(, ) OE B
This is the p.g.v.f we need. Claim.
IIX(P)II <_ Ens(P)IIX,(v)II <2IIdf(p)II,
(df(P),X(P)) = En0(P)(df(P),XPp(O)) > Ildf(P)112.
The local finiteness of the supports of (r
I
/3 E B) implies the C''0
smoothness. Once we have a p.g.v.f. of a function f, we get a decreasing flow by solving the following ordinary differential equation (ODE):
a(t) = -X(a(t)) a(0) = x0 E P4'.
The equation is locally solvable. Along the flow, the function t i- f o a(t) is decreasing: dtfoa(t) _ (f(a(t)),a(t))
-(f'(a(t)), X (a M) -IIf'(a(t))1l2. Our first goal is to prove that if f -'[a, b[nK = 0, i.e., f has no critical value in the interval [a, b], then fa is a deformation retract of fb. The deformation will be realized by the flow. Now the problem is that the pseudo gradient flow so far is only defined locally; we do not know if it could exist as long
as it arrives at f,. A condition on the function f is needed.
R' and c E R1, we say that f satisfies Definition 3.2. Given f : M the (PS), condition if any sequence C M along which f c and df 0 (strongly) possesses a convergent subsequence.
We say that f satisfies the (PS) condition if it satisfies (PS), for all c E R1. We have the following facts:
3.
A Pseudo Gradient Vector Field and Deformation Theorems
21
i
(1) If F : M _. R1 satisfies (PS), V C E [a, b], and if K n f -'[a, bJ = 0, then 3 to, bo > 0 such that Ildf(x)II > eoV x E f -1 [a - bo, b + bol.
Claim. If not, 3 xn E f -'[a - n, b+ 11, n = 1, 2,... , satisfying df 0. According to the (PS)c V c E [a, b], there exists a convergent subsequence xni - x', which implies x' E K n f -' [a, bJ. This is a contradiction. 1
(2) If f : M 2, R' satisfies (PS)c, then Kc := K n f (c) is compact. Claim.
If {xn} C Kc, then f(xn) = c and df(xn) = 0. The (PS)c
condition implies a convergent subsequence.
Lemma 3.2. (Deformation). If f E C'(M,R') satisfies (PS)c, V c E [a, b], and if K n f '' (a, bJ = 0, then fa is a strong deformation retract of fb-
Proof. 1. We consider the pseudo-gradient (semi) flow on f a(t)
= -X(a(t))/IIX(a(t))II'
o(0)
= xo E f[a,bl
[a, bJ :
We want to show that the maximal solvable half interval 10, T 0) satisfies (1)
T:o < +oo,
(2)
f(a(T=o - 0)) = a.
Claim. Since
f(a(t)) - f (xo)
f
< df (a(r)), a(r) > dr < - t4.
0
each initial point xo E f -1 [a, bJ arrives at fa along the flow in a finite time, i.e., T=, < +oo. Noticing II
f
t'
o(r)dr IIS
f
t'
II X(
T) II
this implies a' =
to
It, - t),
lim a(t) exists. If f (a') > a, then by fact(1), the flow t-+T.,-0 a(t) can be extended beyond T=o. This is in contradiction with maximality. We call T=o the arriving time. 2. The arriving time function x TT : f -' (a, bJ R1 is continuous.
Claim. t = Ta is the solution of the equation f(a(t,x0)) = a,
22
Infinite Dimensional Morse Theory
where we write a(t, x0) the flow a(t) emanating from x0. Since
f oa(t,xo)It=T.a = (c(f(a(T=o,xo)), o(T..'xo))
< -4,
the implicit function theorem is applied. 3. Define
(t, x0) =
if xo E fa a(Txot, xo) if x0 E fb\fa, x0
then n : [0, 1) x fb -, fa satisfies id,
77 (1, A) C fa,
77(t, .)If. = id/a V t E [0, 1].
We only want to verify the continuity of n. The verification is divided into three cases: 0
(1)
f.,
1 0,
(2) on [0, 11 x (fb\ fa), the continuity of rl follows from the theory of ODE, (3) on [0,1] x f -t (a).
Vc > 0 we want to find 6 > 0 such that
f(x)=a IIx-yll<6==>Ilx-17(t,y)II<e V t E [0, 1] If Y E fa, then 6 = c.
If y ¢ f, then ll"i(t, y) - xII = Ila(Tvt, y) - xII Tit
Iix -
yII+I
Ila(s,y)Ilds
<6+coTV=6+ o(Ty-T=). By the continuity of the function 7=, 3 60 > 0 such that if Ily - xII < 6o, we have
7y < 7= + 2coc.
Therefore, I]ij(t, y) - xII < e, when IIx - yII < 6 = min { z,6o}
.
3.
A Pseudo Gradient Vector Field and Deformation Theorems
23
Theorem 3.2. (Second deformation lemma). Let M be a C2 Finsler manifold. Suppose that f E C1 (M, 11 F1) satisfies the (PS), condition V c E
[a, b] and that a is the only critical value off in (a, b). Assume that the connected components of K. are only isolated points. Then f,, is a strong deformation retract of fb\Kb. Before giving the proof, we need a fact from point set topology (cf. Kelley (Kel1J).
Lemma 3.3. Suppose that K is a metric compact space and that F1, F2 are two compact subsets. Then either 3 a connected component of K, joining F1 and F2, or 3 compact subsets K1, K2 such that K1 U K2 = K, K1 n K2 = 0, and F; C K,, i = 1, 2. Proof. This is an improvement of Lemma 3.2, so we define the pseudogradient flow as before:
a(t,x) = &0, ,:s a(0, x) =x E f-1(a,b)\Kb. As we have shown that V x E f -1(a, bJ\Kb, there exists the arriving time T= > 0 such that lim f o o(t, x) = a. t-+T,-O
1. We shall prove that the Jim o(t, x) does exist and then
tT.-0
f(o(T. - 0, x)) = a. Claim. (PS), implies K. is compact. Either one of the following cases occurs: (a)
inf
dist(o(t, x), K0) > 0,
inf
dist(o(t, x), K.) = 0.
t E (0,T=)
(b)
t E IO,Tx)
In case (a), again by (PS), V c E (a, b), 3 a > 0 such that inf
t E IO,T=)
IIdf(a(t,x))tI > a.
Thus [t,
dist(a(t2, x), a(t 1 i x)) < e,
t2
.i ,
II dt dt
II
dt
IIX(C(t,x))II
<
ft2 ,
dt
< It2 - t1 J a Jdf(a(t,x))II
Infinite Dimensional Morse Theory
24
Since T is finite,
lim a(t, x) exists. t-.T=-o
In case (b), we shall prove that 3 z E Ka such that a(t, x) = Z.
lim
t --.T1 -0
First, we assert that lira dist(a(t, x), Ka) = 0. t- .T=-O
If not, 3 t; -+ T= - 0, such that
dist(a(t;,x), K0) > Fo > 0. But case (b) implies that 3 t; -. TT - 0 such that
Jim dist(a(t;,x), K0) = 0. I -COO
Thus we have two sequences t; < t? ', both converging to 7=, such that dist(a(t,*, x), K0) = co
dist(a(t;',x), K,) = co, and
Vt E [t;,t;
a(t,x) E
where (K0)6 denotes the 6-neighborhood of K.. Again by (PS), V C E [a, b], we have inf., IJdf ((7(t, x))JI > a > 0.
iElti ,tI Therefore e-o
2
< dist(a(t;',x), a(t;,x)) rt <
.
II
dt II dt
This is a contradiction. It follows that lira df (a(t, x)) = 0, t-DTs-o
provided by the compactness of K0.
3.
A Pseudo Gradient Vector Field and Deformation Theorems
25
/11
(PS)a condition then implies that the limit set A of the orbit {o(t, x)It E (0, T=)} is nonempty and that, for each sequence t; -. t=_o, there exists a subsequence t i such that a( t;, x) is convergent. Next we prove that A is a compact connected subset of K4. The compactness is obvious. We only want to prove the connectedness. If not, 3 open subsets 0 and 0' such that
OnO'=0,A=(OnA)u(O'nA) and OnAO0, O'nA760. Choosing z E O n A, z' E O' n A, 3 t, -+ TT - 0, t; -. T,, - 0 satisfying a(ti, x) - z, a(t;, x) -+ z'. For large i, we have a(t{,x) E 0, or (t;, X) E O' so 3 t j* E (t;, t;] (or It, t;)) such that
a(t;,x) 59 OUO'.
Denote the limit of u(t;, x) by z'. (It exists because of (PS)a.) So z' E A
but a' it 0 U O'.
26
Infinite Dimensional Morse Theory
This is a contradiction. According to the assumption that the connected components of K. are isolated points, we conclude that A = a point z E K0. 2. We shall prove the continuity of the function T. As shown in Lemma 3.2, if o(T=o - 0, xo) V K then the function T,T is continuous at x0. So we restrict ourselves to the case z = a(T=o - 0, xo) E K0.
If Tx is not continuous at such a x0, then 3 x -+ x0 such that IT=r - T 0 I
Co > 0,
so either (subsequence) TT < T=o - co, or T= > T10 + co.
Since
foo(T=-e,x)-foa(t,x) = J c
7!Z
df(a (t, x)) di dt
we have f o (o, (t, x)) ? a + I (T
- t).
But for any fixed c >/ 0, according to ODE, (liSt(C(T10 - e,xn), (a(7mo - e,xo)) -' 0,
3.
A Pseudo Gradient Vector Field and Deformation Theorems
27
so that f oo(T=o - e,xo) = lim f oo(T=o - e,xn) n
oo
1
rr
> limo La+
4(T= -T=o +e)J
a+ I(co+e), if T= >T.+eo. Letting e - 0, we obtain This is a contradiction. Similarly, we prove that Ti,, < T=o - co is impossible. 3. Finally, we define the deformation retract as before
sl(t, x) =
Claim.
x o(T=t, x) o(T= - 0, x)
if (t, X) E 10, 1) x fo, if (t, x) E 10, 1) x (fb\(fa U Kb)), if (t, x) E { 1) x (fb\(f, U Kb)).
Only the continuity of rl has to be verified. Four cases are
distinguished: (a) (t, x) E 10, 1) x fo, (b) (t, x) E 10, 1) x (f-(a,b)\Kb),
(c) (t, x) E {1) x (f -' (a, b)\Kb), (d) (t, x) E [0, 11 x f `(a).
We only want to verify cases (c) and (d). Since their proofs are similar, we only give the verification for (c). If , is discontinuous at (1, xo), then 3 c > 0 and to - T=o - 0, xn xo
such that dist(a(tn,xn), o(Txo - 0,x0)) > C. Let z = o(T=, - 0, x0) (E K,), and let
F, = (z), F2 = (M\B(z, e)) n Ko.
F, and F2 are compact subsets of K0. Provided by the assumption of K0, and the lemma, we have compact subsets K,, K2 C K. such that K, fl K. = 0, F; C K;, i = 1, 2, and K, U K2 = K0. Obviously, we may O take K, C B(z, e), and N = K2 U (M\B(z, e)), then O
or = dist(N, K,) > 0.
28
Infinite Dimensional Morse Theory
K1
K2 x
The continuity of the flow as well as of the arriving time T implies that 3 6 > 0 such that dist(a(t, xo), a(T10 - 0, xo)) < if t E IT,,, - d, T20 ), and 3 61 > 0 such that
T=>T=o-6 if x E B(xo,61). For t E [Tz0 - 6, T 0) n IT.. - 6, T= ), x E B(xo, 61), 3 be E (0, 6) such
that
dist(a(t,x), a(t,xo)) < g. In summary, for such a t, and for any x E B(xo, bt ),
a(t,x) E (Kl) f , the 4 neighborhood of K1. For large n, xn E B(xo, Se), denote t',, = such a t, satisfying
a(t'n,xn) E (K1)f. Reducing 6, and repeating the above procedure, we obtain a subsequence {xn, tn, tn} such that tn, to -, 7=0 - 0
a(t',, xn) E (K1) f I
O(tn, xn)
K
B(2, c).
A Pseudo Gradient Vector Field and Deformation Theorems
3.
29
We may assume to < t,,; then we have In, fn such that
tn<1n
o(tn,xn) E 8j(K1)f] O(fn, xn) E 81(N)?) a(t, xn) 1! (K1) V U (N)? V t E [Pn, tn],
n = 1,2,.... According to (PS)T, V c E [a, b]
p=inf{Ildf(x)Illx E f-'(a,b)\((K1)I U(N)f)} >0. Therefore
< dlst(O(tn,xn), Q(tn,xn))
II
(Idt <
Ifn - LnI -+ 0.
2 This is a contradiction. The proof is completed. Remark 3.1. Theorem 3.2 is due to E. Rothe (Rotl), K.C. Chang (Chal) and Z.Q. Wang (WaZ4). Another deformation lemma is also often used.
Theorem 3.3. (First deformation lemma). Let M be a C2 Finsler manifold. Suppose that f E C' (M,1R 1) satisfies the (PS),, condition. As-
sume that N is a closed neighborhood of K,, = K n f -'(c). Then there exist a continuous map rl
:
10, 1) x M --. M and constants F > e > 0 such
that
(2) 7)(t,')Icf-'lc-r,c+rl = idlcf-ilc-t,c+j, (3) rl(t, ) : M -+ M is a homeomorphism V t E (0, 1], (4) 77(1,ff+,\N) C ff (5) f o 1(t, x) is nonincreasing in t V(t, x) E 10, 11 x M. We would rather prove a more general version of Theorem 3.3, which indicates the fact that E depends on constants derived from the Palais-Smale condition. Namely,
30
Infinite Dimensional Morse Theory
Theorem 3.4. Let M be a C2-Finsler manifold. Suppose f E C' (M, R'). Let N' C N be two closed neighborhoods satisfying dist(N',8N) > 7b, 6 > 0. Suppose that there are constants b and iF positive, such that
Ildf(x)II >- b V x Erff+.\(ff-, U N'),
0<E
We assume Theorem 3.4 at this point; then we give the
Proof of Theorem 3.3. According to (PS),., K, is compact. [fence, for 6 > 0 sufficiently small, N(6) = {x E M I dist(x, K,) < 6) C N. Again, by (PS),., there exist constants b, E > 0, such that (3.1)
Ildf (x)II ? b V X E ff+z\(f,-z U N(6 )).
Since (3.1) remains valid, if E is decreased, we may assume
0<E<Min{4bb2, ,6b}. Theorem 3.3 follows from Theorem 3.4, if we take N' = N($).
Proof of Theorem 3.4. Define a smooth function: P(s) =
f0
fors V Ic-E,c+T),
1I
for s E Ic - E, c + C11
with 0 5 p(s) < 1. Let A = M\(N')i, where (N')6 = {x E Mldist(x, N') < 6), and B = N' be two closed subsets. Let dist(x, B) dist(x, A) + dist(x, B)
We see 0
1
0<s<1,
d
3 >1.
i
3.
A Pseudo Gradient Vector Field and Deformation Theorems
31
Letting X be a p.g.v.f. of f, we define
V(x) = -g(x)P(f(x))q(IIX(x)II)X(s) Thus V E C'-0, and IIV(x)II < 1. Consider the ODE
a(t) { a(0)
= V(a(t)) = x° V xo E M.
Since V is bounded, the global existence and uniqueness of the flow a(t) on R' are known. Let 77(t, x) = a(t), with a(0) = x. Then 77 E C([0,1] x M, M) satisfies (1), (2), (3), and (5). It remains for us to verify (4). From (5), one needs only to verify (4) for x E ff+,\(ff_,UN). More precisely, we shall prove: f 017(
3
6, x) < c- e Vx E
U N)
Indeed, if not, 3 x E f -'[c - e, c + eJ\N such that
c-E< fo,l(t,x)
P(f o n(t, x)) = 1. Noticing that Iin(t, x) - 17(0, x) 11 < t, we have
dist(t/(t, x), (N') f) > dist(t7(0, x), (N') #) - t 7 >(8-8-4)6=0,
so g o tl(t, x) = 1. Now, Wt- f otl(t,x) = (df (77(t, x)), (t, x))
= -q(IIV(il(t,x))II)(df(o(t,x)),V(n(t,x))) < -2q(IIV(n(t,x))II)Ildf(n(t,z))II2.
If IIV(t)(t,x))II < 1, then we have
If
orr(t,x) < -211df(n(t,x))II2 < -262.
Infinite Dimensional Morse Theory
32
Otherwise, d
d f °n(t,x) s -12 IIdf(,7(t,x))II2/IIV(i1(t,x))II 4IIdf(?n(t,x))II <
b.
In summary,
dt
f o r)(t, x) < -Min{ 1 b2,
4b}.
Thus, f o r1(t, x) < f o t7(0, x) 5
bMin{ 262, 4 b}
c+c-bMin{4b2,8b}
This is a contradiction.
Remark 3.2. Theorem 3.3 is due to Palais [Pall]; see also P.11. Rabinowitz [Rabl). For Theorem 3.4, cf. J.Q. Liu [Liu2], and K.C. Chang, J.Q. Liu [ChL2].
4. Critical Groups and Morse Type Numbers Basic Morse theory is set up in two steps: (1) Locally, we define a sequence of groups, which we call critical groups, to describe the local behavior of a function f near its critical point. (2) Globally, we define a sequence of numbers, which count the critical points in accordance with the critical groups. These numbers are called Morse type numbers. We study the relationship between these Morse type numbers and the deformation property of the underlying manifold via the deformation property.
Definition 4.1. Let p be an isolated critical point of f, and let c = f (p). We call
Cq(f,p) = IIq (fc n up, (f,,\ {p}) n up, G)
the qth critical group, with coefficient group G of f at p, q = 0,1, 2, ... , where Up is a neighborhood of p such that K fl (ff fl Up) _ {p}, and Il. (X, Y; G) stands for the singular relative homology groups with the abelian coefficient group G. According to the excision property of the singular homology theory, the critical groups are well-defined; i.e., they do not depend on a special choice of the neighborhood Up. Now we give some examples.
4.
Critical Groups and Morse Type Numbers
33
Example 1. Letting p be an isolated local minimum of f E C(M, RI), then
Cq(f,P)=
C
q=0
0
gi4 0.
In a finite dimensional manifold Mn' if p is an isolated local maximum of
f E C(M",R'), then Cq(f,P) =
q=n Io
q
n.
Example 2. Let M be a 1-dimensional manifold. If p is an isolated critical
point of f E C(M, R'), which is neither a local maximum nor a local minimum, then we have Co(f,P) = Cj(f,P) = 0.
Example 3. (Monkey saddle.) Let f be the function x3 - 3xy2 defined on R2. Then we have
C0(f,e)=C2(f,e)=0, and C,(f,B)=C®C. Let M be a Hilbert-Riemannian manifold. f E C2(M,R'), p E K is called a nondegenerate critical point, if d2 f (p) has a bounded inverse. Since A = d2 f (p) is a self-adjoint operator which possesses a resolution of identity, we call the dimension of the negative space corresponding to the spectral decomposing, the Morse index of p, denoted by ind(f, p). (It can be oo).
Now we are in a position to compute the critical groups of a nondegenerate critical point via its Morse index. The following Morse lemma is a cornerstone in studying the local behavior of a nondegenerate critical point. The proof is postponed to the next section in which a general splitting lemma will be proved.
Morse Lemma 4.1. Suppose that f E C2(M, R') and that p is a nondegenerate critical point; then there exists a neighborhood Up of p and a local diffeomorphism 0: Up - Tp(M) with 0(p) = 0, such that
f o 4 `w = f (P) +
2
(d2f(P)f, f) , d f E 0 (Up),
where ( , ) is the inner product of the Hilbert space H, on which the Riemannian manifold M is modeled.
Infinite Dimensional Morse Theory
34
Theorem 4.1. Suppose that f E C2(M, R') and p is a nondegenerate critical point off with Morse index j; then Cq(f,P) =
G
9=j
0
q 34 j
Proof. According to the Morse Lemma, we may restrict ourselves to a special case where f is a quadratic function on the Hilbert space H : f (x) = 1(Ax, x), where A is a bounded, invertible, self-adjoint operator. Let Pf be the orthogonal projection onto the positive (negative) subspace H± with respect to the spectral decomposition of A. We have
f(x) =
2
(II (AP+)# x112 - II (-AP_)i x112
.
On an c-ball B( centered at 0, it is easily seen that
Bf nfo = {x E H I Ilxll < c, 11y+ 11 5 Ily- Il,y± = (±APP)i x} We define a deformation
n(t,x)=N-+ty+
V(t,x)E10,11X(B,. nfo).
It is a strong deformation retract from (B, n fo, Bf n (fo\{0}) to (H_ n B(, H_\{0}) n BE). Thus Cq(f,P) = Hq (fo n BE, (fo\{9}) n Bc)
Hq(H_ nBe,(H_\{0})nBE) IC q=j Hq (Big 0
if j is finite.
q96 j
4.
Critical Groups and Morse 71Mpe Numbers
35
Nevertheless, for j = +oo, since S°O is contractible, we always have C9(f,P) °-` 0.
0
The conclusion is proved.
Let M be a C2-Finsler manifold, and let f E C1(M, R1). Suppose that f has only isolated critical values and that each of them corresponds to a finite number of critical points; say
...
Kci =
'1
,
i = 0,±1,±2,...
We choose
0<ej <min{c+I -cj,c;-c;_1}, i=0,±1,±2,.... Definition 4.2. For a pair of regular values a < b, we call
Mq(a,b) _ E rank Hq (ff+,!j,fcj_t;,G) a
the qtli Morse type number of the function f with respect to (a, b), q =
0,1,2,....
For functions f satisfying the (PS) condition, Morse type numbers are well-defined according to the deformation lemma, i.e., they are independent of the special choice of (e;). In the case where f E C'(M, R'), assume that
c is an isolated critical value and K,, = {zj}'', then for sufficiently small c > 0, we have
Theorem 4.2. m
11. (fc+c, fc
G) °-` H. (f,, f,\Kc; G) °' 3 C. (f, zj) j=1
Proof. By the deformation theorem and the homotopy invariance of singular homology groups, we have
Hq (ff+., f,, G) °-` Ilq (ff, f,; C) and
Hq (fc\Kc, fc
G) '-` Hq (fc
fc
C)
0.
Infinite Dimensional Morse Theory
36
Applying the exactness of singular homology groups to the triple (fc, fc\Kc, fc_c):
IIg (fc\Kc, fc-c)
IIq (fc, fc-c)
Hq (ff, fc\Kc) - Hq-I (fc\Kc, Ic-c)
...
we find
0 : I1q (ff, Ic-c)
I1q (ic, fc\Kc)
If, (fc, fc-c)
Hq (fc, fc\Kc)
0,
By using the excision property, we may decompose the relative singular homology groups into critical groups m
m
If. (fc,fc\Kc) - H. f,n U B(zi,E),ffn U (B(z,,E)\{zj}) i=I
i=1
m j=1
fore > 0 small enough, where B(x, e) is the ball centered at x with radius D
c.
Ej `, rank Cq(f,
Corollary 4.1. MM(a, b) =
q = 0, 1, 2, ... .
This is the connection between critical groups and Morse type numbers. Now we formulate the Morse inequalities. Let us define (3g = 13q(a, b) = rank IIq (Ib, fa, G),
q = 0,1, ... , where a < b are regular values of f.
Theorem 4.3. Suppose that Af is a C2-Finsler Banach manifold and that f E C' (M, Ilk') satisfies the (P S)c condition b c E [a, b], where a, b are regular values off . Assume K fl f -' [a, bJ = {zi, z2, ... , z,). Then we have 00
(4.1)
00
> Afgtq = >Qgtq + (1 + t)Q(t), q-0
q-o
where Q is a formal series with nonnegative coefficients, Afq = Mq(a, b) =
rank CC(f,z,), and pq = Qq(a,b) = rank Hq(fb,fa), q = 0, 1,2,.... Note. (4.1) is formal: It means
4.
Critical Groups and Morse lype Numbers
and
37
00
00
E(-1)gMq = q=o
E(-1)9,09,
q=o
if all Mg, ,0q, q = 0,1, 2, ... , are finite and the series converge.
Proof. (4.1) follows from the subadditivity of the functions
Sq(X,Y) = t(-I)q-i rank H,(X,Y). i=o
More precisely, for a triple X 3 Y D Z, we have Sq(X, Z) < Sq(X, Y) + Sq(Y, Z),
(4.2)
q = 0, 1, 2, ... .
Indeed, let cl < c2 < ... < c,, be the critical values of f in (a, b). We choose ao, a,, ... , an such that
a=ao c 1 <
rank Hq (fai,fai_1)
I
i = 1,2,... ,n,
q=0,1, ...
are all finite. It follows from (4.2), n
9
EE(-I)g-j rank Hi
i=i i-o q
> >(_1)q-i rank If, (fan,fao) i=o
Consequently, by Corollary 4.1 and Theorem 4.2 q t(_1)q-ifi.
(-1)g-'MM >
i=o
i=o
Since we have assumed the convergence of the series, there exists qo such
that
Mq=lq=0 Vq>qo.
38
Infinite Dimensional Morse Theory
The identity 00
00
D-Irmq = q-O
q=0
is nothing but the two consecutive inequalities above. Now we prove (4.2). In an exact sequence,
Since kerj = Im i, rank A = rank Im j + rank Im i. Let us denote rank Im i by e(A), then rank A = c(B) + e(A).
For a triple Z C Y C X, we have the exact sequence
...- Hq(Y,Z)-HQ(X,Z)-+Ifq(X,Y)thus rank Hq(X,Y) = fq(X,Y) +eq-1(Y,Z), rank Hq(X, Z) = fq(X, Z) + eq(X, Y), rank HQ(Y, Z) = c9 (Y, Z) + fq(X, Z).
It follows that rank HQ(X, Y) + rank Hq(Y, Z) - rank (X, Z)
=eq(Y,Z)+fq-1(Y,Z) Consequently, we obtain
Sq(Y,Z)+sq(X,Y)-sq(X,Z)=fq(Y,Z)>0, (4.2) is proved.
q=0,1,2,... 0
The geometric interpretation of these Morse inequalities is as follows: If
f is a nondegenerate function on a compact smooth manifold M, and if we choose b > maxxE nl f (x), a < min-1E Al P X), then Mq is the number of critical points with Morse index q, and pq is the Betti number of Al.
The following handle body theorem describes a clear picture of the changes in topological structure as the level sets pass through a critical value on which there are only nondegenerate critical points.
4.
Critical Groups and Morse 7)jpe Numbers
39
Theorem 4.4. Suppose that f E C2 (M, Rl) satisfies the (PS) condition, where M is a C2-Hilbert Riemannian manifold, and that c is an isolated critical value. Assume that Kc = {z1, z2,. .. , ze} consists of only nondegenerate critical points with Morse indices {m1, m2, ... , me) resp. Then there exist e > 0 and homeomorphisms hi : B'"i M (into),
i = 1,2,... ,t, such that
satisfying
(Bmi) = f-1(c - e) n hi (Bmi) = hi (8Bm' )
(4.3)
i = 1,2,...,t, where Br"i is the mi dimensional ball, and the notation ^- means a deformation retract. t
Proof. From the second deformation lemma, we have c > 0 such that fC+l
fC .
We choose 6 > 0 small enough such that B(zi, 6) fl B(z f, b) = 0 d i 0 j, and that the Morse lemma in B(zi, d) is applicable, which yields a local homeomorphism ti : B (0, bi) -- B (zi, b) (into) satisfying
fo01(iii)=
(1116112-Ily'_II')+c, 2
where y' E B(9,bi), the ball centered at 9, with radius bi > 0, in the tangent space Ti (M), and y' = y+ + y' is the orthogonal decomposition according to d& f (zi), i, j, = 1, 2,...
, 1.
40
Infinite Dimensional Morse Theory
Let N = Ui=14,B(0,bi). We apply the first deformation theorem to N, yielding a deformation
retract n : n(Ic+,\N) c Ic Since t) decreases the value of f,
(IcnN)uf It suffices to show (f n N) u Ic-c = Ut_1 hi(Bmi) U fc-c We choose el E (0, e) satisfying gel < min {bi,
i = 1, 2, ... , e)
,
and define fX
t(t, x) =
if x E fc_e
(yi +(l+t(Ilv+I
IIIIV...ax{111/'11Z-2e1,0)-1))y+),
where t E [0, 1], y1 = -' (x) as x E B(zi, b), and y' = ,y+ + y'- according to the orthogonal decomposition, i = 1, 2, ... , e. Thus
Ic-el U Ic n U -t(B (0, ai)) i=1
fc-el U U 4, (Bmi). i=1
The restriction 4P(B(o, 2cl)nP Ti(ff) yields the homeomorphism hi, where
Pi- is the orthogonal projection onto the negative eigenspace of d2 f (zi ),
i = 1,2,... J.
Combining y with , we obtain the deformation retract. (4.3) is easily verified.
This completes the proof.
0
The references for Theorems 4.1 through 4.4 are Milnor (Mill], Schwartz (ScJ1], Rothe [Rots], Palais [Pall], Pitcher (Pitt] and Marino Prodi (MaPI). In Theorem 4.4, the handle body theorem is established on Hilbert Riemannian manifolds, where the Morse Lemma holds, and the local behavior
of a nondegenerate critical point is quite clear. In order to extend this theorem to Finsler manifolds, or to Banach spaces, new difficulties arise in two ways. One way is conceptual: The above definition of nondegeneracy does not make sense in Banach spaces because the Hessian d22f(X) is a bounded linear operator from the space to its dual, so that one cannot say that d2 f (x) has a bounded inverse except when the space is isomorphic to its dual. The other difficulty is technical. The Morse lemma is no longer useful because it is not compatible with the Palais-Smale condition,
e.g., the quadratic functional f(u) = ff u2(t) dt does not satisfy the (PS) condition on the space I"(0,1], for p > 2.
4. Critical Groups and Morse Type Numbers
41
The theory can be extended as follows.
An operator L E B(X), the Banach algebra of all bounded linear operators from a Banach space X into itself, is said to be hyperbolic, if the spectrum o(L) is contained in two compact subsets separated by the imaginary axis. Definition 4.3. Let M be a C2 Finsler manifold, f E C2(M,IR1). Let po be an isolated critical point. We say that po is nondegenerate if there is a neighborhood U of po on which T(M) is trivialized to a be U x X, such that there is a hyperbolic operator L E B(X) satisfying
(1) d2f(po)(Lx,y) = d2f(po)(x,Ly), dx,y E X, (2) d f (po)(Lx, x) > 0 d x E X\{ti}, (3) (df (p), Lx) > 0, `d p E U, p = po + x in the local coordinates. The dimension of the negative invariant subspace of L is called the index of p.
According to the new'definition (which coincides with the old if M is Hilbert-Riemannian by taking L = d2 f (po)-1), Theorem 4.4 does hold for Finsler manifolds modeled on Banach spaces with differentiable norms. The reader is referred to Chang (Cha2l and K. Uhlenbeck [Uhlll. A different statement was also given by T. Tromba [Tro3l. Remark 4.1. The above theorems as well as the corollary can be extended to functions defined on a manifold M with boundary OM. The same proof works and the same conclusion holds for functions f under the following assumptions:
(1) KnOM=0; (2) 3 a p.g.v.f. V off such that -VI81y\f_1(Q) direct inward (i.e., the negative pseudo-gradient flow n(t, x) E M, V X E OM\f'1(a), d t > 0 small). For a more general boundary condition, see Section 6.1.
Now we are interested in defining a reparameterized flow in order to make the flow have infinite arriving time if it passes through a critical point. Let
P(r) = -X (P(r)) P(0) = x° E f-1(a,bl\Ko.
Then p = p(r, xo) is a reparametrization of the p.g. flow a defined in Theorem 3.2. We shall prove
Theorem 4.5. If f E CZ-0(M, W), and if a(Tt0 - 0, xo) = z E Ka, then p(+oo,xo) = t Iimop(r,xo) = Z.
Infinite Dimensional Morse Theory
42
Proof. Let
s=f0P(r,x0)
(4.4)
We have
ds
=-(df,X)op(r,xo)
dr
Suppose that T is the arriving time in the new parameter r, and assume that f < +00; we obtain ra+6 dr
6lu rn ,/ ds ds f(xo) f(=o)
= litn
-,+o
j
+6
ds
(df, X) o p (r, xo)
Let O(s) = (df,X) o p(r), where s and r are related in (4.4). We have rf(=o) ds
d()
a
<00.
From O(s)
II X o P(r)11'
< 4 Ildf op (r,xo)
- df(z)112 2
j(rl,xo) dr'
< 4C
11
/DIIP(r''xo)Illdr'Ids,12
<4C1 < 4C
ds'
IIX op(r',xo)II2 (ds'
< 4C'
s,)
J
')12) (s - a)
(s - a) < M(s - a)
J
for some constants, C, C' and M > 0, it follows rf(xo) ds
J
0+6
m(3)
> if-a+6 s-ds 1
rf(xo)
1
+- (fn (f (xo) - a) - £nb) M
+oo
as 6 -p +0. This is a contradiction.
The new reparametrized flow p(t,xo) is well-defined for all (t,xo) E
R' x (M\K).
Gromoll-Meyer Theory
5.
43
Theorem 4.6. If f E C2-0(M, R1) satisfies (PS) and if p is an isolated critical point of f, which is not a local minimum, then Co(f,p) = 0.
Proof. We choose a path-connected neighborhood U of p such that K n U = {p}, and U C fc+E for some a> 0 where c = f (p). Let U+ = Ut>op(t, U), where p is the reparametrized flow defined above. Then we have
C. (f, p) " H. (ff n U+, (ff\{P}) n U+) °-` H.
(ff+E n U+, (fc\{P}) n U+)
because fc n U+ is a strong deformation retract of fc+E n U+ (Deformation Theorem). Since ff+E n U+ is path connected, and (ff\{p}) n U+ j4 0. Co(f,P) = 0.
As an extension of Example 2, we have
Example 4. If f E C"-0(M", R"), and if p is an isolated critical point of f, which is neither a local maximum nor a local minimum, then CO(f,P) = C"(f,P) = 0-
5. Gromoll-Meyer Theory The contributions of Gromoll and Meyer to isolated critical point theory are threefold: (1) a splitting lemma, which is a generalization of the Morse lemma (cf. the previous section); (2) an alternative definition of critical groups; (3) a shifting theorem which reduces the critical groups of a degenerate isolated point to the critical groups of the function restricted to
Infinite Dimensional Morse Theory
44
its degenerate sub-manifold. In this section, we shall rewrite their theory with slight improvements and prove the equivalence of the two definitions of critical groups. Theorem 5.1. Suppose that U is a neighborhood of 0 in a Hilbert space
N and that f E C2(U,RI). Assume that 0 is the only critical point of f and that A = d2 f (B) with kernel N. If 0 is either an isolated point of the spectrum a(A) or not in o(A), then there exist a ball B6i b > 0, centered at 0, an origin-preserving local homeomorphism 0 defined on B6, and a C1 mapping h : B6 fl N -i Nl such that (5.1)
f o O(z + y) = 2 (Az, z) + f (h(y) + y),
V X E B6,
where y = PNx, z = PNlx, and PN is the orthogonal projection onto the subspace N. Proof. 1. Decomposing the space 11 into N ® N', we have d.p (0I + 02) = 01
(01 = PNl©' 02 = PNO)
and
dzf (01 + 02) = AINI.
Because of the implicit function theorem, there is a function h : B6 fl N -+
Nl, 6 > 0, such that d=f(y + h(y)) = 01.
Let u = z - h(y), and let (5.2) (5.3)
F(u, y) = f (z + y) - f (h(y) + y). F2 (10 _
I (Au,u).
Then we obtain
F(0,,y) = 0 du F (01,y) = d:f(h(y) + y) = 01,
dUf (01,02) = d=f(B) = AINj. 2. Define : (u,y) - uo E F2 1 o F(u,y) fl {rl(t,u) I is the flow defined by the following ODE
n(4) = rl(0) = U.
Art(s) 11An(4)II
I
I < IIuII}, where rl
Gromoll-Meyer Theory
5.
45
Claim. rl is well-defined for Itl < Ilull. Since
Iln(s) - ull 5 Isl
,
we have Ili7(t, u)ll ? IIuII - ItI. From this, together with 77(t, u) E N1, it follows that the denominator of the vector field is not zero for Itl < Ilull. Claim. C is well-defined on B6 x B6 L, where B6 = B6 n N, and B6 = B6 n Nl for some 6 > 0. In fact, the following inequalities hold: (a) V c > 0, 3 61 = 6(c) > 0 such that
IF(u,y) - F2(u)I = IF(u,y) - F(0l,y) - (duF (01, y), u) - F2(u)l
j(i - t) 1
=
I
((d2u F(tu,y) - dF(01,02)) u,u) dt<
cllull2,
where (u,y) E B6 x Bi; (b) 110t
F2(n(t,u)) - F2(u)I =
d-F2(q(s,u))ds
= Li0t lltl
(d F2(), ) ds= u) 11 d1
a
C
f
Itl
Ins, u) lids
> c Clittll
2
ltl - i
where C > 0 is a constant determined by the spectrum of A. We conclude
that (c) F2(n(t, u)), as a function of t, is strictly decreasing on (-(lull, IIuII); (d) F2(rl(-t, u)) > F(u, y) > F2(n(t, u)) holds for (5.4)
Ilull-
C there exists a unique t(u, y) with
It(u, y)1 5 l 1-
Ji- C
(lull
Infinite Dimensional Morse Theory
46
such that (5.6)
F2 (i, (t(u, y), u)) = F(u, y)
Thus the function
is of the form:
(u, y)
41
u = 01
rJ (t(u, y), u)
u # 01-
y), y). We shall verify that is a 3. Define a map -+/, (u, y) -. local homeomorphism. That t(u, y) is continuous easily follows from the implicit function theorem for u = uo # 0, provided :
(5.7)
49
F2 o rl (t, uo) = -II Arr (t, uo) II # 0;
and for u = 0, provided by (5.5).
We have used the path n(t, u) to carry a point (u, y) to the point ({(u, y), y); the same path can be used for the opposite purpose, i.e., to define the inverse map 4? = tfr' . The same reason is provided to verify the continuity of 4;. Therefore 4i is a homeomorphism. The equality (5.1) follows directly from (5.6).
Note. The function y - f (y + h(y)) is Cs. In the case where N = {4}, the Morse Lemma is a consequence of this theorem except the conclusion is that 4' is a diffeomorphism.
Proof of the Morse Lemma. We have proved that 0 is a homeomorphism. Now we shall prove it is a diffeomorphism. That t(u), and then £(u), is continuously differentiable for u E B6\{B}, follows from the implicit
function theorem and (5.7). It is also easily obtained that 4(0) = id, by using (5.8)
_V I - C I
11 71 (t(u), 11) -1Il S It(u)I <
11,111 = o(Ilull)
C1
(i.e., rt(t(u),u) = u+o(IIuII) as u -+ 0). Hence Ildd(u)- idll = o(1) remains to be proved. Since rl(t, u) = u
- Jot II Ar7(s, u)ll ds
we write r1(t) = r)(t, u), nu (t) = (5.9)
rou(t) = id -
for
Itl < IIuII
u) for simplicity,
A,7(s) ® ((Anu(s))' Art(s)))
/
IAn(s)II
IIAll(s)l13
J
ds,
5.
47
Gromoll-Meyer Theory
that is Ilv,.(t)II < 1 +C1 f o
I
[Iqu(s)II d9,
1177(19)11
where Cl > 0 is a constant depending on a(A). But (5.10)
Ji -
lal 5 If(u)j.
hull < I177(s)II 5 211u11
Applying the Gronwall inequality,
`
for
0<[tl51(u).
Thus Ilnu (t(u),u) - id 11 5 C3I11 III
=
o(l) as
0.
hull
Since
Ildk(u) - idll = IIt (1(u), u) - id + dut(u) n (t(u), u)I
and by (5.6) and (5.10), we have Ildut(u)II =
IIduF(u) - jru (t(u), u) Arl (1(u), u) 11
IIArl(t(u),u)I[ IIAu + o(Ilull) - (id + o(1))A(u + o(IIuII))II = 0(1) = IIA(u + o(Ilull))II as
u
0.
This proves that IIdk(u)- idl[ -' 0 as I[ull -. 0. The implicit function theorem is employed, and 1; is a local diffeomorphism.
0
Remark 5.1 There is no difficulty extending the Morse Lemma as well as Theorem 5.1 to the case where M is a Hilbert Riemannian manifold. The theorem was first obtained by Gromoll and Meyer [GrMI] under the condition f E C3. However, the C2-Morse Lemma was proved by N. H. Kuiper [Kuil), from whom the idea of the above proof was taken. The C3-Morse Lemma was given by Palais (Pall), Nirenberg [Nil) and Schwartz [Schl) (cf. Appendix). As to Theorem 5.1, for a different proof see J. Mawhin and M. Willem IMaW21 based on [Caml], in which the condition that 0 is either isolated or not in a(d f (B)) is replaced by one in which d2j(6) is a Fredholm operator with index 0. However, our condition seems weaker, because a self-adjoint Fredholm operator A must have a finitely dimensional kernel N(A) and a closed range R(A). In the case where N(A) y6 {B}, the induced operator A : 111N (A)
R(A),
48
Infinite Dimensional Morse Theory
defined by
A[x] = Ax V x E (x), (x] E H/N(A), is invertible. Because of the Banach theorem, A-' is bounded, which implies that 0 is isolated in a(A).
On the other hand, the condition dim N(A) < oo is not assumed in Theorem 5.1. Let Af be a C2 Finsler manifold, and let f E C1(M, IPL') be a function satisfying the (PS) condition. Suppose that V : M\K --+ T(M) is a p.g.v.f.
of f. Definition 5.1. Let p be an isolated critical point of f , c = f (p). A pair of topological spaces (W, W_) is called a Gromoll-Meyer pair with respect to V, if (1) W is a closed neighborhood of p possessing the mean value property,
i.e., V t1 < t2, rl(ti) E TV, i = 1, 2, implies q(t) E W for all t E [t1, t2J, where i7(t) is the decreasing flow with respect to V. And there exists e > 0
such that W n f,_, = f-i(c - e,c) n K = 0, W n K = (p); (2) IV_ := {x E IV I r)(t, x) V 1V V t > 0) is closed in W;
(3) IV_ is a piecewise submanifold, and the flow r) is transversal to lV_, i.e., r) rh IV_.
At this point, the existence of a Cromoll-Meyer pair is assumed. The following theorem claims the motivation of the definition.
Theorem 5.2. Let (H', IV_) be a Gromoll-Meyer pair with respect to a p.g.v.f. V of an isolated critical point p of the function f E C'(M,lIB') satisfying the (P, S) condition. Then we have
H.(TV,W_;C) ^-C.(f,p)
Proof. We now introduce two sets U+ = Uo<<,,.ri(t, W) and U+ = Uo
t E 10, 11, x E U.
defines a strong deformation retract U. --+ f - n U+
n U.+, which
reduces to
II.(f+,nU{,f.- 0, such
that y = r/(t,x) and t = t(y), x = x(y) continuously depend on y. The mean value property implies that U+ = W U Uo«
5.
49
GromoU-Meyer Theory
Let U6 = U6 0, i.e., U6 = {y E U+ I t(y) > 61, where we define t(y) = 0 V y E W, and t(y) = t(y) d y E U+. Hence U6 is open, U6 = U 77(t, W-) C U rl(t, W-) = int (U+) . 0
6
By the excision property, we have H. (U+, U+; G) = H. (U+\U6, U+\U6; C)
Again, there is a strong deformation retract, defined by the reversed flow r)(-t) : ((U+\U6), (U+\U6)) - (W, W-), provided q dh W_, we obtain
H. (U+\06,U+\U6;C) = H.(W,W_;C). On the other hand, by the Deformation Theorem 3.2,
nU+;C) H. (fo n U+, (ff\{p}) n U+; C) = C. (f, p)
H. (ff+, nU+,fc
Combining all these isomorphisms we obtain H. (W, W_; C) S, C. (f, p).
0 We now turn our attention to constructing a Gromoll-Meyer pair for an isolated critical point. For simplicity, we may assume that M is a Hilbert space, and p = 0, f (0) = 0. Choose e > 0, 6 > 0 such that 0 is the unique critical value in [-c, c), and 0 is the unique critical point in the ball B6 centered at 0. Define a function 9(x) = Af(x) + 11z112,
a neighborhood (5.11)
W = f-1[-r,7] n9N
and a subset of W, W- = f -' (-ry) n W,
(5.12)
where A, p and -y are positive numbers to be determined by the following conditions: (5.13)
B612
n f-' (-ry, yj c W C B6 n
f-'
(5.14)
f-1(-y,7jn9-I(µ) C B6\B612
(5.15)
(dg(x), df (x)) > 0 V x E B6/B512.
ej
Infinite Dimensional Morse Theory
50
Due to the (PS) condition, there exists /3 = inf=EB6/B6/2IIdf(x)II > 0 Let M = SUp:EB6 Ildf(x)II A, a and ry are then determined consecutively:
.\ >
, 0 < ry < min(e,
362
), and
a4
+
b2 - J1ry.
Theorem 5.3. The pair (5.11), (5.12) is a Gromoll-Meyer pair with respect to the negative gradient vector field -df (x).
Proof. We only want to verify (1) the mean value property,
(2) W_ = {x E W I q(t, x) f W, V t > 0, for the negative gradient flow '1
Claim (1). For simplicity, we may assume q(0),77(t) E W. We wish to prove q(s) E IV V s E 10, tI. Define 7' = sup{s e 10, t) I q(s') E IV V s' < s).
If T < t; then q(T) if B6\2. But
7?fo17(0)?foq(T)>foq(t)2-?' (gorl)'(T) = -(dg,df)In(T) <0, provided by (5.13) and (5.15). This contradicts the fact that T is a supremum. Therefore T = t. Claim (2). We write IV- = {x E IV I q(t, x) V W, `d t > 0). Obviously
we see W_ C W. Now we prove W- C W_. By definition, W- C OW and OW = W_ U (f-1('Y) n g,.) U (g-' (li) n (W \W--)). If x E f-' (ry) n gr., then x V W-. For x E g-I(µ) fl (W\IV_), according to (5.14) and (5.15), we have (go 77)'(0, x) < 0 and f (x) > -ry. These imply that 3 r > 0 such that go q(r, x) < .t and I(f o q)(r, x)I < -y, i.e., x V IV-. We have proved
0
W_ = W-.
We intend to compute the critical groups of an isolated critical point p, which may be degenerate; the splitting theorem is employed. Let 4i be a local parametrization of M defined in some open neighborhood U of 0 in Tp(M) H with I'(9) = p, such that f o 4i(z, y) = (Az, z) + fo(y), z where A = d2 f (p), 0 is either an isolated point of the spectrum a(A) or not in c(A), and fo is a function defined on N, the null space of A. We call N = 4i(U fl N) the characteristic submanifold of M for f at p with respect to the parametrization 0. The following theorem sets up the relationship between the critical groups of f and those of f := f IN.
Theorem 5.4. (Shifting theorem). Assume that the Morse index of f at p is j, then we have
Cq(f,p) =Ca-i (f.p), q=0,1,...
5.
Gromoll-Meyer Theory
51
First we need:
Lemma 5.1. Suppose that H = H1 ® H2, gi E C2(Hi,R1), 8i is an isolated critical point of gi, i = 1, 2, where H1, H2 are Hilbert spaces. Assume that (Wi, Wi_) is a Gromoll-Meyer pair of 8i with respect to the gradient vector field ofgi, i = 1, 2; then (W1 x W2, (W1_ x W2)U(W1 x W2-))
is a Gromoll-Meyer pair of the function f = g1 + g2 at 0 = 01 + 92 with respect to the gradient vector field df, if 0 is an isolated critical point off . This is easy to check. We omit the proof. Theorem 5.5. Under the assumptions of Lemma 5.1, we have C (f, 8) = C. (91, 01) ®C. (92, 82)
Proof. This is a combination of Theorem 5.2, Lemma 5.1 and the Kiinneth formula.
Proof of Theorem 5.4. This is a combination of Theorems 5.1, 5.5. Remark 5.2. Theorem 5.5 was conjectured by Gromoll and Meyer (GrM1J and was solved by G. Tian (Tiall. In (Danl(, Dancer independently proved the conjecture in the finite dimensional case.
In order to verify that the pair ((W1_ x W2), (W1 x W2-)) is an excision couple, we use the notations: U+, U+, V+, V+, defined as in Theorem 5.2 where U+ = Ut>orl1(t, W1), U+ = Ut>o41(t, W1- ), V+ = Ut>orh(t,W2), V+ = Ut>or12(T,W2_), and p1, rh are the decreasing flows with respect to dg1 and dg2 respectively. Also 06i V6, 5 > 0, stand for the same meaning. Thus H. ((W1_ x W2) U (W1 x W2-), W, x W2_)
H. ((&+ xV+)U(U4 xV+),U+ xV'> °- H. (U6 X V+, 06 X V+)
(excision)
ff. (W1_ X W2, W1_ X W2_) .
Corollary 5.1. Suppose that N is finite dimensional with dimension k and p is (1) a local minimum of f, then Cg(f,p) = SgjC, (2) a local maximum of f, then Cq(f,p) = bv,i+icG,
Infinite Dimensional Morse Theory
52
(3) neither a local maximum nor a local minimum of f, then Cq (f, p) = 0
q < j,
for
and q> j + k.
Finally, we prove the homotopy invariance of the critical groups. It is similar to the Leray Schauder index for isolated zeroes of vector fields. The perturbation should preserve the isolatedness of the critical points.
Lemma 5.2. Let (W, W_) be a Gromoll-Meyer pair of an isolated critical point p of a C2-0 function f, defined on a Hilbert space H, satisfying the (PS) condition, with respect to -df (x). Then there exists e > 0 such that (IV, W_) is also a Gromoll-Meyer pair of g satisfying the (PS) condition with respect to certain p.g.v.f. of g, provided that q has a unique critical point q in W, and If - 9l1C' (W) < E 0
Proof. 3 r > 0 such that B(p, r) C TV. Due to (PS)
13 =inf{Ildf(x)111 xEW\B (iii ) > 0. Define p E C'-'(H, iltl), satisfying
(1 xEB(P,2), 0
x V B(p, r)
with 0 < p(x) < 1, and a vector field V(x) =
3lp(x)dg(x) + (1 - p(x))df(x)].
Choosing 0 < c < 0, we obtain IIV(x)ll < 211d9(x)II, and
(V (x), d9(x)) ? 11d9(X)112
V X E W,
for II9 - filc1(W) < c. Claim. V x V B(p, 2 )
Ildq(x)11 ? Ildf (x) 11 - c > p - c > 3c.
We have
(V(x),d9(x)) >
3 2 2 3
[IId9(x)112 - clld9(x)II] I
- 3IId9(x)1121
11V(x)II < 2 (IId9(x)II + c) < 211d9(x)II.
= Ild9(x)II2
5.
Gromoll-Meyer Theory
53
Since b x E B(p, ), V (x) = dg(x), the verification is trivial. z
0
Notice that V (x) = -df(x) outside a ball B(p, r) C W, and q is the only critical point of g in W. It is not difficult to verify that (W, W_) is a GM pair of g with respect to V. (The mean value property holds because the flow is the same negative gradient flow of f outside B(p, r), particularly in a neighborhood of OW. Similarly, for other properties assumed on W_.)
Theorem 5.6. Suppose that (f, E C2 (H, RI) I a E (0,1() is a family of functions satisfying the (PS) condition. Suppose that there exists an open set N such that f, has a unique critical point p, in N, V or E (0, 1], and that a -+ f, is continuous in C' (N) topology. Then C. U." P.)
is independent of a.
Proof. Due to the (PS) condition, a -. p, is continuous. Applying Theorem 5.3, we may construct a GM pair (W Wo_) for p V a E (0, 11. Combining Theorem 5.2, Lemma 5.2, and the finite covering, we obtain our conclusion directly.
Theorem 5.7. (Marino Prodi (MaP2() Suppose that f E C2(M,IR1) has a critical value c with Kc _ (p1, p2, ... , pe). Assume that d2 f (p,) are Fredholm operators, i = 1, 2, ... , t. Then d e > 0, there exists a function g E C2(M,1R1) such that
(1) 9=1 in M\ Uf-1 B(Pj,e), (2) 119 - f IIC2(M) < e,
(3) g has only nondegenerate critical points, all concentrated in Uj_1 B(Pj, e).
(4) Let ind(f, pj) = mj, and dim ker d2 f (pj) = nj; then the Morse indices of those nondegenerate critical points of g in B(pj, c) are in (mj, mj + nj1, j = 1, 2, ... , t. Furthermore, if f satisfies the (PS) condition, then the same is true for g.
Proof. One may choose e > 0 so small that all balls B(pj, e), j = 1,2,... , P are disjoint. We only change f in small neighborhoods of pj, j = 1, 2, ... , t, so the function g can be constructed consecutively. With no loss of generality, we assume K. _ (p) to be a single point, with Morse index m and nullity n. Because of the splitting theorem, we choose e > 0 so small that under suitable coordinates 0 : B(p, e) - II, the Hilbert space on which M is modeled. The function f is written as f 0 4-1(x) = c +
2
(Ilx+ III - Ilz_ 112) + h(y),
Infinite Dimensional Morse Theory
54
where x = (x+, x_, y) E (11+ ® H_ (D 110) n B(0, b), for some b > 0, and h E C2, possesses 0 as the unique critical point with h(y) = o (IIyII2) for Let us define a smooth function p : llf+ -. IR', 0 < p 1, p(t) t < 6/2, p(t) = 0 for t > b, and Ip'(t)I < 4/b. V b E W', set hb(y) = h(y) + (b, y)p(IIyII), , ) is the inner product on Rn. Since dh(y) # 0 V y E B(0, 6)\B(0, 6/2), one finds that c > 0 such that
where (
Ildh(y)II ? c V y E B(0, 6)\B (0, 612).
Thus Ildhb(y)II = Ildh(y) + bp(Ilyll) + (b, y)P (IIyII)
>c-5b>0, if b
Ilyll
II
't'herefore hb has no critical point in B(0, 6)\B(6, 6/2). In B(0, 6/2), however, hb (y) = h(y) + (b, y). Because of Sard's Theorem, hb(y) has only nondegenerate critical points for a.e. b E R ; we may choose b small. Fixing such a b E IR", define
go
(x) = c+ 2 (IIx+112 - IIx_II2) +hb(y),
and
g=f
outside
#-' B(0, b) C B(p, e).
This is what we need. In fact (1), (3), and (4) follow directly from the construction. (2) holds if IIbil is small.
Furthermore, g differs from f only in the neighborhood &'B(0,6). Since 110 n B(0,6) is compact (dim Ho = dim ker d2 f (p) < +oo) and d2g o &-' I (»+en_)ne(o,6) is invertible, we conclude that g satisfies the (PS) condition if f does.
6. Extensions of Morse Theory In previous sections, Morse theory was studied on a manifold without boundary, except in Remark 4.1, in which we assumed the negative gradient points inward at the boundary. The purpose of this section is to extend the previous study in two aspects: (1) under more general boundary conditions, and (2) when the underlying space is not a manifold but a locally convex closed set.
6.
Extensions of Morse Theory
55
6.1. Morse Theory Under General Boundary Conditions Let M be a Hilbert Riemannian manifold with boundary OM = E, which is a smooth oriented submanifold with codimension 1. Thus the outward normal unit n(x) of E at x E E is well-defined.
Given two real numbers a and b and a function f E C' (M, R' ), set M[a, b] = M n f -1(a, b], E[a, b] = E n f -' [a, b]. In the following, if there is no confusion, we shall simply write M and E in place of M[a, b] and Ela, bJ respectively.
Definition 6.1. f E C' (M, R') is said to be satisfying the general boundary condition (GBC in short), If
(1) K n E = 0, i.e., it has no critical point on E; (2) the restriction f = f (E, as a function on E, has only isolated critical points.
Let E- = {x E E I (f'(x),n(x)) < 0), where (,) is the inner product of the Hilbert space H. Suppose that f has only isolated critical points {x1, x2, ... , xn, ... }; we define the Morse type numbers as follows:
Ma=En 1 rank Ca(xn,f),
q=0,1,2,....
Let {mo, m1, ... , Mn, ... } and {po, p1, ... , µn, ... } be the Morse type
numbers of the functions f and f [E_ on M and E- respectively. We define the augmented Morse type numbers:
Mn =mn+pn
n=0,1,2,....
Our main results are the following:
Theorem 6.1. Suppose that both the functions f and % satisfy the (PS) condition on M and E respectively, and that f has only isolated critical points and satisfies the GBC. Then we have the following compact version of Morse inequalities: En>OMntn = En>opntn + (1 + t)Q(t)
where Qn is the nth Betti number of the manifold M[a, b), n = 0, 1, 2,... and Q(t) is a formal power series with nonnegative coefficients.
Theorem 6.2. Suppose that f E C2, K n f -1 [a, b] = 0, and that j is a Morse function on E- which has the critical set {y1, yz, ... , y,} with Morse indices (M1, m2, ... , mL }. Assume that both f and f satisfy the (PS) condition. Then fb ~ fo Uh1 (Bm1) U... Uh, (Bin'),
Infinite Dimensional Morse Theory
56
where B' is the m ball, and h is a homeomorphism. It is well-known that if the negative gradient vector field -f'(x) satisfies
the inward condition at the boundary E, i.e., E- = 0, then the above theorems hold. The main idea in proving our theorems is as follows:
(1) We first extend the known theorems from E- = 0, to K- := {x E
E- I j'(x)=O}=0. (2) We then define a new function fl which satisfies the following conditions:
ft coincides with f except for a neighborhood of the boundary r, the set K- for f, is empty, and c. the critical points of fl, in excess of those of f, correspond biuniquely to the critical points of f JE_ with preservation of the a.
1).
Morse indices.
Lemma 6.1. Suppose that f E C' (M, l ), and that f satisfies the (PS) condition. If K- = 0, then there exist a vector field X on M' := Af\K, K = {x E M I f'(x) = 01, and a positive number d, such that (1) JIX(x)11< 1,
(2) (f'(x), X (x)) > Min {d, 1/211 f'(x)II}, (3) (n (x), X (x)) > 0
V x E Al, V X E E.
Proof. The proof is an extension of the standard proof of the existence of a pseudo gradient vector field for manifolds without boundaries. We shall pay more attention to the construction of such a vector field near the boundary E. First, we note that 3 d > 0, such that I1f'(x)II > 2d d x E
E-. 0. According to the (PS) Claim. If not, 3 x E E- such that condition for the function f, 3 x' E E-, with f'(x') = 0, i.e., x' E K-; this is a contradiction. Next, let {U I a E Al be an open covering of M' satisfying T(M) ju,, Up x H, V a E A. If either x E int(M'), or x E E+ {x E E I (f'(x), n(x)) H, and a neighbor> 0), then we choose V = f'(x)/jIf'(x)Ij E T=(M) hood 0= C UQ of x, for some a, such that (V=, f'(y)) > 1/2 !I f'(y)II V y E
0t, and 0
fl E= O
(11 (Y), V.) > 0
if
V y E O= fl E
x E int (M'),
if x E E+.
In the remaining case, i.e., x. E E-, one may choose W= E T=(E), i.e., (n(x), WZ) = 0, satisfying II WxII = 1, and (W=, f'(x)) > d, provided by
6.
Extensions of Morse Theory
57
IIJ'(x)II > 2d. After a small perturbation, we have V= E T1(M), satisfying JJV:11 = 1, ( V=, f'(x)) > d, and (n(x),V=) > 0. Again, we have a small neighborhood 0= C Uo for some a, such that (V., f i(ll)) > d
V y E 0=, and
(n(y), V=) > 0
V y E O= C E.
Finally, to the open covering {0= 1 x E M), we have a locally finite refine-
ment covering {C,6 10 E B}, and an associated C1 0 partition of unity
{Op 1,0 E B), i.e., supp 00 C Cp, 0 < Op < 1, and EpEB O,(x) = 1, V x E M. For each /3, 3 x = x(13) E M' such that supp 00 C O. Let us define X(x) = EPEB iOp(x)V=(p). This is just what we need. In fact, (1) is trivial. By the definition,
(X(x),f'(x)) _ E O,(x)(V=(p),f,(x)) iEB > Min {d, 1/2 IJ f'(x)Il}
V x E M',
(2) follows. And (n(x),X(x)) = EpEB(n(x),V=(p)) Op(x) > 0, V x E E, because V x E E, x E Supp 00 C 0=(p) implies x(/3) E E.
0
In the case K- = 0, both the Morse inequalities and the Morse handle body theorem hold under the following assumption:
f E C' (M, lR')
and both the functions f and f satisfy the (PS) conditions on M and E respectively.
Now, we are going to reduce the GBC problem to case K- = 0, by perturbations.
Assuming that {y} E K-, we choose a chart (U, O), 0 : U -+ H_ {C E II (t;, e) < 0) for some fixed e E H\{0}, satisfying K fl U = 0, K- flU = {y}, 0(y) = 0, n(y) = e, and E fl U = 0-1(0(U) fl I1°), where H° = span {e)1. In other words, if we write q(x) = to+z, then t < 0, and t = 0 if and only if x E E. Define w = f o 0-' (te + z) - f o O'(z)
(6.2) then
f
o
46-'(8) (0-')e (0) = (f'(y),n(y)) < 0,
because f (y) = 0, and f has no critical point on E, so f(y) # 0. According to the Implicit Function Theorem, 3 r, e > 0 and a C' function t = t(w, z), which solves (6.2) in (-r, 01 x (B(8, e) fl Ho) C O(U), with t(0, z) = 0. Now, (w, z) is regarded to be a new local coordinate of x = O(w, z) =
Infinite Dimensional Morse Theory
58
0-' (t(w, z)e+ z), for (w, z) E [0, r, J x (B(8, e) fl Ho), where r, > 0 depends on r. Let us define P(w) =
0
w>6,
(6 - W)'/b'
6 > w > 0,
where 0 < d < r, is to be determined later; and let 0
s>E,
1
c/2>s>0,
where X E Coo, 0 < X < 1, and Jx'J < 3/c. We define ((x) = P(w) - X(IzI)
for
x E U, := 7P ([0, r,J x (B(0, c) n Ho))
and equal to zero elsewhere. Then we define
fi(x) = f(x) + ((x). Lemma 6.2. Suppose the assumption (6.1), K n U = 0, and K- flU = {y}. Then we have y' E U, and the following:
(1) K1, the critical set of f,, equals K U {y'}, (K,)-, the set K- for the function fl, equals K- \{y}, (2) f, and f, possess the (PS) condition; (3)
wherever b > 0 small.
Proof. (1) Both functions f, and fi have no critical point in E/2 < Izi < In fact, provided by the (PS) condition, 3 d > 0 such that IIf'(x)II > d V x E U n E, with c/2 < Izi :5,E. Let g(z) = f o 0-'(z); then c.
II9(z)II>d
for
c/2
Thus, IIv=fi(x)II = 11g '(Z) +P(w)X'(IzI)z/IzI II
>d-3b/e>d/2 V (w, z) E [0, bJ x [(B(9, c)\B(9, c/2)) n Ho), if we choose b < do/6.
(2) f, has a unique critical point y' = rJ'(wo, 0), in U. In fact, we only want to restrict ourselves to the neighborhood Uo = 0(J0, bJ x (B(O, c/2) n
IIo). Note that (6.3)
f 1 (x) = f (x) + P(v')X(z) = 9(z) + w + P(w)
6.
Extensions of Morse Theory
59
in this neighborhood, and that
8z fl(x) = 0 8w f i (x) = 0
if and only if if and only if i.e.,
z = 0; 1 + p'(w) = 0,
wo = (I -
)b.
(3) If is a sequence, along which converges to 0 and f, (x"') is bounded, then we want to show that there is a convergent subsequence. In fact, if it has a subsequence outside U1, then there must be a convergent subsequence, provided by the (PS) condition for f; otherwise, according to paragraph (1), there must be a subsequence in Uo. Since fi is of separate variables in Uo, f1 = g(z) + w + p(w), g(z) = j (x), and since f satisfies the (PS) condition, again, we obtain a convergent subsequence. Analogously, we verify the (PS) condition for fi.(fl = g(z)+b in UonE). (4) We shall verify (K1)- n U = 0. As before, we are only concerned with fl in the neighborhood Uo n E, so the critical point there is uniquely {y}. Since
(f' (y), n(y)) = (I + p '(w)) -Sew
I(e,z)=(o.e)
= -2(f'(y), n(y)) > 0 thus, y V (Ki)-. (5) Since fi is of separate variables in a neighborhood of a minimum of w + p(w) we have C. U1, y') = C. (f , y)
and wo is
C. (id + p, wo)
=C.(f,y) provided by Theorem 5.5. The lemma is proved.
Proof of Theorem 6.1. According to the isolatedness of the critical points of f and f, and the (PS) conditions, we conclude that both f and f have only finitely many critical points, say, {zi,z2,... ,zk} and { yl ,
y2.... , y,) respectively. Now we apply Lemma 6.2 1 times, and the new function fi satisfies the condition (K1)- = 0. Both fl, j, possess the (PS) condition, and f, has the critical set 01,Z2, ... , zk, YI , Y2, ... , y, } The Morse inequalities are applied for fl. Since a, b are regular values, 3 a > 0, such that K n f -'[a - 2a, a] _ K n f -1 [b, b + 2a) = 0. If we choose e, b > 0 so small that a - a < f (+/' ([0, b) x (13(0, e) n fro))) < b + a,
Infinite Dimensional Morse Theory
60
for each neighborhood of y,, i = 1, 2, ... , I and b < a; then
fi'[a-2a,b+2a) = f-'[a-2a,b+2a). Let p. denote the Betti numbers of the manifold M fl f, '[a - 2a, b + 2a]. Hence,
A1tn = > Qntn + (1 + t)Q(t). n>O
Ilowever, f
n>O
[a - 2a, b + 2a) ?' f -' [a, b), thus
rank II. (fj'[a-2a,b+2a]) rank H. (f [a - 2a, b + 2a]) rank II. (f-'[a,b]) = p.. The Morse inequalities are proved.
Proof of Theorem 6.2. Similar to the proof of Theorem 6.1., but apply the handle body theorem instead. Remark 6.1. Theorems 6.1 and 6.2 generalize the results due to Morse Van Schaach, cf. M. Morse, S. S. Cairns [MoC1], and E. Rothe, cf. [Rot2,4]. They were obtained in Chang, Liu [ChLI).
6.2. Morse Theory on a Locally Convex Closed Set The variational inequality problem is a different kind of variational problem arising from mechanics and physics. For example, let C be a closed convex
set in a Banach space X, and let f : C -. lle' be a restriction of a C' function. We shall find a point xo E C such that (df(xn), x - xo) > 0
VxEC,
where (, ) is the duality between X' and X. We may even extend the concept of convexity from Banach spaces to Banach manifolds, and study the variational inequality problems via the minimax methods and the Morse theory on the extended convex sets. Definition 6.2. Let M be a C' Banach manifold modeled on a Banach
space X. A subset S C M is called locally convex if, V x E S, 3 a chart 0: U --4 X such that O(U fl S) is convex in X.
6.
Extensions of Morse Theory
61
The local convexity depends on the special choice of the atlas A = {(U., 0.) I x E S). This is an additional structure on S. In the following, when we say that S is locally convex, it always means that there is a structure (M, A), such that S is locally convex with respect to (M, A). Definition 6.3. Let S C M be a closed subset. A vector v E TzM is called tangent to S at x E S, if 3 a chart 0: U -+ X at x such that lim h-ld (O(x) + h4'(x)v, gS(U n S)) = 0,
hI+0
where is the distance on X. A vector field V defined on a neighborhood A of S is said to be tangent
to S if V (x) is tangent to S, b x E S.
Let us denote T=(S) as the set of all tangent vectors at x E S. By definition, T=(S) is a nonempty closed cone, if S # 0. In particular, if C is a convex set in a Banach space X, and (6.4)
T1(C) = (v E X 13 e > 0 such that x + ev E C),
then
T, (C) = T= (C), d x E C
Claim. T?(C) C Tz(C) is trivial, and since TT(C) is closed, Tom) C T=(C). It remains to verify that d v E T=(C) 3 en > 0, vn E X such that x + envn E C, and vn -+ v. By Definition 6.3, we have yn E C, h,, j 0 such that
hn' Ilx+hnv - ynII -' 0. Set v = hn'(yn - x), and en = hn, n = 1, 2,. . . ; these are what we need. Lemma 6.3. Assume that S is a locally convex closed set with respect to (M, A). Then T=(S) is a closed convex cone in TzM V x E S.
Claim. In fact, if (V, 0) is a chart in A at x, 0 : U - X, then T=(S) = 0'(x)-1 . Tm(=) (cf(U n S)).
According to (6.4), TT(z)(O(U n S)) is convex, and since 0'(x) is linear, the convexity of the cone T,,(S) follows from (6.4). Therefore Tm(S) is convex. By Definition 6.3, if v E Tz(S), then tv E T5(S), so that T1(S) is a cone.
Infinite Dimensional Morse Theory
62
Definition 6.4. Let S be a locally convex closed set with respect to (M, A), and let f E 0(0, 1RI) where f2 is an open neighborhood of S. We say that x0 E S is a critical point of f with respect to S if (df (xo), v) > 0
d v E T=o (S),
where (,) is the duality between Tz0M and T=0M. Or, equivalently, we say that x0 is critical with respect to S if (df (xo), O'(xo)-' (y - 0 (xo))) > 0 bay E O(U n S). Now, d x' E T=OM, let
IV*I:o = Sup ( (x', v) I V E Txo(S)
with
II0'(xo)vllx
1}.
Therefore:
xo E S is a critical point of f with respect to S if and only if I - df (xo) I=o = 0.
The Palais-Smale condition (PS)c with respect to S is extended as follows: Any sequence C S, along which and f(xn) - c I - 4. (xn) Ixn -4 0, implies a convergent subsequence.
Applying the same argument employed in Theorem 3.3, we have
Lemma 6.4. Suppose that f E C' (fl,1f8l) satisfies (PS),, with respect to a locally convex closed set S. Then the critical set K, = Kn f -1(c) with respect to S is compact, and for any closed neighborhood N (if Kc = 0, then N = 0) of K., 3 constants b, c > 0 such that
I -df(x)Iy > b,
Vx E Sn f-1[c-e,c+E].
Now we are in a position to establish both the critical point theory as well as the Morse theory with respect to S as in Sections 3 and 4. Since the idea is the same, it suffices to outline the main steps and the necessary modifications.
1. The construction of a p.g.v.f.
From now on, we assume that M is paracompact. First we point out that the function x -. I - df (x)I= is continuous on S.
6.
Extensions of Morse Theory
63
Claim. According to Lemmas 6.3 and (6.4),
I - df (x) I. = Sup { {-df (x), 0'(X)-'(y - O(x))) I Y E O(U n S) with II1l - O(x) II x < 1). The conclusion is obvious.
Second, V xo E S' := S\K, where K is the critical set with respect to S, we choose vo E f.0(S) = 0'(xo)-1Te(=0)(O(U fl S)) satisfying (i) II0'(xo)voII < 1, (ii) (-df (xo), vo) > I - df (xo) 1.0. ; lemma, 3 b = 6(xo) such that B(xo, b) c U, and According to the above d x E B(xo, b) fl S the vector field vo(x) = 0'(x)-' Im(xo) - O(x) + -0'(xo)vo) E T.(S) satisfies II
'(x)vo(x)II <
31
and
(-df (x), vo(x)) >
2
3I - df (x)I =.
We find a locally finite partition of unity {Qe I [ E A) associated with a refinement covering B of the covering {B(xo,b(xo)) I xo E S'}. Set W(x) = >Qe(x)Ve(x) where Ve(x) is the v.f. defined on B(xe, b(xe)), which includes the support of fle(x). Set
V(x) = 3I - df(x)1.W(x). We obtain (1) V(x) E T1.(S), (2) IIo'(x)V(x)II < 21 - df(x)1=,
(3) (-df (x), V(x)) > -df(x)I=
dxES'. 2. ODE on closed sets.
We define deformations by the semiflow for the p.g.v.f. on a closed set S. The question is under what conditions does the semiflow remain in the set S? Definition 6.5. A subset S C M is said to be a (locally) semi-invariant set with respect to a vector field V if the semiflow x(t) remains in S for all (small) t > 0:
x' = V(x(t))
{ x(0) = x E S.
Infinite Dimensional Morse Theory
64
The following local existence and uniqueness theorem for semiflow on a closed set was obtained by Brezis (Brel[ and Martin, cf. [Marl).
Theorem 6.3 (Brezis-Martin). Let A be an open subset of X, and let B C A be closed in A. If V : A -+ X is a locally Lipschitzian mapping, then t1 x E B, 3 d > 0 and r(t) satisfying
r(t) = V(r(t)) { r(0) = x E B;
V t E (0; b)
r(t) E B
if and only if
lii mh-1d(x+hV(x),B) = 0
i.e., V(x)ET=(B) VxE B. Lemma 6.5. Let S be a closed subset of M. Then S is a semi-invariant set with respect to a locally Lipschitzian v.f. V on M, if and only if (6.5)
lii o h-ld (O(x) + h46'(x) V (x); O(U n S)) = 0
V X E S V chart (0, U).
Proof. Assume that (6.5) holds. The following ODE in X W) = 0'(x)V(x)1==m-1(00) Y(0) = O(x)
has a local solution yo. Since O(UnS) is a closed set in q(U) and (0' V)o0` is tangent to 0(U n S), by the Brezis-Martin Theorem, (U n S) is locally invariant with respect to O' o V. Setting xm(t) = 0-1 (yj(t)), it follows that xm(t) E U n S. It is not difficult to verify that xm does not depend on the special choice
of 0 in a neighborhood of t = 0, and it is easy to extend the solution to a maximal solution which remains in S via the method of continuation. Conversely, if S is a semi-invariant set with respect to V, then for any chart ((0, U), O(U n S) is semi-invariant with respect to W V) o 0' . (6.5) follows directly from the Brezis-Martin Theorem.
3. Deformation theorems and critical groups With the above preparatory work, the first and second deformation theorems do hold for any locally convex closed subset S of a C2 paracompact Banach manifold.
The proofs are just the same as in Section 3 and hence are omitted. Similarly, the critical groups for isolated critical points with respect to S are well-defined.
7. Equivariant Morse Theory
65
4. Morse relations for functions with isolated critical points Let S be a locally convex closed subset S of a C2 paracompact Banach
manifold. Let f E Cl (fl, R' ), where Il is a neighborhood of S. Assume that f satisfies the (PS) condition with respect to S and has only finitely many critical points (x1, x2i ... , with respect to S in f -1 [a, b]f1S, with critical value (Cl, c2, ... , c,,, } satisfying a < c3 < b, j = 1, 2, ... , m. Relative to any coefficient field for homology, set m
MQ = Mg(f; [a, b]) = E rank CQ (f,xj), and
Rq = R4 (f ; (a, b]) = rank H a (fy, fa) ,
We have
00
00
E M9t4 = V=o
4 = 0,1, 2,... .
Rot' + (1 + t)Q(t), Q=o
where Q(t) is a formal series with nonnegative coefficients. Remark 6.2. Readers might be puzzled about the different conditions in the two subsections. The functions in the first subsection are under strong boundary conditions, but in the second subsection there are no restrictions. The underlying spaces in the first are manifolds with smooth boundaries, but in the second are locally convex closed sets. After all, we have the same conclusion, i.e., the Morse inequalities. The point is that the meanings of the critical points in Sections 6.1 and 6.2 are different. In Section 6.1 they correspond to the variational equation, i.e., the Euler-Lagrange equation, while in Section 6.2 they correspond to variational inequality. Remark 6.3. Morse theory on convex sets was initially studied by M. Struwe [Str1J and K. C. Chang, J. Eells [ChE1J in the Plateau problem. It was developed in K. C. Chang [Cha7] to study the variational inequality problems. The introduction of local convexity in critical point theory first appeared in T. Q. Wang (WaTI].
7. Equivarlant Morse Theory Let us assume a compact Lie group C and a smooth manifold on which the
group C acts. Equivariant Morse theory studies the Morse relations and the Morse handle body theorem for G-invariant functions. Noticing that for a C-invariant function, if x is a critical point, then the points on the C orbit containing x are also critical points. It is hard to say regarding the
66
Infinite Dimensional Morse Theory
isolatedness of critical points if G is continuous. In this section, we shall introduce the notions of isolatedness and nondegeneracy for critical orbits.
7.1. Preliminaries Let C be a compact Lie group. A C space (or manifold) Al is a topological space (or manifold resp.) with a continuous G action, i.e., 0 : G x M -+ M
with lji(g,x), written as g x, such that e x = x, and (91 92) x =
dxEM.
V x E M, we call O(x) = {g x I g E G} a G orbit. The set of all G orbits is called the orbit space. Endowed with the quotient topology, it is denoted by Al/G or simply M. The subgroup of G, defined by G= _ {g E G I g x = x}, is called the isotropy subgroup at x. If x has the isotropy subgroup H, then g x has the isotropy subgroup
g H g-1. Thus a conjugacy class of isotropy subgroups is attached to each orbit.
Ciscalled afree action ifGx=eVxEM. Let II be a subgroup of G, and M be a C space, we denote
dhEH}, i.e., the set of points fixed under H. MG is called the fixed point set of C, which is also denoted by Fixc. A set A C M is called G-invariant, if g- x E A, V x E A, V g E G. A G pair (X, Y) is a pair of G-invariant spaces (X, Y) with Y C X. A function f : M -+ 1181 is called C-invariant, if f (g x) = f (x); d x E
M,VgeC. A map F : (X, Y) -+ (X', Y') between two G pairs is called G equivariant, if
VxEX, ` 9EG. Thus, a G equivariant map F induces a map
F=FIG: (X,Y)-+(X,Y'). Let n : E -+ B be a fiber bundle. (E, ir, B) is called a G bundle, if V g E G, g : E -+ E is a differentiable bundle map, such that gE= = Eg.r V x E B. Thus, if Al is a C-manifold, the tangent bundle TAI is a G bundle, with
g.X
ET=(M), VxEA1.
(F,, a, B) is called a Riemannian C vector bundle, if the G vector bundle possesses a Riemannian metric, and the C-action is isometric.
In the following, we always assume that M is a Hilhert Riemannian
manifold, with a Riemann metric on TM. Let E C M be a compact connected submanifold; then TE, the tangent bundle of E, is a subbundle of TAI, and then the normal bundle NE, which is the orthogonal complement to TE, is also a subbundle of TAI.
7. Equivariant Morse Theory
67
If in addition M is a G manifold and E in G invariant, then both TE and NE are all G-bundles. Let f E C1(M,JR1) be a G invariant. It gives rise to a C-equivariant gradient vector field df (df (g x), g X) = (df (x), X) V (g, x) E G x M, d X E T(M), i.e., g* c[f g = df.
Since the action g on T=(M) is unitary, dg' = g-1, we obtain df Analogously, the Hessian d2f is also G-equivariant, if f E C2 It is obviously seen that the level sets fc, f -1(c), and the critical sets K, Kc = K n f -1(c) are all G-invariant. And a critical orbit 0 = O(x) is a G submanifold of M. It follows that T=(O) C ker d2 f (x) and that the induced bounded selfadjoint operator d2 f (x) : N=(O) -. N=(O), satisfies g* . d2f(g . x) . g = d2f(x)
7.2. Equivariant Deformation For a G-invariant function, we shall improve the first and second deformation theorems to make the deformations equivariant. Namely, we shall prove
Theorem 7.1. (First C-equivariant deformation theorem) Let M be a C2G-Hilbert Riemannian manifold. Suppose that f E C' (M, R') is Ginvariant and satisfies the (PS)c condition. Assume that N is a C-invariant closed neighborhood of Kc for some c E IR1. Then there exist constants M such Z > e > 0 and a G-equivariant continuous map rl : [0, 1] x M that (1) 17(t,.) (CJ-llc-t,c+1 = id JCJ-'lc-r,c+tl (2) 1l(0, = id, (3) 77(1,fc+,\N) C fc-c (4) V t E 10, 11, 7 7 ( t ,- ) : M - M is a G-homeomorphism, (5) t " f o t)(t, x) is nonincreasing V t.
Infinite Dimensional Morse Theory
68
Theorem 7.2. (Second C-equivariant deformation theorem). Let M be a C2C-Hilbcrt Riemannian manifold. Suppose that f E C' (M, R') is C invariant and satisfies the (PS), condition, V c E [a, b]. Assume that a is the only critical value off in [a, b), and that any connected component of K. is always a part of a certain critical orbit. Then J, is a G-equivariant strong deformation retract of fb\Kb. The proofs of these theorems are just modifications of those given in Section 3. A new ingredient is to construct a G-equivariant locally Lipschitzian p.g.v.f. We carry this out as follows:
Lemma 7.1. Assume that f E C' (M, IR') is C-invariant. Then there exists a C'-°G-equivariant p.g.v.f. Proof. We have already a C1-° p.g.v.f. X (x) from Theorem 3.1. Let us define
,k(X) = fg' . X (9 .
(9 - x)= fG 9 -'X(9-9'-x)dl4
=9 f c(9'9)-'X(9'9 .x)dlt =g'X(x)V9'EG, X is C equivariant.
We shall prove X E Cl-'. It is only required to prove that V orbits C(x°), 3 a neighborhood U, such that X is uniformly Lipschitzian on it. Actually, V x E G(xo) 3 an open ball B(x, b=) and a constant C= > 0 such that IIX(y) - X (z) 11 < C.d(y, z)
V y, z E B(x, 61)
where is the Finsler structure on TM. Since C(xo) is compact we find a finite covering U = U,"., B(xi, b=i) D G(x°). Let
6= Min {(b=i, 1 < i < n} ,
and
C = Max jCZ,,supIIX(y)II,1
b ve U
We conclude
IIX(y)-X(z)II
JJJ
7. Equivariant Morse Theory
69
In fact, if d(y, z) > 6/2, then II X(y) - X(z)II 5 IIX(y)II + IIX(z)II < 2 sup IIX(y)II VEU
5 2 C < Cd(y, z).
Otherwise, d(y, z) < 6/2, y and z must fall into a certain neighborhood B(x;, 6=, ), and hence II X (y) - X (z) 11 <- C 1d(y, z) < Cd(y, z)
Finally, we prove that X is again a p.g.v.f. Since g is isometric I19(z)II 5 Ic I I X (9 - x)II dp
<2
f
c
= 2IIdf(x)II, and
(df (x), X (x)) = f (df (x), 9-'X (9 - x)) dp
= f (df (g x), X (g x)) dp
j
c
Ildf(x)112.
By virtue of Lemma 7.1, the flow defined by the G-equivariant p.g.v.f. X must be G-equivariant. (The uniqueness of the solution of ODE). The rest of the proofs of Theorems 7.1 and 7.2 is exactly the same as in Section 3.
7.3.
The Splitting Theorem and Handle Body Theorem
for Critical Manifolds Definition 7.1. A connected submanifold 0 C M is called a critical manifold, if df (x) = 0 V x E 0 (may not have group action) and f J o = const. A critical manifold 0 C M is called isolated if there is a neighborhood U
of 0 such that U n K = 0.
Infinite Dimensional Morse Theory
70
In the following, we always assume that 0 is compact. Now we shall study the local behavior of the function f near its isolated
critical manifold. The restriction of the tangent bundle TM on 0 now is split into a direct sum TO ® NO, the tangent bundle and the normal bundle of the manifold O. The Riemannian metric on TM reduces a metric on the normal bundle NO. The exponential map exp, regarding the geodesic sprays on M, yields a diffeomorphism from the normal disk bundle NO(c) _ {(x, v) E NOIIIiHI
c} to the tubular neighborhood N(c) _ {x E MI dist (x, 0) < c} of 0, for some e > 0 because 0 is compact. In this sense, we shall not distinguish NO(c) from N(c).
In order to split the normal bundle NO into mutually orthogonal bundles, we need
Lemma 7.2. Let II be a Hilbert space and Q, R be two orthogonal projections. If IIQ - RII < 1, then QR: RH -+ QH is an isomorphism.
Proof. QR is an injection. Otherwise, 3 v E RH\{9) with Qv = 0. Therefore II(Q - R)vII = IIvII, which contradicts IIQ - RII < 1. QR is also a surjection. Otherwise, 3 v E QHn(RH)1\{0}, then (v, QRw) = (Rv, w) = 0 V w E II, which implies Rv = 0. Again, we get IIQ - R)vhI = IIvII; this is impossible.
Lemma 7.3. Let f = (E, 7r, N) be a Hilbert vector bundle on a connected manifold N, with a Riemannian metric (, ). Let P be an orthogonal bundle projection with respect to (, ). Let I be a section satisfying IS = id in C(E=, ES) d x E N. Assume Q = I - P. Then Pf = (PE, aI PE, N) and Q{ = (QE, trIQE, N) are Hilbert vector bundles with
Proof. It suffices to prove that d x E N 3 a bundle chart t/, : U x II
tr't(U), x E U, and an orthogonal projection P _E £(H, H) such that V y E U, ipy : H
Ey, t/iy (v) = t/i(y, v) maps PH, QH isometrically onto
PE,, and QEy resp. where H is a Hilhert space, and Q = I - P. From the definition, we have a bundle chart 0: U x H - 7r'1 (U), x E U, and then define
P=P dyEU.
For small U, IIPy - P-, 11 < 1 V y E U. By Lemma 7.2
PPPy:PV H-.P111 is an isomorphism. Similarly, set Qy = tv 'QOy, we have
Q=Qy : QyII , Q11I is an isomorphism.
71
7. Equivariant Morse Theory
H
Ey
P1
P.JP-
E_
H
IP
,
E.
Ey
Thus the map
By = P=P, + Q.Qy is invertible in C(H, H) fl L(P,H, PXH) fl G(QyH,Q=H). We define A. E L(H, H) by (myv, 0yw)Ey = (AyByv, Byw)H d v, w E H.
Then y,-- Ay is C' and positive definite.
Finally, set 0. = OyoBv 1Ayi, and and P are just what we need.
P. The maps 0(y, v) = >[iy(v),
Now we extend the splitting theorem to isolated critical manifolds. First, we need
Assumption 7.1. f E C2(M,llt'), 0 is an isolated critical manifold, and there exists e > 0 such that (7.1)
o (d2f (x)) fl ((-e, eJ\{0}) = 0,
V x E O.
Furthermore, dim ker(d2 f (x)) = const. V x E O. Noticing that d2 f is a well-defined quadratic form on the normal bundle NO, let X. (* = +, -, 0) be the characteristic functions of the intervals If, +oo), (-e, e) and (-oo, -e) resp. Then the orthogonal bundle projections (7.2)
P. = x.
(d2f(.))
*=+,_,o
are well-defined on the normal bundle NO.
Lemma 7.4. Under Assumption (7.1), f =t+ED E-ED to
where C = (NO,a,O),
C' = (P.NO,aIP.NO,O),
This is a consequence of Lemma 7.3.
_
0-
72
Infinite Dimensional Morse Theory
Theorem 7.3. Under Assumption (7.1), there exist e > 0, a local homeomorphism $ : NO(c) - N(c), and a C' map h : {o(c) -+ such that
f -40) =
2(IIP+(x)V112 - IIP (x)v112)
+ f o exp(Po(x)v + h(Po(x)v))
V p = (x, v) E NO(c), x = irp, v E NOR (c), where .°(c) = ((x, v) E °I x = irx, v E l;°, with Ilvll < c}. P. and ' are defined in (7.2) and (7.3) resp.
Proof. V X E 0, the function f o exp I(_- possesses 0 as an isolated critical point. The splitting theorem (5.1) is exploited to obtain a local homeomorphism mR : R(c) -+ {=(e), and a C' map hR : (=(c) --- fX a) :X such that
f o exp.= s=(v) =
2
(IIP+(x)v112 - 11P(x)v112)
+ f o exp= (Po(x)v + hx (Po(x)v)) dVE1;R(C)
By a standard method, we verify that 4 and h can be defined by -tR = expx 0_, and hi oiei = hR respectively.
Corollary 7.1. Assume that f E C2(M, Il8') is C-invariant, and that 0 is an isolated critical orbit of f. Suppose that 3 xo E 0, such that 0 is either an isolated point in a(A) or not in a(A), where A = d2 f (xo)ICxp. Then the conclusion of Theorem 7.3 holds. In addition, ob and h are all C-equivariant.
Proof. Since d2f(g. x) = g-'d 2f(x)g V X E M, the spectrum a(d2 f (x)I fix) is G-invariant V x E 0, so Assumption (7.1) holds. It remains to verify C equivariance. Since z = hR(y) solves the equation uniquely: d.f oexpx(y+z) = 0, we conclude that gh1(y) = h9.R(gy), provided by the equivariance of df. Similarly, the map nR solves the equation uniquely: 2 (A(x)77.(u, y),'r1.(u, y))
= f o expR (u + hR(y) + y) - f o exp1 (hr (y) + y) , where y E C=(c), u E ({s ®t;= )(c), A(x) = d2 f (x), and 7lx(u, y) E (f= _ ). The map 0x is just the inverse of the map (u, y) F-+ (ii. (u, y), y). Since
A(g x) = g-'A(x)g,
7. Equivariant Morse Theory
73
we obtain rrg.z(9u, 9Y) = 917-(u, Y).
Thus
I'9. (9v)=94':(v) VxEO, Vv Et:(f). Definition 7.2. Under Assumption (7.1), the dimension of the fiber of
the vector bundle t- is called the Morse index of f with respect to the isolated critical manifold 0, and is denoted by ind (f, O). 0 is called nondegenerate if the dimension of the fiber of the vector
bundle to is zero, i.e., to = O. f is called nondegenerate if it has only nondegenerate critical manifolds. Definition 7.3. Let t = (E, x, O) be a vector bundle over a compact connected manifold O. The disk bundle
fi(r) _ {(x, v) E E I V E E_, x = ax with
r),
Theorem 7.4. Suppose that f E t: 2(M, Rl) satisfies the (PS) condition, and that c is a critical value with K. = {01 i 02, ... , Of } consisting of nondegenerate critical manifolds. Then there exist f > 0 and homeomorphisms h, : t;- (e) -. M, j = 1, 2,... , t, such that t fc+E = fc-e U U h, (Sj (f)) J=1
with
fC
n h, (Cj (f)) = f
(c - E) n h, (f; (f)) = h, (oSj (f))
where 8tf (f) is the m, - 1 spherical bundle, m, = ind (f,O,), {j is the associated negative subbundle of NO,, j = 1, 2,... , t. The proof is the same as Theorem 4.4, because we have already established the generalized Morse lemma (Theorem 7.3). Actually, the deformation is constructed along the fibers.
Corollary 7.2. Suppose that the function f in Theorem 7 . 4 is Cinvariant, and O,, j = 1, 2, ... , t, are nondegenerate critical orbits. Then the homeomorphisms h, : f) (f) M, j = 1, 2, ... , t, are G-equivariant, and the deformation is also G-equivariant. In the proof of the corollary, G-equivariance follows from the construction along fibers and Theorems 7.1 and 7.2.
Infinite Dimensional Morse Theory
74
The Gromoll-Meyer pair (W, W_) for an isolated critical manifold 0 is
naturally extended. (Only W fl K = {p} is changed to be W fl K = 0.) The construction of such a pair (W, W_) for isolated critical manifold is similar to the construction in (5.11) and (5.12). We only want to replace the ball B6 by the tubular neighborhood N(b) of 0, and to replace the function g (defined in Section 5) by g(x) = A f (x) + dist (x, 0)2.
In particular, if f is G-invariant, 0 is an isolated critical orbit; then the Gromoll-Meyer pair (W, TV-) may be chosen G-invariant, and the flow in the definition is G-equivariant.
7.4. G Cohomology and G Critical Groups The extension of nondegeneracy of critical manifolds has the following advantage:
Suppose that E " M is a fibring, and f is a nondegenerate function on Af in the sense of Definition 7.2. Then it is easy to see that the pull back ir' f
on E is again nondegencrate. Further, the index of 0 as a nondegenerate critical manifold of M equals the index of it-r0 as a critical manifold of E, i.e., ind (f, O) = ind (ir' f, it-10). Let us consider a compact Lie group action C. If the C action is free, then the orbit space Al = M/C is also a manifold, the projection
ir:M-.M/G is a smooth fibration, with fiber C, and there is not any difficulty in carrying out the Morse theory on the orbit space M.
However, if the action is not free, then M possesses singularities, and one cannot do the same as above. Consider any smooth principal G bundle E over a base manifold I3, and the following diagram:
E
*2
l-
ExAf
I-
E xc M
E/G
T1
'Al
I-
Af/C
q
of the C-actions on Al and E, and G operates diagonally on E x M, i.e., g(a,x) = (ga,gx) V (a, x) E E x Al, where E xc M = (E x M)/G. Since the action on E is free, this diagonal action is also free. On the other hand, a G-invariant f on Al clearly lifts to a G-invariant f on E x Af, and hence to a smooth function f E on E x G M. Now E x G M is itself a fiber space over the base B = E/G with fiber M and structure group G. Noticing that 7r-1 (E xc 0) = nI 1(0)
75
T. Equivoriant Morse Theory
for any critical manifold 0 of f, E xc 0 is the corresponding critical manifold of fE. We have
ind (f,0) = ind (fe,E xcO). There are many principal C-bundles E we may choose; among them we single out a universal G-bundle that Is unique up to homotopy which possesses the following important property: The total space E is contractible. Such bundles always exist cf. IIousmoller [Hou1J. We are satisfied ourselves to give the following few examples:
B=Bc=
E=EG=
G Z2
S°O
S'
S°°
RP°° CP°°
SU(2)
S°O
IHIP°°
real
projective quaternion space complex
In the universal case, we shall write MG = E xc M, Bc = E/G, Ec = E, and fc = fE. MG is called the homotopy quotient of M by G. The advantage of this choice is that the map
q:Mg -.M/G is a homotopy equivalence, if G acts freely on M. In summary, we consider the function fc on MG, the homotopy quotient, which is G-free, instead of f on M/C. However, there is an (equivalently) alternative way to consider the problem, that is the concepts of G cohomology.
Definition 7.4. Given a C pair (X, Y) and a coefficient field K, let
III(X,Y;K) = II' (XG,YG,K), where Xc and Yc are the homotopy quotients of X and Y by the group C. We call IIc the G cohomology. It was proved by A. Borel that the C cohomology enjoys most of the properties of the cohomology. More precisely, the exactness, the homotopy, and the excision axioms hold, but not the dimension axiom:
Hc(pt) = H* (Bc) Furthermore if F : (X, Y) -+ (X', Y') is a G-equivariant map, then F x 1 : (X, Y) x Ec -. (X', Y') x Ec induces a C-equivariant map on the homotopy quotient:
Fc : (Xc,Yc)
(Xc,Yc),
and hence a homomorphism:
Ft. : III (X', Y')
HH(X, Y)
76
Infinite Dimensional Morse Theory
cf. T. Tom Dieck [Diel). Now we are able to define the G-critical groups, and present some computations. At the end, the Morse inequalities are established. Definition 7.5. Suppose that U is a G-invariant neighborhood of an isolated critical orbit 0, such that K fl (ff fl U) = 0, where c = f (O). Then for any coefficient ring lid,
Cc(f,O) = He(f., nU,(f., \O)nU,K) is called the qth C critical group of 0, q = 0, 1, 2, ... . And by the same proof as Theorem 5.2, we have (7.4)
CC(f, 0) = I1 (I-V, W-; K)
where (W, W_) is a C-Gromoll-Meyer pair of 0.
Example 1. Suppose that 0 is an isolated critical orbit, corresponding to a local minimum of a C-invariant function f. Then (7.5)
CC(f,O) =
H'(OG).
Example 2. Suppose that the normal bundle l; = NO is a trivial bundle. Then we have the following formula: 9
(7.6)
Cc(f,0)
®&c'(f,0)®IIc(0) i-o
where CC (f, 0) = Cc(f o exp) I(,, 0), i = 0, 1, ... , V x E 0. Claim. We shall prove that V x E 0, the function f oexp Ie= possesses an isolated critical point 0. In fact, if v E l;=(e) is a critical point of f oexp kkx,
then the normal derivative at v must be zero. On the other hand, f o exp is G invariant, and the tangent derivative is also zero. It follows that d(f o exp)(v) = 0. However, we have assumed that 0 is isolated, therefore
V=0. Denote (IV, IV_)_ = (W, W_) n exp {_ (c) V x E 0, for a suitable f > 0 such that exp-'(IV, W_) C (e). Obviously, exp-'(W,W_)= is a GM pair for the restriction f o exp I(=. Noticing that
(W, W-)9.= = 011, W-). and g is a diffeomorphism, we obtain Cc (f o exp It., 0) = Cc (f o exp IE9 :
B)
7. Equivariant Morse Theory
77
Thus the critical group CC(f, 0) is well-defined.
By the assumption that f = NO is a trivial bundle, we know that (W, W_) possesses a product structure (W, IV-). x 0. Using the Kiinneth formula and (7.4), we obtain Cc (f, O) = H,'gl (W, W ; K) a
= ®H°7 3 (W, WIII) ® He(O) j=o v
= ® CV 1 (f o exp [(_, 9) ®Hc(O) j-0
4) eQq j(f,0)®Hc(0). j=o
Example 3. (Nondegenerate critical orbits) Let us recall the orientation of a sphere bundle and Thom's isomorphism theorem.
Consider a r sphere bundle
_ ((E, E), 7r, B),
(Dr+I,Sr) _. (EVE)
IB
where B is connected, and Dr+1 is the r + 1 disk, BDr+i = Sr
t is
said to be orientable if there is an element 0 E Ilr+' (E, E; 1K) such that,
V x E B, the restriction of 0 to (w-'(p),a-1(p) n E) is a generator of II r+ I (r-1(p), lr- I (p) n E; K). Such a cohomology class is called an orientation class over 1K of the sphere bundle t.
(Thom's Isomorphism Theorem). Let t = ((E, E), a, B) be an rsphere bundle. Then for any ]K module A, there are natural isomorphisms
: H.
(E;A) - H.-r-I(B;9®A),'I(z) = a.(Bnz)
and
V : II'(B;B®A) -. H'+r+r(E,E;A),4s'(v) = ir'vUB where 0 is the orientation bundle oft, cf. Spanier [Spal], p.259.
78
Infinite Dimensional Morse Theory
Theorem T.S. Assume that f E C2(M, R') is G-invariant, and that 0 is a nondegenerate critical orbit, with ind (f, 0) = Ji, and c = f (0). Then we have
CC(f, 0) = II "(0, 9- (& IK),
where B- is the orientation bundle of '.
Proof. We apply Theorem 7.3. There is a G-homeomorphism 4i NO(e) - N(e) such that f 00(p) = I (III'+(x)v112 - III _(x)vI12) +
where p = (x,v), x = np and v E t_, (f). Applying the deformation employed in Theorem 4.1, we obtain CC(f, 0) = He (f,, n fi(r), (ff\O) n E(E),1K)
= He The pair
(C-
is then a A sphere bundle:
(f
(DA,
(f),. (e)\0) 1
0 Thom's isomorphism theorem is applied to obtain
CC(f,O) = H a (0,0-01K). Mirthermore, if - is C-orientable (if G is connected, this simply means that t:- is orientable. Otherwise we assume that G preserves the orientation on C- also), then we have
Cc(f,0) = IIc a(O,K). In all of these examples the computations finally reduce to the computation
of the G cohomology of the C. orbit 0, i.e., Hc(O). Noticing that 0 = O(xo) = GIG=o, where Cxo is the isotropy subgroup at xo E M, the map rl defined in the commutative diagram in Section 7.4 is induced on an orbit space by projection. The mapping E - O(xo) given by v '- Iv, xo[, where [v, xo[ E E x c M is the orbit containing (v, xo), induces a homeomorphism
E/G=o - ri-'(O(xo)). Thus r '(0(xo)) has the homotopy type of the classifying space I3c=o. Therefore
If. (ri-' (0 (xo))) = II' (Bc=o) .
79
T. Equivariant Morse Theory
In case G__0 is trivial, we have H,9(O) = HQ({pt}) = bqo. For example, let M = R2, C = Z2, acting on R2, be reflecting about the
origin 6. Then Ec = S°°, Ba = RP°°, `dxo34 B,
but for xo = B,
G:o={e}, Go = 7L2,
He (O (xo) , Z2) =
69. 7L2
HH(O(O), Z2) = 7L2
d q;
V9-
The advantage in introducing the C homology lies in providing more information about the critical orbits. The Morse inequalities are then apparently extended.
Theorem 7.6. Suppose that f E C' (M, G) is C-invariant and satisfies the (PS) condition in f -' [a, bJ, where a, b are regular values. Assume that
f has only isolated critical orbits (Oi I j = 1, 2,... , t) in f -1(a, b]. Then 00
L
00
EF, rank Cc(f,O,)t'=1: i=oisi
rank
ffc(fb,fa)t'+(1+t)Q(t),
i=o
where Q(t) is a formal series with nonnegative coefficients.
Proof. It follows from Theorems 7.1, 7.2 and the proofs of Theorems 4.2 and 4.3.
Theorem 7.7 (Bott). Under the above assumptions, if, further, f E C2, and has only nondegenerate critical orbits Oi, with ind (f, O,) = Ai, j = 1, 2, ... , t, then we have 00
l
E E rank He (0j, O ®X) tai+i =o j=I 00
_ 1: rank He (fb, fa; W) t' + (1 + t)Q(t), i=0
where 6
is the orientation bundle of f,-, f i = NOi, j = 1, 2 ... , e.
Proof. It follows from Theorems 7.5 and 7.6. Remark 7.1. Equivariant Morse theory for nondegenerate critical manifolds with coefficient field 7L2i was first studied by R. Bott (Botl-2). It was extended to complete C-Riemannian manifolds, and to Hilbert Riemannian manifolds by A. C. Wasserman (Was1J and W. Meyer [Mey1J respectively. Equivariant Morse theory for isolated critical orbits was done by Z. Q. Wang (WaZ1]. The use of G cohomology was started from M. Atiyah, R. Bott (AtB1J, and followed by H. Hingston (Hin1J, C. Viterbo (Vitl) and A. Floer (Flol).
80
Infinite Dimensional Morse Theory
Finally, we shall give an equivariant version of Theorem 5.7.
Theorem 7.8. Suppose that f E C2(M, R') is a G-invariant function, and that d2 f (x) is a F}edholm operator V X E M. Assume that c is a critical value, with K, = {01, 02, ... , Ot}. Then d e > 0 there exists a G-invariant function g E C2(M, IR') satisfying the following: NOj(E); (1) g(x) = f(x) in (2) fig - f IIc2(M) G e;
(3) g has only
nondegenerate critical orbits,
concentrated in
(4) if rnj = ind (f,Oj) and nj = dim ker d2f (0j), then the Morse indices of those nondegenerate critical orbits of g in NOj (e) are in Jmj, mj + nj).
The proof depends on a Meta theorem which is often advantageous in proving a theorem about compact Lie groups. It replaces the technique of doing a double induction on the dimension and number of components.
Meta Theorem. Let P be a statement valued function defined for all compact Lie groups. If, whenever G is a compact Lie group, the truth of P(II) for all compact sub-Lie groups H C implies the truth of P(G) and if P({e}) is true, then P(G) is true for all compact Lie groups G. We need the following preparations. Let it : NO(e) -+ 0 be the projection from the e-tubular neighborhood
NO(e) to the G-orbit 0 = Cx for some x E M. (As before, we shall not distinguish the tubular neighborhood and the normal bundle.) Let S = it-1(x), and let G= be the isotropy group at x.
Lemma 7.5. (1)VyES,G1, cC1. (2) S is a Gs-manifold.
(3) If fo is a G=-invariant function defined on S, then the function f(z) = fo(y)
if
z = gy,
y E S.
is well-defined on GS, and is C-invariant.
Proof. By definition, n is G-equivariant. (1) V y E S, V h E Gy, i.e., by=y, we have x = 7r(y) = ir(hy) = hir(y) = hx. Therefore h E G=.
7. Equivariant Morse Theory
81
(2) We only want to verify that G=S C S. Indeed, V g E G,,, d y E S, W(9y) = 97r(y) = 9x = X.
It follows that gy E S. (3) It suffices to prove that f is well-defined. Indeed, if there exists g E C such that gz = z for some z E S, then x = x(z) = 'r(9z) = 97r(z) = gx.
It follows that g E G. We have assumed that fo is G=-invariant, so f is well-defined.
Proof of Theorem 7.8. By the partition of unity, one may assume that K° is a single orbit O. Assume that the theorem is true for all compact sub-Lie groups H of G. We shall prove that it is true for G. For x V Fixc, let 0 = Gx, 7r : NO(e) -. 0, and S = ir-' (x). According to Lemma 7.5, S is a G=-manifold, where the isotropy group G. is a subLie group of C. By the assumption, we can modify the function f Is to a function Is such that 9Is satisfies (1) - (4) with M = S and G = G. Then we apply Lemma 7.5, and the function 9Is can be extended to GS = NO(e) and then to M as a G-invariant function g with only nondegenerate critical orbits, i.e., g satisfies (1)-(4). If x E Fixc, then 0 = {x}, G= = G, and NO(e) is an a-ball. We choose local coordinates at x, such that x = 0, and NO(e) is the unit ball B in a Hilbert space E. By Sard's Theorem in the nonequivariant case, one may assume that 9 is a nondegenerate critical point of f IFfxc, and also 8 is the unique critical point of f in B. Define 9(y) = f(y) + 2cA (IIvII2) IIPvII2
where .1 E C°°(R .,lRI) satisfies a(t) = 1 as t < ;, and \(t) = 0 as t P is the orthogonal projection onto Fixa, and e > 0 is small such that d2g(y) is also a Fredholm operator. Since g = f on Fixc, 9IFix , has 9 as the unique critical point, which is nondegenerate on Fixc. We shall prove that 9 is also nondegenerate on E. Indeed, d2g(B) = d2f(B) + eP.
If h E ker d2g(9), then (7.7)
(d2 f (0) + eP) h = 9.
82
Infinite. Dimensional Morse Theory
Since 0 is nondegenerate for flFiXC, h E Fix.' = (kerP)1. And on Fix-, equation (7.7) is reduced to P (d2f(O) + cl) Ph = 0. However, d2 f (0) is a Fredholm operator, 0 cannot be an accumulate point of o(d2 f (B)), which implies that for sufficiently small e > 0, P(d2 f (0) + c)P is invertible on C(Fix.'). Therefore h = 0, i.e., 0 is nondegenerate on E.
The case G = {e} is just Theorem 5.7. The proof is completed by applying the Meta Theorem mentioned above. Remark 7.2. Theorem 7.8 was proved by Wasseman (Wasl) and Viterbo )Vitl).
CHAPTER II
Critical Point Theory
In the study of nonminimum critical points, a basic method is the socalled minimax principle. In this chapter we study the connections between Morse theory and a variety of concrete versions of the minimax principle. We point out that the minimax principle for relative homology classes is particularly suitable for Morse theory because certain critical groups for the critical points determined by this minimax principle can be proved being nontrivial; they then have contributions to Morse inequalities. Section 1 is devoted to applying the homology minimax principle to the study of the linking method used initially by Rabinowitz, Benci-Rabinowitz and Ni. In Section 2, we estimate Morse indices for critical points, determined by various minimax principles in which the critical groups are not well-determined. In Section 3, we study the connections between Leray-Schauder degree theory with Morse theory, as well as the LjusternikSchnirelman category with the subordinate nontrivial homology classes. In Section 4 some theorems due to Ambrosetti-Rabinowitz, which were proved using genus, indices or pseudo- indices, are proved using homology methods. Section 5 consists of some abstract critical point theorems, which will he applied to concrete problems in the following chapters. Section 6 deals with perturbation theory and, although relatively independent, is useful in the applications.
1. Topological Link A variety of critical point theorems, studied initially by Ambrosetti-Rabinowitz [AmRI), Benci-Rabinowitz [BeR1) and W. M. Ni [Nil), is based on the following topological idea: There exist two linking sets which are separated by a level set. In this section we clarify the concept of linking from the points of view of the homotopy and the homology, and emphasize that the homology link is more interesting in Morse theory. 'T'hroughout this section, Al stands for a C2-Finsler manifold, and f E C, (Al, RI).
Definition 1.1. Let D be a k-topological ball in M, and let S be a subset
in M. We say that OD and S homotopically link, if 8D n S = 0 and V(D) n S 34 0 for each cp E C(D, M) such that cOIOD = idI8D
84
Critical Point Theory
The reason for the terminology comes from
Theorem 1.1. Assume that OD and S homotopically link. If f E C(M, lR1) satisfies
f(x)>a f(x)
(1.1)
(1.2)
dxES dxEOD
then 7rk(fb, fa) 34 0, where b > Max{ if (x) I x E D).
Proof. Since D C Fb, the class (OD] which includes the map idIBD is trivial in 7rk_1(fb), where k = dim D. On the other hand, 8D C fa; but Max=ED f 0 1p(x) > a
V cp E C(D, M)
provided that W(D) n S 54 0. This means W(D) is not included in fa; hence (OD] is nontrivial in 7Tk_1(fa). We observe the exact sequence a*
Irk(fb,fa) _' Irk- 1(fa) where i : fa
i.
Irk- 1(fb) -'
fb is the inclusion, and 0 is the boundary operator. It
follows
(OD) E ker i. = Im e.. Consequently lrk(fb, fa) is nontrivial. Similarly, we obtain
Definition 1.2. Let D be a k-topological ball in Al, and let S be a subset
in M. We say that OD and S homologically link, if OD n S = 0 and In n S 0, for each singular k chain r with Or = OD where Irk is the support of r. In the same way, we prove
Theorem 1.1'. Assume that OD and S homologically link. If f E C(M,1R') satisfies (1.1) and (1.2), then Ilk(fb, fa) 34 0. There are many linking examples:
Example 1. Let ci be an open neighborhood of a point zo in a Banach space. Set z1 V Il, S = OD, and D = TO-z-1, the segment joining zo and z,. Then S and OD = {zo, z1 } homotopically link. Claim. By connectedness, S n 1 34 0, where f is any path-connecting zo and z1. Example 2. Let X be a Banach space. X = X1 ® X2 is a direct sum
decomposition where dim X1 < +oo. Set S = X2,
and D = B1 n X,
1. Topological Link
85
where B1 is the unit ball centered at 0. Then OD and S link. Claim. Obviously S fl 8D = 0. We only want to show (1.3)
d cp E C(D, X) and (v1aD = id 1aD = V(D) fl S = 0.
Define a projection P : X -. X1. It is equivalent to showing that 3 xo E D such that P o cp(xo) = 0. Define a deformation
F(t, x) = tP o cp(x) + (1 - t)x V (t, x) E [0,1) x D. Since 0 V 8D = F([0,1) x 8D), deg(F(1, ), D, 0) = deg(F(0, ), D, 0) = 1. The equation P o V(x) = 0 is solvable, using Brouwer degree theory.
Example 3. Let X = X1 ® X2 be defined in Example 2. Let e E X2,
flefl = 1, and let R1, R2, p > 0 with p < R1. Set S = X2 n M,,, and D = {x + se I x E X1 f1 BR2, s E 10, R1]). Then OD and S homotopically link.
Claim. Obviously s f1 8D = 0. It remains to prove (1.3). Again we use the same projection P in Example 2 and define a new deformation:
F(t, x + se) = [(1 - t)x + tP o w(x + se)]+
[(1 - t)s + tII(I - P) o cp(x + sel - p]e. It is easily seen that F(0, x + se) = x + (s - p)e,
F(1,x+se.) = Pocp(x+se)+ [11(I- P)ocp(x+se)II -p]e, and
F(t, x + se) = x + (s - p)e 9A 0, d t E 10, 11, Vx + se E 8D.
Thus (1.3) is equivalent to finding Yo E D satisfying I' o'p(Yo) = 0, II'P(Yo)II = P;
i.e., F(1, yo) = 0. Since
deg(F(1, ), D, 0) = deg(F(0, ), D, 0) =deg (idX1, Xl n BR2, 0) deg (id - p, (0, RI), 0) = 1.
86
Critical Point Theory
Consequently, (1.3) is solvable, i.e., S and c3D homotopically link.
Actually the above three examples are of homological linking as well. We have
Theorem 1.2. Suppose that the boundary 8D of the k-topological ball D and S homotopically link. Assume that (1) S n D = single point, (2) S is a path-connected orientable submanifold with codimension k,
(3) there exists a tubular neighborhood N of S such that N n D is homeomorphic to D. Then 8D and S homologically link.
Proof. (N, 8N) can be regarded as an orientable sphere bundle over the base space S with fibre (Dk, S"), ), i.e.,
(Dk,SI-I)
(N,BN) S
According to Thom's isomorphism theorem (see Chapter 1, Section 7),
H.(N,ON)
H.(S) 0 H. (Dk Sk-l)
Since S is path-connected, we have Ho(S) -' G. It follows that
Hk(N,BN) = C. Let ]r] be the generator of Hk(N, ON). We choose a singular chain r E ]T), such that Or = ON n D, according to (3). Since N is a tubular neighborhood of S, we have
H.(M,M\S)
II.(N,N\S)
provided by the excision property. Let us apply a deformation retract along the exponential map. It follows that
H. (N, N\S)
If. (N, 8N).
From assumptions (1) and (3), we see that 8D and Or are homologous in H.(Ilf, M\S). Therefore (DJ = (r] is nontrivial in Ilk(M, M\S), where )D] is the relative singular homology class containing D. The nontriviality of )D] implies that 8D and S link homologically. Thus the above three examples are also homological links. Remark 1.1. In order to prove the homological links for the above three examples, only the Kiinneth formula is needed.
1. Topological Link
87
Theorem 1.3. (Minimax Principle). Suppose that 7 is a family of subsets of M. Set
c = inf sup f (x). FEf =EF
Assume that (1) c is finite, (2) f satisfies the (PS)c condition, and (3) 3 £o > 0, such that the family F is invariant under the family of maps 4i'o = {w E C(M, M) id, API fc-,o = id fc-e, }. Then c is a critical value of f.
Proof. We prove it by contradiction. If c is not a critical value, then there exists e E (0,eo) such that K fl f'1 [c - e,c + eJ = 0. One finds Fo E Jr with Fo C f,,+,. There exists 77 E C(M, M) satisfying rl - id, nI fc_e, = id fc_ fo and rl(fc+,) C fc theorem. Set Fl = +l(Fo); we have
according to the first deformation
F1EF and F1Cff_,. This contradicts the definition of c.
Theorem 1.4. Let a < b be regular values of f. Set c = inf sup f (x) ZEa =EJZI
with a E irk (fb, fo) nontrivial
(and
c' = inf sup f (x)
with a E Hk (fb, f0) nontrivial).
Assume that c > a (and c' > a resp.), and that f satisfies the (PS), (and (PS),.) condition. Then c (and c' resp.) is a critical value off . Moreover, we have c' < c.
Proof. It suffices for us to verify that the families F = {IZI I Z E a}
(and f' = {IrI I T E a}) are invariant with respect to 0,o, where Co E (0, c - a) (and Co E (0, c' - a) resp.). This is just the homotopy invariance of the homotopy (and homology) classes. The inequality c' < c follows
from F C F. In particular, let a be the class in 7rk(fb, f,) (or Hk(fb, f,)) with 8.a = [OD] in Theorem 1.2. Then
c = inf sup f o V(x) eE1'xED
where r = {rp E C(D, M) I W18D = idSD} (and (1.5)
c' = inf t sup f (x) I r is a k rel. singular chain in (fb, fQ) with Or = 8D } ).
I
sE I * I
JJJ
Critical Point Theory
88
The following assumption: f(x) > d> a
(1.6)
V X E S,
guarantees that c > d > a (and c' > d > a), under which c and c' are critical values.
Remark 1.2. The separation conditions (1.2) and (1.6) may be weakened to (1.2) and
f(x)>a
(1.7)
dxES.
Claim. We shall prove that if c = a then & 54 0. Otherwise we assume & = 0. Since OD is compact and OD n S = 0, 3 two open neighborhoods Ul D Ul of 8D such that U2 n S = 0. According to the first U2 deformation theorem, there exist E > 0 and a homeomorphism r) such that
n : (-f)- +E\U2 - (-f)
-E
and
n Jul = idu1,
77: f'-''\ U2 rl Jul = idu1,
f`+e
and
where f{x E M I T (x) > a}. Therefore 77(S)Cf`+E
Let S1 = rl(S); then we have
S1 n 8D =0 and S1 n W(D) = n (S n rl-1 W(D))
0
v W E r,
i.e., S1 and OD link (homotopically). Since f (x) > c + E V x E S1, we conclude that
c = inf sup f o V (x) > inf f (x) > c + e. WErxED
=ESI
This is a contradiction. The above minimax principle (1.4) under condition (1.6) was obtained by Benci-Rabinowitz [BeRI[, and Ni [Nil]. The condition (1.6) was weakened
to condition (1.1) by Chang (Cha8), and to condition (1.7) for k = 1 by G. J. Qi [Qil). Actually, under a more general linking definition (8D may
89
1. Topological Link
not be compact), the above conclusion remains true, cf. Y. 11. Du [Dull and N. Ghoussoub (Ghol].
Theorem 1.5. Assume that a E Hk(fb, f,) is nontrivial, and
c' = inf sup f (x). ZEa XEIZI
Suppose that f satisfies the (PS)c. condition and that K. is isolated. Then there exists xo E K. such that Ck(f,xo) 34 0. Proof. It suffices to prove that 0 using Theorem 4.2 in Chapter 1. Let y = 8Z be the boundary `d e > 0 3 Z E a with support IZI C
of Z; then Iyi C f, C fc.. We shall prove that [yJ E Hk_i (fc
is
nontrivial. Indeed, if Iy) is trivial, then there exists a singular k-chain r and (8r] = (y]. Due to the second deformation such that IrI C theorem, 3 ri E C(M, M), satisfying rl - id and ,i(fc C f,,- -, for which is in some c > 0. Let rl = p(r), we have (riI = (r] and Ir1I c contradiction with the definition of c'. We observe the exact sequence
-
80
Hk
Ilk-1
where i : is the inclusion. Since i.((y]) = 0, i.e., (y] E ker i. = Im (9., we conclude Hk
(fc +,,, fc-
9' 0.
Corollary 1.1. Under the assumptions of Theorem 1.5, if, in addition, all critical points in K.. are assumed to be nondegenerate, then there exists xo E Kc. such that ind(f, xo) = k.
Theorem 1.5 can also be extended to C-invariant functions. Let C be a compact Lie group, and let M be a C2-Hilbert Riemannian C-manifold. Suppose that f E C' (M, R 1) is C invariant, and that a, b are regular values of f . Assume that (ZJ E Hak (fb, f,) is nontrivial. One defines
c = inf sup f (x). ZEIZJ rEIZI
By the same proof, we have
Corollary 1.2. Suppose that / satisfies (PS),, and that K. consists of isolated critical orbits. Then there exists a critical orbit 0 C Kc such that Crk,. (g, O) 54 0.
Critical Point Theory
90
As a consequence, we get an estimate of the Morse index for certain critical
orbits in K. Corollary 1.3. Under the assumptions of Corollary 1.2, we assume further that f E CZ, d2f(X) is a Fredholm operator V x E M, and that Kc fl Fixc = 0. If H' (BG,, 0-) = 0 for * 54 0, d x V Fixc, where 0is the orientation bundle of C-(Gx), then there is a critical orbit 0 C Kc satisfying
m
where m = ind(f, 0) and n = dim l;0(0).
Proof. According to Theorem 7.8 in Chapter I, we consider the approximation function g, which has only nondegenerate critical orbits. Let
Kc = {01,02.....01}, and let mi = ind (f,0,), ni = diml;°(0i). Then g has critical orbits 0,, with ind(g, 0,,) = q;,, E [mi, mi + n,],
s=1,2,...,.71.
One concludes that there exists at least a pair (i, s) such that Cc' (g, O,,) # 0. Indeed
t
i,
®® CC (g, Die) = HC (gc+E, ge-e) = HC (fe+c, fc-E) i=1 8=1
so there exists a pair (i, s) such that Cc(g, O,,) j4 0k-°t° According to Theorem 7.5 in Chapter I, Ckc(g, Oia) = HC (Uia, B (Oia)) Ilk-9is(BCxo,0-) where x0 E 0,,. By the assumption that H'(BC,,9-) _ 0 for * 34 0 V x V Fixc; it follows k = qi,. Therefore
rn,
Let us return to Example 1, in this case c = c' = inf sup f(x), tEr" =E[10,11
where F = It E C([0,11, X) I f(i) = xi, i = 0, 1}. According to Theorem 1.5, there exists a point xo E Kc such that (1.9)
C1 (f, xo)
0.
Due to the geometric meaning, we call a point xo satisfying (1.9) a mountain
pass point. The following theorem provides more precise information on mountain pass points.
91
1. Topological Link
Theorem 1.6. Suppose that f E C2(M, RI) and that xo is a mountain pass point. Assume that d2 f (xo) is a Fredholm operator and that (1.10)
dim ker (d2 f (xo)) = 1
if
O E a (d2 f (xo))
then
rank Cq(f,xo) = 6p1
Proof. Let k = ind (f, xo). 1. If xo is nondegenerate, then CC(f,xo) = 6gkG. From (1.9), it follows k = 1. and then CC(f,xo) = 69 1G. 2. If xo is degenerate, then we have Cq (f, xo) = Cq_k (f, e)
d q,
where f = f IN, and N is the characteristic manifold of f, according to the shifting theorem. From (1.9), it follows that k < 1. Case a. If k = 1, then C0(1, 9) 34 0, which is equivalent to 0 being an isolated local minimum of j; then Cq (f, 9) = 6goG.
Thus Cq (f,xo) = 69 1G.
Case b. If k = 0, then Cq(f,xo) =Cq
(f,e),
dq.
By assumption (1.10), the function f is defined on a 1-manifold. And the only possibility for C1(f , 0) 36 0 is the case in which xo is a local maximum
of f. Thus Cq (f, xo) = Cq (f, 0) = 6,, G.
The proof is completed. Remark 1.3. The condition (1.10) is essential. For example, the Monkey
saddle is a mountain pass point, with critical groups Cq = 6g1G ®G, VgEH. Remark 1.4. The idea of the homological link was introduced by J. Q. Liu [Liull, from which he proved Theorem 1.5. C. Viterbo independently gave an estimate of the Morse indices for critical points defined by equivariant homology classes, i.e., Corollary 1.3, cf. [Vit1J. Theorem 1.2 is more or less related to V. Benci [Ben2J and X. F. Yang [Yan1J.
Critical Point Theory
92
2. Morse Indices of Minimax Critical Points Minimax critical points, by definition, are Independent of Morse theory. However, the Morse indices of these critical points play a role as a connection. We shall pay more attention to the estimate of the Morse indices of critical points as determined by some standard minimax methods. All notation appearing in Section 1 will be used without change.
2.1. Link Theorem 2.1. Suppose that f E C2(m, Il81) satisfies the (PS), condition, where c > a is defined in (1.4). Assume that KK = {XI, X2.... , xt} with Morse indices {k1, k2, ... , kt}, and that 0 is not an accumulate point of a(& f (xi)) whenever 0 E o(d2 f (xi)), i = 1, 2,... , t. Then (2.1)
Min {k, I
i = 1, 2, ... J) < k
Proof. We prove (2.1) by contradiction.
Suppose that k < Min 1k. I1
6-ball in Hi and if, where Hi = ker(d2 f (x,)) respectively, and r, b > 0, such that in each Ni, the function f is written in the form f (X) = c +
11Y+ III
-
11Y_112
+
1, (yo),
where x = 4>,(y), y = y+ + y- + yo, yl = (y+, y-) E Br and yo E B6, i = 1.2,... J. One may assume that f, I ao < r2. `d e > 0, 3 Z E a, where a E lrk(fb, fh), such 2that IZI C Using the deformation theorem, we have Z1 E a satisfying IZ1I C f,-,, U UL=1 Ni for some el > 0. One may assume that El < Zr2. In order to simplify the notation, we omit the subscripts i, and the transformations 4i,, and regard Ni as BT x B6, if there is no confusion. Let g(yl) = 11y+ III 11y_ 112; then d go E B, *5, the sections ff No=vo = g_ j(LO), and fc-f, Iyo=yo = ji9o>. Let Br C B; be the r-ball {y1 = (y+,y-) E B,i I y+ = B}. We shall prove that there exists Z2 E a satisfying
-
t
t
IZ2I C
U Ni u U'bi (Q:)
-
where Q, = Bki X {B+} x (Bb fl II,), and Bpi is the k,-dimensional r-ball,
i = 1,2,...,t. Indeed, let us denote = -e1
-
f (yo),
b
2
Max{-a,0} +
2
al
,
2. Morse Indices of Minimax Critical Points
if 0<s
0 cp(s, b) =
where ctg a =
93
{ (s - b)ctg a if b < a < 2-°
2(r2 + a)/(
-
''z °
Max -a, 0 ). We define deforma-
tions as follows: if y E fa_r1 \G 771 (t, y) =
Y y-
+ yo +
[1
+ t (nw(Iy-I, b)
- 1) ] y+
if
y E G,
where G = {y E B, x B6 I Iy+I ? W(ly- 1, b)), and
if
y E fc_e1 \ Vi=1 Ni
11++yo+(t r - 1y+ +(1-t)Iy-I) n2 (t, y) =
if y E (B,1 x B6) \G Y+ + y0 + t
0fly+J,bT
if
y E C, with Iy+I <-
r 2
where t/i(s, b) = tg a a + b, t E [0, 1). The composition of rh and 171 yields Z2 = (7ho7)1)(1, Z1), which satisfies
IZ2I C E := Cfc_,1 \ U -1
Ni/
U U c(Q) i=l
This means that there exists a continuous map a : Dk -. E such that Z2 = a(Dk). One may modify a to be C1 on a-l$i(Q1) by the partition of unity because Qi is convex, i = 1, 2,... , t. Since we have assumed that k < minki, the projection iriIZ2I of IZ2I onto Bri cannot cover Bkl. There must be at least one point pi E I3ki
missing. Again, we project iriIZ2I onto BBri from pi, 1 < i < t. Then IZ2I is deformed into fc_, UU{=1 4i(BBki x {B+} x (Bo Hi)), which gives Z3 E a such that
sup f (x) < C. xEIZ31
This is a contradiction. By the same proof, we have
Theorem 2.1'. Let M be a Hilbert-Riemannian manifold. Suppose that f E C2(M, Dtl) satisfies the (PS) condition in f -1 [a, b], and that
Critical Point Theory
94
Kn f-I[a,b] = {x1,x2,... ,xt}, with ki = ind (f,xi) and a< f(xi) < b,
1 < i < t. If 0 is not an accumulate point of a(d2 f (xi)) whenever 0 E a(d2 f (xi)), 1 < i < t. Then (2.2)
7rk(fb,fo)=0 for k < Min {kiIi=1,2,...,t}.
There is a dual way of defining critical values for Example 2.
Instead of the set family F = {V(D) I ip E C(D,X) I
[F C X I compact, d a E C(F, X 1) with a I FOOD
idFn&D,
we have 0 E a(F)} .
Since.F C J' (using Brouwer degree theory), F is nonempty. Let us define (2.3)
e = inf sup f (x) FE.* zEF
then c < c. We conclude c > d if (1.6) holds. In fact, by taking a equal to the projection onto X1, Fn X2 # 0 V F E .F, which implies snp=EF AX) > inf=Ex2 f(x) > d.
If, further, we assume (1.2) and that f E CI (X, IRI) satisfies the (PS)e condition, then c is again a critical value.
Claim. By choosing co E (0, d - a), it is not difficult to verify that the family F is invariant under the class of maps -tEo (cf. Theorem 1.3). The conclusion follows from Theorem 1.3. In order to obtain a counterpart of Theorem 2.1 for c, we need
Lemma 2.1. Let A C IR" be a compact set, and let a E C(A,Rk) with k > n, such that 9
a(A). Then a can be extended continuously to
a E C(IR",IRk\{9}).
Proof. Since A is compact and 0 a(A), there exists E > 0 such that a(A)nB,(0) = 0. According to'I'ietze's theorem, we have a continuous extension & of a, o : 1R"
IRk. Moreover, we may assume o E CI (IR"\A, II8k).
By the assumption k > n, it follows from Sard's theorem that &(IR"\A) is measure zero in IRk. Therefore Be(9) is not included in o(lR"), i.e., there exists x0 E Bf(0)\&(1R"). We project o(IR") n B,(0) onto OBe(0) from xo. This gives the extension &. Theorem 2.2. Let X be a Hilbert space with the direct sum decomposition X = X1 ®X2, dirnX1 < +oo. Assume that f E C2(X,IRI) satisfies the (PS)e condition, where c is defined in (2.3), and that (1.2) and (1.4) hold. Suppose that K4 = {x1, x2, ... , xt} with Morse indices (k1, k2, ... , kt) and
2. Morse Indices of Minimax Critical Points
95
nullities {n1,n2i... ,nt} respectively, and that d&f(xi), i = 1,2,... ,t, are Fredholm operators. Then Max {ki + ni I 1 < i < t} > dim X1.
Proof. V e > 0, 3 F E . such that F C ft+c. Applying the first deformation theorem, the splitting lemma, and the argument in Theorem 2.1, there is a deformation v' satisfying tl 18D = Id8D and t
t7(F) C
l
(fe_\JNi)
t UU
(Qi) .
On the one hand, we have q(F) E .f. On the other hand, let t
t
1\`
A= q(F)\ UNtJu U.i(BBrcix{9+}x(B6nHi)). i=1
i=1
0
We have A C fi, therefore A that a IBDnA = idoDnA Suppose that
.F'. Consequently, 3 or : A -' X1\{9} such
dimX1> Max{ki+ni,1
0i(OBki x {9+} x (B6nHi)) has a continuous extension
o : U 0i (Qi) - X1\{9} i=1
according to Lemma 2.1. Let us define
o(x)
or (X)
{ &(x)
VxEA d x E Y7(F)\A.
Then o : sl(F) -+ X1\{9} is continuous. If we choose r > 0 so small such that f 1,7(F)\A > a, 1 < i < t, then
I.7(F)n8D = & I An8D = a I AnOD = IdAnOD
= id,I(F)nBD,
Critical Point Theory
96
which implies that rl(F) V 7. This is a contradiction.
2.2. Genus and Cogenus Let X be a Banach space. Let E be the family of compact symmetric subsets of X. Two integer valued functions ry+, -y: E --+ H U {+oo} are defined as follows:
ry+(A) = sup (n E N 3 W: S"-1
A odd and continuous)
and
-y(A) =inf{nENj3v: A.S"-1 odd and continuous}, in which, if no such cp exists, then we define -y' (A) = +oo. -t+ (A) is called the genus of A and -y- (A) is called the cogenus of A. We have the following properties: (1) V (p: X X odd and continuous, (2.4)
7}(A) 51'}(V(A))
(2) 7+(A)
7-(A)
Claim. If not, 3 A E E such that k = -y+(A) > -f-(A) = m, then there exist continuous odd maps tp1 and 02 satisfying
According to the Borsuk-Ulam theorem, it is impossible. Let us define two classes of families of subsets in E:
.F ={AEEI-y (A)>k}
VkEN.
Set
ck = inf sup f (x)
(2.5)
V k E H.
AEFF xEA
If c E (ck * = + or -, k = 0, 1, 2, ... ) is finite, and if f E C1(X, lFY1) is odd and satisfies (PS), then c is a critical value of f, according to Theorem I
1.3.
The following relations are obvious:
ck
k=0,1,2,...,
2. Morse Indices of Minimax Critical Points Ck < Ck+1
97
* = t, k = 0, 1, 2,... .
As a counterpart of the theorems in the previous subsection, we have
Theorem 2.3. Let X be a Hilbert space, and let c be one of the * = ±, k = 0,1, 2, ... , defined in (2.5). Assume that f E C2(X, R') satisfies the (PS)c condition. Suppose that KK = {±x1 i ±x2, ... , ±xt} with M o r s e indices {k1, k2, ... , kt} and nullities {nl, ns, ... , nt} respectively, and that 0 V K,. If d2 f (xi), i = 1, 2,... , t are Ftedholm operators, ck,
then
Max{ki+n1Iik-1
as * = -,
Min {ki I1
as*=+.
and
Proof. We prove them by contradiction. Assume that
k>Max{ki+niI1
as*=-,
k<Min{kiI1
as*=+.
and
V e > 0, 3 Ao E .Fk such that Ao C f,,+,. As in the proof of Theorem 2.2, we have a deformation 17 satisfying
t n (Ao) C
f.(_ ,\
U (Ni u(-NN)) U U ($i (Qi)) U (-4i (Qi)) i-1
i=1
where Qi = Bri x (0+) x (B6 n Hi) and r > 0 is small enough such that Ni n (-N,) = 0, 1 < i < t. Thus 77(A0) C .Fk and let A = (77(Ao)\(j(NiU(_Ni))) i=1
t
u U (f-ti (BBri x (0+) x (B6 n Hi))) C
f0
r.
i=1
For -. -y-(A) < k - 1, i.e., there exists an odd continuous map cp : A - Sk'2. By the Dugundji Theorem, we may extend
Critical Point Theory
98 u,i_,f0:(8/3.i4i
(B+)x(BsnHJ
to cpl : U;_1 Rki+ni -. Bk-1 continuously,
where Bk-1 is the unit ball in k - 1-dimensional Euclidean space. Let cP(x) = 2 (WI (x) - W1(-x))
Then cp is an odd extension of cp Jut_, (± , (BBk; x {9+} x (B6 n Ili))) with cp : U;_I Rki+ni Bk-'. By the use of the same argument as in 0
Lemma 2.1, we have a pair of points fpi E Bk-I\O(Rki+ni) Let r} be the projection onto 8Bk-1 from ±p,; then r Bk-1 \{fp, } Sk-2 is continuous, and r; (x) _ -r;+(-x), i = 1, 2,... l. Define :
x -{l( V(x) r}cp(x)
xEA
x E (±N,) n q (Ao).
Then cp : 17(Ao) -+ Sk_2 is odd and continuous. Therefore ry-(il(Ao)) <
k - 1. This is impossible.
For * = +. We have -y+(A) < k - I and ry+(n(Ao)) > k. From the rl(Ao) odd and continuous. Denoted by latter, there exists cpo : Sk-I P; , * = +, -, 0, the orthogonal projections onto the positive, negative and 0-subspaces of d2 f (x,) respectively, we conclude that P; c,o(Sk-1) cannot
cover Bk', because k - I < Min {k,
I
I < i < P}, f = 1,2,... f. Thus,
3 pi E Bk' such that { ±p, } x (Pi+ + P°)X n Vo(Sk-1) = 0, Let ri+ : Br' \{pi } -. 8Bk' be the projection onto the sphere from pi, let T,-(z) -7-,+(z), and let
ri(x)
rf (Pr x) + (P+ + P°) x if x E ±(BI'\{pi}) x ((P± + P°) X) x
ifx V ±B,, x ((P,++I;°)X),
for i = 1, 2,... , f. The functions Ti : X\Ui_1({fpi} x (Pi+ + P°)X) -, X, i = 1,2,... ,f are odd and continuous. 't'herefore rr o . 0 r1 0 cpo : Sk-1 -+ A is odd and continuous which implies that -y+(A) > k. Again it is impossible. .
Remark 2.1. The same conclusion holds if f E C' (S,118), where S is the unit sphere of a Hilbert space, and if .Fk are modified to be families of subsets on S. Remark 2.2. Theorems 2.1 and 2.2 were obtained by Lazer-Solimini (LaSI], and Theorem 2.1' improves a result due to J. T. Schwartz (ScJI]. Lemma 2.1 can be found in L. Nirenberg [Nirl]. Theorem 2.3, under the assumption that f satisfies the Palais-Smale condition (not only (PS),), was given by A. Bahri [Bahl] for genus, and A. Bahri, P. L. Lions (BaLlJ for cogenus. For some extensions, see for instance, Coffman (Cofl,2], S. Solimini (Sol2J and N. Ghoussoub [Ghol].
3. Connections with Other Theories
99
3. Connections with Other Theories 3.1. Degree Theory Let M and N be oriented n-dimensional manifolds without boundary and let
T:M-.N
be a smooth map. If M is compact and N is connected, then the Brouwer degree deg(T) is well-defined INir 11.
First consider an open set U C R', and a smooth vector field
V:U-+1Qn with isolated zero at the point p E U. The function
V(s) = V(x) )It
maps a small sphere centered at p into the unit sphere Sn-1. The degree of V is called the index of V at the zero p, denoted by the index (V, p). For a smooth vector field V on an arbitrary manifold M (finite dimensional) with isolated zero at the point p E M, the index of V at the zero p is defined as equal to the index of the corresponding vector field dg-1 oV og
at zero 9-'(p), where g : U -* M is a parametrization of a neighborhood of p in M. The celebrated Poincare-Hopf Theorem studies the relationship between the indices of zeros of a smooth vector field on M and the Euler characteristic of the manifold M. Namely,
Theorem 3.1. (Poincare-Hopf). If M' is a compact manifold with boundary 8M, and if V is a smooth vector field on M which points outward
at all boundary points and has only isolated zeros, then the sum of the indices at the zeros pi, i = 1, ... , n, of such a vector field is equal to the Euler characteristic of M, i.e., n
m
E index (V,p;) = X(M) _° E(-1)9 rank IIQ(M). i=1
9=1
We shall extend this theorem to bounded domains on an infinite dimensional Hilbert space for a special case of vector fields, the gradient vector
fields. Let H be a real Hilbert space, and let D be a bounded open set in H. For a vector field V : b H, with 9 V V(81)) and V = id - V compact, the Leray-Schauder degree deg(V, D, 8)
Critical Point Theory
100
is well-defined, cf. (Lio1J. Suppose that p is an isolated zero of V; then there exists a ball B,(po) with radius e > 0 such that V has no zeros in BE (p) other than p. Hence it is possible to define index (V, p) = deg (V, B, (p), 0)
which is independent of c for e > 0 sufficiently small in view of the excision property of the Leray-Schauder degree. The number index(V, p) is called
the index of V at the point p. In case dim H < +oo, the definition of index(V, p) coincides with the previous one. We start with a local result in which the connection between the index of the gradient vector field of a function f at its isolated critical point and the critical groups of that point is studied; namely:
Theorem 3.2. Let H be a real 11ilbert space, and let f E C2(1f, RI) be a function satisfying the (PS) condition. Assume that
df (x) = x - T(x), where T is a compact mapping, and that po is an isolated critical point of f. Then we have 00
(3.1)
ind (df,po) = E(-1)Q rank Cq (f,po). q=o
Proof. 1. First, we assume that po is nondegenerate. Since T is compact, we see that the Hessian d2f(po) = id - dT(po) has only finite index j. By definition, and by Leray's formula
index (df,po) = (-1)j. In view of Theorem 4.1 in Chapter 1, we have Cq (f, po) = bq,jG.
Thus,
index (df,po) =
J(- 1)q rank Cq(f,po) 00
q=o
is proved in this special case. 2. For an isolated degenerate po we may assume for simplicity that po = 0 and f (po) = 0. Let (W, W_) be a Gromoll-Meyer pair constructed
3. Connections with Other Theories
101
in Chapter 1, (5.11) and (5.12), and let 6 > 0 be sufficiently small such that 0
B6 c W n f -11-2, J I, where -y is the real number appearing in Chapter I (5.11).
We shall define a function j satisfying the (PS) condition such that (1)
If(x) - f(x)I < , d x E H.
(2) d f (x) = df (x) for x in a neighborhood of W. (3) In W, f has only nondegenerate critical points {pj )1 `, finite in number, contained in B6. Once the function f is constructed, we obtain immediately
W_ = f_,. n W c f_ j,y n W C f_ j n W C
fjnWcf4,nwcf,nw=W. However, there are strong deformation retracts:
finW and f_jnW-9 f_,nW
f,nW
provided by the Gromoll-Meyer property. We have (3.2)
H.(W,W_)=H.(ft,nw,/_,nw)
due to the exactness of the homological group sequence. Thus, index (df, 0) = deg(df, W, 0) = deg(df, W, 0)
(by (2))
M
_
1:
index (df,pj)
(by (3))
J=1
M 00
= E E(-1)q rank Cq (f, pj)
(by 1).
j=1 q=0
Applying Theorem 4.3 and Remark 4.1 of Chapter I to the function f on W, we have m 00
00
E E(-1)q rank Cq (f, pi) _ E(-1)q rank Hq (f,, n w, j=1 q=0
q=0 00
_ E(-1)q rank Hq(W,W_) q=0
0
= 1: (-1)q rank Cq(f, 0). q=0
n w)
Critical Point Theory
102
This is due to the fact that the negative gradient flow of f directs inward on 8W\W_, and hence also on 8W\(W fl f-1(-37)). 3. Finally, we shall construct a function f, satisfying the (PS) condition as well as conditions (1)-(3). We define f (x) = f (x) + p(Uxll) (xo, X), where p E C2 (R ,1181) is a function satisfying
p(t) =
(1 0
06,
with 0 < p < 1 and Ip (t)I < 6, xo E II is determined later. Let Q = inf{Ildf (x) II Ix E B6\B612};
then 6 > 0. We choose xo E II such that -1 0
Then we have
If(x)-f(x)I <3 VxEI! IIdf(x)II
g V X E B6\B612
df(x)=df(x) tlx¢B6. For smaller IIxoII, df is a k-set contraction mapping vector field with k < 1 (cf. Lloyd [Hol]). Therefore deg(df,W,©) is well-defined. The (PS) condition for the function f is verified directly. On account of the SardSmale theorem, a suitable xo can be chosen such that f is nondegenerate. The proof is complete. Corollary 3.1. Under the assumptions of Theorem 1.6, if po is a mountain pass point, and if the smallest eigenvalue Al of d1 f (po) is simple when-
ever Al = 0, then Al <0, and index (df,po) = -1.
Proof. If a1 > 0, then po must be a local minimum, which implies C, (f, po) = 0. This contradicts the assumption that po is a mountain pass point. Therefore Al < 0.
3. Connection., with Other Theories
103
In the case where Al = 0, we have assumed dim ker d2 f (xo) = 1. It follows from Theorem 1.6 that Cq(f,po) = bg1G. In the case where Al < 0, the Morse index ind(f,po) = k > 0. From the mountain pass point assumption, it must be k < 1. Consequently, k = 1. Again, we apply Theorem 1.6, Cq(f, po) = bg1G In both cases our conclusion follows from Theorem 3.2. We generalize Theorem 3.2 as follows:
Theorem 3.3. Under the assumption in Theorem 3.2, suppose that W is a bounded domain in H on which f is bounded. Assume that
(1) W_ °_(xEOW(ri(t,x)VWdt>0}=Wflf`(a)forsome a, where ri(t, x) is the negative gradient flow off emanating from x;
(2) -dflow\w_ directs inward. Then we have
(3.3)
Proof.
deg(df, W, 0) = X(W, W-).
Due to assumptions (1) and (2), 0 V df(OW), the Leray-
Schauder degree deg(df, W, 0) is well-defined.
If f is nondegenerate on W, then the critical set K consists of finitely many isolated point {p, } i ` since f is bounded on W, and assumption (2), as well as the (PS) condition holds. According to Theorem 3.2, we have m
deg(df, W, 0) _ F, index (df, pj) i=1 o0
in
_
>(-1)g rank Cq (f,pj) l=1 g=o 00
_ >(-1)g rank Hq(W,W_). g=0
The last equality follows from assumption (2), Theorem 4.3 of Chapter I, and the fact that f(ps) > a, which is a consequence of assumption (1). If f is degenerate, we perturb it as in Theorem 3.2. Since the critical 0 set K is compact in W, we construct a C2- function p(x), satisfying: {
0
x E f16,
1
x 0 f126,
where b > 0 such that dist(K,OW) _> 26, K is the critical set of f in W, and 126 = {x E W I dist (x, 8W) < b}.
104
Critical Point Theory
We may assume that IP(x)I < 1 and
Ip'(x)[ < M2 < +oo.
Let
M, = sup{ IIxII I x E IV 1,
b = inf {f(x) I T E W\116}
and
p = min 1-0126 inf Ildf(x)II, I } J
Then, by our assumptions, b > a and 8 > 0. One defines AX) = f (x) + P(x) (xo, x)
for suitable xo E II, with
0
Ali
'3'3M1M21
By the Sard-Smale Theorem, x0 can be chosen such that f is nondegenerate. Thus, Ildf(x)ll > Ildf(x)II - IIxoII - I111(x)II IIxII
.
Ilxoll
1
> Q - 3/3 - All M2 Ilxoll
> 3Q>0
VxEf126.
Now the function f satisfies the (PS) condition and assumptions (1) and (2). Indeed, the (PS) condition is easily verified by the above estimate,
assumption (2) is trivial, and assumption (1) is verified by the following inequality:
f(x) ? f(x) - 11X0 11 IIxII ? f(x) -
6Alia
.M>a
V x E 1.V\ 16.
Since
deg(df,1V, 0) = deg (d f, W, 0)
(the RIIS is the generalized Leray-Schauder degree for a k-set contraction mapping vector field with k < 1), the proof is completed. Remark 3.1. Theorem 3.2 was given by E. Rothe [Rot II tinder stronger assumptions, and Corollary 3.1 was obtained by Ilofer 11Iof4] and Tian [Tiall by a combination of the Poincare-Nopf Theorem with the splitting theorem.
3. Connections with Other Theories
105
3.2. Ljusternik-Schnirelman Theory Let X be a topological space. The category of a closed subset A of X is defined by Catx(A) = inf{m E N U {+oo} 13 contractible closed subsets F1,... , F,,, in X, such that A C Um 1 F; } .
In the following let M be a C1-Finsler manifold. We write simply Cat (M) = CatAf(M).
Ljusternik-Schnirelman Theorem. Suppose that / E C' (M, R') is a function bounded from below, satisfying the (PS) condition. Then f has a least Cat(M) critical points. The topological invariant Cat(M) can be estimated by other topological invariants, for instance, the cup length of M. Namely, we have
Cat(M) > CL(M) + 1. For a proof and a more general result, see Corollary 3.5 below. Now we extend this relation to estimate the number of critical points.
It is known that for a nontrivial class jr] E H. (fb, f,), a < b, one can determine a critical value (3.4)
c = rmf l supI f (x) EI-XEII
if c E (a, b) and if f satisfies the (PS)c condition. It is natural to ask, if we have two homology classes [rl1, [r21 E H. (fb, fa),
both nontrivial, and if c1, c2 are defined in the same way as in (3.4), are there two distinct critical points? Generally speaking, no, but we have
Lemma 3.1. Suppose that f E Cl (M, liF' ), has regular values a < b. Let [rut, 1r2J E H.(fb, f,) be nontrivial classes. Suppose that 3 w E H'(fb), with dim w > 0, such that In J = [r21n w. Set (3.5)
cf = inf sup f (x), i = 1, 2. r,EIr.) xEIr,I
Assume that f satisfies (PS)1,;, i = 1, 2, and that 3 a neighborhood N of Kc2 and a singular cochain cw E w such that supp w n N = 0. Then cl < c2 are two distinct critical values.
Proof. By definition cl < c2. It remains to prove cl < c2.
Critical Point Theory
106
dE > 0, 3 a singular relative closed chain r2 E Ire] such that Ir21 C f'2+1 We choose a neighborhood N' of K121 such that N' C N C N, and subdivide r2 into r2' + rz , such that Irsl C N and Ire"I C fc2+F\N'. According to the assumption, supp ro n N = 0, and we have
r1 =r2(1iw=rZ fl w, which implies Ir1I C
ff2+1\N'.
Since a and b are assumed to be regular, then a < cl < c2 < b. According to the first deformation lemma, there exist constants 0 < e < E < Min {b - c2, c1 - a) and 77 E C([0, 1] x M, M) satisfying rl (l, fc2+e\N') C fc2-e, and
rl(1, ) - id
in
(fb, f.).
Therefore
77 (1, 17-11) c fC2
However, 77(1, 7-1) E (7-11, and it follows
Cl
We emphasize that the positive lower bound of the difference C2-cl depend
on the behavior of df near K'2.
Corollary 3.2. Suppose that there are positive constants d, E, and b satisfying (3.6)
II df (x)II > d d x E fc2+e\
(h2-r U (Kc2)6)
and (3.7)
E < Min {26d2, 6d} .
where (K,)6 = (x E M dist (x, Ke) < 6). Then Cl < C2 - E/3.
the firs' Let -
The proof is the same as above. The only thing changed is that formation theorem is replaced by Theorem 3.4 from Chapter I. ecall the notion Ire) < (r2], cf. Definition 1.1, Chapter I.
Corollary 3.3. Suppose that f E Cl (Al, l)ql) and that a < b are regular values. Assume that [rl] < (r2] are nontrivial classes in H.(fb, fa), and
3. Connections with Other Theories
107
that (PS),,, holds, where ci is defined in (3.5), i = 1, 2. If #Kc2 < oo, then c1 < c2 are two distinct critical values.
Proof. We follow the proof of Lemma 3.1. Since #Kr2 < oo, there exist two contractible neighborhoods of Kr2, N' C N' C N. We write r2 = r2' + rz where [r2'1 C N and Irz [ C fc2+,\N'. Let w E H'(fb) be the cohomology class satisfying
dimw>0 and [rl]=[r2Jnw. The cap product does not change if we shrink N to a point because there exists rv E [wJ which applies to any chain having support in N gives 0. Therefore [r2'1 n w = 0.
The remainder of the proof is the same as Lemma 3.1.
Theorem 3.4. Suppose that f E C' (M, IR') and that a < b are regular values. Assume that f satisfies the (PS) condition, and that f has only isolated critical points in f -' [a, bJ. If there are m-nontrivial homology classes jr, ] < [r2) < ... < [r ) in H. (fb, f.), then f has at least m-distinct critical values.
Proof. It follows directly from Corollary 3.3.
Corollary 3.4. Suppose that f E C' (M, R') is bounded from below and that f satisfies the (PS) condition. Then f has at least CL(M) + 1 critical points.
Proof. According to Theorem 1.1 from Chapter I, L(X, Y) = CL(X, Y)+ 1 for any topological pair (X, Y). We have m = CL(M) + 1 nontrivial homology classes [rl] < [rsJ, ... , (rm], by taking Y = 0, and X = M. (f0 = 0 if a < inf f). The conclusion follows from Theorem 3.4.
Theorem 3.5. Suppose that f E C' (M, R i) and that a < b are regular values. Assume that [r1J < [r2) <
... <
ITm] are nontrivial homology classes
in H. (fb, f,). Let (3.8)
ci = inf sup f (a), i = 1, 2, ... , M. ,1E['']=E['i[
If c = cl = C2 = ... = c,,, and if f satisfies the (PS), condition, then we have
CatM (Kr) > m. Proof. Since Kr is compact, we may choose neighborhoods N' C N' C N of Kc satisfying the following conditions:
(1) Catm(N) = CatM(Kc);
(2) there exist constants 0 < e < i' < Min {b - c, c - a} and rl E C(M, M) such that
Critical Point Theory
108
rlllc-e =
ld fE,
n - id, and
r! (ff+E\N') C
according to the first deformation theorem. We prove our theorem by contradiction. If CatM(K,) < m - 1, then
there are (m -1) contractible closed sets {B}T' cover N, i.e., U'-' B, 3 N. Since IT;] < [r2) < ... < [Tm), there exist w2, w3, ... , w,,, E H' (fb), dimwj > 0 such that
[Ti-I) = [ri)nwi,
j = 2,3,... ,m.
Since dimw, > 0 and B; is contractible, we always may choose wj E wj, with li'',I nB; = 0, j = 2,3,... ,m. We choose r E [rm) which has support IrI C ff+E. Subdividing r into r = ai + 0`2 + ... + Cm such that
Ia,ICf,,+,\N', and lajICB;, j=2,3,...,m, one has
T'=Tn(''2U...U(Dm) =ai n(t'2U...U(vm) Hence IT'I C fc+E\N' and rl(Ir'I) C f,,-,. However T' E [ri]; therefore rl(r') E )r1), which implies Ci
This is a contradiction.
Corollary 3.5. Suppose that f E C' (M, la') and that a < b are regular Assume that [ri) < [r2) < ... < )rm) are nontrivial homology
values.
c l a s s e s in H. (fb, f.). Let c, be defined as in (3.8), i = 1, 2, ... , m. Suppose
that there are two open neighborhoods N' C N of U;_, K,, satisfying disc (N', 8N) > 6, b > 0 and that there are constants b, f > E > 0 and c such that (1) Ildf(x)II > b V x E ff+r\(ff-EU N'), (2) 0 < E < Min { bbl, 1 bb}, (3) C - E < C < C2 <4 ... < Cm < C . + E. Then Cat1 f (N) > in. Proof. This is a combination of the proofs of Theorem 3.5 and Corollary 3.2.
3. Connections with Other Theories
109
3.3. Relative Category Definition 3.1. Let (X, Y) be a topological pair, and let Y be closed. A closed subset A of X is said to be of m category relative to Y, and we write Catx,y(A) = m, if m = inf{n E N U {+oo} 13 closed subsets Fo, Fl, ... , Fn, and 3 hi E Q0, 11 x Fi; X), such that
(1) hi(0,x)=x, VxEF1, 0
(3) hi(1,x) = xi, V x E Fi, for some xi E X,
ordinate nontrivial homology classes L(X,Y), L(X,Y) > CL(X,Y) + 1. What is the relationship between Catx,y (X) and CL(X, Y) ?
Theorem 3.6. Catx,y (X) > L(X, Y). Proof. Let m = Catx,y(X). Next, assume that there are (rj] E H.(X,Y), 0 < j < in, with (ro) < (r1] < ... < (rm]. We have to prove (ro) = 0. Assume, by contradiction, that (ro] y6 0. By definition we have
(r,_1] = (rjJflwj, j = 1,2,... m, withWj E H'(X), dimw, > 0. Moreover there is a class wo E H'(X,Y) such that o jA ([7-o],wo) _ ([r1) nwi,wo) _ ... _ < (Tm J, Wm U Wm _ I U ... U W0 > .
Consequently, (3.9)
WmUWm_iU...Uwo00.
Consider the exact sequence
44H'(Fi)-... for i = 1, 2,. .. , m, and
We claim that rt, is surjective for * > 1, i > 1 and that no is surjective for * > 0. Indeed, provided by assumption (3), the injection ji: F, - X is homotopic to the constant map F; '- x,, and we conclude that j; :
Critical Point Theory
110
11'(X) 11 (F1) is trivial for * > 1, and then n? is surjective for * > 1 and i > 1. Next we define g(t'x)=
j
ho(t, x) x
if (t, x) E (0, 11 x Fo
if(t,x)E10,11xY;
because of assumption (2), g is well-defined and continuous. Then the injection Y) --+ (X, Y) is homotopic to the map g(1, ), which maps FO UY into Y. Again, by exactness, r)o : H' (X, Fo UY) H' (X, Y) is surjective for * > 0. This proves the claim. Combining with the commutative diagram
H'(X,F0uY)® H'(X,Fi)®
®H'(X,F-) -"+H' CX,YUUF, I
lei H'(X,Y)
® H'(X)
We find that w,,. U finishes the proof.
jnT
®
®
H'(X) -
{9}
1
H'(X,Y)
U... , U wo = 0, which contradicts (3.9). This
Corollary 3.6. Cat(X) > CL(X) + 1. Claim. By definitions, Catx,O(X) = Cat (X), and CL(X, 0) = CL(X). It is obvious that the following relative properties hold. (1) If Y = 0, then FO = 0, and Catx,O(A) = CatX(A). (2) Let A, B he closed subsets of X, then
Catx.y(AUB) < Catx,y(A)+ Catx(B). (3) Catx,y (A) = 0 a 3 h : (0,1] x A -' X such that (i) h(0, - ) = idA, (ii) h(1,A) C Y, (iii) h(t, )[Any = idAny. (4) If 3 h E C((0,11 x A,X) such that h(t, . )[Any = idAnY, then Catx,y(A) < Catx,y(B), where B = h(1, A). Remark 3.2. The relative category was initially studied by E. Fadell (Fad1J. The notion was consequently used by G. Fournier and M. Willem [FoW11. Theorem 3.5 was independently proved by Chang, Long, Zchnder [CLZ1J and Fournier-Willem (F0W2J. A different definition was given by A. Szulkin [Szul]. Similarly, but with a different idea, see J. Q. Liu [Liu2J.
4. Invariant Functions
III
4. Invariant Functions For a G manifold M, generally speaking, invariant functions possess more critical points than functions without symmetry. That is, if the action is free. As a matter of fact, an invariant function is regarded as a function defined on the quotient manifold M/C, and, in many cases, the quotient manifold M/G has a richer topology than the manifold M, e.g, M = Sn,
C = Z2, which contains the antipodal map as an action, M/G = P^, the real projective space. If the action is not free, then things are not so simple, but the fact that there are more critical points remains true. In this respect there are many interesting theories, among them, LjusternikSchnirelman category theory, genus and cogenus theory, cohomology index theory and pseudo index theory. The purpose of this section is to present a different approach which is consistent with our Morse Theory. That is, in replacing index theory, we
use the subordinate relative homology classes to obtain multiple critical points. Furthermore, a Galerkin approximation method is employed to
avoid the pseudo index theory. For the sake of simplicity, we are only concerned with the two groups Gl = 72 and G2 = S1. It is well-known that E = S°° is the universal total G-space for G = G;, i = 1, 2, and that the classifying spaces B are P°° and CP°° respectively. The cohomological rings He (P°°, Z2) and He (CP°°, Q) read as follows:
He (P°°,Z2) = Z2 (w),
dimw = 1,
H' (CP°°, Q) = Q(w),
dim w = 2,
and
where R(w) is a polynomial ring generated by w with the coefficient ring R, and Q is the rational field. Let X be a Hilbert space, and let G = G1 and G2 act on X orthogonally (in the case where G1i X is real) or unitarily (in the case where C2, X is complex).
For G = 7L2, one can find orthogonal subspaces X 1 and X2 = X1 such that X 1 = Fixc and that the G action has the representation x -x V x E X2. Thus G acts freely on the invariant subspace X2. For G = S', according to the Stone Theorem, d g = e'B, 0 E [0, 2n], the unitary representation has the following spectral decomposition: T (g) =
J 00 e"aedEA, 00
where Ea is the spectral family with spectrum o C Z. Let X1 = (E+o E_o)X, which is the fixed point set Fixc, and let X2 = Xi . It is worth noting that V x E X2\{0}, the isotropy group G= is a finite group. Indeed, let V C X2 be a finite dimensional invariant subspace.
Critical Point Theory
112
Then V is isomorphic to the unitary space Ck, and the group action is represented by the diagonal matrices
T(g) = diag {e''1 a, eia2B,. .. e;ake} where \ 1, ... , Ak are nonzero integers. Let m be the greatest common factor of al, ... , Ak; then G= = 7,,,,, V x E V\{B}.
Theorem 4.1. Assume that (fl) f E C1(X, R1) is G-invariant, G = C1 or C2. (f2) 3 two regular values a < b, such that (PS), hold for all c E la, b). (f3) There exist Cd-invariant subspaces X+ and X_ with
j = d codim X+ < m = d dim X_ < +oo,
d = 1 or 2,
where j and in are integers, such that (1) Fixp C X+, Fixc n X_ = {B},
(2) f(x)>a VxEX+,
(3) f(x) 0,
(4) Fixc n f
1 [a, b]
= 0.
Then f has at least m - j distinct critical orbits. The proof is separated into two cases: G = 71.2 and G = S1. Case I. G = 71,2. From (f3) (2), (3) and (1), it follows that fa\X1 C X\X f and X_ n S. C fb\X1. We have the injections i, j and k as follows:
(x- n S,, 0) /C
.
(f b\X l , fa\X l) /G
\'I
(X\X1,X\X+)IC.
From (f3) (1) and (4), the three pairs (X_ n S.,0), (fb\X1,fa\X1) and (X \X 1, X \X.#) are G-free. We shall figure out the homology groups of these pairs. We have
H. ((X- n So) /G, 0) = II. (P'"-1 and
H. ((fb\X 1) /G, (fa\X 1)/G) = H. (fb/G, f./G)) provided by the excision property.
V x E X, we have the orthogonal decomposition x = y + z, where
yEX+,zEX+. Let 7i(t, x) = y + tz
V t E 10, 11
4. Invariant Functions
113
X+\{9}, which is G-equivariant. We obtain Then r) (X\X+) H.((X\X1)/G, (X\X+)/G) = H.((X\X1)/G, (X+\{O})/G). Similarly, :
H. ((x\X1) /G, (x+\{9}) /G) = H. ((x2\{9}) /G, (x+\{9}) /G). In summary, the following commutative diagram holds:
H. (P--1, 0)
H. (fb/G, f./G)
I j. H. (Pn-1, Pi-1) where n = dim X2 > m. As to the cohomology rings, we have
H'
(P--1)
H' (fb/G, X1/G)
H' (Pm-1) It is easily seen that k'w2 = (k *W)2
where w is the generator of H'(Pn-1) = En-1 wj, and that dim k'w # 0. Thus, we have nontrivial classes (zt] E Ht(Pm ) for t = 0, 1,... , m -1, satisfying (zr_ 1 J = [zej n k'w, t = 1, 2, ... , m - 1. 0 for t = j, j + 1, ... , m - 1. Indeed, it We shall prove that k. (zt] suffices to prove that Izel n X+ 36 0, d zt E (zt), t = j, j + 1, ... , m - 1. If this is not true, i.e., there exists to E (j, m - 11 such that )zjo I n X+ = 0, then Izto I is deformed in X+ n S1. It is impossible. Since k' =
we have
i. (ze-1J = i. ((zeJ n k'w)
= is (Z[) n j'w,
and dim j'w 96 0. Then we obtain m - j subordinate classes i. (Z.J < is 1zj+1) < ... < i. (Zm_11.
Theorem 3.4 is applied to provide in - j distinct critical orbits of f with values in [a, b], because f -1[a, b] /G is a manifold. The proof is complete.
Case II.C=S1. If we follow the proof given above, then the argument is stuck by the
fact that the pairs (X_ n S,,, 0), (fb\X1, fa\X1) and (X\X,iX\X+) are no longer G-free.
114
Critical Point Theory
Although f -I [a, b]/C is not a manifold, the minimax principle is still valid, due to the C-deformation theorem, cf. Chapter I, Theorem 7.1. More algebraic topology is needed to derive the chain of subordinate classes. In the following, fl* denotes the Nch-Alexander-Spanier cohomology functor. It is known (cf. Spanier [Spall pp. 340 and 420) that fl* (Y) = H' (Y), the singular cohomology if Y is a CW complex. We shall
figure out II'((X\Xi)/G) and H'((X_ nSp)/C) via their counterparts in
Jr.
Let us recall the following ([Spal) p. 344).
Theorem 4.2. (Vietoris Begle). Let f : X' X be a closed continuous surjective map between paracompact Ilausdorff spaces. Assume that there
is n > 0 such that II*y(f-1(x)) = 0d x E X, and for q < n. Then
f' : Ilo(X)
II°(X')
is an isomorphism for q < n and a monornorphism for q = n. From this we relate II' (Y/C, Q) with the G-cohomology III (Y, 0) where
is the rational field and Y = X \X r or X- n Sr. Lemma 4.1. Suppose that Y is a paracompact HausdorffG space, and that V x E Y, the isotropy group G= is finite. Then the map j : E xG Y Y/C induced by the projective j : E x Y Y induces isomorphisms II°(Y/G,Q) -+ IIG(Y,Q) in all dimensions q, where E is the universal total G space.
Proof. We make use of the filtration
El CE2C...CE"'CEm+IC... of the universal total space E, and consider the diagram for each rn:
F. "' x Y r Y 1
1
El xG Y
Y/G
Note that j' is closed because E'" is compact and that ()'")-rx = E'°/G= V x E Y, where G,, is the isotropy group. Since Ho(Em/Cx,Q) = 0 for q < m, applying the Vietoris Begle Theorem,
q<m
4. Invariant Fjnctions
115
are isomorphisms. Then 3' is just the composition of (3')' and the isomorphism J (E'" xG Y,Q) °-` R (E xG Y,Q), q < m. Let p : C -+ B be an orientable r-dimensional vector bundle, and let DC, S be the associate disk and sphere bundles. According to Thom's isomorphism theorem, HT (Dt, SC) °-' Z possesses a generator t(t;), which induces the isomorphism
H'(Df) -+ H'+'(Df, St) where U is the cup product. Since the disk is contractible, H'(Dt) 25 fl-(B), let e({) E H'(B) be the image of t(e) under this isomorphism. The exactness of the cohomology sequence reads as follows:
(DC)-i!'(SC)-.... It turns out to be the following Gysin sequence:
-, H'(B) -. H'+t(B) -P. H.+r(St:) -, ... , e(()U
where p is the projection S -' B.
Lemma 4.2. If H.9,
S2"-I
(S2"-I,
C X2, then
Q) =
Uli
q=2j, O<j
0
otherwise
and
dim X2-I
HSI (X2\{o},Q) = j=o
Proof. Consider the following diagram:
ExS2n-'
f-+
1
1
B = CP°°,
Stn-I
E xSI
E=S°°
p
where p is the projection induced by the projection p. We look at the vector bundle 7r : t = E x C" -+ B = CP°°. The associate 2n - 1 sphere bundle
St turns out to be p : E xSI S2n-' SE) 1I
Hq-2n(B)
B. We have the Cysin sequence
N°(B)
f e(()
Critical Point Theory
116
Thus, for q < 2n - 1, we have the isomorphisms HQ(B) °5 HQ(SC); and for
q = 2n - 1, from 0 -. H2n-1(B) l(
H2n(B)
II H°(B)
0 0.
H2n-I(S1;)
it follows
/e(()
For q > 2n - 1, from Lemma 4.1, Hq(5C Q) = HQ (E xS1 S2n-1,Q) =1151
(S2n-I,Q)
= Hq (S2n-1/$' Q) = 0. Therefore, Hq
(S S'
2n-1 ,
wj,
q=2j, 0<j
0
otherwise;
Q)
and then 1151 (S°°, 0)
= 10
q = 2j,
j=0,1,2....
otherwise.
The conclusion follows. After these preparations, the previous arguments can be modified to suit
the case C = S'. Indeed, since (X2\{B})/S' and (X_ n SP)/SI are CW complex, the singular cohomology and the Gech-Alexander-Spanier coho, Q), mology are isomorphic. We have H' (X_ n SP)/S', Q) H' (CP'n-I
H'(CPn-',Q).
and H'(X2\{B})/S',Q)
Again, from the commutative diagrams
Jr
Jr ((X_ n S.) IS') k'
(CPm-I) !
11' ((fb\X1) IS')
If
11' (X\X1)"H' (CP"-') and
if. ((X- nSo)/S',0)
11. (Jb/S',la/S') 1j.
H. ((X\XI) IS', (X\X+) IS') we find nontrivial classes (zt1 E Ht((X_ nSo/S'), t = 0,1,... satisfying (zt_ I J = [zt] n k'w, e = 1, 2, ... , 7n - 1 and k' Jzt4
- 1, 0 for
4. Invariant Ainctions
117
C = j, j + 1,... m - 1. Thus i.(z,) < i.Jz,+lJ < ... < i.[z,n_1J are the required m - j subordinate classes. The proof is complete. Corollary 4.1. Under the assumptions of Theorem 4.1, there are critical orbits {Ot I t = j, j + 1,... , m - 1) such that Cc(f,O1)3&0,
t=j,j+l,...,m-1.
A dual form of Theorem 4.1 can be formulated as follows. In replacing (f2) and (f3), we state (f2) There is a sequence of Gd invariant finite dimensional vector subspaces:
X1 CX2C...CX"C..., dimX"=nd, with UOD_1 X" = X, satisfying
(1) ((PS)" condition) 3 no E N such that d n > no, the restriction f" = f Ixn satisfies the (PS), condition V c E (a, b). (2) ((PS)' condition) For any sequence xn E X", n = 1, 2,... ,
a< f (xn) 5 b and df" (x") --. 9 imply a convergent subsequence.
There exist Gd invariant subspaces X+ and X_ with
j = d codimX+ <m=
ddimX_ <+oo, d= 1,2,
where j and m are integers such that (1) FixG C X_, Fixc n X+ = {B},
(2) f(x)>a VxEX+nS,
for some
p>0
(3) f(x)
Theorem 4.3. Under the assumptions (fl), (f2), and (f3) where a, b are regular values, the function f has at least m - j distinct critical orbits.
Proof. First, we change f to -f, -a to b and -b to a. Assumptions (2) and (3) of (f3) are changed to be the following:
(2') f(x)0,and (3') f(x)>a `dxEX_.
Critical Point Theory
118
According to (fz), we choose n large enough, such that both X+ and X_ are included in X", and then restrict ourselves on the finite dimensional Gd-invariant subspace
Xn. Thus
fn(X)
f"(x)>a VxEX_. Let X" = X+ n X" and X+ = X_; then
codirn X+ _ (n - m)d and
dim X"_ = (n - j)d.
Also, X" n Fixc C X_ = X+, (X" n Fixc) n X" = Fixc n X+ _ {9}. Now we apply Theorem 4.1 to f" and conclude that there are (m - j) subordinate homology classes
which correspond to critical values
a < cJ < Let
ct= lim cl, n-oo
cm_1 < b for n large.
e=j,j+l,...,m-1.
Then
a
C=Ck=Ck+1 =...=CC, j e - k + 1. We prove it by contradiction. If if, = {Ol, 02, ... , Os }, s < e - k + 1, then we choose neighborhoods UQ of OR, 1 < a < s, such that (1) UQ n Up = 0 for a 54 Q, (2) Ua-1 U. C f -' [a, b), (3) Each U. is in a G-tubular neighborhood of OQ in f -l [a, b].
We omit the subscripts a. Let n : U
0 be the projection, and let
S = it-1(p) t1 p E 0. It is well known (cf. Chapter 1, Lemma 7.15) that S is a Gp-manifold, where Gp is the isotropy group at p. The orthogonal projection onto (X')1 is denoted by P,, L. Let i X be the injection; then U/Gp
X"nS={xESIPi(x)=9}
4. Invariant Functions
119
is a finite dimensional submanifold, provided n is large enough. Again, one may assume that the geodesic segment connecting p and the nearest point
p' E X' fl S is in S. We choose suitable coordinates: Y = Y1 ® Y2, dim Y1 = nd, W : Y n B1 -+ S, such that W(Y1 n B1) c S n Xn, 0 such that W(V) C S, where V = {(y1, y2) E Y1® Y2 I IIy1II < b, IIy2 - wil < b, 3t E (0,1 ], w = t112 ). Thus N = V(V) is an open neighborhood of p with the following prop-
erty: 3 a deformation retract
q:N/Gp-XnnN=Nn. Indeed, , = cp o 1r o cp-1, where r is the projection onto Y1. Noticing that Cv is a finite rotation group with fixed point p, we may choose b > 0 so
small that Nn = Nn/GP. Let us introduce a
s
N=
and N"= a=1
a=1
We choose b > 0 small enough such that there exists G-invariant open We conclude 3 b > 0, e` > 0 such that
neighborhood N' of K. with dist (ON,N') > II df"(x)II
Bb.
b d z E (f")e+r \ ((fn),,_, U (N")') and
0<(< Min f 1662,866where
(N")' = Xn n N'.
Indeed, if not,
3 x" E X"\(N")'
satisfying
f (xn) - C and II df n (xn) II -+ 0, then by (PS)', 3 x' V N' with f (x') = c, df (x') = 9. This is impossible. Therefore, `d e E (0, C/3), 3 no large enough such that for n > no
c
U
C (Nn)'.
t=k
We exploit Corollary 3.5 and conclude that
E Catpa (R.) > t-k+1, a=1
Critical Point Theory
120
because NQ = NQ /GpQ = G NQ /G.
However, Cat,,, (R) = 1. Indeed VQ is contractible, and so is I. Hence NQ is contractible in NQ/GpQ, i.e., 3 £Q : 10, 11 x 1VQ - NQ/GpQ satisfying Q (0,) = idRQ, Q (1,) = pQ E NQ/GpQ . Let us define rQ = 11Q o fQ. Then rQ is a contraction in N. n.
Hence, we obtain a contradiction:
3=> Catjq (Na)>t-k+1. 0=1
Corollary 4.2. Suppose that f E C1(X, R1) is even with f (0) = 0, and that (1) 3 p, a > 0, and a finite dimensional linear subspace E such that f IElnso > a,
(2) 3 a sequence of linear subspace Em, dim Em = m, and 3 Rm > 0 such that
f(x)<0 VXEEm\BR,,,, m=1,2,....
If, further, f satisfies (PS)' with respect to {Em I m = 1, 2.... }, then f possesses infinitely many distinct critical points corresponding to positive critical values.
Proof. If not, f has at most a critical points. Let dim E = j; we choose m - j > t. Now, (fl) is obviously true, with FixG = {9}. Let
X" = E ; the (PS)" condition holds, using (2), so (f2) is satisfied. Let X.. = El, X_ = Em, a = a, and b = Max=EEm f (x) + 1 then (f3) holds. We apply Theorem 4.3. There are at least m - j pairs of critical points. This is a contradiction. Moreover, all critical values, obtained by the minimax principle, are greater than a, so they are positive. Remark 4.1. Corollary 4.2 was given by Ambrosetti-Rabinowitz [AmR1I,
where the function f is assumed satisfying the (PS) condition rather than the (PS)' condition. Remark 4.2. Theorems 4.1 and 4.3 were proved by many authors via index theory and pseudo index theory, and the index for G = 7L2 is genus. The associated theorems were given by Clark [Cla1J, Ambrosetti-Rabinowitz [AmRU], and V. Benci [Ben4J. S1-index theory was first introduced by Fadell-Rabinowitz [FaR2J; see also V. Benci [Ben3], L. Nirenberg [Nir3), Costa-Willem [CoWI), Fadell, Husseini, Rabinowitz [FIIR1). Pseudo index theory was introduced by V. Benci [Ben4J; see also K. C. Chang [Chal3). Other approaches can be found in Fadell [Fadl] (relative category) and Liu [Liu4) (pseudo category). The new result is Corollary 4.1.
5. Some Abstract Critical Point Theorems
121
5. Some Abstract Critical Point Theorems In this section, we shall give several abstract critical point theorems using Morse theory. Their applications will be studied in subsequent chapters.
Theorem 5.1. Let M be a C2 Hilbert-Riemannian manifold, and let f E C2(M,lg1) satisfy the (PS) condition. Suppose that Hk(fb, fa) 0 for some k E N, where b > a are regular values, and that {x1, x2, ... , xt } C K n f _I [a, b] with Fredholm operators d2 f (xi), i = 1, 2, ... , t. If either (5.1)
ind(f,xi) > k or ind (f, xi) + dim ker d2f (xi) < k
f o r i = 1, 2, ... , t, then f has at least one more critical point xo with Ck(f,xo) 34 0. 0,
Proof. If not, K = {x1, x2, ... , xt}. From (5.1), it follows Ck(f, xi) _ i = 1, 2, ... , t, provided by the shifting theorem. We apply the kth
Morse inequality:
t
rank Ck(f,xc)?/ik(a,b,f)
0=MMk(a,b,f)= i=1
= rank Ilk (fb, fa) > 0. This is impossible.
Corollary 5.1. Suppose that the boundary OD of a k ball D and S homologically link in M, and that {x1, x2, ... , xt} are critical points of f satisfying (5.1). Then the conclusion of Theorem 5.1 holds. This is a combination of Theorems 1.2 and 5.1. Remark 5.1. Corollary 5.1 includes a theorem due to Lazer Solimini [LaSI] as a special case, in which S and OD are as in Section 1, Example 2.
Let H be a real Hilbert space, and let A be a bounded self-adjoint operator defined on H. According to its spectral decomposition, H = H+ ® Ho ® H_, where H f, Ho are invariant subspaces corresponding to the positive/negative, and zero spectrum of A respectively. Let Pt, PO be the projections (orthogonal) of these subspaces. The following assumptions are given:
(111) At := A I y} has a bounded inverse on H. (H2) -y := dim(H_ (D Ho) < oo.
(H3) g E CI(H,1RI) has a bounded and compact differential dg(x). In addition, if dim Ho # 0, we assume
Critical Point Theory
122
9(Pox)-+-oo
as
We shall study the number of critical points of the function
P x) = I (Ax, x) + g(x), or equivalently, the number of solutions of the operator equation Ax + dg(x) = 0.
Lemma 5.1. Under the assumptions (HI ), (112) and (H3), we have that (1) f satisfies (PS) condition, and (2) IIq(H, fa) = b9.yG for -a large enough, as fQ n K = 0.
Proof. 1. First, we verify that f satisfies (PS). For {xn } j° C H, df (xn) 0, and f (xn) bounded, we shall find a convergent subsequence. In fact, from
df (xn) -i 0, it follows d e > 0, 3 N = N(e) such that for n > N I(Axn,xn)+(d9(xn),xn)I
EIIx'II,
where xn = Pfxn. Hence Ilxn II, and then (Axnxn) are bounded. Since 19 (Poxn)1 < 19 (xn) - 9 (Poxn)I + 19 (xn) I <m(IIxnII+Ixnl)+I9(xn)I
where in = sup{IId9(x)II I x E H). If f (xn) is bounded, then Ig(xn)I, 11Poxnll is bounded. Since dg is and therefore Ig(Poxn)I, is bounded. Thus compact there is a subsequence xni such that dg(xni) is convergent. By df (xni) = A+xni + A_xni + dg (xni)
0
and by the boundedness of A-' we conclude that xni is convergent. Since dim 110 is finite, there is a convergent subsequence Poxni . The (PS) condition is verified. 2.
R+
Denote c± = inf {(Ax+, x+)
I
Ilxt II = 1 } which is positive, and
let
M = (H+nBR+) x (Ho ED H-). From
(df (x), x+) _ (Ax+, x+) - (d9(x), x+) > E+
11X+ 112
- m IIx+II
5. Some Abstract Critical Point Theorems
123
we know that f has no critical point outside M, and that -df (x) points inward to M on OM. Noticing that - 2IIAII IIx-II2 - m(IIx-ll +R+) +g(Pox)
Sfx)5 ZIIAIIR+--c-IIx-II2+m(IIx-II+R+)+g (Pox), we obtain
-oo e=# IIx_ + Poxll -' oo uniformly in x+.
f (x)
Thus, V T > 0, 3 a1 R2 > O such that
(H+nBR+) x ((Ho(D H_)\BR1) c fal nM c (H+nBR+) x ((Ho®H_)\BR2) c fat nil Also we find T>0such that Knf_T=0. The negative gradient flow of f defines a strong deformation retract
rl:Mnjag-Mnfa,. Another strong deformation retract in fat n M
r2: (H+nBR+)x((Ho(D H_)\BRZ)
'
(H+nBR+)x((Ho®H_)\BRi)
is defined by r2 = {(1, .), where
(t;x++xo+x-) x+ + xo + x_ x+ + =o+=_ (tRI + (1 - t) Ilxo + x_ II) ,
if Ilxo + x_ II > Ri if llxo + x_ II < R1
We compose these two strong deformation retracts, r = r2 o r1, and then obtain a strong deformation retract
r:Jetnfat
(H+nBR+)®((Ho®H_)\BR,)
and, again, the following deformation: rl (t, x+ + x_ + xo)
-
( x++xo+x_ St
(tR+ + (1 - t)llx+Il) + xo + x_
if 11x+11 :5 R+
if IIx+II > R+
Critical Point Theory
124
is a strong deformation retract of the topological pair from
(H, fat)
to
(M,M n fat)
provided by (5.2).
3. Finally, we have
Hq (M, m n fat)
Hq (H, fat)
HH((H+nBR+) x(Ho®H_), (H+nBR+) x (Ifo ®H_) \BRi )
~HH(Ho®H+,(Ho®H-)\BR1) l "Hg ((ffo®H_)flBR1,O((Ho®H-)nBrt+)) N g7G.
Theorem 5.2. Under the assumptions (I11), (112) and (113), if f has critical points {pi)!-, with n
U Im- (pi), m-
'Y
i=i
(Pi) + mo (POI
where m_ (p) = index (f, p) and mo(p) = dim ker d2 f (p), then f has a critical point po different from p,.... ,pn, with Cy(f, po) 0. Proof. Directly follows from Lemma 5.1 and Theorem 5.1. Remark 5.2. In Lemma 5.1 and Theorem 5.1, if dim Ho = 0, the boundedness of dg(x) can be replaced by the following condition: (5.3)
Ildg(x)II = o(IIxII)
as
IIxii
oo.
Proof. Condition (5.3) implies that f has no critical point outside a big ball BR(B), R > 0. Now we define a new function f (x) = 2 (Ax, x) + p(II xII)g(x), where
P(t)=
1
0
0
t>R2
,0
and satisfies the estimate t < IP ()I
3
-2R2-R, 1
where R2 > 1 > R are suitably chosen.
5. Some Abstract Critical Point Theorems
125
The new function f possesses the same critical points as the function f, and satisfies the (PS) condition as well. In fact, f (x) = I (x) for IIxII S R1, and df(x) = Ax for IIxII > R2, we only want to verify that lldf(x)II 96 0 for x E BR2(0)\BR,(0). Let E = 3IIA-' ll ' by assumption (5.3); 3 Ro > 0 such that Ildg(x)II < EIIxII
V x V Bfto.
The compactness of dg(x) implies that 3 Me > 0 such that Ildg(x)II < EIIxII + MM d x E H.
Thus Ig(x)I < EIIx112 + A'fellxll + lg(0)I
Let
R, > max{R,Ro,E (4Me+3) 111
R2 = max {2,1+lg(0)1}Rl; we have
Ildf(x)II = II Ax + P (IIxiI)g(x)
IIxII
+ P(IIxII)dg(x)II
IIA-'II-' IIxII - (EIIxII + Me) 3
1
2R2-R, (EIIrII2+MMIIxII+Ig(0)I) >1 d x E BR2\BRI. As to the (PS) condition, suppose that df(xn) --+ 0; then {xn} C BRI except for finitely many points, according to condition (5.3) and the invertibility of A. Since dg is compact, there exists a convergent subsequence Comparing this with the assumption 0, and the boundedness of A-', we obtain a convergent subsequence.
Remark 5.3. In the case H = IRN, n = 1, pl = 0, and dim IIo = 0. This theorem is due to Amann Zehnder [AmZ1J. The above lemma, and the general statement with the condition -y < m_ (pi), i = 1, ... , n, is due to Chang [ChalJ. The above version is due to Z. Q. Wang IWaZ21.
Corollary 5.2. Under the assumptions (H1), (H2) and (H3), if f has a nondegenerate critical point po with Morse index m_(po) ry, then f has a critical point p, j4 po.
Critical Point Theory
mo(PI) <
Im-(Po)-7I
then f has one more critical point P2 96 PO,PI
Proof. The first conclusion follows directly from Theorem 5.2 and then we have
C,
(f,P1) 54 0.
However, by the shifting theorem
7 E [m- (Pi) , m- (Pi) + mo (P1)] Condition (5.4) implies one of three possibilities: (1) m-(PO) V [m-(P1), m_(Pi) + MO(PO) and 7 E (in-(p1), m-(P1) + mo(Pi )), (2) -y = m-(P1), (3) -y = m- (P1) + mo(pj).
Using the splitting theorem and the critical group characterization of the local minimum and the local maximum, we see in both cases (2) and (3), C'v (f,Pi) = 69,C. The Morse inequalities in combination with the Betti numbers for the topological pair (H, f6) (see Lemma 5.1), gives the existence of the critical point P2 In case (1), again by the splitting theorem and the critical group characterization of the local minimum and the local maximum, we obtain Cm_(P1) (f,Pi) = C'._(P1)+m0(P1) (f)P1) = 0 and
!39 (H,fa)=ho7 for -a large enough, by Lemma 5.1. However, case (1) implies either m_(po) < m_ (p1) orm_(po) > m_ (p1)+ mo(P1) If there were no other critical points, then in the first case, the m_ (po) + lth Morse inequality would read as -1 > 0. This is a contradiction. And in the second case, both the m_ (p1)+mo(P1)lth and the m_(p1)+mo(p1)th Morse inequalities would imply the equality m- (P, )+mo (P, )
(-1)Q (rank CQ (f,Pt) - ba7) = 0. 9=m- (P, )
5. Some Abstract Critical Point Theorems
127
Again, the m_(po) + 1th Morse inequality would read as
-1>0, and this is also a contradiction. To sum up, we have proved the existence of the third critical point. Now we turn to a variant of Lemma 5.1 which provides more information
on the number of critical points if the function f is defined on H attached by a compact manifold V.
Lemma 5.2. Suppose that (H1) and (112) hold. Let V" be a C2 compact manifold without boundary. Assume that 9 E CI (H x V", R') is a function having a bounded (if dim Ho = 0, Ildg(x, v)II = o(IIxII) V V E V) and compact dg, satisfying
g(Pox,v)
-oo as
IIPoxII
+oo,
if dim Ho 54 0.
Let
f (x, v) = 2 (Ax, x) + g(x, v). Then
(1) f satisfies (PS) condition,
(2) H9(H x V", fa) °` H9-7(V") for -a large enough, with K n fa = 0. The proof is similar to the proof of the previous one. Now define
M = (H+nBR+) x (HoED H_) x V". By the same method, we eventually obtain H. (B-f, S''-1)®H.(V"),
by the Kenneth formula. Thus Hq (H X V", fa)
H9 (M, fa n M) - H9-7 (V") .
Theorem 5.3. Under the assumptions of the above lemma, the function
f has at least CL(V") + 1 critical points. If further, g E C2, and f is nondegenerate, then f has a least E,"_0 0, (V") critical points, where (V") i s the ir" Betti number of V", i = 0, 1, ... , n. Proof. Since Hq (H X V",fa)'° H9-7 (V"),
Critical Point Theory
128 and
HQ (H x V")
HQ (V"),
we obtain P+1 nontrivial singular homology relative classes (Z1+1] < (Zt] <
< (Z1), with Z1, ... , Zt+1 E Hq(H x V", f0), where e = CL(V"). The first conclusion follows from Theorem 3.4. Moreover, if f is nondegenerate, by the same argument, we may assume
K fl fa = 0 for -a large enough. The Morse inequalities now read as
mq > rank HQ(H x V", fa) = rank Hq_.r(V") = Qq_7(V"),
q=0,1,.... Therefore there are at least E7.,=o,0j(V") critical points. The same idea can be applied to study functions bounded from below. We have
Theorem 5.4. Suppose that M is a CZ-Finsler manifold. Assume that f E C' (M, 1181), satisfying the (PS) condition, is bounded below. Suppose
that there exists a critical point po, which is not the global minimum of f, with finite _q of -1)Q rank Cq(f,po) # x(M) - 1. Then f has at least three critical points.
Proof. According to the (PS) condition and lower semi-boundedness, f has a global minimum p1. Let c; = f (pi), i = 0, 1. If f had no critical points other than po and p1 then for arbitrary b > co there would be no critical point in M\fb, and the following identity would hold: X (fb) = X (fb, fco-e) + X (f.0 -c)
where 0 < e < co - c1. Since there exists a strong deformation retract deforming M into fb, and fc0_f into p1, we would have X (fb) = X(M), and
x (fco-,,) = x ({p1}) = 1. But
00
X (fb,fco-E) = >(-1)Q rank Cq (f,po) q=0
because PO is the unique critical point in f -- [co e,
b). This is a contra-
diction.
Corollary 5.3. Let H be a Hilbert space, and f E C' (H,1181) he bounded below with the (PS) condition. Suppose that df (x) = x - T(x)
5. Some Abstract Critical Point Theorems
129
is a compact vector field, and po is an isolated critical point but not the global minimum with index (df, po) = f 1. Then f has at least three critical points. Remark 5.4. In the case H = R", the corollary was proved by Krasnoselskii via degree theory, but was rediscovered by Castro Lazer in [CaL1] by a homology method. The above version was given by Chang [Chal]. A little later, Amann presented a degree theoretic proof [Aural] for Corollary 5.3. Theorems 5.3 and 5.4 are due to Chang [Chal,5].
Now we turn to the study of bifurcation problems. Let H be a Hilbert space and Si be a neighborhood of 9 in H. Suppose
that L is a bounded self-adjoint operator on H, and that G E C(12, H) with G(u) = o(Ijull) at u = 0. We assume that G is a potential operator, i.e., 3 g E Cl (fl,1R1), such that dg = G. Find solutions of the following equation with a parameter A E A&1: (5.5)
Lu + G(u) = Au.
Obviously u = 0, for all A E IR1, is a solution of (5.5). We are concerned with the nontrivial solutions of (5.5) with small hull. Because (5.5) is the Euler equation of a function with parameter A, the bifurcation phenomenon has its specific feature. We shall prove the following theorem due to Krasnoselskii [Kral] and Rabinowitz [Rab2] via Morse theory, cf. [Cha6].
Theorem 5.5. Suppose that f E C2(S,1R1) with df(u) = Lu+G(u), L being linear and G(u) = o(Ijull) at u = 0. If p is an isolated eigenvalue of L of finite multiplicity, then (p, 0) is a bifurcation point for (5.5). Moreover, at least one of the following alternative occurs: (1) (µ, B) is not an isolated solution of (5.5) in {p) x Si.
(2) There is a one-sided neighborhood A of p such that for all A E A{p}, (5.5) possesses at least two distinct nontrivial solutions.
(3) There is a neighborhood I of p such that for all A E I\{p), (5.5) possesses at least one nontrivial solution.
The proof depends upon the Lyapunov-Schmidt reduction. Let X = ker(L - pI), with dim X = n; and let P, P1 be the orthogonal projections onto X and X', respectively. Then (5.5) is equivalent to a pair of equations (5.6)
px+PC(x+x1) = Ax
(5.7)
Lx1 + P1G (x + x1) = Axl
where u = x + x.1, x E X, xl E X'. Equation (5.7) is uniquely solvable in a small bounded neighborhood 0 of (p, 9) E 1R1 x X, say x1 = W(A, x)
Critical Point Theory
130
f o r (A, X) E 0, where W E C'(0, X 1). Substitute x1 = V(A, x) into (5.6), and px + PG(x +,p(A, x)) = Ax,
(5.8)
which is again a variational problem on the finite dimensional space X. Let
J,, (x) _ !(x + W(A, x)) (5.9)
2
(Ilxll2 + IIv(A, x)112)
=
2
II,O2+g(x+s')
where dg = G, with g(0) = 0. It is easy to verify that (5.8) is the Euler equation of J, and that W(A,x) = o(Ilxll) at x = 0. The problem is reduced to finding the critical points of Ja near x = 0 for fixed A near it, where Ja E C1(01, l ' ), 11 is a neighborhood of 0 in X.
Proof of Theorem 5.5. Clearly x = 0 is a critical point of J.,, V A such that (A, 0) E O. If 0 is not an isolated critical point of J,, which corresponds to case (1) in the theorem, then there are only two possibilities: (i) x = 0 is either a local maximum or a local minimum of J,,; (ii) x = 0 is neither a local maximum nor a local minimum of J. In case (i), suppose that 0 is a local minimum of J,,. For some e > 0, W = (J,,)E = {x E Stl I JA(x) < E} is a neighborhood of 0, containing 0 as the unique critical point. The negative gradient flow of J, preserves W, and therefore the negative gradient flow of J,, preserves W for IA - p1 small.
Since W is contractible, X(M) = 1, x = 0 is a local maximum of JA, for A > p and J,, is bounded from below on W; we obtain two nontrivial critical points, according to Theorem 5.4, in particular, Corollary 5.3. Therefore, for each A in a small right-hand side neighborhood of it, there exist at least two distinct nontrivial solutions of (5.5). Similarly, we prove that there exist at least two distinct nontrivial solutions of (5.5) for each A in a small left-hand side neighborhood of it, if 0 is a local maximum of J,,.
In case (ii), 0 is neither a local maximum nor a local minimum of J,,. We see that (5.10)
Co (J,,, 0) = C. (J,,, 0) = 0,
according to Example 4 in Section 4 of Chapter I. Since (5.11)
CO(JA,0) = 1,
(5.12)
C,, (Ja,0)=I
for for
A < it,
and
A>p,
we conclude that there is a neighborhood I of p such that for A E .I., possesses a nontrivial critical point. If not, 3 A,,, -+ it, say A,,, > µ,
6. Perturbation Theory
131
such that J.,,, has the unique critical point 8, then C,i(Jam, 8) = 1, m = 1, 2, ... , implies C (Jµ, 0) = 1 by Theorem 5.6 of Chapter I. This contradicts (5.10). Similarly for An < p. This completes the proof. Remark 5.5. A weaker result that (p, 8) is a bifurcation point was proved by a simpler argument, cf. Berger [Berl).
More information on the number of distinct solutions can be obtained if we assume, in addition, that the function f is G-invariant on some C manifold. For C = Z2 the reader is referred to E. Fadell and P. H. Rabinowitz [FaRI]; for C = S1, we E. Fadell and P. H. Rabinowitz [FaR2], and A. Floer, E. Zehnder (FIZZ]. As to the general compact Lie group G, see T. Bartsch and M. Clapp [BaC1].
6. Perturbation Theory We study two problems in this section: (1) Given a Cs-function f, let E be a nondegenerate critical manifold of f. What becomes of E if we perturb f to f + g where g is small? In the first part of this section, we shall study this problem under the various metrics of g : CO, C1 and C2. (2) For a given function f, which does not have the (PS) condition, we
perturb it to ff = f + eg such that for each e > 0, ft possesses the (PS) condition. Under what conditions can one extend the critical point theory for the perturbed functions to the original one? This will be studied in the second part of this section.
6.1. Perturbation on Critical Manifolds We start with the C°-perturbation, i.e., g is assumed to be small in the C°-norm. Because of the very flexibility of g, one cannot expect any strong conclusion.
Lemma 6.1. Let A C Y C B C A' C X C B' be topological spaces. Suppose that H. (B, A) H. (B', A') ^_- 0. Then h. : H. (A', A) H.(X,Y) is an injection. Proof. Observing the following diagrams:
Hq+1(B',A')
Hq(A',A)
Hq(B',A) H9 (X, A)
Hq(B',A')
132
Critical Point Theory
and
Hg(B, A)
Hq(X, A)
(a,)
/
H, (X, B)
Hq-1(B, A)
(Q,).
Hq(X,Y)
where i : (A', A) -' (B', A), i1, a, /3, al, 0 1, are incursion maps. From the exactness of these sequences, and the assumptions H. (B, A)
H.(B',A') = 0, i. and (i1). are isomorphisms. However, i. = ,0. o a. and (il). _ (/31). o (al).. Therefore (a1). and a. are injections, so is h. = (al). o a.. Theorem 6.1. Suppose that f E C' (M, R1) satisfies the (PS) condition, with an isolated critical value c. Assume that (a, b) is an interval containing c. Then there exists an e > 0 such that for (6.1)
Sup{I9(x) - f(x)I I x E f-'[a,b]} < e/3.
We have an injection h.: H.(fc+e,fc-e)
fI.(9c+,9c-)
Proof. Choose e > 0 such that c is the only critical value of f in [c - e, c + El C (a, b). (6.1) implies
fc-e C 9c-f5 C fc-6 C fc+t C 9c+I C fc+e Applying Lemma 6.1, we obtain
h.: H. (fc+f,fc-e) - H. (9c+i,9c-l) is an injection. Hence H. (f +e, fc_e) --+ H. (gc+ , gc_ 2
is an injection. 2)
Theorem 6.2. Suppose that f E C' (M, fft' ), satisfying the (PS) condition, has only finitely many critical points in f `[a, b], where a, b are regular values off. Then there exists an e > 0 such that Mq(f) 5 Mq(9)
q=0,1,2....
for all g E C' (M, IR') satisfying (6.1) and the (PS) condition, where Mq( ), V q, are the Morse type numbers with respect to (a, b).
Proof. Straightforward. Theorem 6.2 implies that the Morse type numbers are lower semi-continuous
under C°-perturbation. As a direct consequence, we have a result due to Arnbrosetti-Coti Zelati-Ekeland [ACEI].
6. Perturbation on Critical Manifolds
133
Corollary 6.1. Assume that f, g E C1(M,lR1) satisfy the (PS) condition, with c = inf f > -oo, d = inf g > -oo, and that there exists q > 0 such that Hq (K. (f)) 34 0
where K,,(f) is the critical set off with critical value c. Suppose that there exists e > 0 such that
K(f) fl f-1(c,c+e) = 0 and Sup {19(x) - f(x)J I X E fc+e} < 2
Then g has at least two critical points.
Proof. Obviously Kd(g) 0. We prove by contradiction that if K(g) _ Kd(g) = single point, then Mq(g) = 0. But
Mq(9) ? Mq(f) = rank Hq (fc+e) = rank H. (K. (f )) > 0. This is impossible.
Next, we turn to C2-perturbation. It is equivalent to the C'-perturbation of the variational equation df (x) = 0. The inverse function theorem is applied.
Theorem 6.3. Let M be a C2-Hilbert-Riemannian manifold, and let f : [-1, 1) x M -. 1R1 be a C2-function. Let E be a compact nondegenerate
critical manifold for f° = f (0, ). Assume that d2 f°(x) is a Fredholm operator V x E E. Then there exist an i= > 0 and a neighborhood U of E such that d 0 < (el < g. The function f e = f (e, ) has at least Cat(E) critical points in U.
Proof. We regard fe as a family of functions defined on the fibers of a normal disk bundle over E. The function f ° has a nondegenerate critical point on each fiber. We shall prove by the inverse function theorem that f e has the same number of critical points. 1. We choose a tubular neighborhood W of E, which is diffeomorphic to a normal disk bundle NE(r), r > 0, in the following sense: V x E W, 3 a unique decomposition, z = Px E E and v = Qx E N=(E)(r) such that x = exps V. Suppose that M is modelled on a Hilbert space H, b z E E. We consider the orthogonal projection 7r,
: H - Im d2 f °(z).
Therefore V x E W n:dfa(x) = 0,
(6.2)
(I - 7rs) dfe(x) = 0.
(6.3)
dfa(x) = 0 4=t.
Critical Point Theory
134
2. Let 9G(e,z,v) =7rzdfE(x).
We have 0(0, z, 0) = 7rZdf°(z) = 0, `d z E E, and Ip;,(0,z',0) = 7r2,d2f°(z') = d2f°(z'),
which is invertible from Im d2f°(z) into Im d2 f°(z'). By the implicit function theorem, one has 1< > 0, a neighborhood U° C NE(r)1 and a C'-map a : (-e, e) x E -+ Uo such that 'O(E,z,a(6,z)) = 9
and a(o, z) = B, which solves equation (6.2) uniquely in U°. We shall prove that the section z .-+ a(e, z) of NE(r) provides the critical points of f E.
3. Indeed, let
EE={x=exp=a(e,z) IzEE}. Then EE is a compact connected manifold, with Cat(EE) = Cat (E) since exp(a(e, )) is a diffeomorphism between E and E. Note that H T=M LY T. (E) ® N=(E) and
H=kerd2f(z)® Imdf2(z), where ® denotes orthogonal direct sum, we have
kerd2f(z) = TZ(E) and Im d2f(z) = N;, (E).
Let U C W be the pull back of U° C NE(r), then U fl K (f-) = {x E EE I in which (6.3) holds} = {x E EE I dfe(x) E TT(E)} _ {x E E. I dfe(x) E T. (EE)}
=K(fEIEE),
where K (f) denotes the critical set of f. This is due to the fact that as jel > 0 small, TL(E) is closed to T=(EE).
4. V 0 < JeI < i<, the function fel EE has at least Cat (EE) = Cat (E) critical points, according to the Ljusternik-Schnirelman Theorem. The conclusion follows.
Finally, we study the C'-perturbation.
6. Perturbation on Critical Manifolds
135
Theorem 6.4. Let M be a Hilbert-Riemannian manifold, and let f, g E C' (M, R1). Suppose that both f and f + g satisfy the (PS) condition, and that E is a connected nondegenerate critical manifold off with finite index. If g is sufficiently small in the C'-norm in a neighborhood U of E in M, then #(K(f + g) n U) > CL(E) + 1. Furthermore, if f, g E C2, and f + g is nondegenerate in U, then
0 #K(f +g) ? EQi(E) i=o
Proof. According to the (PS) condition, E is compact. With no loss of generality, we assume K. = E. There exist 6 > 0 and two tubular neighborhoods V, W of E satisfying
(1) VCVCWCWCU, (2) V = NE(rl ), W NE(r2), 0 < r1 < r2 < ri + 1, where NE(r) is the normal disk bundle of E,
(3) WC int (f,,+I\f.-I) and I -'(c-6,c+6(nK = E. According to Theorem 7.4 of Chapter I,
(fc+6, fc-6) " CS (r),
(r) ,
where is the negative sphere bundle over E. Let us define ip E C2(M,RI) satisfying {p(x) = I for x E V, w(x) = 0 for
x¢W, 0<w(x)<1VxEM,and ll4(x)II1. According to the (PS) condition, there exists co > 0 such that
Ildf(x)II?eo VxEW\V. Fixing 0 < e < Min (1, 2c }, we obtain IId(f +wg)(x)II >- co - Ildg(x)II - Ild(v(x)II Ig(x)I > 0 Vx E W\V, if IIglIclcu> < C.
Thus the function F(x) = (f + wg)(x) satisfies the (PS) condition and possesses the same critical points as f + g in W. Moreover, we have (Fc+6, Fc-6) = (fc+6, fc-6) Using Thom's isomorphism theorem,
H. `f-(r), (r)) = where k = ind (f, E)
.
`
Critical Point Theory
136 Since
H. (Fc+6, Fc-6) ^-' H. (fc+6, fc-6)
= H.
(f+o\_6, f-' (c - b))
and 0
H-
H' (E-(r)) '=' H'(E),
preserving the ring structure, there exist w1, w2i ... , we E H' (E), I = CL(E), with dim w; > 0, i = 1, 2, ... , e, such that w1 U W2 U ... U Wt 34 0, from which we obtain [ZoJ, [Z1], [Z1] < ... [Zt].
...
,
[ZeJ E H. (Fc+6, F,:-6) such that [Zo] <
Therefore there are at least t + 1 distinct critical points of F, and then, of f + g, in U, provided by Theorem 3.4. Furthermore, if f + g is nondegenerate, then F = f + cog is also. Since 11. (F,7+6, Fc-6)
H._k(E), we have at least E_o f3,(E) critical points,
provided by the Morse inequalities. Remark 6.1. Theorems 6.1 and 6.2 were given by Marino Prodi [map 11. Theorem 6.3 was obtained by Reeken [Reel]. A special case of Theorem 6.4, in which Al is assumed to be finite dimensional, is a version of a theorem under the name of Conley Zehnder by A. Weinstein [Wei2J, cf.[Clial5].
6.2. Uhlenbeck's Perturbation Method Lemma 6.2. Let M be a C2-Finder manifold, and let f, g E C' (M, 1181) .
Suppose that f is bounded below and that g > 0. For C E (0, 1], let f' = f + eg. Assume that (1) IIdgjj is bounded on sets on which g is bounded, (2) f" satisfies the (PS)c condition, for some c.
Then the function h = g(c - f)-' satisfies the (PS)E_1 condition and K,,-, (h) = Kc(f E), where Kb(f) = the critical set of f at level b. Proof. By computation
dh = (c- f)-'dg+(c- f)'2gdf = (c - f)-2g(df + h-'dg). Thus
xo E KE-i (h) e
c'
f
h(xo) = dh(xo) = 0
6. Perturbation on Critical Manifolds
qJ
137
fe(xo) = f(xo) + h(xo)-19(xo) = c
l df`(xo) = df(xo) + h(xo)-1d9(xo) = (c - f(xo))'dh(xo) = 0
a xo E K. (f`) Suppose that {xi } is a sequence along which
h(xi) -4 e-1
and
dh(xi) -. O.
By the same algebraic computation, we have f`(xi) -+ c
and
Ildf`(xi)II =II(c-f(xj))2g(xi)-1dh(xi)+(e-h(xi)-1)d9(xi)II <- I9(xi)h-2(xi)I Ildh(xi)II +o(IId9(xi)II) Since 0 < g(xi) < (1 + 1)(c - f (x,)) is bounded, provided by the fact that f is bounded from below. Combining this with assumption (1), we obtain df`(xi) -' 0. Again, using assumption (2), it follows the (PS)e_1 condition for the function h. Corollary 6.2. Under the conditions of the above theorem, if, further, Kc (f e) = 0 V e E (0, eo], then (f eO ), is a strong deformation retract of
Proof. Since g > 0 and co > 0, (f o ) C
From the theorem,
Kc(f`)=0eaKe_1(h)=0 deE(0,eo], there is therefore no critical value of h between co 1 and +oo. The (PS)E_1 condition for the function h `d e E (0, co), implies that (f 'o )c = 1 is a o strong deformation of f, = ham. Definition 6.1. Let M be a Finsler manifold, and let f E C' (M, IR' ). We say that f satisfies the e-deformation property, if
(1) f(K(f)) is closed; (2) for any interval [a, b] C lil:' on which f has no critical value, i.e.,
K(f) n f -' [a, b] = 0, there exists a family of functions f e E C' (M, R') with f c > f V E E (0, 11 such that the level set (f e)a is a strong deformation retract of fb V e E (0, 1], i.e., 3 if : [0, 11 x fb fb, satisfying l` (i) r(0, ) = id,
(ii) if(l,fb) C (fl)., and (iii) if (t, ) IW)a = id(fe)0.
Critical Point Theory
138
Theorem 6.5. (E-Minimax Principle) Let Jr be a family of subsets of M, and let f E C1(M,R1) satisfy the E-deformation property. Set
c = inf sup f (x). FEY WE F
If (1) c is finite, (2) F is invariant with respect to if `d e E (0, 1] where i f is the strong deformation retract satisfying (i)-(iii) with any interval [a, b] containing c as an interior point. Then c is a critical value of f.
Proof. Suppose that c is not a critical value provided by the closeness
of f (K (f )), there exists 6 > 0 such that K(f) fl f -1 [c - 6, c + 6) = 0. Choosing Fo E F such that Fo C ff+6, we have rif (1, Fo) c f,1_6, but
f(x)< ff(x)
Theorem 6.6. Let M be a C2-Finsler manifold modeled on a separable Banach space with differentiable norms. Let f, g E C2(M, W), satisfy the following assumptions:
(1) f > -m > -oo, g > 0 and f E = f + eg satisfies the (PS) condition,
Ve>0. (2) IJdg(x)II is bounded on sets on which g is bounded. (3) Uo 0.
(4) a, b V f (K(f )). And d xo E K(f) C f -1(a, b), 3 a neighborhood U such that K(f `) fl U consists of a curve x(e), with x(0) = xo, E E [0, 61, and x(E) are nondegenerate (in the sense of Section 4, Chapter 1) and of the fixed index.
Then fa with handles, adjoined in a corresponding fashion to the critical points of f with values in (a, b), is a deformation retract of fb. The dimensions of the handles correspond to the dimensions of the indices at these critical points.
Proof.
First, assume b < +oo. Then 3 6 > 0 such that a, b
ff (K(fe)) `d e E [o, 61.
We claim that (f) -1 [a, b) n K(f) consists of all these families of curves x(E). (ffj)-1[a,bJ; but xj do 0, xj E K(f`j) n In fact, if not, then 3 ej not lie on any families x(e) defined in (4). However, by assumption (3),
xe E K(f) n f -1 [a, b). This contradicts assumption (4). Then Corollary 6.2 is applied: (f6)a t-- f., (f6)b ti A. Since the Morse handle body theorem for f6 is known (Chapter 1, Section 4)
xj
`f 6)a U U hj (Dm') '=' lf6)b' j=1
6. Perturbation on Critical Manifolds
139
where hj (Di) denotes the attached handle, j = 1, 2, ... , s. We obtain s
foU6hj(D-j)'fb. i=1
Second, if b = +oo, then 3 bj - +oo such that bj V f (Kf), and E j 1 0 with the following properties. (1) d e E (o,ej], bj_i, bj are not critical values of
(2) V ao E K(fl) n (f`)-lla,bj], it lies on the unique one parameter family of nondegenerate critical points of f` near the critical points of f. Then (f `i )bj can be retracted to (f `j )bj_1 with handles corresponding to critical points of f`i with values in (bj_1ibj). In addition, (f`i-1 )b,_1 is a deformation retract of (f `j )bj . The sequence of retractions gives the desired result for f+,,. Remark 6.2. The material of this subsection is taken from K. Uhlenbeck (Uhl2].
CHAPTER III
Applications to Semilinear Elliptic Boundary Value Problems
Semilinear elliptic boundary value problems have attracted great interest in the applications of critical point theory because they are good models to deal with multiple solutions problems with respect to both results and methods.
1. Preliminaries Let us turn to some notation and basic facts in the theory of partial differential equations. Let 52 C R" be a bounded open domain with smooth boundary 852. For a nonnegative integer vector a = (al, ... , a") we write
a^ =
8x'1...8x""
to denote the differential operator, with Ial = al +... + a". Let D(52) be the function space consisting of C°° functions with compact support in f2, and let D'(52) be the dual of D(52), i.e., the Schwartz distribution space. For each integer m > 0, we denote C'"(52) = (u: S2
IR1 18°u is continuous on It, 10I < m}
,
with norm Ilrlllm =
sup Il7'u(x)I. "Ell
For p > 1, and an integer in > 0, we denote Wp (52) = {u E LP(Q) 18"u E Lp(52), lal < rn), where LP is the p-th power integrable Lebesgue space, and o", is the differential operator in the distribution sense, with norm 1
Iluplip =
110
ll
P(n
.
P
1. Preliminaries
141
WD (i2) is called the Sobolev space. In particular, if p = 2, Hl(fl) stands for Wz (it). The closure of D(1l) in the space WD (0) (Hm(l) and Cm (f2) is denoted by Wn (f2) (Ho (i2), Co (?) respectively). 0
The dual space of WI(Q) (i2) (and Ho (f2)) is denoted by
(and
H-m(12) resp.), where v + = 1. The following inequalities are applied very frequently.
Poincarr Inequality
(JIUIPdz)*
C(2) (1 JDuj° dx)' n
d u E W1)
where Vu denotes the gradient of u, and C(fl) is a constant independent of U.
Sobolev inequality. Suppose that for 1 < p, r < oo and integers l > m > 0, we have (1) If < r + t n then the embedding Wp (it) .-+ W;" (fl) is continuous. If the inequality < is replaced by a strict inequality <, then the embedding v is compact.
(2) If v < t-n , then the embedding Wp(A) `- CM) is continuous. If the inequality < is replaced by a strict inequality <, then the embedding is compact.
For a function f E C(ifi x R1,IR'), suppose that there exist constants C > 0 such that If (x, t)I < C (1 + (t(°) .
Then the following nonlinear operator:
u '-- f (x, u(x))
maps boundedly and continuously from Lp(i2) to L9(f2), with p = aq, for example, cf. (Berl). The operator is called the Nemytcki operator. Applying this result, in combination with the Sobolev inequality, we see that the functional J(u) = f(x,u(x))dx Jn is well-defined and continuous on the Sobolev space Ho (f2) if <
2n
n-2
If we further assume that f E C' (i'2 x 1R, R') satisfies the following growth condition:
+2 , 1 fi(x, t)I < C (1 + (t(°) with a < n
Semilinear Elliptic Boundary Value Problems
142
then the functional J is C' on the space Ho (S2), with differential
(dJ(u), v) = fn ft (x, u(x))v(x) dx,
V v E Ho (Il).
Furthermore if f E C(?! x R' ,1l ' ), satisfies
]fit(x,t)] < C(1 + Itl°) with a <
n
2,
4
then J is C2 on Hol (it), with d2J(u)(v, w) =
dx `d v, w E
H(i2).
As for the Laplacian -0 defined on L2(1), with domain D(-0) _ H2(I) n I1o(Sl), it is a self-adjoint operator. The operator -A can be O
extended from WP n 1V1(0) to LP(S2), I < p < oo, continuously.
It is known that ker(-0) = {B}, and that K = (-0)'1 maps L"(Sl) into itself continuously, and is a compact operator. Also the operator K maps L2(1) into Ho(1), such that
in
u v dx = (Ku, v),r01
V v E 110'(f ), u E L2(1l).
The following boundary value problem will be considered:
j -Du = g(x, u(x)) in f2 l Ulan
=0,
where the following growth condition on g E C(S2 x D81,IIF1) is assumed: ]g(x, t)] < C (1 + fit,°)
(1.2)
a< n+
2
- n-2'
if n > 3
-
(and if n < 2, a has no restriction) for constants C, a > 0. We say that uo E Ho(f1) is a weak solution of (1.1) if
it
(vuo Vv - g (x, uo(x)) v(x)] dx = 0 `d v E Ho(Sl).
If we define the functional (1.3)
J(u) =
f (vul2 -G(x,u(x))) n
dx,
1. Preliminaries
143
where (1.4)
G(x, t) = J t9(x, ) dd, 0
then J is C' on Ho (f2) with
(dJ(u), v) = Jivu Vv - g(x, u(x)) v(x)) dx d v E H(f2). This means that the differential equation (1.1) is just the Euler equation of the functional J and the weak solutions of (1.1) are critical points of J, and conversely. Since it is well-known that weak solutions of (1.1) are classical solutions
of (1.1), if the function g is smooth enough (cf. (GiTI)), it is enough to look for weak solutions of (1.1), i.e., the critical points of J. The notion of sub- (or super- ) solutions of the equation (1.1) is also
important. We say u E C'(f2) n C(?) (or fi) is a sub- (or super- resp.) solution if
j -Au < 9(x, u(x)) in fl l ulen < 0
(or - Afi > g(x, fi(x)) in fl Ulan > 0).
If furthermore u < 0, then we say they are a pair of sub- or supersolutions. They are called strict if they are not solutions. For second order elliptic operators, (especially for -0) the maximum principle plays an important role in both qualitative and existence studies.
Theorem (Maximum Principle). Suppose that u E C2(11) satisfies the equation (1.5)
-Du < 0 in f2.
If u attains a maximum M at a point of 0, then u =- M in 0. Theorem. (Strong Maximum Principle) Suppose that u E C2(fl) satisfies (1.5). Suppose that u < M in f2, and u = M at a boundary point P. Assume that P lies on the boundary of a ball K, in D. If u is continuous in fl U P and an outward directional derivative Ou/On exists at P, then
Ou/8n>OatP unless u _= M.
Corollary 1.1. The operator K = (-0)-' is positive, in the sense that it maps nonnegative functions to nonnegative functions. Particularly, K : LP (f2) - CC(rl) for p > 2, maps nonnegative functions to the interior of the positive cone in CO(f7). For positive operators, we have the Krein-Rutman Theorem which asserts that the first eigenvalue (-0) is simple. More generally, we have
Semilinear Elliptic Boundary Value Problems
144
Theorem. (Kato-Hess [KaH1)) Suppose that m E C(17), and that there is a point xo E 0 such that m(xo) > 0. Then the equation
r
Du(x) = Am(x)u(x)
x E f2, A E R'
. 1Ulan =0
admits a principle eigenvalue Ai (m) > 0, characterized by being the unique positive eigenvalue having a positive eigenfunction. Moreover, Al (m) has the following properties: (1) if A E C is an eigenvalue with Re A > 0, then Re A > A, (m). (2) pl (m) := 1/A, (m) is an eigenvalue of the operator K L2(f)) L2(fl) with algebraic multiplicity 1.
In the applications, sometimes we would consider the restriction J of J on a smaller Banach space Co (fit), where J is defined in (1.3). The functional j may lose the (PS) condition (on Co(f2), even if J has on 110 (f2)). However, by a bootstrap iteration, the following is proved in [Cha3).
Theorem 1.1. Under assumption (1.2) with a < n±2, ifn > 2, suppose that g E C', and that J satisfies the (PS) condition; then the functional .1 possesses the following properties:
(1) J(K) is a closed subset. (2) For each pair a < b, K n J-i (a, b) = 0 implies that JQ is a strong deformation retract of Jb\Kb, where K is the critical set of J ( and also
J). Thus for any isolated po E K, we have
Corollary 1.2. C.(J,po) = C.(J,po) with integral coefficients. Claim. For any open neighborhood U of po, let V = UeES' q(t, U), where q is the negative gradient flow of J. We have C. (J, po) =11. (Jo n V, (Jo\{po}) n V; Z) = H. (.c+e n V, J,.-,..V; Z = 11. (5c+e n v,3c_e
nv;7l, I =C. (.7, P.),
using the Palais Theorem at the end of Chapter I, Section 1, where c = f (po) and e > 0 is suitably small.
2. Superlinear Problems The classification of the semilinear elliptic BVPs into superlinear, asymptotically linear, and sublinear is very vague. Roughly speaking, it describes
2. Superlinear Problems
145
the growth of the function g(x, u) with respective to u in (1.1). But sometimes g(x, u) is superlinear in one direction, but sublinear in the other, so that it is not easy to classify them very clearly. Nevertheless, we follow the customary notation in the literature.
In the following, (1.2) is assumed (a < 9, subcritical, a =
is
called critical). Our first result in this section is the following.
Theorem 2.1. Assume that the functional J defined in (1.3) satisfies the (PS) condition on the space Ho (it), and that J is unbounded below. Moreover, if there exists a pair of strict sub- and supersolutions of equation (1.1), then (1.1) possesses at least two distinct solutions.
Before going into the proof, we recall a well-known result (cf. Amann [Amal]) that if there is a pair of sub- and super- solutions u < u of (1.1), then there is a solution uo E C of (1.1). One asks whether we can characterize the solution by the corresponding functional J? Now we shall prove that J is bounded from below on Cx = C fl Co (a), where C = (u E H01(f)) I u(x) < u(x) < fi(x) a.e.}, and then attains its minimum, which is the variational characterization of uo. Applying Example 1 from Chapter I, Section 4, we obtain the critical groups of uo:
Ck(J,uo)=
(2.1)
G
k=0
0
k
0,
if it is isolated.
Lemma 2.1. Suppose that u < u is a pair of strict sub- and supersolutions of (1.1). Then there is a point uo E Cx which is a local minimum of the functional J = JIcplrjl. Moreover, if it is isolated, then
Ck(J'uo)={
(2.2)
G
k=0
0
k 36 0.
Proof. One may assume that u(x) < fi(x), without loss of generality. Define a new function
g(x, fi(x)) A (-Du(x)), 9(x,f) =
e u(x)
g(x,.),
g(x, u(x)) V (-Du(x)),
C
> u(x) < f < u(x)
< u(x)
where a V b = max{a, b}, and a A b = min{a, b}. By definition, g(x, {) E C(? x R1) is bounded and satisfies: g(x,t) = &, t)
for
u(x) < .
< u(x).
146
Sernilinear Elliptic Boundary Value Problems
Let E
C(x, ) = f
9(x, t) dt.
0
Then G E C' (St x R), and the functional
j(u) = rn f
2
- G(x, u)1 dx
defined on III (f2) is bounded from below and satisfies the (PS) condition. Hence there is a minimum uo which satisfies
dJ (uo) = 0, i.e., uo satisfies the equation 1.-Duo = 9(x, uo)
uolan=0. According to the LP regularity of solutions of elliptic BVP and the strong 0 maximum principle, we see that uo E CX, the interior of Cx in the Co(S2) topology. (See Remark 2.1 below.) However, JIcX = JIcX = JIcX; therefore uo is a local minimum of J. (2.2) follows from Example 1, Chapter 1, Section 4. _ Under condition (1.2) J is well-defined on Ho (ft). Since Co (St) is dense
in IIH (S2), uo must be also a local minimum of J. In the case when it is isolated, (2.1) holds. Remark 2.1. We verify uo E CX. Claim. Since u is a strict sub-solution,
0 (uo - u) (x) > 0, but not identical to 0, in Il,
1 uo-1 ou>0. It follows from the strong maximum principle, T0(uo
that uo > u, and
- u)Ian < 0, where 0 is the outward normal derivative. Simi-
larly, we have uo < u, and 8(u - uo&W, < 0. Therefore uo is an interior point of CX in Co topology.
Proof of Theorem 2.1. We already have a local minimum so that it suffices to find another critical point. Since J is assumed to be unbounded
below, 3 ul E Ho(f2) such that J(ui) < J(uo). A weak version of a link (mountain pass) is easy to see.
JIlliua,ai > J(uo)
max{J(ui),J(uo)} < J(uo).
for b > 0 small.
2. Superlinear Problems
147
Exploiting Theorem 1.2 (or Remark 1.2) from Chapter II, there exists a different critical point. We present an example for the application of Theorem 2.1. Assume that
(g1) (1.2) with a < 2; ($2) 9 9> 2 and M > 0 such that BG(x, t) < t g(x, t) d x E Il, for Itl > M;
(g3) h E LN (il) is nonnegative, but not zero. Theorem 2.2. Under assumptions (g1), ($2), and (g3), the equation
f to = g(x, u) - h inn l u1en = 0 possesses at least two solutions, if g(x, t) > 0 V (x, t) E A x 1R1, 3 to > 0, g(x,to) > 0 and g(x,0) = 0.
The proof is just a verification of Theorem 2.1.
Lemma 2.2. Under assumptions ($1) and (g2), for any h E L' (fl), the functional
J(u) =
(2.4)
1
J Jn
[IVuI2 - G(x, u(x)) + h u(x)J dx
satisfies the (PS) condition on 1101P).
Proof. Let {uk} be a sequence along which IJ(uk)I < C1 and dJ(uk) 0.
First, {uk } is bounded. In fact, 3 C2, C3, C4 > 0 such that C1
1 IIukII2
2
IUkk(=)I>M
JIukIl2
2 1
2
G(x,uk(x)) dx - IhJ . IIukll -
C2
- 10 JLk(2)I?M uk(x)9 (x, uk(x)) dx - Ihi . IIuk11 - C2
- 6 J IIukII2 + 1
1
(VukVuk - 9(x,uk)uk) dx
- IhI - IIukII - C3 12
-B-c) IIukII2+(dJ(uk),uk)-C4,
11, I and (, ) stand for Ho norm, L norm, and the HH(fl) inner product respectively. Since dJ(uk) - 0, J(dJ(uk),uk)I < efluk11 if we choose 2e < z - 8, then IJukil is bounded.
where 11
148
Semilinear Elliptic Boundary Value Problems
Let p = a + 1, and consider the following maps:
'
Ho ( H)
-0)-f
H- '(n) LP (n)
LP ( Q) 9C=,)
1. i is a compact embedding, as is i'. Both (-A)-' and where v + g(x, ) are continuous. The boundedness in I101(n) of {uk} implies a convergent subsequence (-A)-1 - i* - 9(., uk-). Since dJ (uk,) = Ilk' - (-A)-1 - i* - 9 ( Ilk') -' 0 in Ho, finally, we obtain a convergent subsequence {uk' }.
Proof of Theorem 2.2. It suffices to verify (1) J is unbounded below. (2) 3 a pair of strict sub- and supersolutions for (2.3).
Claim (1). Since g(x, t) > 0 and g(x, to) > 0, so G(x, t) > 0 `d t > to. Fort > Max {to, A11 we have (2.5)
g(x,t)
0
G(x, t)
t
Hence G(x, t) > Cte for some constant C > 0. There exists a constant C, > 0 such that
J(u)<
j111
{,VuI2 - Cu° + h . u} dx+C1
for any nonnegative It E Ho (S2).
Noticing 0 > 2, say, if we choose It = tWj, where V> > 0 is the first eigenvector of -0 with 0-Dirichlet boundary data, and t > 0,and let t +oo, then J(tV,) -00
Claim (2). The equation (2.3) has a strict supersolution 0, and a stric subsolution u:
-Au = -h
in S2
u1pn = 0.
By the Maximum Principle u < 0. All conditions in Theorem 2.1 are fulfilled. The proof is complete. Example 1. The equation (2.6)
-Du=u2-h infl
ubtxt = 0 pOS3esses at least two solutions, if (g3) is satisfied.
2. Superlinear Problems
149
Theorem 2.3. Suppose (gi), (g2) with G(x,t) > 0, Itl > M, and
(g+)gEC'(11 xR') with 9(x,0)=9t(x,0)=0. Then equation (1.1) possesses at least three nontrivial solutions. We need
Lemma 2.3. Under the assumptions of Theorem 2.3, there exists a constant A > 0, such that J. ^-- S°O, the unit sphere in Ho (0)
for -a > A where J is the functional (1.3). Proof. By the same deduction, but by assuming G(x, t) > 0 V t, Iti > M, we conclude G(x, t) > CItle b t, Itl > M. Thus d u E SO°,
J(tu)-.-oo
as
We want to prove: 3 A > 0 such that d a < -A, if J(tu) < a, then jJ(tu) < 0. In fact, set
A = 2MIf1I
Max
(x,t)Eflx i-M,Ml
Ig(x, t)I + I.
If J(tu)(= 3 - fn G(x, tu(x)) dx) < a, then dt J(tu) = (dJ(tu), u)
= t - ju(x).g(xtu(x))dx t t
<2 t
G(x, tu(x)) dx
2
jtu(xYY(xtu(z))dx+a}
JJ
(!_!)J 0 2u( 1
1( 9
- 21> CO
tu(x)g(x,
tu(x)) dx + (A - 1) + a
x)1>M
f
Itlelu(x)Iedx - 1
< 0.
The implicit function theorem is employed to obtain a unique T(u) E C(S°°, R') such that J(T(u)u) = a d u E S°°.
150
Semilinear Elliptic Boundary Value Problems
Next, we claim that IIT(u)Il possesses a positive lower bound e > 0. In fact, by (g4), g(x, 0) = gi(x, 0) = 0, J(t, u) = 2 - o(t2) V U E SOD. The conclusion follows.
Finally, let us define a deformation retract r< 10, 11 x (H\B,(0)) H\Bf(0), where H = Ho(S2), and Be(0) is the a-ball with center 0, by
rl(s, u) = (1 - s)u + sT(u)u. Vu E ll\B((6). This proves H\B,,(0) -- Ja, i.e..1,, ^-- S.
Proof of Theorem 2.5. 1. Provided by (g4)
(2.7)
J(u) =
2I
I1u112 + 0 (11u112) ,
so, 0 is a local minimum, and Cq(J,0) = boo G. 2. We find two nontrivial solutions. Let us define 9+(x,t) _
g(x,t)
t > 0
0
t<0
and
J+ (u)
=J
[IVul2 -G+(x,u(x))J dx
where
G+ (x, t) = f g+ (x, s) ds. r 0
Again, J+ E C2(H01(Sl),1181) satisfies (PS), using Lemma 2.2. As in the proof of Theorem 2.2, we also have
J+(tcpl)- -oo
as
t -'+oo
where Cpl > 0 is the first eigenvector of -A with 0-Dirichlet data. On the other hand 3 b > 0 such that J+18B6(e) >
4bz
provided by (2.7). The mountain pass lemma (Theorem 1.4 from Chapter II) is applied to obtain a critical point u+ E 1101(Q), with critical value c+ > 0, which satisfies
f -Au+ = g+ (X, u+) 1
U+ lost = 0.
2. Superlinear Problems
151
By using the Maximum Principle, u+ > 0, so it is again a critical point of
J. Analogously, we define ( g(x,t)
t < 0
0
t > 0,
and obtain a critical point u_ < 0, with critical value c- > 0. Chapter II (Theorem 1.7) and the Kato-Hess Theorem imply that Cq(J±, U :E) = bq1 G. According to the Palais Theorem (cf. Section 1), we have
Cq (Jt, uf) = C. (Jt, ut) = Cq (J, ut) where J = JIco; and again, Cq(J,u±)
,
Cq(J,u.+). Therefore
CC(J,u±) = 6Q1G.
3. Suppose that there were no more critical points of J. The Morse type numbers over the pair (Ho (fl), J.) would be
Mo=1, M1=2, Mq=0, q>2, but the Betti numbers
Qg=0 dq=0,1,2,... since Hq(Ho(n),J.) = H9 (Ho(0),S°°) = 0. This is a contradiction.
Example 2. The function g(t) =
t°1
t>0
-It102
t<0
satisfies all assumptions in Theorem 2.3, with 1 < a1, a2 < . For odd g (in t), because of the symmetry of the functional J, one gets more solutions. Namely,
Theorem 2.4. Under assumptions (gi), ($2) with G(x, t) > 0 for It) > M, and
(gs)g(x,t)=-g(x,-t) V(x,t)Ef1xJR1. Equation (1.1) possesses infinitely many pairs of solutions.
Proof. The functional defined in (1.3), J E C1(Ha (fl), R'), is even, and satisfies the (PS) condition.
Semilinear Elliptic Boundary Value Problems
152
According to (gl ), we have
J(u) > 2 IIutI2 - C (1 +
(2.8)
IIuIIie+1)
where C > 0 is a suitable constant. By using the Gagliardo-Nirenberg inequality, (2.9)
IINIILQ+1 5 CI IIuIIyoIIuIIL,2
where C1 is a constant, and 0 < Q < 1 is defined by
-L=0 a+1
1
(2-1
1
+(1-Q).2
Substituting (2.9) in (2.8), for u E c9B,(0), we have
J(u)
2! p2 - Cep(°+1)aIIuII(2A)(°+1) - C3. t
Let Al < A2 < A3 < ... , be the eigenvalues of (-0), associated with eigen, and let Ej = span (WI, W2, ... , Ws), j = 1, 2, ... . vectors (P1, 'p2, 1P3, The variational characterization of the eigenvalues provides the estimates IIuIIr2 2 < (j +1IIuII
V u E EjL, j = 1, 2, ...
.
[fence
J(n) > 2 (1 -
p2 - C3 V u E OB,(0) n EL
where 6 = -z(1 - 0)(1 + a) < 0. Since A -. +oo as j - oo, we choose p,jo such that
1 - 2C2p°''A1o+1 > 2, P2 > 8C3.
Thus
J(u)> . p2>0 VuE8Bo(0)nEo. Since all norms on a finite dimensional space are equivalent, and since it was already known that G(x, t) > CItIB
V t, ItI > Al,
there exists R; > p such that
J(u)<0 VuEE,\BR, (0)
3. Asymptotically Linear Problems
153
According to Corollary 4.2 in Chapter II, the proof will be finished if the (PS)' condition with respect to the subspaces {E,b = 1,2,...) is verified. However, the verification is similar to that of the (PS) condition given in Lemma 2.2.
Remark 2.2. Theorems 2.1 and 2.2 are taken from K. C. Chang (Cha34), and Theorem 2.3 from Z. Q. Wang (WaZ2). Remark 2.3. There is a beautiful application of the Morse index estimates to the following perturbation result: The equation
Du = JuIP-' u - f (x) l Ulan = 0
has infinitely many solutions,
if f E
in 1
L2(Sl), and I
< p < n1
cf. A. Bahri, H. Berestycki (BaB1), A. Bahri, P-L. Lions (BaLI,2), Dong Li [DoLl), M. Struwe (Stri) and P. H. Rabinowitz (Rab4).
3. Asymptotically Linear Problems
Nonresonance and Resonance with the Landesman-Lazer Condition 3.1.
First, we assume that the function g is of the form g(x, t) = At + W(x, t),
(3.1)
where W E C(3l x 1R', Ht'), satisfying
V(x, t) = o(ItI)
as Itl
oo uniformly in x E ft.
We study the BVP (1.1) via the abstract theorems of Section 5 in Chapter
11. Set H = Ha(ft), A = id t) _ fo tW(x, s) ds and
F(u) = - / 4;(x, u(x)) dx.
in
Problem (1.1) is equivalent to finding critical points of the functional
J(u) = (Au, u) + F(u). 2
Semilincar Elliptic Boundary Value Problems
154
Theorem 3.1. If A V a(-0), the spectrum of (-A), then (1.1) possesses a solution. If we further assume that
W(x,0)=0
c0EC1(?2xD81,II81),
{ v't(x,O) = Ao - . 1
NO,
and 3 A E o(-0) such that either A < a < \o or Ao <
(3.3)
<
then (1.1) possesses a nontrivial solution.
Theorem 3.2. If A E o(-0), and cp(x, t) is bounded, and we assume the Landesman-Later condition that (3.4)
no
r
Jin
4i
no
tjc0j(x)
x,
dx -. +oo
as Ltd . oo,
j-1
j-1
where ,ipn0} = ker(-0 - Al). Then (1.1) possesses a solution. Furthermore, if (3.2) holds, and if 3 A E o(-0) such that A <
< A0, then (1.1) possesses a nontrivial solution.
Proof. Let Al < A2 < A3 < ... be the eigenvalues associated with the eigenvectors W1, X02, W3.... of (-0) with 0-Dirichlet data. Let
H+ = span {Wj, I Aj > i},
Ho=ker(_L
- al),
and
11- = span {'O, 1 '\j < A}
.
Then (II1) and (112) of Theorem 5.2 of Chapter 11 hold. Since dF(u) = (-0)-1c0(x,u(x)), F E C1(11,1181), and then IIdF(u)II _ of flull) if (3.1), and IIdF(u)U < C, a constant, if c0(x, t) is bounded. Note
that Ho
L2
Ilo.
L2
So dF is compact. Therefore (113) also holds. We apply Theorem 5.2 of Chapter II (with Ho = 0) in the case n = 0, to obtain the existence of a solution in Theorem 3.1, and, in the case n = 1, to prove the existence of a nontrivial solution.
In fact, -y = dim(Ho ® II_) = dim H_ = Max {j
I
\j < A}, and
m_ (O) = Max f j I aj < A0), mo(9) = dim ker(-A - Aol). Condition (3.3) is equivalent to ^y
171L_ (B), nL- (O) + mo(o)].
3. Asymptotically Linear Problems
155
Similarly, Theorem 3.2 is proved by the fact that the Landesman-Lazer condition (3.4) implies
F (Pou) -. -oo
as
IlPoull -. 00,
where PO is the orthogonal projection onto Ho. Remark 3.1. Problems in which A V o(-A) are called nonresonance, and in which A E o(-A) are called resonance.
Theorems 3.1 and 3.2 are due to Amann-Zehnder JAmZIJ and Chang (Cha2J respectively. See also Liu (Liu3]. For a similar problem from the two point boundary value problem in ordinary differential systems, we can get a better estimate of the number of solutions. Let us assume that G E C2(10, 7r) x R", R') satisfying
(I)G(t,6)=0,
(2) 3 integer k such that
k2ln < A., (t) < (k + 1)2 I",
where In is the n x n identity matrix and the limit
limiul
d2uG(t, u) exists.
-
(3) 3 k E Z+ such that 1)2
k2In < Ap(t) = d2uG(t, 9) < (k +
In
where the notation B < A means that the matrix A - B is positive definite.
Theorem 3.3. [WaZ3J Under assumptions (1)-(3), if k 96 k, then the equation
u=Cu(t,u) { u(0) = u(n) = 0
possesses at least two nontrivial solutions in C2((0, W], R").
Proof. Again, we use the abstract framework, H = Ho((0,,rJ,R"), A =
id - (-
H+ = span {e;sinjt I i = 1,2,... n, j > k+ 1}
and H_ = span {ei sin jt I i = 1, 2,... , n, 0 < j < k}, where {ei} i is the basis of 1R".
Condition (2) implies that Gu(t, u) is asymptotically linear. Indeed, for any uo E R" we have
IIG,.(t, u) - A.(t) ulls. <-
IIGu(t,uo) - A.(t)uoIls..
+ IIGu(t, u) - Gu(t, uo) - A,,.. (t) (u - uo)Ilsn
< IIGu(t,uo) - A.(t)uoIls+ Sups E 10, 1) Ild2G(t, su + (1- s)uo) - A,o(t)II . Ilu - uolls..
Semilinear Elliptic Boundary Value Problems
156
Thus
IIGu(t,u) - A.(t)uIIin = o(IIuIIS-) Let F(u) _ - fo G(t, u(t)) dt + f o
as IIuIIa^ -, 00.
u(t)) dt; then
z IIdF(u)IIH = o(II1LIIH)
and dF(u) is compact. The following functional is considered:
J(u) = 2 (Au, u) + F(u). By simple computations, we have
(d2J(O)u,u) = 2 J * [Iu(t)I2 - (Ao(t)u(t),u(t))] dt.
Condition (3) implies that 0 is nondegenerate with m_(O) _ (k + 1)n.
But 'y = dim H_ = (k + 1)n # m_ (0). We conclude that there is a nontrivial solution uo(t). Moreover, note that
ker (d2J(uo)) _ (u E Ifo ((0, 0,118") I u(O) = u(ir) = © and
-ii = duG(t, u(t))u in (0, ir) }
,
which is of dimension < n. The condition I'v - m_(8)I ? mo(no)
is fulfilled. Corollary 5.2 in Chapter 11 can be applied to obtain the second nontrivial solution.
3.2 Strong Resonance Now we study a new class of resonance problem, called strong resonance, in which the Landesman-Lazer condition is dropped so that the (PS) condition does not hold. The function g is again written in the form (3.1): g(x, t) = At + cp(x, t)
A E of -A).
The Landesman-Lazer condition (3.4) placed on W is replaced by the following:
(H) V {j E R'"O,
Ifil
oc,
Vu, -+ u in Ho (11), and V v E Ho(l);
we have MO
(3.5)
litre J 3 'O° n
x, ui(x) +
l;,e1(x)
t-t
v(x) dx = 0,
3. Asymptotically Linear Problems
157
and
(3.6)
lim
j -+oo
Jn
it (xu1(x) +
v(x) dx = 0 1=1
where {ei(x)}m0 is an orthonormal basis of the eigenspace ker(-0 - AI),
and fi=(fj,Cf,...,Cj0).
Again we study the critical points of
J(u) = (Au, u) + F(u) 2
as above. The only difference is that the assumptions F(Pou) -. -oo as oo, and the boundedness of IIdF(u)II are replaced by F(u) -. 0, IIPouII and dF(u) -+ 0 as IIPouII --+ oo, where P0 is the orthogonal projection onto
ker(-0 - .V). Lemma 3.1. Under assumptions (H1) and (H2) of Theorem 5.2 of Chapter II, if we assume (H3)
(N3) F E C' (H, R1). dF is bounded and compact.
F(un) - 0, and dF(un) - 0
as IIPounII
oo.
Then J satisfies the (PS), condition V c # 0. Moreover, if J(un) -+ 0, dJ(u,,) -- 0 along a sequence un, then 3 a subsequence (still denoted by un), with the property that either un converges, or II(I - Po) unII - 0
as
IIPou, II - oo.
Proof. Suppose that (3.7) (3.8)
J(un) = 2 (Au, un) + F (u,,) -+ C dJ (un) = Aun + dF(un) - 0.
Decompose u, into un + un + un, where un is the orthogonal projection of un onto H±, and un = Poun. Then (3.9)
J (Au' , un )1 - 1(Aun, un )1 = I (dJ (un) - dF (un) , un) I
C Ilun 11
Since At has a bounded inverse on Hf, (3.9) implies the boundedness of un . If un is bounded, so is un, and then it has a weakly convergent subsequence. By the compactness of dF, and the finite dimensional conditions on H0, we get a strong convergent subsequence. On the contrary, if
158
Semilinear Elliptic Boundary Value Problems
- oo (we ignore a subsequence) then .II un - 0. Finally, (3.7) implies that (3.8),
0. From
0,
IIu
0, i.e., c = 0.
A new idea is employed to avoid the difficulty arising from the lack of compactness (the (PS)o condition). We compactify the space H by adding some oo points, extend the critical point theory (in particular, the deformation theorem) to the enlarged space, and distinguish the genuine and the pseudo critical points that come from the oo points. We proceed as follows:
1. Compactify the subspace 110 = kerA. Let E = H o U {oo} ^- S'"0,
mo = dim Ho,
and
E = H0 x E where Ho = II®®Lf-. Define
J(x) - {
J(V+s)
(v,s)EHo xHo
z (Av, v)
(v, s) E Ho x {oo}, _
where x = (v, s) E E.
The assumption J(v + s) - 0 as IIsII -- oo implies that J is continuous (but not differentiable in general).
Although we cannot comment on the critical set of J, we call k = K U { (0, oo) } the pseudo critical set, where K is the critical set of J. Points in K are said to be genuine, and the single point (0, oo) to be pseudo. Thus
K,,:=KfIJ-1(c)
t( & Ko U {(0, oo)}
if c 34 0,
if c = 0.
2. The first deformation theorem can be extended to study the pseudo critical set. Namely, we have
Lemma 3.2. Under the assumptions (H1), (H2) of Theorem 5.2 of Chapter II, and (H3), if KO is bounded in 11, then V c E R1, V closed neighborhoods N of K, in E, there exist rl E C((0, 11 x E, E) and constants 0 < e < i` such that (1) 1)(0, ) = id, id Ic.i-llc-e,cfel, (2) +7(t, ) (3) r)(t, ) : E -+ E is a homeomorphism V t E [0, 1], (4) q (l, !,.+,\N) C J,, and if Kc = 0 then 7)(1, Jc+() C Jc-e.
Proof. The proof is similar to the standard one (cf. Theorem 3.3 of Chapter 1). The main point is to construct a flow. Let C(t,v,s) be. the flow defined on 110 x Ho by the vector field V (x) (cf. Theorem 3.4 of
3. Asymptotically Linear Problems
159
Chapter I). Let the functions p, q have the same meaning as those in the referred theorem. Let W be a vector field on Ho x {oo} defined by
W(v) = -p
((Avv)) q(IIAvII)Av.
The associated flow (= W (C), ((0, v) = v, is denoted by C(t, v). Set rl(t, v, 8) = Sl
if (v, x) E Ho x H0 if (V,.9) E Ho x {oo).
tr(t, v, s) (t, v), oo)
Obviously this mapping satisfies all the conditions stated above. It remains to verify its continuity at points on Ho x {oo}.
Claim. First, we show that
IIV(v + s) - W(v)II - 0 as v remains on a bounded set and IIsII
oo.
This is due to the facts
(J(v + s) - I (Av, v) = IF(v + s)I - 0, I
IIX(v+s)-AvII =
rrp(v + s)dF (xp)
and
- 0,
$
where {rlp} is the partition of unity in the construction of p.g.v.f. X, in combining with the following inequalities:
((Avv)) I < C J(v + s) - 2 (Av, v)
I p(J(v + s)) - p
I
0
Iq(IIX(x)II) - q(IIAvID)I s CIIX(x) - AvII - 0.
t, v, -+ v, sn
Next, assume that t,
oo, where to E (0,1J, vn E Ho,
and sn E Ho. We want to show that P0. (tn, vn + sn) -+ oo and
(I - Po)t (tn, vn + sn) -' ((t, v).
Consider the equation
Jf=V(f) l
(0,vn + sn) = vn + an.
Since IIVII S 1, II{(t,v. + sn) - l;(0,vn + s.) 11 < 1, it follows II(I - Po) E(t,vn + an)II < Iivnll + I < C, IIPof(t,vn + an)II > Isni - I -. oo. We conclude 11 V(.(t, vn + sn)) - W((I - P0)e(t, vn + an)) 11 -. 0 from above.
160
Semilinear Elliptic Boundary Value Problems
We turn to the differential inequation: Wt IK(t, v) - (I - PO){ (t, vn + sn)II
IItiV(((t, v)) - (I - Po)V V (t, V. + SO) 11 11 IV (((t,v))(t,vn+sn))II+o(1) < C II((t, v) - (I - Po)SW((I``-P0)f (t, vn + 8.)11 + o(1).
According to Gronwall's inequality, we have II((t, v) - (I - PO) f (t, vn + sn)II C II((0, v) - (I - P0) ( (0, vn + SO 11 + o(1) =CIIv-Vnll+o(1).
On the other hand II((t,vn+sn)-S(tn,V.+sn)II < It - tnI
follows from II V II < 1. Thus II( (t, Vn) - (I - Pp) ( (tn, vn + SO 11 -i 0.
Theorem 3.4. Under assumptions (H1), (112) of Theorem 5.2 of Chapter II and (H), equation (1.1) possesses a solution. If, further, we assume that W(x,0)=0, 'EC1(S2x1IR1,1R1)
and that either (1) $(x,0) = 0 or (2) 4'(x,0) > 0, and Vi(x,0) $ I-J1 + A,0), where
max(A, IE
0(-0), .j < A), or (3) -t(x,0) < 0, and 'e (x,0) V I0,.1- A), where
= min(aj IE a(-0),
>A1), then there is a nontrivial solution. Proof. One may assume that Ko is bounded, since otherwise we would
be done. Now Theorem 5.3 of Chapter II, is applied to assure at least CL(StO) + 1 = 2 pseudo critical points. Therefore there is at least one genuine critical point, which is the desired solution.
Suppose v(x, 0) = 0; then 0 must be a critical point. However by the minimax principle for subordinate classes, 3 a critical value c # 0. In case (1), I(x,0) = 0, 0 is on the level J-1(0), so a point uo in K is a nontrivial solution.
3. Asymptotically Linear Problems
161
If c < 0, we may choose uo to be the one, which corresponds to a m_ relative homology class. According to Theorem 1.5 of Chapter II, Cm_ (J, uo) 54
0, and if c > 0, we may choose it to be the one, which corresponds to a m_ + mo relative homology class. Therefore Cm_+mo(J, uo) # 0. However,
C, (J, 0) = 6,
=0
if Ak < A + W'(x, 0) < Ak+1
gO[m,MI
if
where Ak < Ak+1 is a pair of consecutive eigenvalues of -A, and k+1
k
dim ker(-A - A, I), m =
m=
dim ker(-A - A,I). i=1
S=1
Thus Cm_ (J, (9) = 0
if V't (x, 0) < -A + A or
r
(x, 0) > 0
and
G',.,_+mo (J, B) = 0
if WI (x, 0) < 0 or
a p e (x, 0) >
The proof is finished. Several sufficient conditions can be given to assure (H). Namely, (1) V(x, t) - 0 and o(x, t) 0 as Itl -. oo, if A is simple and the nodal set of the associate eigenfunction has measure zero.
(2) W(x, t) =po(t) + h(x), where h E ker(-A - Al)-'-, and spo(t) together with its primitive Oo(t) = ff cpo(s)ds are bounded and uniformly continuous on R1. And dim(fl) = I As a special case of (2), we assume that WO is a T-periodic function, with fo Wo(t)dt = 0. Indeed, the verification of (1) is trivial. (2) needs a little real analysis, so we refer to J. Mawhin [Mewl]. See also Solimini [Sol3] and Ward [Warl]. Theorem 3.4 is due to Chang and Liu [ChL2].
3.3. A Bifurcation Problem For simplicity, the function g(x,t) in (1.1) is replaced by g(t). We assume
that (1) limig, Opt < A1i the first eigenva111e of -A with 0-Dirichlet boundary value. (2) g(0) = 0, and g E C1(IR1).
Theorem 3.5. Let A = g'(0), then (i) Fbr A > A1i the BVP (1.1) has at least two nontrivial solutions.
162
Semilinear Elliptic Boundary Value Problems
(ii) For A > A2i or A = A2 with _> A in a neighborhood U oft = 0, (1.1) has at least three nontrivial solutions. (iii) For A > A2, we assume that if A E o(-L) either
g(t) > A or t -
t
- A holds
for t 76 0 in a neighborhood U of 0, then (1.1) has at least four nontrivial solutions.
Proof. By condition (1), there exists an a E (0, A1) and a constant
C. > 0 such that g(t) < at + C,, if t > 0, and g(t) > -at - C, if t < 0. Let 'o he the solution of the following equation
Awo=c o+CQ
in 11
cook=0. Then, by the maximum principle, WO > 0, and hence, -WO <
J(u) =
(
fn
I
(V2)2 - C(u)J dx
on
Mo(il),
which is bounded from below so that the P.S. condition is satisfied. (i) Let V, be the first eigenfunction, with max=Ea', = 1, and gyp, > 0. We may choose e > 0 so small that
-vo < -ecpl and ecP1 <'Po are two pairs of strict sub- and super-solutions of (1.1). According to Lemma 2.1, we have two distinct solutions z1, z2 E III (f1), satisfying /
Ck (J,zt)=
C
k=0
0
k340'
1
_ 1, 2,
if they are isolated. (ii) We may assume that there are at most finitely many solutions. The weak version of the mountain pass lemma is employed. We obtain a third solution z3. According to the Kato-Hess Theorem and Theorem 1.6 in Chapter 11, we have
Ck(J,z3) _
G 0
k=1 k#1.
3. Asymptotically Linear Problem,
163
We shall prove that z3 34 0. Indeed, if A > A2, then
Ck(J,9) = Ck_i(J,9),
where j = indd2J(9) > 2.
So z3 0 0. If A = A2, with RLtl > A2 for Itt 76 0 small, then ind(J, 9) = 1, 0 is
degenerate, and is a local maximum of Jon the characteristic submanifold
N at 0. Since dim A( = dim ker(-0 -AI) = mo, we have Ck(J,0) =
G
{0
k=1+mo, 1+mo.
k
Again, the critical groups single out z3 from 0. (iii) In the case A E o(-0), 9 is a degenerate critical point. But, g(t)/t > A for t E U \ {0} implies that J = JIN < 0, where N is a neighborhood in
the characteristic submanifold at 0, dimN = dim ker(-0 - AI), equal to m0. Let m_ be the Morse index of J at 0. We have m > 2, and
Ck(J,0)=
k=m_+mo
G
l0
because 0 is a local maximum of J.
Similarly, in the cases either A $ a(-0) or A E o(-0) but g(t)/t < A for t E U \ {0}, 0 is a local minimum of J; thus Ck (J, 0) =
G
k=m-
0
k 34 m_.
If there were no other critical points, then a contradiction would occur due to the Morse inequalities: /30 = 1, pk = 0, k 54 0. In fact, for k > j, one would have
Mk-Mk_1+...+(_1)kMo=
3k-13k-1+...+(_1)koo.
The LHS is even, but the RHS is odd. Therefore there are at least four nontrivial solutions. The theorem is proved.
A special case of this problem is due to the fact that the function is of the following form: (3.10)
g(u) = Au - h(u),
where A is a real parameter and h(u) satisfies the following conditions: h E C'(R'), h(0) = h'(0) = 0 and liml i-,,. hu = +oo. In this sense, we call it a bifurcation problem.
164
Semilinear Elliptic Boundary Value Problems
Example 1. The special form (3.10) of Theorem 3.5 has been studied by many authors. Cf. Ambrosetti (Ambl], Struwe (Str], for at least three solutions, IIofer [Hof], Tian (Tial] and Dancer [Danl] for at least four solutions in cases (i) and (iii).
3.4. Jumping Nonlinearities Elliptic equations with jumping nonlincarities were first studied by Ambrosetti and Prodi [AmP1] and followed by many others: cf. Amann and Hess [AmH1], Berger and Podolak [BeP1], Fu 6k (Fu 1), Kazdan and Warner [KaWI], Hess (Hess], Dancer [Danl], H. Berestycki and P. L. Lions (BeLl) and author (Cha4). After an observation due to Lazer and McKenna (LaMI), more solutions were obtained. In this respect, the reader is referred to Solimini [Sol]), Ambrosetti [Ambl], liofer (Hofl) and Dancer [Danl). We consider the following BVP with a real parameter t E ll8'. (Pt)
Du = f (x, u) + tcpi
in tl
I. Ulan = 0
where Bpi is the first eigenfunction, with cp, (x) > 0 dx E Q. Assume that f E C'(S! x lltl), satisfying the following conditions:
(1) limE-+. fE(x,l;) = y uniformly in x E S!, and 'y E (Aj,Aj+1) for some j > 1, where {A; (j = 1, 2, ...) = a(-0). ti Q < A, - 6, uniformly in x E D, for some 6 > 0. (2) (3) There exists a constant M such that
lff(x,ol <M(1+1w ) We note that condition (1) implies that lim
f (x, )
7
Theorem 3.6. Suppose that the conditions (1)-(3) are fulfilled. Then there exists t' E R' such that (Pt) has (1) no solution, if t > t'; (2) at least one solution, if t = t'; (3) at least two solutions, if t < t'. If further, we assume j > 2, i.e., -y > A2r the second eigenvalue of -0, then
there exists t" < t' such that (Pt) has at least four solutions if t < t". The proof depends on the following lemmas.
3. Asymptotically Linear Problems
165
Lemma 3.3. Assume conditions (1), (2), and f E C(3F x RI). Let
J(u ) =
[IVuI2
n
1
- F(x, u) - tcolu] dx u E
Ho (S2)
where F(x, ) = fo f (x, s)ds. Then for all t E R1, Jt satisfies the P.S. condition.
Proof. For each function u E LJ (S1) we denote u+ = max{u, 0}, and u- = u - u+. Assume that {un} c Ho(ft) is a sequence satisfying (3.11)
J in
where II '
Jn
II
(Vu, Vv-f(x,un)v-t'P1v)dx=o(IIvII) vEHo([)
is the Ho (st) norm. Then we obtain
[Vunov - f(x,un)+v - (tip1)+v] dx = o(IIvII) d v E Ho(st).
Let Pn = en - pn , where
Pn = un -
(-0)-1 [f (x,un)*
+ (tit)}] -+ 9 in Ho(st).
By condition (1), IIf(x,un)+ -TunIIL2 =°(IIunIIL2)i but
tt,+i = (td- Y(-0)-1)-1 ( (-A)_'I(f(x,un)+-Tun)+t+i11 +Pn); it follows that {IIunII} is bounded. From conditions (1) and (2), we have 6 > 0 and C > 0 such that (3.12)
f(x,f) - alf > 611- C.
Let us choose p < Al such that Al -,u < 6 then we have
(-A - A)(un - Pn) = f(x,un) + tlpl - {t(un - PO > -C + tip1 + l1Pn By the weak Maximum Principle, one deduces (3.13)
un - Pn > (-A - µ)-1I-C + til + {tpnj
noticing pn -+ 0 (HO (0)). Combining (3.13) with the boundedness of un, we obtain that is bounded. IIun II L,
Substituting this fact in (3.11), we get that { IIun II } is bounded. After a standard procedure, the (PS) condition is verified.
Semilinear Elliptic Boundary Value Problem,,
166
Lemma 3.4. Under conditions (1) and (2) there exists a subsolution ut for the BVP (Pt) such that for each solution ut of (Pt) we have ut > ut. Proof. According to (3.13), if we define ut to be the solution of the following BVP:
-Au - pu = -C + tcoi {
in fl
fu lbf2 =0
the conclusion follows from the weak Maximum Principle, and the inequality (3.12).
Lemma 3.5. Under conditions (1) and (2) there exists to E 1R' such that if (Pt) is solvable, then t < to. Proof. By (3.12), we have b > 0, C > 0 such that
f(x,f) -ate? 5KI -C. Thus, if ut is a solution of (Pt), then multiplying by Wt on both sides of the equation, and by integration, we obtain dx =
a, f
1
Jn f (x, tut )VI dx + t stf co dx.
From this one deduces
tJ caidx+b J
n
n
/r
that is
t < (J tpi dx
\n
n
C
J
W, dx.
Lemma 3.6. Under conditions (1), (2) and (3), there exists to E Il8' such that (Pt) possesses a positive solution ut which is a nondegenerate
critical point of Jt with index d2Jt(ut) = hj for t < t1, where hj _ >k<j dim ker(-0 - Akl). Proof. Let 9(x,0= --YC + f(x,0
' ? 0.
oo. We extend the function g to be a function g such that g E C(0 x R'), with Ig(x, C)I _ We have g(x,C) = o(Ifl) uniformly in x E f2, as C
o(ICI) uniformly in x E f2. According to Theorem 3.1, the equation
-Au = ryu + (x, u) + {
uien=0
possesses a solution tut. Define (3.14)
vt = ut _
twu
At - 7
in ft
3. Asymptotically Linear Problems
We obtain
Avt = 7vt + 9(x, i'It)
167
in Cl,
`vtIesl=0.
Thus, the LP a priori bounds for vt are employed to deduce
Uvtllca = oWD as It, -- 00. Substituting the estimate in (3.14), we obtain
ltt>0
t
for
where -t1 is a real number large enough. This proves that Ut is a solution of the problem (Pt). Again, we have
ut(x) = vt(x) +
t W1(x) - +oo, Al - ry
a.e. as t --. -oo.
This implies that
ff(x,ut(x)) -- ry a.e. as t - -Co. Since
d2J(ut) = id -
Ut(x)),
by the Holder inequality as well as the Sobolev embedding theorem, we see
that (Id2J(ut) - (id - 7(-A)-1)Ilc(H
(n))
1/p
rr
5 (j I ff
(x,
ut(x)) - 7Ipdx)
where p = z . Applying the Lebesgue dominance theorem, we arrive at
IId2J(Ut) - (id - y(-A)-1)jIc(H (n)) = o(1).
o(-A) is assumed, id Since y nondegenerate with Morse index hj.
is invertible. Thus Ut is
Proof of Theorem 3.6. Define t' = sup{t E R1 I (Pt) is solvable}. Combining Lemma 3.5 with Lemma 3.6, t' E DF1, hence (1) is proved.
Now we come to look for a strict supersolution for t < V. In fact, choose t' E (t,t') and let ut be a solution of (Pt,). Then ut, is a strict supersolution of (Pt) which satisfies ut, > ut' > ut. Hence [ut,ut'] is a pair of strict sub- and supersolutions of (Pt). Lemma 2.1 is employed to deduce a solution ut of (Pt) which is a local minimum of the functional
it =
JtIca(n), so that
Ck(Je,ue)
-(
G
k= 0,
0
k9k 0.
168
Semilinear Elliptic Boundary Value Problems
Noticing that the functional Jt is unbounded from below along the ray u, = sWI, s > 0, Theorem 2.1 is applied. We find a second solution ut with critical groups C k = 1, Ck(Jf,ttt) 0
k#1.
The conclusion (3) is proved. As for conclusion (2), we prove, by the same method as in Lemma 3.3,
that the set {ut ] t E [t' - 1, t']}, where ut is the solution of (Pt) obtained by the previous sub- and supersolutions, is bounded in Ho (1). We obtain a sequence t; - t' such that ut, weakly converges in H01(11), say to W. Then u' is a solution of (P1.). Finally, we assume y E (A3, Aj +1), with j > 2. According to Lemma 3.6,
there is a t" < t' such that there exists a third solution ut of (Pt) such that ut is nondegenerate, with G
tt_
Ck (Jt, t)
0
k=hj, k#h,.
One more solution will then be obtained by a computation of the LeraySchauder degree. In fact, by Lemma 3.3, we conclude that all solutions of the equation (3.15)
u=(-A)-I(f(x, u)+tv1) O
are bounded in an open ball Bu,, where Rt, the radius, depends on t continuously. By the homotopy invariance of the Leray-Schauder degree, one has O
deg(id - (-0)-1Ft, I3R,,0) = const. V t E IR', where
Ftu = f (x, u(x)) + tcp1(x).
But, from conclusion (1), if t > t', (3.15) has no solution. It follows that O
deg(id - (-A)-' Ft, Lit, 0) = 0, `d t E IR'. If t < t", suppose that there are only three solutions ut, ut and ut, then by Theorem 3.2 of Chapter 11, the Leray-Schauder degree would be deg(id - (A)
0 Fe, B1,0)
1)h,.
This will be a contradiction. Remark 3.2. Lemmas 3.4 and 3.5 are due to Kazdan and Warner [KaW 1 ],
and Lemma 3.6 is due to Ambrosetti [Ambl] and Lazer and McKenna
3. Asymptotically Linear Problems
169
[LaM1(. The idea of the proof is taken from Hofer (Hof1J, Dancer (Dan1J and Chang [Cha4J. An extension, in which limE-_ LEO < As, i > 1, has been studied by many others. The reader is referred to the survey paper by Lazer [Lazl]; see also Lazer and McKenna [LaM2] and Dancer [Danl].
3.5. Other examples Suppose that g E C' (R1) satisfies the following conditions:
(1) 9(0) = 0, 0:5 9'(0) < Ai; (2) g'(t) > 0 and strictly increasing in t for t > 0; g'(t) exists and lies in (A1, A2). (3) g'(oo) = Theorem 3.7. Under conditions (1), (2), (3) the equation Du = g(u)
(3.16)
in ft
1.ulan=0
has at least three distinct solutions.
Proof. 1. It is obvious that 0 is a solution, which is also a strict local minimum of the functional
J(u) =
j
[2(Vu)2
- G(u)1J dx on H.' (11),
where G is the primitive of g, with G(0) = 0. 2. Modify g to be a new function
g(t)=
f g(t)
t>0
0
t < 0,
and consider a new functional
J(u) = I {(vu)2 - G(u)] dx, n
where G(t) = fo g(t)dt. It is easily seen that 9 is also a strict local minimum of J, which is a Cl-functional with a (PS) condition.
Since J is unbounded from below, along the ray u, = sco (x), a > 0, Theorem 2.1 yields a critical point uo 34 0 of j which solves the equation {
Au=g(u) xEf2, Ulan 0.
Since g(u) > 0, by the Maximum Principle, uo > 0, hence uo is a solution of (3.16).
170
3.
Semilinear Elliptic Boundary Value Problems
Now we shall prove that -A - g'(uo(x)) has a bounded inverse
operator on L2(Sl;), which is equivalent to the fact that Id-(-i)-'g'(uo(x)) has a bounded inverse on Ho (f2), i.e., uo is nondegenerate. Since uo satisfies (3.16), it is also a solution of the equation -DUO - q(x)uo(x) = 0,
uoIet2
= 0,
where
q(x) = f 9 (tuo(x))dt. 1
0
Let p, < 142 < ... be eigenvalues of the problem
f -Aw - 1'g'(uo(x))w = 0, 1
WI00=0.
We shall prove that ;z < 1 < 112. This implies the invertibility of the
operator -0 - g'(uo(x)). In fact, according to assumption (2), we have
q(x) < 9 (uo(x)) V x E Il
so that
f(VW) 2
µ1 =min 1.
)
'u w2
2
< min f 9(
xw2 ))
< 1.
Again, by assumptions (2) and (3), we have g'(uo(x)) < A2
V x E Q.
According to the Rayleigh quotient characterization of the eigenvalues z
µ2 =
I
2
sup in
E, -EEi f (uoo(x))w2 > \2 El wEE _ f w2)
1
where El is any one-dimensional subspace in !f01(1). 4. The Morse identity yields an odd number of critical points. Therefore there are at least three solutions of (3.16). Finally, we turn to the following example.
Theorem 3.8. Suppose that g E C' (R) satisfies the following conditions: (1) g(0) = 0, and A2 < g'(0) < A3;
(2) g'(oo) =
g'(t) exists, and g'(oo) ¢ a(-0), with g'(oo) > A3;
(3) Ig(t)I < 1 and 0 < g'(t) < A3 in the interval [-c,c], where c = max=E? e(x), and e(x) is the solution of the BVP: De = 1 e+afi = 0.
in Sl
3. Asymptotically Linear Problems
171
Then equation (3.16) possess at least five nontrivial solutions.
Proof. Define
g(t)
if t > c if It) < c
g(-c)
if t < -c
g(c)
NO = and let
J(u) =
1[(v11)2_a()], n
where d(t) = foe g(a)ds. The truncated equation
Du = 9(u) in fl
(3.17)
Ulan = 0
possesses at least three solutions B, u1, u2, because there are two pairs of sub- and supersolutions [EvI, e) and [-e, -evi), where rpr is the first eigenfunction of -0, with pr(x) > 0, and E > 0 a small enough constant. By the weak version of the Mountain Pass Lemma, there is a mountain pass point u3. That u3 # 0 follows from the fact that Ck(J,u3) =
{G
k=1
0
k36 1.
But from condition (1)
_
G
Ck(J,B)-{0
k=mr+m2 k36 mi+m2,
where m; = dim ker(-0 - A; I), i = 1, 2, .... By Lemma 2.1, one has
Ck(J,ui)= j
0
k#p, i=1,2.
Noticing that J is bounded from below, we conclude that there is at least another critical point u4. Obviously, all these critical points u,, i = 1, 2, 3, 4, are solutions of equation (3.17). On account of the first condition in (3), in combination with the Maximum Principle, all solutions of (3.17) are bounded in the interval 1-c, c). Therefore they are solutions of (3.16); moreover, all these solutions u, because of their ranges, are included in [-c, c), and we conclude: 2
ind(J, u) + dim ker(d2J(u)) < r n:= dim ®(-o - AII), !_1
Semilinear Elliptic Boundary Value Problems
172
provided by the second condition in (3). Because of condition (2), we learned from Theorem 3.1, Theorem 5.2 of Chapter II is applicable, with 7 > in, because g'(oo) > A3. Therefore there exists another critical point us, which yields the fifth nontrivial solution for the equation (3.16). Cf. Chang (Chal2J.
4. Bounded Nonlinearities 4.1. Functionals Bounded from Below The functionals J associated with equation (1.1) in this section are considered to be bounded from below. We shall study several cases which occurred in PDE about numbers of solutions. First we assume (ge) 3a < A1/2, and 0 > 0 such that G(x, t) =
Jo
t
g(x, s)ds < &2 + Q
where Al is the first eigenvalue of -0 with 0-Dirichlet data; (g7) lgi(x,t)l < C(1 + Itl)7, -y < n42, if n > 2.
Theorem 4.1. Under assumptions (g6) and (g7), suppose that (4.1)
g(x,0) = 0, and 3m > 1 such that a,,, < gi(x,0) < Am+1
where {A1, A2, ...
} = o(-A). Then (1.1) has at least three solutions. Proof. Again, we consider the functional (1.3)
J(u) = f {IVuI2 - G(x, u)J dx n
which is well-defined and C2 on !fo(ul) provided by (g7). (g6) implies that J is bounded from below: (4.2)
J(u) >
1( 2
1-
2[Y Al
J
)
n
IVtl12dx -,(3 mes(S2).
And 0 is a nonminimum, nondegenerate critical point with finite Morse index of J provided by assumption (4.1). In order to apply Theorem 5.4 of Chapter II it suffices to verify the (PS) conditions. In fact, the coercive condition (4.2) in conjunction with the boundedness of J(un) imply that {u,,} is bounded, and hence is weakly compact. From (g7), we see that 19(x. tW5 C1(1 + 1tD"
IL <
n+2 n - 2'
if it > 2.
4. Bounded Nonlinearities
173
We use the same argument as in Lemma 2.2, from
dJ(un) = un - (-z)-lg(x,un(x)) - B. We conclude that the sequence un is subconvergent.
4.2. Oscillating Nonlinearity The following technique is often used to reduce problems such as (1.1) to problems with bounded nonlinear terms.
Lemma 4.1. Suppose that the function g in (1.2) satisfies (gs) 3 C > 0 such that g(X, l:) < 0 d x E 37, and let (g(x,f) 9(x, t)
if t > e
Sl
g(x,t)
if t < C.
Assume that u E Ho (fl) is a solution of the following equation:
j .
Du = g(x, u(x))
x E fl
ulert-0.
Then u(x) < t, so it is also a solution of (1.1).
Proof. By the standard regularity theorem u E WW (12) V p < +00 so u E Cl (SZ). Considering the domain D = {x E 111 u(x) > t; }, we have {
Du(x) < 0 d x E D,
luleo«'
By the Maximum Principle, we have u(x) < !; in D, and hence D = 0, i.e.,
u(x) < f. Corollary 4.1. If in the above lemma, (gs) is replaced by
(g8)' 3t>0such that g(x,t:) <0dxE Then u(x)<. VxEfl.
.
Similarly, we introduce the assumption
(g8) 31; < 0 such that g(x,t:) > 0 (or > 0) V x E i2, and consider the truncation Nx,t)
We have
t >
g(x,t)
if
g(X, t;)
if t < t;.
Semilinear Elliptic Boundary Value Problems
174
Corollary 4.2. Under (1.2) and (ge)', if u E Ho (f1) is a solution of
f -tu(x) = 9(x, u(x))
x E f1
ulan = 0,
then u(x) >
V x E Sl, and is also a solution of (1.1).
By the same trick, if one looks for positive solutions, the function g defined on il x R+, is extended continuously to be g:11 x RI , R1, with nonnegative for t < 0. Keeping this in mind, we consider some examples. For the sake of simplicity, we assume g(x, t) = g(t), and study the eigenvalue problem (4.4)
I
Au(x) = Ag(u(x)) ulan = 0.
X E S1
Theorem 4.2. Suppose g E C1(1i8+) and g(0) > 0. Assume that
(g9) There exists 0 < at < a2 <
1,2,... ,m.
< a,,,, such that g(a,) = 0, i =
(910) G(t) = fo g(s)ds satisfies
max{G(t) 10 < t < ai_1} < G(a,),
i = 2,3,... ,m and G(al) > 0.
Then 3 Ao > 0 such that for A > Ao, (4.4) possesses at least (2m - 1) nontrivial solutions. Furthermore, if g(0) = 0 and g'(0) < 0 then 3 Al > 0 such that for A > A1, there are at least 2m nontrivial solutions for (4.4).
Proof. By the truncation trick, we consider the functions
gi(t)=
g, (0)
t<0
g,(t)
0
0
t > a,
gi E C(R'), and the functionals with parameter A
Ji(u, A) = J
o
I
1 IVti12 2
- AG(u(x))J dx,
i = 1, 2, ... , in.
Given the above explanation, we know that the critical set K,(A) of A) must be a subset of the critical set K;+1 (A), i = 1,2,... 'M - 1,
and by Lemma 4.1,VuEK,(A),0
i = 1,2,...,rn.
Second, u, (A) ¢ K,_ 1(A) for A large, i = 2, 3, ... , m.
4. Bounded Nonlinearifies
175
Indeed, we only want to show 3 Ai > 0 and w E Ho (f2), with 0 < w(x) < a,, such that Ji(w,A) < Ji_1(ui_1(A),A) VA > A,.
Let a = G(ar) - Max{G(t) I 0 < t < ai_1}. By (gio), a > 0. Let i26 = {x E 0 I dist(x, 8l) < b} for b > 0, and let w6ECp (f2),
0<w6
Thus
IinG(w6)dx = J2\f1, G(ai)dx + J , G(w6(x))dx
/
G(ai)If2I - 2CilS16l
jG(u,_i(A))dx + alfll - 2CIS26I, where C1 = max{G(t) 10 < t < ai}. Hence
Ji(w6, A) - Ji-f (ui-1(A), A)
=2I 1
2
[IOw6I2 - IVui-1(A)12] dx -
A inI IG(w6) - G(ui-1(A))I dx
in IVw6I2dx - A(alfll - 2Cilf26I) < 0
for b = b; > 0 small, and A > Ai large enough. The function w = w6 is just what we need. One may assume Al < A2 < ... < Am. Third, from C(al) > 0, we have
VA>A1, using the above argument, so ul(A) 76 0. One may assume #K,,,(A) < +00. Then the Morse equality is applied to the bounded from below function Ji. 0, Noticing that H01(0) is contractible, we have 60 = 1, (31 = 32 and
where Mi(A), , = 0,1,... are the Morse type numbers for K,(A). But the Morse equality also holds for Ji+1 i and we have known that u;+l (A) E Ki+1(A) \ Ki(A) for A > Ai+1 r and that u;+1 is the global minimum of J;+1, so Cq(Ji+1, ui+1) = SQ,G, i.e., the contribution of ui+1(A) in the alternative
summation E,(- 1)!M,+(A), is 1. If there were no other critical point in Ki+1(A) \ K1(A) for A > Ai+1, then the equality would lose the balance. Therefore, we concludes
# (Ki+1(A) \ K,(A)) ? 2
if A > Ai+l,
i = 1,2,... , m - 1.
176
Semilincar Elliptic Boundary Value Problems
In the cases g(O) = 0 and g'(0) > 0, 9 has no contribution in critical groups. This is proved by the standard perturbation technique in combining with the homotopy invariance property (cf [Chal6J). In summary, we have #(K,. (A) \ {9}) > 2m - 1, if A > An. The first conclusion is proved. Assume that g(O) = 0, and g'(0) < 0; then V A, Jl (u, A) > 0 = J1(9, A) for QQuOO small. 9 is a local minimum of J1(., A), but not the global minimum
ul(A) V A > A1. We apply the Morse equality to J1, that there must be one more point in K1(A) for A > A1, i.e., #(K1(A) \ {0}) > 2, A > Al so is #(K,,,(A) \ (9)) > 2m, if A > A,,,.
4.3. Even Functionals Theorem 4.3. Suppose that g(x, t) is of the form a(x)t + p(x, t) where a E C(D), and p E C'(11 x IR', R'). Assume that a > 0 in Ti, and that (gs), (g, 1) p(x, t) = o(JtI) uniformly with respect to x E S2, and (g12) p(x, t) = -p(x, -t) V (x, t) E Sl X R', hold. Then the equation -Du(x) = Ag(x, u(x)) {
in Il
ulml = 0
has at least k distinct pairs of solutions, if ,\ > Ak, where Ak is the kth eigenvalue of the eigenvalue problem
-Ov(x) = pa(x)v(x) in f { vl80 = 0.
Proof. VA, the functional is written as
Ja(u) = I 2IVu12 - A (a22 + P(x, u(x)) 1 dx where P(x, t) = fo p(x, s)ds is an even function with respect to t, provided by (g12). Thus JA is an even functional. According to ($e) and Lemma 4.1, Ja is bounded from below. And a > 0 plus (g, I) imply that there exists p > 0, such that Ja I SpnEk
< 0 for A > Ak, where S, is the sphere with radius p centered at 0 in
HH(1k), and Ek is the direct sum of eigenspaces with eigenvalues < Ak of the problem (4.5). The verification of the (PS) condition is omitted. Now we apply Theorem 4.1 of Chapter II. There are at least k pairs of distinct solutions.
4. Bounded Nontinearities
177
4.4. Variational Inequalities A variety of variational problems with side constraints arising from mechanics and physics are called variational inequalities. They have been extensively studied since the 1960s. See, for instance, Duvaut and J. L. Lions [DuLll. A typical example is as follows: Given a closed convex set C in Ho (f2), a continuous g: D x 1R' - IR' and hEL (f)), find uo E C such that (4.6)
r [Vuo V(u - uo) - (g(x, uo(x)) - h(x)) (u - uo)(x)] dx >_ 0 du E C.
In
In fact, the variational inequality is attached to the following variational problem: to find uo E C, which is the critical point of the functional
J(u)
2
f
[IVuJ2
- G(x, u(x)) + h(x)u(x)] dx
with respect to the closed convex set C (cf. Definition 6.4 of Chapter I). In this sense, all the critical point theories, including the Morse inequalities on closed convex sets, are suitable for the applications. In contrast with the well-developed variational inequality theory, in which g is assumed to be nonincreasing in t so that the solution is a minimum of the functional J, the restriction on g is avoided in this subsection. Indeed, one can find minimax points. We are satisfied to study the following two examples mainly by explaining the differences.
Example 1. Assume that g satisfies (ge), (g3) and (g3). Let C = P be the positive cone in Ho(fl); then there are at least two solutions of (4.6), if g(x, 0) = 0 and g(x, t) > 0 V (x, t) E fl x lR+, and if 3 to > 0, such that g(x,to) > 0. Claim. We follow Lemma 2.2 step by step to verify the (PS) condition
with respect to P. Note that
I-dJ(u,t)I,,,, :=sup{(-dJ(uk),v-uk) I V E P,Ily - ukIIHi < I) where (
,
0,
) is the inner product in 1101 (f2). It implies V e > 0 3 ko E TL+
such that
(dJ(uk),uk)
V k ? ko.
ellukll,
(WARNING: This is only a one side inequality! Not like that in Lemma 2.2 in which we got I(dJ(uk), uk)I < This is enough to assure the boundedness of llukll, as shown in Lemma 2.2. ellukll).
178
Semilinear Elliptic Boundary Value Problems
Now we prove the subconvergence of uk. As shown in Lemma 2.2, we obtain a subsequence, which we still write in Uk, such that
(-0)-1 0 i' 0g(., uk) -+ u' Since g is positive in t > 0, Uk E P, and (-0)-' preserves the positive cone (Maximum Principle), u' E P. Again, from I 0, it follows that V e > 0 3 ko E Z+, such
that (-Uk + (-o)-' o i' 0
uk), V - Uk) < 1211V- uk Il, Vv E P, V k > k0.
Consequently, 3 kl E Z+, such that
(-Uk+u',V-uk)<Elly-ukll VvEP, Vk>k1. In particular, set v = u', this proves Uk
U.
To study the multiple solutions, it is easily seen that 9 is a local minimum
for J in P. Since the first eigenvector 'p, E P, J is unbounded from below in P. A weak version of the Mountain Pass theorem with respect to P is applied to obtain the second solution. For the same functional J, but we change to a different closed convex set, one has Example 2. Suppose V; E H' (S2), and C = {u E Ho (S2) I 0 < u(x) <
O(x) a.e.}. Under the same assumptions on g and h in the Example 1, assume that (4.7)
inf{J(u) I u E C} < 0.
Then the variational inequality (4.7) possesses at least three solutions.
Claim. The (PS) condition with respect to C can be verified as above. Now, ul = 0 is a local minimum, and J has a global minimum u2. Assumption (4.7) implies ul 36 u2. We apply the Morse equality which provides the third critical point of J.
Remark 4.1. Theorem 4.1 is taken from K.C. Chang (Cha1J. For an extension of it see K.C. Chang [Cha2J. Theorem 4.2 is an extension of the results due to K.J. Brown and II. Budin [BrBlj and P. Uess [Hes2J, in which only the case g(0) > 0 was discussed. Section 4.4 was studied in K.C. Chang [Cha7J.
CHAPTER IV
Multiple Periodic Solutions of Hamiltonian Systems
0. Introduction In this chapter, we shall apply Morse theory to estimate the numbers of solutions of Hamiltonian systems. Let H(t, z) be a C' function defined on It' X R2n which is 27r-periodic with respect to the first variable t. We are interested in the existence and multiplicity of the 1-periodic solutions of the following Hamiltonian system: { .
q = -Hp(t;q,p) P = Hq (t; q, p),
where q, p E R", z = (q, p). The function II then is called the Hamiltonian function. Letting J be the standard symplectic structure on R2n, i.e.,
-In) j= (0 `In 0
'
where In is the n x n identity matrix, the equation (0.1) can be written in a compact version
-Jz = Hs(t,z).
(0.2)
Equation (0.2) is very similar to the operator equation considered in Chapter II, Section 5. Indeed, let X = L2 ((0' 1),1R2' ), and let
A: z(t) '-- -Jz(t) with domain
D(A) = H; ((0 21r) It2n) = {z(t) E H'((0,21rl,1R2") I z(0) = z(27r)}. For the sake of convenience, we make the real space R2" complex. Let
C" = R" + iR", and let {el, e2i ... , e2n} be an orthonormal basis in R2n. Let Bpi = ei + iej +n,
j = 1,2,... , n,
Multiple Periodic Solutions
180
which defines a basis in C'1. The linear isomorphism 1R2n -. (Cn 2n
n
Z = E zjej
-. Z = E(zj - izj+n)Wj, j=1
j=1
is called the complexification of 1182'1, which preserves the inner product. Namely, n
[Z, w] = Re J(zj - izj+n)(wj - iwj+n) j=1 n
2n
= E(Zjwj + zj+ntt'j+n) _ E Zjwj = (z, W), j=1
j=1
, ] is the inner product on C". We introduce the complex Hilbert space L2([0, 2ir], Cn) to replace the real Hilbert space L2([0, 2ir], R2n), whose inner product reads as
where [
(Z, w)
= / 2 *[Z(t), w(t)]dt 0
= 0f2,,(z(t),Ti'(t))dt = (z, w).
From now on, if there is no confusion, we shall not distinguish between these two. Sometime, we only write z but not Z. One important thing is that Jz - iZ. Thus, if we expand z E L2 ([0, 27r], C'1) in Fourier series: n
00
z(t) = [ [
L L rjmP j=1 n,=-oo n
D(A) =
_;mt
Vj,
00
z E L2 ([0, 27r], Cn) I E E (1 +
]rn])2]cjn,12
< +00
j=1 m=-00
and A is self-adjoint with the following spectral decomposition:
L2(]0,ir],Cn)_ ®m(m), mEL
where M(m) = span {e_Imt pl, e-'mto2, , e-""<
A+ = AP+,
A- _ -AP-, A° = AP°, A = A+ - A-.
1. Asymptotically Linear Systems
181
Let
H = D
(IAI1/2)
oo
= IZ E L2 (10,2x],C") I E E (1 + Im()Ic,mI2 < +oo j=1 M-00 with the following scalar product:
(z,w)H = (z,w)L3 + ((AI'/2z, IAIh/2w)L,
.
Then II is a Hilbert space. There are few simple properties often used: (1) The embedding 11
LP ((0, 2.r1, C") is compact, d p < +oo.
Lp ((0, 27r(, C") is compact and Ht/2 ((0,2a),C") is continuous. (2) 3 a compact linear operator K: LP' ([0, 27r), C") - H, such that
Claim. D(IAI'12)
Since H1/2 ((0, 2irj, C")
2w
(K f, z)H = J
+ - = 1.
(f (t), z(t))dt,
-
Claim. K = (I + IAI)-'/, the rest follows from (1). (3) Let H' = P'H, where * is +, -, or 0. There exists a bounded linear operator A E L(H, H) satisfying (Az, w)H = ((A+ )112Z, (A+)1/2w) L, - ((A )1/2z, (A-)1/2w) Moreover, AIH* has bounded inverse with ker(A) = H°.
Claim. A = A(I + IAI)'1. For a given polynomial growth Hamiltonian function H(t, z), the functional 2yr
fH (t, z(t)) dt
g(z)
is well-defined and is in C' (H, Rl ). Then the Hamiltonian system (0.2) is rewritten as
Az + dg(z) = 0.
(0.3)
This is the problem we studied in Chapter I, Section 5 and Chapter II except now we assume dim H- = +oo. We look for critical points of the functional on H:
J(z) = (Az, z) + g(z), 2
(0.4)
182
Multiple Periodic Solutions
where dg = K o d,, 11(t, ) o i : H -. H is compact, with
H
d, H
H.
L2
1. Asymptotically Linear Systems The Morse index plays an important role in distinguishing critical points. For functionals defined by Hamiltonian systems (0.4), Z
d2J(z)(w, w) = (;I W, w),, - f
r d2H(t, z)w2(t)dt.
o
The Morse index ind(J, z) depends on A or A and the matrix dsH (t, z(t)),
but since dim H- = oo, it is very possible that ind(J, z) = oo. So, it is necessary to introduce some replacement of the Morse index. First of all, let us study the meaning of nondegeneracy.
Definition I.I. Let Sym(2n, l) be the set of all symmetric 2n x 2n matrices. Let B E C (]0, 2ir], Sym(2n, ll8)), with B(0) = B(2ir). We say that A is a Floquet multiplier of
-Jt(t) = B(t)w(t)
(1.1)
if A is an eigenvalue of IV(2ir), where W(t) is the fundamental solution matrix of (1.1). By definition, z is nondegenerate for J(z) (see (0.4)) if and only if the linear system (1.2) -Ju,(t) = d 2 H (t, z(t)) w(t) has no nontrivial 27r-periodic solution, or equivalently, 1 is not a Floquet multiplier of (1.2). In order to classify this continuous loop of symmetric matrices without a Floquet multiplier 1, we study the associated fundamental solution matrices Sym(2n, IR) is continuous, then {Lis(t)}. Noticing that, if B(t): [0, 27r]
d
(WT(t)jW t)) =
-7JW+WTJW
=WTBW-WTBW=0, and from WT(0)JW(0) = J, it follows that WT JW = J. Conversely, if W7 (t)JW(t) = J, then TV(t) is invertible. (W-')7JW-1 = Let B = -JWW-1; then BT = B. Therefore, B E C ((0, 21r], Sym(2n, IR)).
1. Asymptotically Linear Systems
183
Recall that a matrix M is called symplectic if MT JM = J. We denote by Sp(n, R) the set of all real 2n x 2n symplectic matrices. Let P = (-y E C ([0, 2ir], Sp(n, R)) [ y(0) = id, y(27r)has no eigenvalue 1). Definition 1.2. Let yo,-y1 E P. We say that yo is equivalent to yl, denoted by yo - y1, if 3 b E C ([0,1) x [0, 27r], Sp(n, R)) such that 6(i, t) _ y; (t), i = 0, 1, 6(s, 0) = id, and I it a(6(s, 2w)) V a E [0, 1]. The following classifying theorem is applied [CoZ2], [LoZ1].
Theorem 1.1 (Conley-Zehnder-Long). Every path Y E P belongs to one and only one equivalent class Pk, k E Z. Moreover, d k E Z, one finds a distinguished "standard path" Qk(t) as follows.
In the case n > 2, let Xk = diag(0,(k - n + 2)/2,1/2,...,1/2),
Y = diag (1n 2, 0, 0, ... , 0),
Zk = diag((k - n + 1)/2,1/2,1/2,... 1/2), be n x n matrices, and Qk(t) = exp(tJBk), 0 < t < 2,r, where
if (-1)n+k = -1, Bk =
if (-1)n+k = 1.
In the case n = 1, set w(t) = 1(1 + cost), w1(t)
=
w(t),
to,
and let
K= 1
0
,
and W2(t)
-
(0, w(t),
0
1),R(O)=(tH a = I/\a100)a' sing
cos9
a
'
If k is odd, then pk(t) = R(tk/2), 0 < t < 27r, and Bk = k/2. If k = 0, then /3o(t) = H (2t/2*), 0 < t < 2ir, and Bk = ,1(ln2)K; if k even, but not 0, then ak(t) = H (2'''2 <
t < 27r, and
Bk(t) _ -Zw1(t)id+ 127r w2(t)K. Definition 1.3. We define a map j: P --+ Z as follows: If y E Pk, then j (-f) = k. The number k is called the Maslov index of y and is denoted by j(y) or j(B), if y(t) is the fundamental solution matrix of (1.1). In the following, we shall show that the Maslov index of y = W (t), which is associated with dsH (t, z(t)), is a candidate for the replacement of the Morse index.
Let us denote E,,, = ®Ijl
184
Multiple Periodic Solutions
Theorem 1.2. Assume that B E C ((0, 2a], Sym(2n, R)) has no Floquet multiplier 1 and k = j(B). Let Qm = A - PmBPn, E £(Em Em), and m', * = -, 0 be the Morse index and the nullity of Qm. Then 3 N > 0 such that m = 2 dim Em + k and mo = 0, V m > N. (1.3)
Proof. We have seen the one-to-one correspondence between the set of loops in Sym(2n, R) without the Floquet multiplier 1 and P: B(t) = -J y(t)ry(t)-1. In order to transfer the deformation on ry to B, we claim that the above equivalent classification may be restricted to a smaller class P* = {7 E Cl (I0, 2a], Sp(n, !)) 17(2x) = j(0)-y(21r),'Y(0) = id, I It o(,y(2ir))}. Claim. Given 'Yo, 71 E P*, 70 71 in P, i.e. 3 b: [0, 1) x (0, 27r) -. Sp(n, IR), continuous, such that b(i, r) = 'y; (t), i = 0,1, b(s, 0) = id and 1 It c(b(s, t)) V s E (0, 11. By reparametrization of s, we may assume that b(s, t) = ryo(t) for 0 < s < e, and b(s, t) = ryl (t) for 1-e < s < 1. Moreover, by reparametrizing in the t variable along s = 0 and s = 1, we can achieve that b(s, t) is differentiable in t in the neighborhood of the boundary of
[0, 11 x [0, 2a] and satisfies, in addition, the condition y(27r) = -y(0) 7(21r). It remains to smooth out e in the interior. This is readily done by smoothing the generating functions of the symplectic matrices in the parameter t. Then we obtain a continuous family B,(t), S E [0, 1], such that Bo(t) _
B(t), Bi(t) is the standard path in the above theorem, and A - B,(t) is invertible in f- (L2([0, 27r], R2n)) (or equivalently, A - KB,(t) is invertible in £(H)) V s E 10, 11. Therefore (by small perturbation), 3 e > 0, 3 N1 > 0 such that, for m > N1, II(APm - PmB,Pm)2II J EIIPmXII V s E [0, 11, d X E Em.
This enables us to figure out the Morse index and nullity of B(t) by the counterparts of B1 (t), the standard path. Obviously, we have mo = 0, because Qm = APm - PmBiP", is invertible. As to m-, we only want to verify the standard paths, which are constant matrices except for n = 1 and k even but not 0. It satisfies us to verify the simplest case and leave the other cases as exercises because they are elementary linear algebra, cf. Long-Zehnder [LoZ1]. Indeed, the representation of Qm on the invariant subspaces M(j) ® M(-j) under the basis (real) {cos jtet, sinjtee I P = 1 , 2,... , 2n} (and M(0) under the basis {el, e2, ... , e2"}), reads as follows:
(Bi
-Bl
as
j=
1, 2.... , m, and - B1 as j = 0.
1. Asymptotically Linear Systems
185
_Bl I , we split them again according to the
For the 4n x 4n matrix I - Bl
standard path. In the simplest case (n > 2, n + k even), B1
Zk
0
0
Zk
and Zk = ding{(k - n + 1)/2,1/2,... ,1/2}, where k = j(B1) = j(B). To those invariant subspaces, in which the elements in Zk equal to 1/2, we have
j j -/
_j
-1/2
ind
1
=9 f o r j = 12 , ,... , m ,
2
-1/2 1
and
indl
- /2) =2 for
02
0.
There are (n - 1) such elements, and/the total contribution in the index is 2(m + 1) (n - 1). The remaining element in Zk is (k - n + 1)/2. In the case
k>n,
-j
-(k - n + 1)/2
ind
-(k - n + 1)/2
j
j
-(k - n + 1)/2
-j and
ind
-(k - n + 1)/2 2,
if j > k-n+l
4,
if j < k-n+1 '
(k - n + 1)/2
-(k - n + 1)/2
2
if j = 0.
The contribution to the index is 2 (m - k2n)+4 (k2n)+2 = 2m+k-n+2. Therefore ind(Q n) = 2(m + 1)(n - 1) + 2m + k - n + 2 = (2m + 1)n + k, i.e.,
m- = 2 dim E,,, + k. Similarly, we verify the case k < n. This finishes the proof. We turn to study the existence and multiplicity of solutions of asymptotically linear Hamiltonian systems. A Hamiltonian system is called asymptotically linear if 3 B E C ((0, 21r], Sym(2n, IF)) such that B,,,, (27r) and IId=H(t, z) - Boo(t)ztIlz^ = o(IIzII)sz..
uniformly in t, as IIzlI vn - co.
186
Multiple Periodic Solutions
Lemma 1.1. Assume that H E C' ([0, 2n] x R2n, R1) satisfies (1100), where B00 has no Floquet multiplier 1. Then the functional (0.4) J satisfies
the (PS)' condition with respect to {Em I m = 1, 2.... }, and Jm = JI E,,, satisfies the (PS) condition V m large.
Proof. 1. We claim that IIdJm(z) - QmzIIH = o(IIzIIH) as IIzII -+ oo, for z E Em, where Qm = APm - PmB0Pm. Indeed, IIdjm(z) - Qm zIIH = II P,,, (d:H(t, z) - B00z) II H < IIdzH(t,z) - B00zIIH = o(IIzIIH)
2. Since KB00 is compact, and Pm strongly converges to the identity, we have IIK(PmBOO - B00)II -+ 0 as n -+ oo. The operator K(A - B00) = A - KB00 is invertible, so by the Banach inverse operator theorem, APm PmKB00Pm has also a bounded inverse. Moreover, there exists a constant C such that II
(APm
- P0KB0Pm)
1II
< C form large enough.
Combining the above two conclusions, we obtain from dJm(zj) -+ 0 that IIzillEm is bounded. Thus the (PS) condition for Jm holds. 3. Assume IIdJm(zm)IIH --i 0, as m , oo for Zm E Em. Then 11 z,,, 11 it is bounded, provided by the same reason. Noticing that PO is of finite rank, we have a subsequence of zm, which is still written by zm, such that KdZH(t, zm) -+ u and POZm - v as m oo. Consequently, Azm + POZm = dJm(zm) +
(t, zm) + POZm --+ u + V,
zm -4 (A+Po)-1(u+v) in Has m--+oo, since A+ PO is invertible in H. This proves the (PS)' condition. Theorem 1.3. Suppose that II E C' ([0, 2ir] x 1 R R2 , 1 ' ) being 2ir periodic in t satisfies (1100), where B00 has no Floquet multiplier 1. Then the Hamiltonian system (0.2) possesses a 2a-periodic solution. Further, we assume (Ho) 3 Bo E C ([0, 2n], Sym(2n, IR)) such that BO(0) = Bo(27r) and IIds11(t,z) - Bo(t)zII$zn = o(IIzIIZ2..) uniformly in t, as IIzIIvvn -+ 0. If BO has no Floquet multiplier 1, and if
j(Bo)
j(B00),
(1.4)
then (0.2) possesses at least a nontrivial 27r-periodic solution.
Proof. 1. First, we use the Galerkin method and study the restriction Jm of J on E,,,, m = 1, 2,.... For large m, we learn from Theorem 1.2
1. Asymptotically Linear Systems
187
and Lemma 5.1 in Chapter II that dim HQ (Em, (Jm)am) = 6,n ,q for some
am, where m- =
so there exists a critical point z,,, E Em of
Jm. According to (HO), H is C2 at z = 0 on R2,; therefore, the functional fo" H(t,z(t))dt is C2 at z = 0 under the topology C ([0, 2x],) 0
and N > 0 such that i
II (APm - KPmB,0Pm) KPmBoPm)_
ll c(H) < R, 1
II (APm -
for m > N.
ilk(,,) < R,
Take 6 > 0 so small that II dJ(z) - (A - KBo)zII H = II K(dsH(t, z) - Boz)IIH
< 2RIIzIIH
V IIzIIH < 6.
Then, for large m, we have II dJ(z) - (APm - KPmBOPm)zII H <-
2R IIzIIH
<- 1 JI(APm -
V IIzIIH < 6.
But if we write QOM = APm - KPmBOPm, then IIdJ(z)II
IIQmzIIH - 2II(Qm)-111'(H)IIzIIH
II(Qm)-'IIC(H)IIzIIH - 2II(Qm)-'IIL(H)IIzIIH
= 2II(Qm)-'II,(H)
114-
Consequently, if z,,, is a critical point of Jm with IIzmIIH < 6, then zm = 0. Thus IIzmIIH >- 6 for m large enough.
2. Applying the (PS)' condition, there exists a limit point z' of {zm}. Therefore, z' is a critical point of J, with IIz']I > 6. This is the nontrivial solution. The proof is finished.
If one wishes to extend the above result to the degenerate case the Maslov index should be extended. In this respect we refer the reader to Y. Long (Lon], Li Liu [LiLl] and Ding Liu [DiLi).
188
Multiple Periodic Solutions
2. Reductions and Periodic Nonlinearities We have seen that for indefinite functionals, the Morse indices of critical points could be infinite (e.g., the functionals arising from the Hamiltonian systems), the Galerkin method plays an important role. Nevertheless, there is a kind of Lyapunov-Schmidt procedure, called the saddle point reduction, which reduces the infinite dimensional problems to finite dimensional ones. The later method has the advantage of simplicity. We shall introduce this method in Section 2.1, and apply it to the study of (0.1) in Section 2.2. We shall also investigate a class of Hamiltonian systems in which the Hamiltonian functions are periodic in some of their variables. It is interesting to note that it causes multiple periodic solutions. This is Section 2.3.
2.1. Saddle Point Reduction Let H be a real Hilbert space, and let A be a self-adjoint operator with domain D(A) C H. Let F be a potential operator with ' E C' (H, R'), F= 1(8) = 0. Assume that (A) There exist real numbers a < Q such that a,/3 ¢ a(A), and that a(A) n [a, j31 consists of at most finitely many eigenvalues of finite multiplicities.
(F) F is Gateaux differentiable in H, which satisfies II
dF(u)
2 a+/3I I
2
VuEH.
The problem is to find the solutions of the following equation:
Ax = F(x) X E D(A).
(2.1)
With no loss of generality, we may assume a = -/3, Q > 0.
A Lyapunov-Schmidt procedure is applied for a finite dimensional reduction. Let
0
Po= f dEa, A
P+=/
+oo
dEA,
P_=/
Q
0
dEA, oQ
where {Ea} is the spectral resolution of A, and let
Ho=PoH, Hf=PPH. According to (A), there exists e > 0 small, such that -e ¢ a(A). We assume further the following condition:
2. Reductions and Periodic Nonlinearities
189
(D) 0 E C2(V,1R1), where V = D(IAIli2), with the graph norm IIxII V = (II IAI1/2xIIU +
We decompose the space V as follows: V =VoCD V_®V+, where Vo = I A1I -112Ho,
VV = IAEI -1/2 Ht, and A. = eI + A.
For each u E H, we have the decomposition
u=U++Uo+u_, where uo E Ho, ut E H±; let x = x+ + x0 + x_ E V, where xo = IAEI-1/2U0,
x± = IA, I- 112U±.
Thus we have IIxdIIV} = IIu±IIH*,
IIx0IIVo = IIuOIIHo,
and that Vf, VO are isomorphic to H± and Ho, respectively. Now we define a functional on H as follows: f(U) = 2 (IIu+112 + IIE+UO1l2 - IIE_UOII2 - IIu_II2)
-,tE(x),
where E+ = fo dEA, E_ = f °. dEa, and 411(x) = 2IIxIIy + 4'(x). Let
F1=cI+F.
The Euler equation of this functional is the system
ut =
±IA1I-112P±F'E(x),
EfuO = ±lArl-112E±PoFf(x).
(2.2) (2.3)
Thus x = x++xo+x_ is a solution of (2.1) if and only if u = u++uo+u_ is a critical point of f. However, the system (2.2) is reduced to AEx± = P±Fe(x+ + x_ + xo),
which is equivalent to
xf =A,'PPF1(x++x_+xo).
(2.4)
190
Multiple Periodic Solutions
By assumption (D), F E C' (V, V), and by assumption (F)
IIFE(u)-Fe(v)IIuS(e+Q)IIu-vIIH
/u,vEH.
Furthermore, there is a y > ,0 + e such that IIA,-'IH+®H_II s y
by assumption (A). We shall prove that the operator F = AC-'(P+ + P_ )Fe E C' (V, V) is contractible with respect to variables in V+ ® V_. In fact, `d x = x+ + x_ +z, y = y+ + y_ +z, f o r fixed z E Vo, IIF(x) - .17(y) 11 v = II
IAeI_112(I'+
+ P_)(Fe(x) - Fe(y))II H
II IA,I-',2(P+ - I'-)IIB(,r)IIFE(x) - Fe(y)II H
< (e+13)11IA,I-',2(P++P-)Ile(H)II(x++x-)
-(y++y-)IIH Since
IIx±IIH II IA,I-'"2u±IIH <
1
1
IIU±IIH =
IIx±IIv,
and
IIIAEI-'12(P++P)IIB(H) s we obtain 11F(x)-F(y)11V <
e
,
01Ix-YIIV
The implicit function theorem can be applied, yielding a solution x± (xo), Since dim Vo is finite, all topologies on Vo are equivalent. We have
for fixed xo E Vo, such that x± E C' (Vo, V±).
ut(xo) = IA,I'"2x±(xo) E C'(IIo,II), which solves the system (2.2). Let
a(xo) = f (u+(xo) + u-(xo) + uo(xo)), where uo(xo) = IAEI112xo and let z = xo. We have a(z) _
(IlA1/2x+112 + IIA1/2E+2112 - II(-A,) '/2x-II2 - II(-Ae)1/2E_ZII2)
-
2 (Ax(z), x(z)) - D(x(z)),
2. Reductions and Periodic Nonlinearities
191
where x(z) = C(z) + z, C(z) = x+(z) + x_(z) E D(A). Noticing that de(z) = A,-1(P+ + P_)F,' (x(z))dx(z)
by (2.4), one sees that de(z) E D(A) and that
Ade(z) = (I - Po)F'(x(z))dx(z). Thus
da(z) = (dx(z))'[Ax(z) - F(x(z))] = Az - PoF(x(z)) = Ax(z) - F(x(z))
(2.5)
and d2a(z) = [A - F'(x(z))]dx(z)
= Alit. - PoF'(x(z))dx(z) In summary, we have proven
Theorem 2.1. Under the assumptions (A), (F) and (D), there is a one-one correspondence:
z -. x = x(z) = x+(z) + x_(z) + z, between the critical points of the tom-function a E C22(Ho,1R1) with the solutions of the operator equation
Ax = F(x) x E D(A). Now we turn to the asymptotic behavior of the function a.
Lemma 2.1. Under the assumptions (A), (F) and (D), we assume further that there is a bounded self-adjoint operator F,,. satisfying
(F,)
(i)
PoF. = F,0Po;
(ii)
IIF(u) - F,ouII = o(IIull) as lull
(iii)
0 it o(A - F.).
oo;
Then we have that (1) .(z) = o(IIzII) as llzll - oo, and (2) the function a(z) is asymptotically quadratic with asymptotics A-
F,Iyo, i.e., Ilda(z) - (A - F,)zll = o(IIzll) as IIzII -' oo.
192
Multiple Periodic Solutions
Proof. By (2.4), we obtain
At;(z) = (I - P0)F(x(z)). Since Po commutes with F,, we have
(A - FF){(z) = (I - Po)(F(x(z)) - FFx(z)J. Hence, V r > 0 there exists R > 0, such that
II F(x(z)) - F,x(z)II < cC (IIzII + IIC(z)II), if IIzII > R,
where C = II(A
- F... )-III; it follows that IK(z)II = o(IIzII)
By (2.5) we have
II da(z) - (A - F.)zII = IIAz - PoF(x(z)) - (A - FF)zII II F(x(z)) - F,x(z)II + II F ,x(z) - FFzII = o(IIx(z)II) = o(IIzII) as IIzII - 00.
Lemma 2.2. Under the assumptions (A), (F) and (D), we assume that F(O) = 0.
(1) If there is a self-adjoint operator Co E ,C(H, H) which commutes with PO and P_, such that
min(a(A) fl la, pl)I < Co < F'(0), then
a(z) <
2
((A - Co )z, z) + o(IIzII2) as IIzII -- 0.
(2) If there is a self-adjoint operator Co E C(II, II) which commutes with PO and P+, such that F'(9) < Co < coax (a (A) fl
I,
then
a(z) ? 2 ((A - CO ) z, z) + o(IIzI12) as IIzII
0.
Proof. By the definition and (2.5),
a(z) = 2 (Ax(z), x(z)) - D(x(z)) =
2
(Av, 9) -t(q) +
2
(Ax+(z),x+(z)) - (4'(x(z)) - ID(a)),
2. Reductions and Periodic Nonlinearities
193
where q = x-(z) + z. We shall prove that 2 (Ax+ (z), x+ (z)) - (4(x(z)) - 4'(q)) S 0,
that is,
a(z) : 2 (Aq, q) - 4'(q).
(2.6)
In fact,
2 (Ax+(z), x+ (z))
7'(x(z)) -
=
2
(Ax+(z), x+ (z)) +
> 2 IIx+(z)II2
f
1
(F(tx+(z) + q) - F(x+(z) + q), x+ (z)) dt
- f 1 p(1- t)dtllx+(2)112 = 0. 0
However,
,t(q)
-
(F'(9)q, 9) I =
I f(F(tq) - F'(0)tq, q)dt
< 1 sup II F'(tq) - F'(9)IIc(v,v)Il qll v, 2 o
that is,
-1'(q) <- -2(F''(O)q,q)+o(IIgII,).
(2.7)
Note that x_ E C1(Vo,V_); this implies that if IIzII - 0, then IIx_(z)IIv_ = O(Ilzllvo) because x_(9) = 9. Thus IIgIIv =O(IIZII)
Substituting (2.7) and (2.8) into (2.6), we obtain a(z) < <
((A - F'(0))q, q) + o(Ilzll2)
((A - Co)q,q)+o(IIzII2)
= 2 ((A - Co )x-(z), x-(z)) +
((A - Co )z, z) + o(IIz1I2)
as IIzII -4 0. Let
a- = min{o(A) n [a, #11, and, by the assumption,
a_I
Multiple Periodic Solutions
194
This implies
((A - Co)x-(z),x-(z)) < ((A-a-I)x-(z),x_(z)) <0; therefore
a(z) < 2 ((A - Co )z, z) + o(IIz112) as z
9.
We prove the second assumption in a similar way . Finally, we apply Theorem 5.2 of Chapter II to solve the operator equation (2.1) under conditions (A), (F), (D) and (F,,,,). For a symmetric matrix B, let ml (B) be the dimension of the maximal positive/negative subspace.
Theorem 2.2. Under assumptions (A), (F), (D) and we assume F(9) = 0. If one of the following conditions holds: (1) There exists a bounded self-adjoint Co, commuting with PO and P_, such that min(o(A) n (a,f3J}I < Co < F'(9) and
m- (A - Coljfo) >m-(A-F.IHo); (2) There exists a bounded self-adjoint CO +, commuting with PO and P+,
such that
F'(9) < Co < tnax{a(A) n (a, #I) i and
m+(A-CGIH0)>m+(A-F,IH,); then there exists at least one nontrivial solution of the equation (2.1).
Proof. By Theorem 2.1, problem (2.1) is reduced to finding critical points of the function a E C2(Ho, lI '). According to Lemma 2.1, a is an asymptotically quadratic function with a nonsingular symmetric matrix as asymptotics. By Lemma 2.2, condition (1) means that d2a(9) is negative on the subspace Z_ on which A - Ca is negative. Thus
m-(d 2a(9)) ? m- (A - Co IHo) > m- (A - FOI Ho) . Similarly, condition (2) means that
m+(d2a(9)) ? m+ (A - Co Iluo) > m+ (A - F.I Hu) In this case,
m-(A-F,,. dim HO - m+(d2a(9)) = m-(d 2a(B)) + dim ker(d2a(9)).
2. Reductions and Periodic Nonlinearities
195
Both cases imply that
m- (A - F.IHo) ¢ [m-(d 2a(9)),m-(d2a(O)) + dim ker(d2a(O))] . The conclusion follows from Theorem 5.2 of Chapter II.
Remark 2.1. The finite dimensional reduction method presented here is a modification of a method due to Amann [Ama1J and Amann and Zehnder (AmZ1J. Avoiding the use of monotone operators and a dull verification of the implicit function theorem, we change a few of the assumptions and gain a considerable simplification of the reduction theory.
2.2. A Multiple Solution Theorem We apply the saddle point reduction to Hamiltonian systems. Let H =
A = -Jd, with D(A) given in the preliminary. For
L2
H E C2(R' X 1R2",lRI) being 27r-periodic in t, we define
F(z) = d,H(t,z(t)). Suppose that there is a constant C > 0 such that
IId2. H(t, z)< C; then
I(z)
= J 2,. H(t,z(t))dt E C1(H,I81). 0
Again, the derivative F(z) = d4'(z) is Gateaux differentiable with IIdF(z)IIc(H) 5 C V z E H, so conditions (A) and (F) are satisfied. By observing the continuous imbeddings
D (IAI1/2) - H1/2 (I0, 2ir), R2n)
- La ([0, 21rl, l2n) , V p < +00,
condition (D) is also easily verified. When we study Hamiltonian systems under condition (2.9), the equation is reduced to da(z) = 0,
where
a(z) = 2 (Ax(z), x(z)) - t(x, (z)).
Multiple Periodic Solutions
196
Lemma 2.3. Suppose that xo is a nondegenerate 27r-periodic solution of (0.2), i.e., the linearized equation
-Ji = d2I1(t,xo(t))z,
z(0) = z(21r),
(2.10)
has no Floquet multiplier 1, then the correspondence zo E Ho is a nondegenerate critical point of a(z).
Proof. Since dF(xo)z = d2H (t,xo(t)) z(t) d z E H,
0 l! a (A - dF(xo)), because I is not a Floquet multiplier of (2.10). And since a (A - dF(xo)) consists of eigenvalues, (A - dF(uo))-' exists and is bounded. However, d2a(zo) = (A - dF(xo))dx(zo) where dx(zo) = idrfo + A(zo),
hence d2a(zo) must be invertible, i.e., zo is nondegenerate. Now we turn to a result which is concerned with the existence of at least two nontrivial periodic solutions. Theorem 2.3. Suppose that H E C2(1l!t' x 1182n, R') satisfies the following conditions: (1)
There exist constants a < 0 such that
aI < d2II(t, z) < /3I `d (t, z) E Ilk' X Il82n.
(2) Let jo,jo + 1,... J, be all integers within [a,#] (without loss of generality, we may assume a, /3 ¢ 7L). Suppose that there exist -y
and C, such that j' < 7 < 13 and H(t, z) >
17IIZI12
- C V (t, z) E R' X
(3) H=(t,O) = 0. 3i E (jo, jl) n7G such that
jI
R2n.
2. Reductions and Periodic Nonlinearities
197
Then the Hamiltonian system (0.2) possesses at least two nontrivial periodic solutions.
Proof. According to the finite dimensional reduction, we turn to the function
a(z) = 2 (Au(z), u(z)) -
J 2* H(t, u(z))dt,
where u(z) = z + u+(z) + u_ (z), z E Z -°- Ho, and u+(z) E Hf. Since 2* a(z) =
2 (Aw, w) - J
H(t, w)dt
o
+ { 2 [(Au(z), u(z)) - (Aw, w)] - (2*[H(t, u(z)) - H(t, w)]dt .,
where w = z + u_ (z), and the terms in the bracket are equal to
-
f2* 0
(F(su+
j2W f2W
- w), u+) ds + 2 (Au(z), u+)
(dH(t, au+ + w)u+) s ds dt - (Au, u) < 0
by condition (1), we obtain 12w
a(z) < 1(Aw, w) <
H(t, w)dt
2(27rj1-7)IIwII2+C- -oo as IIzII -' oo
using condition (2). Thus the function -a(z) is bounded from below and satisfies the (PS) condition.
In order to apply the three critical points theorem we claim that 9 is neither a minimum nor degenerate. In fact, using condition (3), it follows from Lemma 2.2 that 2 ((A - XI)z, z) + where (A,
0(II2112) < a(z) < 2 ((A - 31)z, z) + o(112112),
C (j, j + 1), as IIz1I --+ 0. The theorem is proved.
Remark 2.2. Saddle point reduction was first applied to Hamiltonian systems by Amann Zehnder [AmZI]. Theorem 2.3 is due to Chang [Chal].
Multiple Periodic Solutions
198
2.3. Periodic Nonlinearity A special class of Hamiltonian systems (0.2), in which the function H(t, q; p), q = (ql, q2, , qn), P = (p1, P2, , Pn), is periodic in some of its variables, , 1 < r < s < T, provides a possible way say q1, ... , q P1, , Pr, Pa+1,
of gaining more solutions. This is due to the fact that the Hamiltonian function H, and then the functional J, is invariant under certain translation groups; therefore, the quotient space contains certain tori. We start with an abstract theorem:
Theorem 2.4. Under assumptions (A), (F) and (D), we assume that (P) 3e,,... , er E ker A, they are linearly independent, and 3 (Ti,... Tr) E R1, such that
(x+E7niTiei)
V(m1i...,mr)EZr, VxEH.
4t(x),
=1
(LL) lt(x) -+ ±oo if Jxj -- oo V x E ker A fl span{e1, ... , er}1. Then the equation
Ax-V(x)=0
has at least r + 1 distinct solutions; and if all solutions are nondegenerate, then there are at least 2' distinct solutions.
Proof. A saddle point reduction procedure is applied. Consider the function on the finite dimensional space Vo defined below:
a(z) = 2 (Ax(z),x(z)) - 1(x(z)). We shall prove that
1. xf (z+E=1Tie,)
X± (z),
V z E Vo.
In fact, r
P± F,
(x++x_+z+ei)
=P±FF(x++x_+z),
j=1
and therefore
xf(z) = xf
(z+Eiei). i=1
2. a (z + >j=1 Tjej) = a(z).
2. Reductions and Periodic Nonlinearities
199
Claim.
a
(z+Tiei)
=
(Ax(z+Tjei)ix(z+Tiei)) f=1 1=1
-+(x(z+±Tjei)) j
=
r
= 2(Ax(z),x(z)+Tjej
I
-i(x(z)+1: jej)
j=
\\
j=1
= 2 (Ax(z), x(z)) - 4(x(z)) = a(z). 3. a satisfies the (PS) condition on T' x (Y ® N(A)1) fl Vo where Y = N(A) fl span{ei,... , e,) 1-, and T' is the r-torus defined by
IRr/(T1Z1 x ... x TrZ'). Claim. Suppose that {zk} is a sequence along which {a(zk)} is bounded, and Ilda(zk)II = o(1). According to (2.5),
I Ax(zk) - F(x(zk))IH = o(1) Let Q be the orthogonal projection onto Y, which is considered to be a
subspace of the Hilbert space K = Y ® N(A)1. Thus on the space IC,
(1- Q)x(zk) = A-1(1- Q)F(x(zk)) + o(1), and since F is bounded, 11(1- Q)x(zk)II is bounded. Noticing
-t(Qx(zk)) = 4(x(zk)) - J (F(xe(zk)), (1- Q)x(zk))dt 1
0
= a(Z11)
- 2 (Ax(zk),x(zk)) - J
0
i(F(xt(zk)), (1 - Q)x(zk))dt,
where
xt(z) = ((1 - t)1 + tQ) x(z), and
-
(Ax(zk), x(zk)) = (Ax(zk), (1 Q)x(zk)) = (F(x(zk)) + o(1), (1 - Q)x(zk)),
200
Multiple Periodic Solutions
4i(Qx(zk)) must be bounded. According to condition (LL), Qx(zk) is bounded. The compactness of zk now follows from the boundedness of x(zk) and the finiteness of the dimension of Vo. 4. If we decompose Vo into span{el,... , er} ® (Y ® N(A)1) n Vo,
z = v + w, (v, w) E span{el,...
G)
and let g(w, v) = 2 (A£(w + v), £(w + v)) - 4?(x(w + v)), where
t; (z) = x+(z) + x-(z),
then g is well-defined on Tr x (Y ® N(A)1) n Vo, and
dg(w, v) = PoF(x(w + v)), which is bounded and then is compact on finite dimensional manifold. The function a(z) now is written in the following form: a(w, v) = 2 (Aw, w) - g(w, v).
Noticing that F is bounded, 11f(z)II is always bounded. If we denote by y the projection of w onto Y, we have + v) + y + v) - 4,(y))
g(y, v) = 2 (Af (y + v), (y + v)) -
The first term and the third term are bounded, therefore g(y, v) -* ±oo as IlyU - oo. The function a(w, v) satisfies all the assumptions of Theorem 5.3 of Chapter
II. Theorem 2.4 is proved, provided cuplength (Tr) = r, and the sum of the Betti numbers of T' is 2r. Now we study the periodic solutions of the Iiamiltonian systems in which the Iiamiltonian functions are periodic in some of the variables. We use the following notations: p, q E Rn, p = (pl, P.),
q = (q1,...
P = (PI,... ,Pr),
qn),
1 < r < s < t < n,
9 = (ql,... ,qr),
P= (Pr+1, ,Pa),
q= (qr+1, ,q.),
P=
9 = (q,+1,.. ,qT),
P+ (PT+1,... p.),
(q,7,+
qn).
2. Reductions and Periodic Nonlinearities
201
We make the following assumptions: (I) A(t), B(t), C(t) and D(t) are symmetric continuous matrix functions on S1, of order (s - r) x (s - r), (T - s) x (T - s), (n - T) x (n - T) and
(n - T) x (n - T) respectively. Let A = f S, A(t), and B = fs, B(t) be invertible. (II) H E C2(S' x R2n, R') is periodic in the following variables 13, ij, p, and d2H is bounded. (C(t) ® D(t))) where (III) Let span{rp1i ... , rp,n} = ker
.=
0
In-T
-In-T 0
and W1, ... V n are linearly independent, and m
H
t, L.
I'm
±00
as
--+00.
j=1
(IV) c, d E C(S1, RT ), with c = (c1j ,... , CT), d = (d1, ... , dT) and
f c; (t) =
d1(t) = 0,
1,...,r,s+1,...,T, j=1,...,s. We define a Hamiltonian function as follows:
H(t, p, q) = 2 A(t)p p'+ 2 B(t)q- q + 2 (C(t)p }3 + D(t)4-4) T
+ > (c; (t)pt + d; (t)q,) + F1 (t, p, q). :=1
Theorem 2.5. Under conditions (I)-(IV), the Hamiltonian system (0.2)
-Jdtz=H2(t,z), tES1, has at least r + T + 1 periodic solutions, and if all solutions have no Floquet multiplier 1, then (0.2) has at least 2"+T periodic solutions.
Proof. Let 0
A(t) =
C(t) 0 0
B(t)
D(t) /
202
Multiple Periodic Solutions
and let
(_J
at
- A(t)) _ (_i) ® (-,id - (A(t) o)) ®(-Jdt
A=
e (-J dt
-
(C(t)
- (0 B(t)))
D(t))) '
where the subscripts on J coincide with those on p. We have
(P,9 E ker
(-Jdt
- (A(t) 0))
q = A(t)p
p=0, q = fo A(s)ds c`+ d,
{p=F
with 4(2ir) = q(0),
(i.e., with A F = 0). According to assumption I, F = 0. We have ker
(-J
dt - (A(t) o)) = {(0'd) I
WE R°-' } °-° R°-r.
Similarly,
- (0 B(t)) _ {(6,0) I C E D8T-°}
ker(-Jdt
RT-°.
Thus
ker(A) = RZr ®R°-r ®RT-° ® span{coi,... , pm }. Let
T Js
j H(t, x(t)) + F fci(t)p1(t) + d;(t)gi(t)] } dt.
l
i=1
J
Then all the assumptions (A), (F), (D), (P) and (LL) are satisfied. The proof is complete. Example 2.1. If the Hamiltonian function H E CZ(S' x RZ",R') is periodic in each variable, then (IIS) has at least 2n + 1 periodic solutions. This is the case where r = s = T = n. This result, related to the Arnold conjecture (cf. Section 5), was first obtained by Conley and Zelinder [CoZl], see also Chang [Cha5].
203
3. Singular Potentials
Example 2.2. If H E C2(S' x R2",R'), where H is periodic in the
components of q, and that there is an R > 0 such that for IpI > R,
H(t,p,q) = I M a symmetric nonsingular time independent matrix, then the corresponding (HS) possesses at least N + 1 distinct periodic solutions.
This is the case where r = 0, 5 = T = n. This is a result obtained by Conley and Zehnder (CoZ1); see also P.H. Rabinowitz [Rab6J.
Remark 2.3.
Periodic nonlinearity has been studied by many
authors: Conley-Zehnder (CoZ1], Franks (Fla1J, Mawhin [Maw2], Mawhin-
Willems [MaW1], Li (Lil], Rabinowitz (Rab6), Pucci-Serrin (PS1-2). Fonda-Mawhin [FoM1] and Chang (Cha9). Theorems 2.4 and 2.5 are due
to Chang. The condition H E C2 can be weakened to H E C1, cf. Liu [Liu4].
3. Singular Potentials Most Hamiltonian systems interested in mechanics have singularities in their potentials. Let 11 be an open subset in 1R" with compact complement C = 1R" \ fl, n > 2. Find X(-) E C2([0,27r], fl) satisfying
{
x(t) = grad,, V(t, x(t)), x(0) = x(2ir),
x(0) = :b (27r),
(3.1)
where V E C' ((0, 2ir) x 11, R1) is assumed to be 2a-periodic in t, with additional conditions: (A1) There exists Ro > 0 such that sup {IV(t,x)I + JIV.(t,x)IIE^ I (t,x) E (0,2x) x (R" \ B&)} < +oo.
(A2) There is a neighborhood U of C in R" such that V (t, x) >
d2 (A
C)
for (t, x) E (0, 21r) x U,
where d(x, C) is the distance function to C, and A > 0 is a constant. The condition (A2) is called the strong force condition, according to W. B. Gordon (Gorl]. For the sake of simplicity, from now on we shall denote the subset of C2((0, 2n), fl), satisfying the 27r-periodic condition, by C2(S', SZ). Similar notations will be used for other 27r-periodic function spaces.
204
Multiple Periodic Solutions
We shall study the problem (3.1) by critical point theory. Let us introduce an open set of the Hilbert space H1(S1,Il.') as follows:
A11 = {x E H1(S', R") I x(t) E 1, `d t E S1 } . This is the loop space on S2. Let us define
J(x) = f f2I Ili(t)112. +V(t,x(t))} dt
(3.2)
on A'Sl, the Euler equation for J is (3.1). In order to apply critical point theory on the open set A'1, one should take care of the boundary behavior of J, i.e., we should know what happens if x tends to O(A'0).
Lemma 3.1. Assume (A1) and (A2). Let {xk} C A'! and xk - x weakly in H1(Sl, R"), with x E O(A'cl). Then J(xk) -' +00Proof. It suffices to prove 2a
V (t, xk (t)) dt -+ +oo. 10
Moreover, since V(t,x) is bounded from below, it remains to prove that there is an interval (a, b) C (0, 27r] such that f, V(t,xk(t))dt - +oo. By definition, x E O(A1 ft) means that there is t' E 10,27r] such that x(t') E Oft. According to (A1) and (A2), there is a constant B > 0 such that V (t, x) >
(x,
C)
d2
-B
hence hence t'+6
V (t, x(t))dt > L
(Ox(t)
B dt
-A
V 6 > 0. However, we have 2w
Ux(t) - x(t')IIB- < It - t'1"2 Uo
II1(t)II ffidt
l
1/2
from the Schwarz's inequality; thus
fV (t, x(t)) dt = +oo.
(3.3)
Since the embedding H1(Sl,ff T) ti C(S1,llk") is compact, we have Max { Ilx(t) - xk(t)IIB^ I t E S1 }
0
after omitting a subsequence. Consequently, t'+6
V (t, xk (t)) dt - +oo,
provided by Fatou's Lemma and (3.3).
as k -t oo,
3. Singular Potentials
205
Lemma 3.2. Assume (A1) and (A2); then there is a constant co depending on the C' norm of the function V on S' x (1Rn \ BR, ), such that J satisfies the (PS)c condition for c > co. Proof. Assume that {xk } C A41 satisfies
J(xk) - c,
(3.4)
and
dJ(xk) = xk +K(grad.V(.,xk(')) - xk) -+ 0 in H'(S',IR°),
(3.5)
where
K = (id - dt
: L2(S', iR") - II' (S', IlF") is Compact.
We shall prove the subconvergence of {xk} in A'fl. Since V is bounded below, (3.4) implies a constant Cl > 0 such that 12*
(3.6)
IXk12dt < Cl. 0
Let lk = 2* fo !xk(t)dt. If we can prove that {l;k} is bounded, then {xk} is bounded in H'(S',l ' ). Hence, there is a subsequence xk - x (weakly in H'). Applying Lemma 3.1, we have that x E A'fl and that is bounded. Hence, the strong convergence of {xk} follows from the compactness of K and (3.5). It remains to prove the boundedness of {Ck}. If not, we may assume oo; then for large k, we have that Ilk I 02a
Ilxk(t)II1^ >- Ilfklll^ - (27r
J0
Ilyk(t)Il2dt/ J 1/2 >- Io,
which implies 2w
V (t,xk(t))dtl <_ 27rsup {IV(t,x)I I (t,x) E S' x (R" \ BR.)) 110
From (3.5), we obtain 2*
J0
[Ilxk(t)II2 + V.(t,xk(t))yk(t)] dt < Ill/kIIH'
where Ilk = xk - t;k, for k large.
.
(3.7)
206
Multiple Periodic Solutions
Since ff"yk(t)dt = 0, we have and IIYkIIL3 < IIYkIIn'i
IIykIIH' = IIxkIIL2, hence 27r
f0
-
IIxk(t)II19ndt < 114110 + IIVx(t,xk(t))II L2IIxkII L2.
It follows that IIxk1IL2 s 1 + IIV.'(', xk('))11L2
1+21r
sup IIV=(t,x)IIi,.. (t,x)ES' X (!"\BRp)
(3.8)
Substituting (3.7) and (3.8) into (3.2), and letting 2
co = -
2
sup
1 + 21r
(t,x)ES'x(b"\BR0) + 21r
IIV.(t,x)IIi
IV(t,x)I,
sup
)
(t,=)ES' X (ID° \BRp )
we have J(xk) < co. This is a contradiction.
Lemma 3.3. There exists a sequence of integers qt < q2 < ... < qJ < ... ,
qj / +oo,
such that HQ, (A 19) # 0. O
Proof. Pick a point Po E C and choose R > 0 such that C C BR, we have
IIRn\BRCf CIR"\{P0}. Since IR" \ BR is a deformation retract of IR" \ {po}, A' (IRn \ BR) is a deformation retract of A'(JR" \ {po}), and then A'(Il&n \ BR) is a retract of A' 1. We obtain
H.(A'f2)
H. (A' (R" \ BR)) ® H. (A'S1,A'(IRn \ BR))
H.(A'Sn-') ®II.(A'1l, A'S"-1) from Chapter I, Section 10. According to Bott [Bot], the Poincare series
of A'S"-' is written as to-1
Pt(A'Sn-1) = (1 + t") +
1-
(I + tn)(1 + t2(n-1
with Z2 coefficients. Our conclusion follows.
to-1)
207
3. Singular Potentials
Lemma 3.4. For each b > 0, there exists a finite dimensional singular complex M = Mb such that the level set Jb = {x E A1f) I J(x) < b} is deformed into M.
Proof. According to (A1) and (A2), we have b1 > 0 such that 2,.
IIt(t)II2dt < b,
d x E Jb.
From Lemma 3.1, there exists co = e(b, b,) > 0 such that
d(x(t),C)>co bxEJb VtES1. Let us choose an integer N = Nb > 2a o , and let
ti= tai N, i=0,1,2,...,N. Define a broken line
Y(t) = l 1 - I N(t - t;_1)) x(ti_,) + 2aN(t - tl-1)x(ti), d t = [t;_l, ti], i = 1, 2, ... , N, f o r any x E Jb, and let M = {!E(t) I x E Jb}. The correspondences '- (x(t1), x(t2), ... , x(tN)) defines a homeomorphism between M and a certain open subset of the N-fold product fl x ft x . . . x f2. We shall verify the following. (1) M C Alf2. Indeed, V x E Jb, t/ t1 > t2i 11x(ti) - X02) 111- <-
Jty
Ilx(t)Ildt (f2*
< < M,
1/2
IIx(t)112dt) /21t,
Itl - t2I1/2
- t2I.
Therefore,
d(s(t),C) > d(x(ti),C) - I 1 -
I N(t - ti-1))
> Co - 2aN-lbi\/2 > 0
V t E (ti-1, ti), i = 0, 1, 2,... , N.
11X(ti) - x(4-1)II1^
208
Multiple Periodic Solutions
(2) There exists rl E C ([0,1) x Jb, A' fl) such that 77(0, ) = id, and
tl(1,Jb)=M. We define n as follows:
x(t) rl ( s,x )(t)
for t > 27rs
I 1 - t-t'-' et
=
x t'-
+2xa +t; 1 x(2n3)
for ti-1 < t < 2rrs for t < t;_1 < 2irs < t;
Wt)
then q is the required deformation. We have proven that Jb is deformed into M in the loop space Alit. The proof is finished.
Lemma 3.5. For each q > nN, where N = N 0 is as defined in Lemmas 3.2 and 3.4, set
c = inf max J(x), zEa zElzi
where a E IIq(A'ft) is nontrivial. Then c > co and then c is a critical value of J.
Proof. If not, c < co, then there is a [zJ E a such that Jz(C Jc.+i According to Lemma 3.4, there exists a deformation rl: [0, 11 x J,o+i - A'Sl, nN,,. This implies that such that 17(1, J,,,+ i) C Mao+t, with dim
77(1, I z[) C Mao+,. But rl(1, [zj) E a, and a E Hq(A'St), with q > nN1.. This is impossible.
Theorem 3.1. Under assumptions (A1) and (A2), (3.1) possesses infinitely many 27r-periodic solutions.
Proof. We prove the theorem by contradiction. Assume that there are only finitely many solutions: K = {x,, x2, ... , xi}. Noticing that the nullity dim ker(d2J(xj)) < 2n, y j, let q' > max {nN 0, ind(J, xj) + dim ker(d2J(xi)) I 1 < j < e) , and
b>max{ro,J(x,)P1<j<e}. It follows that
C0(J,xi)=0 bq>q', j=1,2,...,e,
(3.9)
Hq(A'S1,Jca)=0 Vq>q',
(3.10)
and
Consequently,
4. The Multiple Pendulum Equation
209
provided by the Morse inequalities. But
i.:Hq(A1ft) - Hq(A'fl,Jco) is an injection for q >
and the conclusion of Lemma 3.3 contradicts (3.10). The proof is finished.
For autonomous systems, i.e., the potential V is independent of t, in order to single out nonconstant 27r-periodic solutions, we have
Corollary 3.1. Under the assumptions of Theorem 3.1, if, further, V is independent of t, then (3.1) possesses infinitely many 2w-periodic nonconstant solutions, if V" is bounded from below on the critical set K of V.
Proof. For any constant solution x(t) = xo, the Hessian of J at xo reads as
d2J(xo)x = -x + V"(xo)x with periodic boundary conditions, and hence, the Morse index and the nullity must be bounded by a constant depending on a, where V"(x) >
V x E K. We conclude that all constant solutions have a bounded order of critical groups. Therefore there must be infinitely many nonconstant solutions.
Remark 3.1. Problem (3.1) was studied by Gordon [Gorl]. The critical point approach was given by Ambrosetti-Coti-Zelati [AmZ1-2] and CotiZelati [Cotl]. Theorem 3.1 improves the results in [AmZ1-2] considerably, where assumption (A1) was replaced by much stronger conditions: I V (t, x) I , Ilgrad= V (t, x) II -' 0
uniformly in t, as [IxI) -. +oo; and there exists R1 > 0 such that
V(t,x) > 0 V x,
IIxI[ > Ri.
Theorem 3.1 is due to M.Y. Jiang [Jia1-2]. Some related problems of the three body type were recently studied by A. Bahri and P.H. Rabinowitz [BaRl]. By avoiding condition (A2), Bahri and Rabinowitz introduced the concept of generalized solutions. The existence and multiplicity results for generalized solutions were studied in [BaRI]. A most important problem is to ask when the generalized solution is a regular solution.
4. The Multiple Pendulum Equation The Problem. The simple mechanical system consists of double mathematical pendula having lengths (1,12 > 0 and masses m1,m2 > 0, as illustrated in the following figure.
210
Multiple Periodic Solutions
The positions of the system are described by two angle variables Cpl,
2
(ml + m2)elcl + (mi + m2)ele2 cos((pl - W00102 +2m2e2 2+V(co),
where W = (WI, W2), and the potential energy -V(9) is given by
V('G) =9((ml +m2)elcosWI +m2e2cosW2) Let 2.\ be the smaller eigenvalue of the matrix (ml +m2)Qi (ml +m2)Qle2 A > 0. m2e2 ' (ml +m2)ele2 For the sake of simplicity, we assume the constant of gravitation g = 1. We shall add an additional forcing term f = f (t) E R2, which is assumed to depend periodically on time t with period T > 0, and which, moreover, has mean value zero, i.e.,
f(t +T) = f(t)
Vt
rT
f E L2([0,T],R2),
and
J
o
f(t)dt = 0.
(4.1)
We look for periodic solutions W(t) = (cpl (t), cp2(t)), having period T, of the Euler Lagrange equation d
dt Lv(co, 0) - L,v(W, 0) = f (t), or, equivalently, the critical point in H.'(10, T], R2) of the functional J(W) =
f T [L() + f (t)co] dt,
where
H:((0,T],Bt2) = {gyp E H'(10, TI, R2) V(0) = cp(t)) We shall prove the following I
(4.2)
4. The Multiple Pendulum Equation
211
Theorem 4.1. Under (4.1), equation (4.2) possesses at least three periodic solutions having period T. Furthermore, ifryl := (ml +m2)tl -m212 > 0, and if rye := (ml + m2)t1 satisfies }2
T{
tie + IIIIIL2
< lsireky1,
then (4.2) possesses at least four distinct periodic solutions.
The Solution. The first conclusion is not surprising: it follows from the following simple observation:
J(Vi + 2k7r, cp2 + 2tir) = J(Wl, W2) d (k, t) E Z2. Let Wi = (ipi, c ), where T
1
ipi = T, f cpi (t)dt,
and
0
A = dpi -;Pi,
i = 1, 2. Then J is well-defined on M := T2 x H1([O,T], R2), where 1 (I0,TJ,fle2) = S cp E H:([O,T),R2) I
TfT
w(t)dt = 6h}
Lemma 4.1. The functional J is bounded from below. Proof. Indeed, (ml + m2)t210i + (ml + m2)tlt2 cos(tpl - W00102 + 2m2e c4
\(w1+02)
>
It follows that T
J(V) 2! AJO IbI2dt -
T
CT - f @ f dt,
o
where C = (ml +m2)tl +m2t2, and 4i. f = iplfl +@p2f2, f = (fl, f2). The first eigenvalue characterization provides the following estimate: inf IIwIIL II'PII2ta
_- 1=1 2a 7,
2 I
so
IIsoIIL2 s
2 II0IIL3
Consequently, J((p)
?
This finishes the proof.
'\11011 2.
- CT - 2 IIf IIt2lMIIL2.
(4.4)
212
Multiple Periodic Solutions
Lemma 4.2. The functional J satisfies the (PS) condition. Proof. Let {con} be a sequence in M such that J(con) is bounded, and
dJ(
>_ (A(coo)(On - c'o), On - 00) + ((A(Vn) - A(coo))cpn, cbn - 00)
-o(IIWn-cvolIL2) > alIco - (POIIL2 - 0 (IA(wn) - A(coo)I) - 0 (Ikon- cooIIL2)
where (, ) and (, ) denote the duality on T(M) and the inner product on L2 respectively, and
(mI +
(m1 + m2)e,
A(w) = 1
cos(co1
\ (ml + m2)tle2 cos(ci - c2),
- c2)
m2
Since dJ(cpn) --' 0, con - coo - 0 in H; and con -+ coo in C, as n -' oo, we conclude from the inequality that I10n - 0oIIL2 - 0 Therefore, con is convergent in M. This completes the proof.
Lemma 4.3. The circle Sl =
{coEMIi
_
2 = 0,
j
118 1
= 7f,
2E
(T
)
C J7,T
2n
Proof. Directly compute, Vip E S1, we have T
J((o) =
f[-(nl +
MA cos'32) dt < -y1T.
Lemma 4.4. 3 s E (0, 1) such that
S2=1cpEMIp1=0,72E T
i
(2/ )
nJ-,71T0.
4. The Multiple Pendulum Equation
213
Proof. V V E S2, j(W)
T
? . II0II2. + f (7111 + m2)fJ (COs - cos71)dt 0
+T-11 - Ill IIV
IIwIILT
T
L2+T71-72JI i(t)Idt-Ilfllt. 11C311 t.
AIIOII2
0
all0llL2 + T71- (VT72 + IllIILa) IIWIIL2 aIIwIIL. +T71-
2 ('/T72+IIfIILa) 110110-
Hence, if there is a rp E J...,,, T fl S2, then a11011ia + T71-
2 (V72 + IllllL2) II04a < -871T
But, by the assumption on 72 in the theorem, it cannot be true b s E (0, 1). The contradiction proves the lemma.
Note that 7: M = T2 x H --. 72 x {9} defined by 7: (VI,V2)'-' ($1,;V2)
is a deformation retract. We need
Lemma 4.5. Let X be a topological space which contains subsets satisfying
XDYD U
U
X' D Y' D
Z U
Z'.
If Z is a strong deformation retract of X, and if Z' is a strong deformation retract of X', then the inclusion map j: (Z, Z') - (Y, Y') induces a monomorphism
j.:H.(Z,Z') -' H.(Y,Y')
in homology and an epimorphism
j':H'(Y)
H* (Z)
in the cohomology ring.
Proof. We consider the commutative diagram
H' (Y', Z')
H.+1(X,Z) - H.(Z,Z') 1.1
H.(Y,Y')
-.
H.(X,Z') 71
H.(X,Y')
H. (X', Z')
--
H.(X,Z) (4.7)
214
Multiple Periodic Solutions
where the longest row is the exact sequence of the triple (X, Z, Z') and the longest column is the exact sequence for the triple (X, Y', Z'). The indicated maps are induced by inclusions. By the assumptions H.(X, Z) = 0 and H.(X', Z') = 0.
is a zero map. To prove that j. is injective, assume a E H. (Z, Z') satisfies j. (a) = 0. Then by the commutativity of the rectangle in diagram (4.7), -y o 3(a) = 6 o j. (a) = 0. Therefore /3 is an isomorphism and
Therefore, by the exactness of the longest column in (4.7) there exists an a E If, (Y', Z') such that i7(u) = /3(a). By the commutativity of the triangle in (4.7) and the property of , we have q(N) = t; o £(a) = 0, and since is an isomorphism, we conclude a = /3-1 o ij(a) = 0 as claimed. In order to prove (4.6) we consider the commutative diagram
H'(X,Z) --- H'(X)
--. H'(Z) '''j
11'+'(X,Z) (4.8)
HA(Y)
where the longest row is the exact sequence for the pair (X, Z) and /3 and ry are homomorphisms induced by inclusions. Since H' (X, Z) = 0, 0 is an isomorphism. If a E H'(Z), then by the commutativity of the triangle in
(4.8) j'(7 o #-'(a)) = a, so that j' is indeed surjective. Proof of Theorem 4.1. 1. The first conclusion follows directly from Corollary 3.4 of Chapter II, because
CL(T2 x H) = 2. 2. As to the second conclusion, we consider two separate pairs: (Al', J_ey,T) and (J-,iT, 0), and we want to prove that there are at least two distinct critical points in each pair. For the pair (J-,, T, 0), Lemmas 4.3 and 4.4 yield
(S' \ {0}) x S' x N D J_.y,T D {7r} x S1 x {B}. Construct a strong deformation retraction
77:[0,11 x(S'\{0})x S' xP-+{7r)xS' x{B}, by
17 (t, (iP1, w2, IPJ, ;2)) = ((1- t)iP1 + tir, 72, (1 - t)P1, (1- t)iv2) .
Apply Lemma 4.5. Then there are a monomorphism j.: H. (S') -, H. (J-7iT)
and an epimorphism j':II'(J-y,T) -+ II'(S').
5. Some Results on Arnold Conjectures
215
We pick two homology classes, 0 # [ail E Hi (S'), i = 0,1, and a cohomology class, 0 34A E H' (S' ), such that
(ao] = [ail no.
Let (zi] = j.[ail, i = 0,1, and w = j'-1((3). Then 0 # [zi] E HO-,,T) for i = 0,1 and 0 w E H' (J_.y,T). Since 1Z21 n w =
(J [a2]) n w= j. ((a2] n j'w)
=j.([a2]no) =j.[a1] = (z1], Corollary 3.4 of Chapter II is used to give at least two critical points in J-7, T. To the pair (M, J_,.y,T), we observe that M D M D T2 x (0), and
(S'\{0})x S' xIi J_,-,,TD{Tr}xS' x{0}. Again, applying Lemma 4.5, there are a monomorphism k.: H. (T2, S')
H.(M,J_,.y,T) and an epimorphism ke:H'(M) -' 11* (r). Similarly, we pick two relative homology classes [bi] E Hi (T2, S'), i = 1, 2, and a cohomology class 0 0 0' E H'(T2), and [b1] = 152] no'. Similarly, let [wi] = k. [6j, i = 1, 2, and w' = k'-' (Q'); we have [w1 ] = [w2] n w'. Then we use Theorem 3.4 of Chapter II to obtain at least two critical points in (M, J-.,y, T) In summary, we have proven that there are at least four distinct solutions.
Remark 4.1. The conclusion of Theorem 4.1 was first obtained by Fournier and Willem [Fowl] by a relative category method.
The above method enabled Chang, Long and Zehnder [CLZ1] to extend Theorem 4.1 to a n-pendulum problem. Under suitable parameters, they obtained 2" solutions. For a more general consideration, cf. Felmer [Fell].
5. Some Results on Arnold Conjectures
5.1. The Conjectures Let M be a compact symplectic manifold with a symplectic form w, i.e., a closed nondegenerate 2-form. Let h: ]ff1 x M ]R' be a time dependent smooth function. We call it the Hamiltonian. Supposing h is 27r-periodic in t, we associate a family of vector fields Xi on M, defined by
Xi) = dhi,
Multiple Periodic Solutions
216
where Xt is called the Hamiltonian vector field associated with ht. We consider the Hamiltonian system Ot = Xt(cot),
(po = id,
(5.1)
which defines a family of symplectic diffeomorphisms.
Arnold's first conjecture is concerned with the fixed point of the symplectic diffeomorphism W2,,. Namely,
(AC1) W2,, has at least as many fixed points as a function on M has critical points. Let CR(M) be the minimum number of critical points that a function on M must have, and CRN(M) the minimum number if all are nondegenerate. Clearly,
CR(M) > CL(M) + 1 and
CRN(M) > SB(M), where CL is the abbreviation of cuplength, and SB the sum of Betti numbers.
According to Conley-Zehnder [CoZ1], (AC1) is rewritten as #
CL(M) + 1 SB(M)
if V2,,(M) rh M at Fix(V2x).
This conjecture is somewhat related to Poincarc's last theorem: Let W: D1 x
S1 -+ D' x S' be an area preserving homeomorphism such that W(p,q) = (f(p,q),q + g(p, q))
(p, q) E D' X S1
(5.2)
where f, g are 27r-periodic in q, and for all q E S1, f (±1, q) = ±1, g(1, q) > 0 and g(-1, q) < 0. Then w has at least two fixed points, or, geometrically speaking, for an area preserving homeomorphism on an annulus, if it twists on the boundary, then it has at least two fixed points. Indeed, the symplectic diffeomorphism W2a preserves area (in the case M = D1 x S1). If W2 is written as (5.2), then the condition hq(t, ±1, q) = 0
(5.3)
php>0 for p=±1
(5.4)
implies f(±1,q) = f1, and
implies g(-1,q) < 0 and g(+1,q) > 0 V q.
217
5. Some Results on Arnold Conjectures
The symplectic diffeomorphism W2,r induced from a Hamiltonian h E C2(1R1 x D1 x S1, 1R1), 27r-periodic in t, satisfying (5.3) and (5.4), satisfies the hypotheses of Poincare's last theorem.
On the other hand, there is a one-to-one correspondence between the fixed points of W2w and the periodic solutions of the Hamiltonian system
z(t) = Xt(z(t)) { z(0) = z(27r)
This relationship provided that rp2,. is the Poincar6 map z(21r) = enables us to reduce our study of (AC1) to an estimate of the number of periodic solutions of Hamiltonian systems.
We turn to the second conjecture. A submanifold L C M is called a Lagrangian submanifold if wx(t,tl) = 0 V x E L, V C, 17 E T=(L), the tangent space of L at x, and dim L = dim M (a symplectic manifold M is of even dimension). Arnold conjectured: (AC2) For any Lagrangian submanifold L,
CL(L) + 1
# (L n V2*(L)) >_ { SB(L)
if L fi 02,,(L)
where WU is defined in (5.1). We now give some examples of the Lagrangian submanifolds. Example 5.1. M = 1R2n, n
wo=EdxjAdxn+j. j=1
Then (M, wo) is a symplectic manifold, and
L = i (x1, x2, ... , x20 E
12n
I xn+1 = xn+2 = ... =x2n = 0)
is a Lagrangian submanifold.
Example 5.2. M = Ten, the 2n torus, with the canonical symplectic form (5.6). Then (M,wo) is a compact symplectic manifold, and L = (X 1, x2, ... , x2n) E R
2n/Z2n
I xn+1 = xn+2 =-= X2n = 0)
(= Tn) the n-torus, is a Lagrangian submanifold.
Example 5.3. M = Cl" = S2n+1/S1, the complex projective space. It is defined as follows: First, we imbed S2n+1 into the complex space Cn+1 Stn+1
l (zl, z2,
, zn+1) E Cn+1 I IZ1 I2 + Iz2I2 + ... + Izn+1
I2
= 11.
218
Multiple Periodic Solutions
A group action S1 on Cn+t is defined:
5p: (zl,z2,... ,zn+I)'-' a µ(zl,z2,... ,zn+l) V p E 1181/27rZ' -- S1. The complex projective space is just the quotient space of S2n+1 under the group action S'. However, Cn+l has the canonical Hermite form n+1
z,wj dz,wECn+t
(z,w) _ j=1
It induces a symplectic form WO (Z' w) = -Im(z, w).
In real coordinates x, y, u, v E D8n+1, z = x + iy, to = u + iv, where WO is
just the canonical symplectic form on l2(n+t) Noticing that S2n+1
Cn+ l
1-
CPn
where rr: S2n+t - CPn is the projection z '--+ [z], the equivalence class under the group action S1, and is S2n+t -' Cn+1 is the imbedding, we define a symplectic form w on CPn as follows: ir'w = i'wo. It is well defined, because wo is equivariant under the group action S'. Looking at the symplectic manifold (CPn, w) in this way, the submanifold
L= {[z] E CPn I z E [z], z= x+ iy, y= B) is diffeomorphic to the real projective space 118Pn, and is easily verified to be a Lagrangian submanifold.
5.2. The Fixed Point Conjecture on (T 2n, WO) Theorem 5.1. (AC1) is true for (T2n, wo), i.e., there are at least 2n+ 1 fixed points for cp2ir i and at least 22n fixed points if co2,, (T2n) is transversal
to T2n at all its fixed points.
Proof. As we mentioned before, the problem is reduced to finding the number of 2a-periodic solutions of equation (5.5). Since WO is canonical, and R2n is the covering space of T2n, one may extend the Hatniltonian from T2n = l2n/21rZ' to R2n by II E C2(I1 x R2n,R1), function satisfying ( H(t, z) = II (t, z + 2rrej), S` 11(t, z) = h(t, z)
j = 1, 2,... , 2n, `d (t, z) E 1181 x T2n,
5. Some Results on Arnold Conjectures
219
where {e j I I < j < 2n} is the orthonormal basis in R2n.
Noticing that the Hamiltonian system induced by H and the canonical symplectic form w reads as
-Ji = H.(t,z),
(5.7)
this is the equation we have studied so far. Each solution of (5.7) with the boundary condition z(27r) = z(O) + 2,rka for some ko E 7L2n,
corresponds to a 27r-periodic solution [z] of (5.5) on Ten. Moreover, two such solutions zl, z2 are in the same class [z] if and only if there exists k E Z2" such that z2(t) = zi(t) + 27rk.
Therefore, if there are two distinct 27r-periodic solutions z2 of (5.7) satisfying z, (0) = z, (2ir), j = 1, 2, then they must correspond to distinct classes
(zi], j = 1,2, in Ten. Now since H is of periodic nonlinearity, we apply Theorem 2.5 and conclude that there are at least (2n + 1) (or 22n) distinct 2ir periodic solutions of (5.7) (if all these solutions do not have Floquet multiplier I respectively). The proof is complete.
Let us return to the extension of Poincare's last theorem.
Theorem 5.2. Let It E C2(Ri x B" X R",R1) be 27r-periodic in t and q E R". Assume that hq(t, p, q) = 0 and (p, hp(t, p, q)) > 0 whenever p E OB", where B" is the unit ball in R". Then the Hamiltonian system
-Jz = h2 (t, z),
z = (p, q)
(5.8)
possesses at least n + 1 (or 2") distinct 2w-periodic solutions (if all these solutions do not have Floquet multiplier 1 respectively). Proof. Since h is 27r-periodic in t and q, we may restrict ourselves on the compact set (t, q) E Si x T". Rom (p, hp(t, p, q)) > 0 whenever p E OB", we have 0 < b < e/2, such that (pi, hp(t, p2, q)) > 0, and (pi, p2) > 0 for 1 - e < Ip; I < 1, i = 1, 2 and IPi - P2I < e, and that I hq (t, p, q) I < z
for I - 26 < IpI < 1 and (t, q) E Si x T". Let us define a Hamiltonian H E C2(Ri X R2n,Ri) being 27r-periodic in t and q, as follows: H(t, p, q) = (1 - P(IPI )) h(t,p, q) +
where p E C°°(R+) satisfies 0 < p < 1, P(s)
1
if s > 1
0
if 8 <1-6,
P(IPI)CkIPI2,
220
Multiple Periodic Solutions
and 0 < P' (s) < 6 , and a ¢ Z is chosen such that a >
Max h(t, p, q).
Consequently,
(pi,Ifv(t,p2,q)) = (pi, (I -P(IP2I)hp(t,P2,q) + [P (IP2I) (aJP2I2
- h(t,p2, q)) + 2aIp2I2P(IP2I)) ICI) > 0
Vi-E
-Ji(t) = Hs (t, z(t)) ,
z = (p, q),
(5.9)
and claim that (5.8) and (5.9) have the same 21r-periodic solutions. Indeed, we conclude that 1. (5.9) has no 27r-periodic solutions z(t) = (p(t), q(t)) such that z(t) ¢ B" x lE&" for some t. If not, with no loss of generality, we may assume p(0) ¢ W n. In a neighborhood of (p(0), q(0)) we have
p=-HQ=0 1 q-Hp=2ap, so the solution must be (p(0), q(0) + 2apt), which cannot be a periodic solution. Moreover,
2. (5.9) has no 27r-periodic solution z(t) = (p(t),q(t)) such that z(t)
Bl6 x 1l8" for some t. Otherwise, from 1 - 6 < Ip(0)I < 1 and IjjI = IH9I z , it follows that I - 26 < Ip(t)I < 1 and Ip(t,) - p(t2)I < E. Thus 2 27r
0 < (P(0), \\
Ifa(t, p(t), q(t))dt) _ (P(0), q(27r) - q(0)) =
0.
o
This is a contradiction. However, according to Example 2.2, (5.9) has at least n+ 1 (or 2") distinct 27r-periodic solutions (if all solutions do not have Floquet multiplier I respectively). This proves the theorem.
5.3. The Lagrange Intersections for ((CP",1P P") We turn to (AC2), where the symplectic manifold M = CP" and the Lagrange submanifold L is taken to be RP", as in Example 5.3. Since
CL(RP") = SB(IRP") = n + 1, it is not necessary to consider the transversal case. We have
5. Some Results on Arnold Conjectures
221
Theorem 5.3. (AC2) is true for (CPn, RPn), i.e., there are at least fl RP".
(n + 1) intersections of
We reduce the intersection problem to a critical point problem by several steps:
Step 1. Reduction to a boundary value problem. By definition, pi E if and only if 3 po E RP" such that pl = SP2, (po) E RP",
lltPn
i.e., the equation
ti(t) = Xt(w(t))
{
w(0) = po, w(21r) = p1,
po,pi E IRPn
(5.10)
possesses a solution w(t). Obviously, there is a one-to-one correspondence between the intersection points and the solutions of (5.10): w(t) = wepo = w1 o cpz* pl. The problem is transferred to find the number of distinct solutions of the BVP (5.10).
Step 2. Reduction to Hamiltonian systems on Cn+l. Note that
cn+1 L s2n+1 * 4 S2n+1/Sl = Cpn where 71 is the Hopf fibration. We can associate with h: R' x CP" --+ R' a
function H: R' x C"+' -. R' satisfying (1) H(t, efl'z) = H(t, z) d z = (Zl, Z2, ... , zn+1) E C"+1;
(2) H(t,
h(t, ) o 7r;
(3) H(t, ) is positively 2-homogeneous in a neighborhood of the unit ball; (4)
is C' and C'-bounded.
With no loss of generality, we may assume that h(t, ) > 0, so that H(t, z) > 0 V z 76 0, and also that H(t, 0) = 0 and Hs (t, 0) = 0. We turn to consider a new Hamiltonian system:
J1(t) = Hs(t, z(t)) + Az(t)
{ z(0), z(2w) E Rn+1 n S2n+1 ,
(5.11)
where __
0
-In+1
In+l 0 and A is a Lagrange multiplier. It plays a role here as an eigenvalue.
Lemma 5.1. Let z be a solution of (5.11). Then z(t) E S2"+', and a(e'Atz(t)) solves (5.10).
Proof. We consider the derivative of the norm: d Iz(t)12 = 2(z(t),1(t))
= 2(z(t), JIf: (t, z(t))),
222
Multiple Periodic Solutions
, ) are the norm and the inner product on C"+' = nt2(n+1) respectively. But 0 = d;-- H(t,e'µz)lµ_o = (HZ(t),z(t)),-Jz(t)) because H is S' invariant. Therefore Iz(t)I = const. Here and in the following, we
where I I and (
write either z = (x, y) E 1P4n+1 x lkn+1 or z = x + iy E C"+1, if there is no confusion.
As to the second conclusion, we observe that the symplectic structure w on CPn is defined by ir*w = i*wo,
where wo is the canonical symplectic structure on Cn+1 Thus dht, and
Xt) _
where Y = JHZ(t, ). Since d(ht o ir) = w(., Xt), therefore Xt = ir.YY.
On the other hand, letting z(t) = e'Atz(t), (5.11) is rewritten as
-Jz(t) = lfZ (t, i(t)) . By the uniqueness of initial value problems, we have w(t) = 7rZ(t) = 7r[e' At Z(01'
where w(t) is defined by (5.10). Therefore x(e'Atz(t)) solves (5.10). The proof is finished.
Lemma 5.2. Let (z1,.\,) and (22,.X2) be two solutions of (5.11). Then 7r (e'A'tz'(t)) = 7r (e'A2tz2(t)) implies Al = .\2 (mod 7r).
Proof. First, we claim that if z',z solves
-J,i = HZ(t,z)
(5.12)
and irk' = irz , then 3,u E 11R' /21rZ, such that z`1(t) = e''`z (t).
Indeed, by definition, 3 a function u(t) such that z'1(t) = e"'(t)z (t). Substituting into equation (5.12), we have e+a(t)z (t)+ie'N(t)µ(t)z (t) = e'v(t)Jlls(t,Z-2(t))
223
5. Some Results on Arnold Conjectures
which implies either that 3 to such that r (to) = 0, so that i2(t) = 0, d t, and we can choose µ(t) to be constant, or that µ(t) = 0 V t. And again we have p(t) = const. Next, we have
e'µz2(0) = z1(0), ei(v+a2)z2(2,r)
= e"z1(2ir).
Since z{(2jir) E Rn+1 fl S2n+1 (i = 1, 2, j = 0,1), e+'A must be real. Consequently,
and then ei(.% 1-a2)
A2 = A1(mod 7r).
Let us define an operator on L2(I0,2iJ,Cn+1),
with domain
D(A) = {z E H1(I0, 21rl, Cn+1) I z(0), z(27r) E Rn+1 }
.
Lemma 5.3. The operator A is self adjoint, with spectrum a(A) = !7L; has multiplicity (n + 1). each eigenvalue zk Proof. Indeed, we have the following spectral decomposition: L 2(I(), 27t), Cn+1) =
®s pan{cos 2 kte7+isin 21 ktej+n+1 1
j = 1,2,... ,n+1},
kEZ
where {ej + ie j+n+l I j = 1, 2,... , n + 1) is the basis of Cn+1
Step 3. Reduction to a variational problem. According to the spectral decomposition, we decompose A into the positive, zero and negative parts:
A = A+ + A° - A-, where A+ and A- are positive operators on their associated subspaces. Let us introduce a Hilbert space E = D(IA11/2), the domain of the square root of IAI, with norm 3)12
IIzIIE = (IzI 2 + II IAI1h12zII
In the following, if there is no confusion, we denote by (, ) and by II' II the L2 inner product and norm, respectively. Let us define J(z) = Z (II(A+)1h12zII2 - II(A )1'2zII2)
on the manifold S = (z r= E I IIzII1,2 = 1).
-
j2W
H(t, z(t))dt
224
Multiple Periodic Solutions
Lemma 5.4. Suppose that zo is a critical point of J on S with Lagrange multiplier A0. Then (zo, Ao) solves (5.11), and J(zo) = A0.
Proof. Letting (zo, A0) be the critical point and the Lagrange multiplier, we have ((A+)1/2zo, (A+ )1/2Z)
- ((A )1/2zo,(A )1/2z) - (II:
f
(t, zo) + 2Aozo, z)dt = 0 (5.13)
V z E E, and it follows that Po(I1=(t, zo) + Azo) = 0,
where PO is the orthogonal projection onto the space associated with eigenvalue 0. Consequently, (Af )1/2zo = ±(At)1/2(IAI-1)(II=(t, zo) + Azo), and therefore
zo E D(IAI) C D(A).
Then the weak solution equation becomes Azo = I1 (t, zo) + Aozo, so that zo solves (5.11).
In particular, if we choose z = zo in (5.13), then 2w
(IIA+)1/2zoII2
- f (H. (t, zo), zo) dt = 0.
- II(A )1/2zoII2) - 2(Aozo, zo)
0
Since zo(t) E S2n+1 (Lemma 5.1) d t E 10, 27r), and 11 (t, z) is positively 2-homogeneous in a neighborhood of the unit ball, we have (11=(t, zo), zo) = 211(t, zo).
Thus 1\0 = 2 (II(A+)1/2z(,II2 - II(A )1/2zoII2)
- f 2, H(t,zo)dt = J(zo).
Remark 5.1. We take the working space E = D(IAI1/2) with norm IIIAI"2zIIi2)1/2
IIZIIE = (IIzIILT +
rather than the space 111/2(10,27r),C"41)
since the trace operator is not well-defined on H1/2, so that the boundary value condition cannot be formulated in H1/2.
5. Some Results on Arnold Conjectures
225
Finally, we observe that there is a natural symmetry for the functional.
Namely, J(-z) = J(z). Indeed, the function H(t,z) is S1-invariant, so that H(t, -z) = H(t, eiwz) = H(t, z). Moreover the boundary value condition D(A) is also invariant with respect
to this group action. Consequently, J is well defined on the space P = S/Z2, where S is the unit L2 sphere in E. Returning to the original problem, we point out that we are not concerned with how many distinct critical values of J there are, but how many distinct critical points there are associated with critical values in an interval with length < 7r. After the preparation, we shall give a proof of Theorem 5.3. The Galerkin approximation will be applied. Let Ek =
span { cos 2 ttej + i sin 2ltej+n+1
I
1 = 1, 2, ... n + 1
l
IlI
111
Pk=PnEk, and
Jk = Jlp`. Then dim Ek = (2k + 1)(n + 1), dim Pk = (2k + 1)(n + 1) - 1. Lemma 5.5. J satisfies (PS)' with r e s p e c t to {Pk I k = 1, 2, ... Proof. Suppose that zk E Pk is a sequence satisfying
H(t, zk)dtl < C1
IJ(zk)I = I I2 (Azk, zk) -JO
(dJk(zk), w) E = (Azk, w) = o(IIwIIE)
10
(5.14)
(H:(t, Zk) + Akzk, w) dt
0
d w E E.
(5.15)
We decompose zk into zk +zk+zk , according to the spectral decomposition, to positive, zero, negative eigenvalues, respectively.
First, setting w = zk in (5.15), we have IIZk IIE 5 I'kI+o(Ilzk IIE)+C2 where C2 is a bound for I H. (t, z) 1. Set w = zk in (5.15). Then by (5.14),
141 = (AZk,Zk) - fo 2 *(H:(t,zk),zk)dtI +O(IIzkIIE) < 2C1 + 27rM + o(IIzk1IE),
226
Multiple Periodic Solutions
where M is a bound for IHl on the sphere
S2"+1.
Since
IIzkIIE = IIzkIIL2 < 1,
k and Ilzkll are bounded. Next, we prove that {zk} possesses a convergent subsequence in the E topology.
By the construction of the space E, we see that the injection is D(JAJ) -. D(IAI1/2)
+, -, 0 be the orthogonal projection onto the
is compact. Let 11',
subspaces with positive, negative and zero eigenvalues. Again, from (5.15), we obtain llzk - IAI-111t(Hz(t,zk) + \kZk)IIE = 0(1). Thus zk are subconvergent, and, because fI°E is finite dimensional, we con-
clude that
zk
possesses a convergent subsequence.
The proof is
finished.
Let Pk = Pk n E+, where E+ = I1+E, dim Pk = k(n + 1) - 1. We find singular relative homology classes in H.(Pk,Pk \ PI ,+). Fom the special structure of Pk, there are nontrivial classes: 0 # [ZI] < [z2I < ... < [Zk("+l)I
Lemma 5.6. Set
ck = inf sup Jk(z),
i = 1, 2, ... , k(n + 1).
aElz,l zEIaI
(5.16)
Then
(1) ck are critical values of Jk,
(3) 4(e - 1) - 21ifrr < c,k < qe, for (e - 1)(n + 1) < i < t(n + 1), e = 1, 2, .. , k.
-
-
-
where i11 is the maximum of 11(t, ) on [0, 27r) x S2n+1
Proof. (2) is trivial. We prove (3). On the one hand, `d a E lzi),
IalnP,+1 #0,sothat (e - 1),
sup (Az, z) >
zElal
2
which, combined with 0 < 11(t, z) < Al, yields the left hand inequality. On the other hand, 3 a E [z;) such that lal C PEI, provided by the special structure of Pk, and it follows that (Az, z) 5
e,
2
V z E lal.
5. Some Results on Arnold Conjectures
227
Hence the right hand inequality holds. From (3), [z,), [z2), ... , [zk(n+1)) are nontrivial in (Pk, (Jk)-2M.), (1) follows from the minimax principle. Now we take limits, letting
c; = lim c . Provided by the (PS)' condition, they are all critical values satisfying
-2Ma < cl < ... < ct(n+,) < 4t, t = 1, 2,... .
Moreover, according to the argument in Theorem 4.3 of Chapter II, they correspond to t(n + 1) distinct critical points of J. Finally, we come to the proof of our theorem.
Proof of Theorem 5.3. As we have mentioned before, it remains to prove that there are at least n + 1 distinct critical points associated with critical values in an interval with length < 7r. Indeed, if the conclusion is false, the critical values
-2Ma
k=1,2,...,
so obtained, satisfy #{ci 1 ( 1 - 1 ) / 4 < ca 1.
Suppose T = #{c; I -2Mir < ci < 0}. T is finite, because c;` > 4 t - 2M7r
for
i > t(n + 1), k > t.
It follows that
ci > 4t - 2M7r. Hence, we have the estimate T < 2M(n + 1). Then
m(n + 1) < #{ci I -2M,r < c; < m/4}
dm, which is impossible for m large. This contradiction shows that
#(Lnv,(L)) > n+1, where L = 1RPn. The proof is finished. Remark 5.2. The above method applies equally well when proving
228
Multiple Periodic Solutions
Theorem 5.4. (AC1) is true for (CP", w), i.e., there are at least n + 1 fixed points for
We are satisfied with pointing out the modifications: (1) The boundary value condition in (5.11) is replaced by a periodic condition, i.e., z(O) = z(27r). (2) If two periodic solutions (z',A1) and (z2,A2) on Cn+1 correspond to one solution on CP", then Al = A2 (mod 2ir). (3) Let Tµ be the orthogonal representation on H1/2([0,2rrJ,Cn+1) (= D([A[1/2) for the periodic case) of the Hopf S' action S,,:T,,z =
S is S1 invariant, so is well defined on P = S/S1, and therefore Pk = P fl Ek,
Ek= ®span (cosfte., fsin etej+"+1 Ij=1,2,...,n+1), ltl
is the (2k+ 1 ) (n + 1 ) - 1 dimensional complex projective space, k = 1, 2, ... .
Remark 5.3. The Arnold conjectures were formulated in [Arnl-21. The breakthrough is the pioneer work of Conley-Zehnder [CoZ1J, in which The-
orem 5.1 was proved. Soon after, a series of papers devoted to these problems appeared, cf. Floer [Flol-41, Hofer [Hof3J, Sikarov [Sik1]. The far-reaching contribution of Floer enabled him to solve (AC1) and (AC2) under the conditions ir2(N1) = 0 and ir2(M, L) = 0, respectively. Unfortunately, we have no space in which to introduce his theory. For Theorem 5.2, cf. Szulkin (Szu1J. Theorem 5.3 is due to Givental and Chang-Jiang 1ChJ11, and Theorem 5.4 to Fortune (For1J.
CHAPTER V
Applications to Harmonic Maps and Minimal Surfaces
Geometric variational problems are of one of the most important parts of applications of infinite dimensional Morse theory. The closed geodesic problem, the minimal surface and the constant mean curvature problems, the harmonic map equation, the Yamabe problem and the Yang-Mills equation are not only interesting in themselves, but also for motivations in the development of infinite dimensional Morse theory. Here, however, we are only concerned with a few examples showing how Morse theory is applied, because each topic deserves to be treated in specific books. Readers who are interested in these problems are referred to Klingenberg (KlilJ, M. Atiyah and R. Bott (AtB1], H. Brezis and M. Coron (BrC2J, R. Schoen (ScR1J, C. Taubes (Taul,2] and A. Bahri and M. Coron [BaC1].
1. Harmonic Maps and the Heat Flow Let (M, g) and (N, h) be two smooth compact Riemannian manifolds with
m = dim M and n = dim N. For a map u: M - N, e(u) =
11 (Vu(Z is
called the energy density. In local coordinates, it is written as e(u) =
19'j (x)h,.#(u(x))u°ua, where x = (xi,xz,... ,x,"), u = (u1,u2,... ,u"), u° a" O = 1, 29... , n, i, j = 1, 2,... , m, (ha,#(u)) is the metric h on N, (g`I(x)) is the inverse of the metric g on M, and the convention summation is used. The energy of the map u is defined to be
E(u) = j e(u)d,
(1.1)
where V. is the volume element over M. For a CO° map u, we introduce a CO° deformation Wt t E (-E, c) for some e > 0 with co = a and consider the equation d 0. dt E(Vt )1 a=o = The equation is expressed as follows:
fM (r(u), v)dV, = 0,
H v E COD (u'1T(N)),
Applications to Harmonic. Maps and Minimal Surfaces
230
where r(u) = 'Irace9 Vu, or, in the local coordinates, r(u)°(x) = AMU°(x) + g'j(x) NI A7(u(x))u13(x)u7(x), where Nl pr(u) denotes the Christoffel symbol of the manifold N, AM is the Laplace-Beltrami operator with respect to the metric g, and C1(u_1T(N)) is the set of smooth sections of u-1T(N), i.e., the set of vector fields along It.
In the following we use the shorthand notations:
F(u)(Du, vu) = 9`J Nr#7upu7 and
All = r(11);
then we get the Euler-Lagrange equation Au = 0.
(1.2)
The solution of (1.2) is called a harmonic map. It is easily seen that in the case N = R', M = a domain in W', harmonic maps are harmonic functions.
In the case OM 34 0, harmonic maps are determined by the boundary value conditions. For instance, given 0 E COO(VM, N), we find maps u E CO°(M, N) satisfying (1.2) and the boundary value condition uIenf ='G'
If we want to study the existence (and multiplicity) problem for the harmonic maps from the critical point of view, then we should first find a suitable framework, Ililbert Riemannian manifold or Banach Finsler manifold, on which to develop our theory. The Sobolev function spaces Wt(f2) are extended to spaces of maps as follows: Provided by the Whitney-Nash embedding theorem, the target manifold N can be isometrically imbedded into 1l&k for some k, and then one naturally defines
Wt(M,N) _ {u E W'(M,IPik) I u(x) E N a.e. x E M), where 1 < p < oo, and a is any integer. Since the energy functional is quadratic, it seems that Wz (M, N) is a natural candidate. Unfortunately, WZ (M, N) is not a Banach manifold for m > 2. Another reason for rejecting this choice comes from regularity. According to special nonlinearity, one can prove the smoothness of the solution only if we know it is continuous, but WZ (A1, N) is not sufficient to provide a continuous function.
1. Harmonic Maps and the Heat Flow
231
In this sense, we are forced to take a second choice: Cl (M, N) or Wn (M, N), p > 2. A severe problem occurs: The Palais-Smale condition is missingl
Carefully analyzing the role of the Palais-Smale condition in the proof
of the deformation theorems, we find out that it is strongly tied to the gradient flow. We observe, however, that the heat equation for harmonic maps 8ef (t, ) = Of (t, ) (1.3)
f (t, -)Iem = rG( )
produces a flow f (t, ), which depends on the initial data V and reduces the energy: dt E(f
(t, )) = Im (Of (t, ), OBtf (t, ))dV9
- fM (Of (t, ), 8tf (t, ))dV9 +
g'Jho#8efe(t, )f° . ni dS9
)f(t.
)IsdV, < 0.
One expects to replace the gradient flow by the heat flow f (t, ) (= f,,(t, ), to indicate the initial data). This is possible if one can prove the following conclusions: (1) the global existence of the heat flow, i.e., f (t, ) is defined on the whole half-axis t > 0; (2) the limit of f (t, ), as t - oo, should in some sense be a harmonic map;
(3) the flow f,,(t, ), as a map depending on t and the initial value cp, is continuous. Before going into these statements, we introduce some notation:
N) _ {u E C2+,(M, N) I weal = tG},
0 < 'y < 1,
QT = 10,T) x M, for T > 0; Cl+(,/2),2+,(QT, IRk) equals the completion of C°° (QT, DFk) functions under the norm
IllIlc,+
Illllc +
sup
0
(lotf(t, x) - atf(t', X) IC
It - t'I,/'
+ I8 f(t,x) - e=f(t,y)Ic d(x, y), 1
Applications to Harmonic Maps and Minimal Surfaces
232
6VP'2(QT,lil:k) has the same meaning, but under the norm m
I
IIfIIwp.7 = IIfIIL=+Il19tf1lL,+1: Ilax,fIILP, j=1
C1+(7/2),2+7(QT,
N) _ { f E C1+(7/2),2+7 (QT, Rk) I f (t, x) E N }
has the same meaning for WP,2(QT, N).
The proof is fairly long. It is sufficient to point out the main steps and some cnlcial estimates. Step 1.
Local Existence. There exist e > 0 and a unique f E
C1+},2+ry(Qe, N) satisfying (1.4) in QE, and then there is a maximal exis-
tence interval [0,w).
The proof is standard. One uses the inverse function theorem on the map A:
f .-. (atf - Of, f (0, .), f (t, -)Ilo,TJxonf) Bp2(1-1)
iii1.2 (QT, Rk)
LP(QT,
P
x
X
(M, Il8k)
Bp-'°'2 ° ([0, T] x OM, lflk).
for p > 4, where n2(1-1/P)(11'f,lRk) and B1-(1/2P).2-(1/P)([0,T] x OM,llBk) are Besov spaces, the trace spaces of WP,2 on {0} x Al and [0,T] x OM, respectively.
Step 2. Global Existence. If the following a priori estimate holdsFor solutions of (1.3), there exists a constant Co depending on gyp, tb, (M,g)
and (N, h) such that sup IVI(t,x)I < C'o
(1.5)
[0,W) x Af
-then global existence follows.
Indeed, suppose w < +oo. Combining (1.5) with the Schauder estimate for linear parabolic systems (cf. Ladyszenskaya, Solonhikov, Ural'ceva [LSU1]), we get immediately IIf IIc,.3.2+,(QW)
- C., (w), a constant, for any 0 < y < 1.
Thus the solution of the evolution equation (1.3) may be extended beyond the maximal interval [0,w). This is a contradiction. Consequently, w = +00. In order to prove (1.5), we need a blow-up analysis and the following LP estimates.
1. Harmonic Maps and the Heat Flow
233
Step 3. The LP Estimates. Lemma 1.1. Suppose that 1 < p, q < oo and that t i-+ g(t, ) E LP(M) for a.e. t E 10, T]. Assume that
j and
Ilg(t, )IIi,(M)dt < 00,,
Otf - LMf=g in QT,
f (0, ) = 0 on M, f(t,-)IBM =0. Then we have a constant C = Cp,q such that T Ilf(t, )Ilw;(M)dt < C rT IIg(t, .)IILo(M)dt.
(1.6)
Proof. The linear equation is considered an evolution equation associated with the analytic semigroup, whose generator AM is sectorial on the space LP(M) with domain D(OM) = {u E WW(M)IuIaM = 0}.
First we prove (1.6) in the case p = 2. With no loss of generality, we may assume T = +oo, and we may define f and g as 0 fort < 0. Let f '(r, ) and g '(r, ) be the Fourier transforms of f (t, ) and g(t, ) with t respectively. We have
- AM)f (r, ) = g (r, Am
=Am (:r - AM)-' 9 (r, )
The vector-valued Mihlin multiplier theorem (see N. Dunford, J.T. Schwartz IDS1]) provides the inequality
rT J0
IT
r
T
Next, in combining the LQ estimates for the parabolic equation with the above special case, the interpolation inequalities provide (1.6).
234
Applications to Harmonic Maps and Minimal Surfaces
Lemma 1.2. There exists a positive number eo > 0 such that for a solution of the system (E) in a domain 10, TJ x D, where D = B,(xo) n M, for some xo E M, and p > 0, if
<eo
sup tElto.t,lJD
for some to, t i E (0, T), then for any p' E (0, p) and (to, ti) C (to, t l), we have some a = 1 - 1 > 0 and a constant C depending on co, a, p', p, and to, tl, to, t1 only such that sup tEIto',t'J
IIf(t,')IIc'+o(D') < C[1 +
IIV'IIC2+v(HbfnD)
+C
Ito,t,IxD
I Vf I2pdtdV9
for p> 4, where D' = Bo,(xo)nM. Proof. Define a cutoff function VI E C°°(QT), satisfying 0<
x) = {0
(t,x) E [to,t1] x D'
(t, x) 0 [to, i l l x D.
atF - OAfF = r(f)(VF, Vf) - r(f)(fV 1, Vf) - 2Vf - VV1 + f(Ot - Onf)v,, F(to, ) = 0, F(t, )Ionf = WP1
tG
According to the Sobolev embedding theorem (cf. Nikol'ski INik1J) and linear Lt' theory, we have a = 1 - 4/p > 0 such that sup III(t, )Ilc,+.,(D') tEltb.t'l
< sup
IIF(t,.)IIcl+-(D)
tE Ito,t, I
< CQIIFIIwo.2(It(,.t,JXD)
< C[1 +
II0IIC2+-,(a11fnD) + IIVfIILP(Ito,t,JXD)
+ IIVF. VfIILD(Ito,t,JxD)].
1. Harmonic Maps and the Heat Flow
235
However, provided by the Sobolev imbedding theorem together with Lemma
1.1, letting pt = 2p/(p + 1), we have
rt, IIVFIIL,,(D)dt eo
IIF'IIW,,(D)dt eo
< C[1 + II0IICa+-v(BMnD)
+ i:' IIVfII(D)
t,
+ f IIVF ' VIIILP (D)dt] eo
Applying the Holder inequality,
f
t,
11 VF- VfIIi',(D)dt <
eo
r IIVJIIL3(D)IIVFIIL,,,(D)dt eo
< eo f tf IVF(t, x)I2 dV9dt. D
to
For sufficiently small co > 0, we put inequalities (1.8) and (1.9) together and obtain i f I
t IIVF(t,')IIL2,(D) 2
(OMnD) +fo,
IIVAt,')IIL.I (D)dt]. e
Again by the Holder inequality,
IIVF'
:5 IIVFIIL- ' IIVfIIL2 < Ceo [1 + II 1IIC2+,(OMnD) + IIVfIIL2.(Ito,t,lxD)] X
IIVfIIL2P(Ito.t,lxD)
Returning to (1.7), we have sup
IIf(t,')IIC'+o(D') < C[1 + II+'IICa+,(8MnD)
tE Ito,t',I
+
Lemma 1.3. Let w > 0 be finite or infinite. Assume that VT < w, f E Wn'2(QT, N), p > 4, is a solution of (1.3). If there is a relatively open
set D C M and a sequence of intervals Ij c 10,w) with mes(1,) > 6 > 0 such that suPf IVf(t,')I2dVg <eo. tEIj D
236
Applications to Harmonic Maps and Minimal Surfaces
Then for any open subset D' CC D, for any sequence {t,} with tj E Ij and tj -+ w, there is a subsequence tj, such that f (tj,, ) is C' (V, N) convergent to some it E W2 (At, N).
Proof. Since
j IVf(t, )I2dVg < E(W), t
the family of maps { f (tj, )Ij = 1, 2.... } is weakly compact in Ws (M, Q(;k), so that there is a subsequence {t,.} along which f (t,., ) u weakly in
N 'W, Rk) . Starting from (1.10) with p = 2, we obtain a constant, which depends on co, 0 and b, dominating the norms IIVFIIL4(1,xo) Vj. Applying (1.7), is also dominated. Then, the Sobolev embedding theorem implies the boundedness of IIVFIIL2,(, xn) dp > 4. Thus, we have V t E 1j,
II f (t, )IIC'+-(o') < const.
provided by Lemma 1.2. This implies a subsequence It,,} such that f C'-converges to U.
Lemma 1.4. Suppose that f E WW'2(QT, N) V T < w is a solution of (1.3), where p > 4. Then there is a sequence ty- -- w - 0 and a finite number of points {x1, ... , xt} C M such that f(tj,,.) --4 it(.) in C'+a'(A1 \ {x,,... ,x,}, N)
for some ii EH,2(M,N),and 0
(a) M C U° , Br/2(y,) (b) V x E Al, there exist at most h disks B,-(y;) covering x, where h is independent of r. Then I V f (t, _)12 dVg < hE(f (t, )) < hE((p). Hence d t, 3 at most F = 12h o ° I + 1 disks Br(y,), i = 1, 2,... , F, on which
!J,(v,) IVf(t, )12dVg >
.
Fixing t such disks, there is a sequence t; j w - 0 such that 2,
di>F.
1. Harmonic Maps and the Heat Flow
237
However, let cp E Co (Br(y;)) be a cutoff function, 0 < W < 1, rp = 1, on Bak (yi), IVWI < 1. Multiplying the equation (1.3) by h°p(f)8tf°rps, it
follows from Young's inequality that
J
ha9(f)8tf°8t flSrp2dV9ds + JQ
dte(f)w2dV9ds
<- C f IVfl IBtfl IowipdV d9 Q'
< f h°p(f)8ef°8tfocp2dV9ds+ C' JQ, IVfI2d1' ds, J
Q
where Qt, = Qt \ Qt'. Therefore,
L
IVf(t, )I2dV9 < 2 f e(f (t', <
C1
r2
E(f (s, .))ds
t') E(W),
fB. (v, )
which assures a uniform bound b > 0 such that
sup t-t;I
J *.(v,) (Vf(t, )I2dV9 < co
d > t.
We apply Lemma 1.3 to these C1+a, remaining disks. Then there is a sequence
tj, T w-0 such that f (tp, ) is
convergent on M\Uei-1 Bj(y,). Letting r = 2-k, k = 1, 2, ... , by the diagonal process, there is a subsequence, still denoted by {t,-}, so that f (t,,, ) C'+*'-converges on M \ {xl,... , xt}, because the upper bound of the number of exceptional disks is independent of r.
Step 4. On Asymptotic Behavior. Now, we derive conclusion (2) from (1.5) and the a priori estimates.
Lemma 1.5. There exist a harmonic map u and a sequence t2 T +oo, such that f (tj, ) --+ in CI+°' (M, N) for 0 < a' < a.
Proof.
We cover M by small balls Lfi-I Br12(x;) such that Co
mes(Br(x,)) < co. According to (1.5) sup
tElk,k+Il
IIVf(t, )IIL2(B.(=,)) < Eo,
k = 1, 2, ... , i = 1, ... , p. It follows from Lemma 1.2 that sup tEll,00l
Applications to Harmonic Maps and Minimal Surfaces
2..38
From (1.4),
f
Jai so there is a sequence tj / +oo such that
8tf (t,, )
0
in
E(v),
L2(M, Rk).
On the one hand, according to Lemmas 1.3 and 1.4, with IVf(t;,x)I2dVy < co,
it follows that f (t,, ) i1(.) in C'-*°'(r1 \ {x, , ... , xt}, N) with some {xl,... , x,} C M. On the other hand, 1(1,,.) is bounded in C'+°(M, N), ca' < a. We conclude that f (t,, ) - ii in C'+°'(M, N). Thus
J , [g(Vu, Vp,) + r(ii)(Vi , Vii)V,]dV9 = 0 V W, E Co (111, N). By applying the elliptic regularity theorem again, we conclude that i E C2+ry(M, N), and
Step 5. Blow up Analysis. Let m = inf{E(u) I U E IV2 (M, N), inf { E(v) I V : S2 -. N b { +oo
-
Ulam = tu}
harmonic, and nonconstant} if there is no such map.
Lemma 1.6. Suppose that BT = max(t,x)EQ, IVf(t,x)I is not bounded. Then
E(cp) > m + b.
Proof. We may find sequences Tk / w and ak E M such that IV!(Tk,ak)I = m xIVf(Tk,x)I =OTk, k = 1, 2,.... From now on, we write °T simply as Bk. Neglecting subsequences, we may only consider the following two possibilities:
(1) 0k dist(ak,8AI) -. +00, (2) 0k dist(ak, (9M) -. p < +00;
239
1. Harmonic Maps and the Heat Flow
in both cases, we may assume ak --+ a E 7GI.
Take a local chart U of a. Let
Dk=(yER2Iak+k EU) and
Ik = [-BkTk, 02(c.,
Tk)).
Define a function on Ik x Dk as follows:
vk (r, U) _ [ Tk +
r
k , ak + Bk
k = 1, 2, .... Then we see a,uk = Ovk,
(1.11)
and
IVyvk(r, y)I < 1,
max
k = 1, 2, ...
(1.12)
Let
hk(r) =
Il0,vk(r,Y)I2d1/. JD
Then hk(r)>0,andVe>0 10
rj
hk(r)dr <
Tk-e///e
a
dtJ IBtf(t,x)I2dVg M
E(f Tk e- E(f(Tk, )) \\ (
0
as k -+ oo.
Thus, neglecting a subsequence, we may assume
hk(r) -. 0
a.e.
r E [-e, 0[,
i.e., for almost all r E I-e, 0], Iarvk(r,ll)I2dy _ 0.
(1.13)
ID,, 0
In case (1), a E M, and Dk -. R2 in the sense that `d R > 0, 3 ko > 0, the ball BR centered at 0 in R2 is included in Dk for k > ko. On the one hand, by (1.13) , arvk(r*, y)
0,
L2(BR, Rk),
V R > 0,
(1.14)
Applications to Harmonic Maps and Minimal Surfaces
240
for almost all r' E J-e, 0J. On the other hand, by Lemma 1.2, we have sup
(1.15)
C[1 + (e4trR2)1/n].
rEI -e,al
This implies a subsequence, where we do not change the subscripts, so that C1+(:'(R2)
1'k (r ,y) -. v(y),
for some r' E I-e, 0) (actually in a countable dense subset of [-e, 0J). We conclude that
Ov=0 in
R2.
According to the singularity removable theorem due to Sacks-Uhlenbeck N. (cf. [SaUIJ), v is extendible to a harmonic map -v: S2 We are going to show that v is nonconstant. Indeed, IVyvk((),0)I =
9-IDxf(Tk,ak)I = 1,
since vk satisfies (1.11) on Ik x Dk with the condition (1.12). The Schauder estimate applies to obtain an estimate: (1.16)
I1 t'k(T,Y)11C1+1.2+, ((- e,ol x(ae(e)nDk)) - C{1 + [e7r(26)2J1/p}
for some 6 > 0 small depending on U. The right hand side of the inequality is a constant independent of k. According to the embedding theorem (cf. [Nik1I),
IlVyvk(T, 11)IIC(, ,)/2.I "(l-e.olx(I (o)nDk) < C1,
where Cl is a constant independent of k. Hence
IDyvk(T',0) - Vyvk(0,0)I < C1r'. We may choose r' > 0 small enough so that IVvk(T',0)I > 2
(1.17)
.
It proves that v is nonconstant. Let Tk = Tk +'*, since. Tk -+ la J, JVf(Ik, )I blows tip at at most finitely many points {x1,... ,xt}, which includes the limit set of {ak}, according to Lemma 1.4. We choose 6 > 0 small enough so that
E(f(Tk, ))
= If IOf(Tk,x)I2dV9 I
=
IVf(7 ,x)I2d%
+ If
\v;-1De(=i)
j=1 Bs(x,)
.
1. Harmonic Maps and the Heat Flow Since
241
t
f (r, )
C1 +o' M \ U B6(xj), Rk
u( )
and there exists at least one jo such that a = xi., we have 1im
k-' M\uj'.,Bs(xj) J
IV f (7, x)12dVy = J
IVu(x)I2dV,
and IVvk(r*,y)I2dy
IVf(TJk,x)I2dV9 > J
j/3.k.B)
.(=fo)
for k large. First let k -. oo; by definition
lim k-.oo
J .(x10) IVf(T,x)I2dV> b,
and then, because 6 > 0 is arbitrary, E(w) ? kimo J IVf(r, x)I2dVp M
M
(1.18)
IV (x)j2dV9 + b > m + b.
This is the desired conclusion.
In case (2), a E OM f1 U. We choose a suitable coordinate (Y1, Y2) in R2, such that the y2-axis is parallel to the tangent at a of 8M, and the y1-axis points to the interior of U. Thus Dk tends to the half plane R+ = {(yl,y2) I yl > -p}, and for each point on the boundary, yl = -p, ak + ak --* a.
As in the proof of (1.15), now we have `d R > 0, SUP rE 1-e,01
Ilvk(r, )IICl+°(BRnD&) S C I I + (e4,rR2)1/n
+ II' 1 ak + B-) Since on the right hand side, there is a constant control independent
of k, we find a function v'' on R+ and a subsequence vk(r', ) such that
vk(r', y) -.v (y)
Cl+o (R+ ),
Applications to Harmonic Maps and Minimal Surfaces
242
and then
Ov = 0 in R+, V 18R+ = 0(n)
On the one hand, similar to the proofs of (1.16) and (1.17), we see that v' is nonconstant; and on the other hand, let us define a complex function
n(z) = h(v.', :) where h is the Riemannian metric on N, and
v: = 1(ay,
- iay,) i
,
Z = yl + iy2.
Therefore,
Y7(z) = h(vy,,vy,) - h(ziys,vyz) - 2ih(vy,,vy,). The harmonics of 17 implies the analyticity of the function r). The boundary i implies that the function ,j can be analytically extended to the whole complex plane. From the condition
condition on
r)(-P + iy2) = 0,
we conclude that n(z) - 0, and hence that i is a constant map. This is a contradiction, so Lemma 1.6 is proved. In the following, we assume ir2(N) = 0. We shall expand the conclusion of Lemma 1.6 to the following:
E(W) > my + b,
where r is the homotopy class of cp, and
mf = inf{E(u) I u E .r}. Only the inequality (1.18) should be fixed. It is known that f (Tk, ) ii(-) in C'+°'(Af \ Ut=, B6(x,),Rk). We only want to show ii E F. Let b > 0 be small enough so that B6(xi)f1B6(Xi) = 0, if i 34 j. Combine ii(xi) with the map f (Tk, )IOB,(z,) by the following map: v x V U1-1 B6(xi),
f(Tk,x) fk(x)
exp;,(=,) (17(
x-
J
exp-'j) f (Tk,x)),
d x E B6(Xi),
1. Harmonic Maps and the Heat Flow
243
where r) E C°°(R') satisfies n(r) =
1
r
0
r<
IkI oB.(=,) = f(Tk, *)1eB.(=,)9
we see that Ik remains in the same homotopy class F. And from Ik C(M, N), we conclude u` E F.
u
Step 6. Continuous dependence. From the point of view of PDE, the continuous dependence to s-+ fw (t, x)
from C.+'(M, N) to C';+ '2+ry(QT, N) does hold. The proof depends on the locally uniform boundedness of the heat flow
W '-- f,,, i.e, VWo E E° = {u E
I
E(u) < c}, where c <
m7 +6, 36>0andC1>0such that Sup(t.=)E(o,oo))xMIVf ,(t,x)I 5 C1, VW E B6(loo),
(1.19)
where B6 is the 6-ball in CC+'(M, N). Indeed, if (1.19) does not hold, then 3 Wk -' Wo in C2+"(-M, N) and 3 Tk with TO = iimT4, satisfying
IIVf,,.(Tk,')IIL-(M) -' 00 and V T < TO, 3 C2 (T) < +oo such that IIVff.(t, )IIL-(1o,T1xM) 5 C2(T).
By local existence, we may assume Tk > e > 0, and we shall prove oo, which contradicts Lemma 1.6 because
IIVfp,(T cpoEEE.
For simplicity, we write f k = f o , k = 0, 1, 2 .... It is sufficient to prove that {fk} is a Cauchy sequence in W""s(QT), VT < TO. Since IIV fk(t, )IIL-(M) < C2 (T) < oo, d k d t < T < TO, for functions pk = dk, where dk(t, x) = dist (fk(t, x), fo(t, x)), we have a constant C3(T) > 0 satisfying
AN 5 Apk +C3(T)pk Thus, by the Maximum Principle, IIdkIIL-(Q,) < C4(T)dist (ck, co).
244
Applications to Harmonic Maps and Minimal Surfaces
Again, letting f = fk - fo we write
A(r(f)(vf, vf)) = r(fk)(vfk, vfk) - r(fo)(vfo, vfo) = (r(fk) - r(fo))(vfk, vf) + r(fo)(vf, vfo) + r(fo)(vfo, v f). We have the following equations:
etf = DAff + o(r(f)(vf, vf)), f (0, ) _ Wk - VO,
(1.20)
f [Io,TjxoAf= 0
We apply the LI estimates to (1.20), p > 4 and obtain 1If[IWo.'(Q,T) < CS(7')IlVk - W016+,(7,N).
(1.21)
This proves the conclusion.
Once (1.21) is established, T' = +oo, so (1.19) holds and then coninuous dependence follows, provided by a bootstrap iteration.
Nevertheless, this is not exactly what we need. As a family of maps f (t, ), t > 0, depending on cp, it is no longer a continuous flow under the strong topology on the Banach manifold C2+-, (M, N). (The problem occurs at t = 0!). But it is continuous if we use a weaker topology, e.g., WP 2, 1 - P > ry. In the following, we shall employ the heat flow f,,(t, ) as deformations, under a weaker topology Wp (M, N), on the incomplete manifold N). For details, cf. Chang [Chat.
Step 7. The First Deformation Lemma. The critical set
& _ fit EC241(M,N) IDu=0} is compact in C2+' as well as in the W, -topology. In extending critical point theory, we have the modified first deformation lemma: For a closed neighborhood U of K, in the TVp topology, 3 e > 0 and a Wp continuous deformation rt: [0, 11 x Ec+e -+ EE+E satisfying rl(0, ) = idE,+., 17(1, EE+, \ U) C E,,. The deformation is constructed by the solution of the following evolution equation:
atf(t,) f ( 0, . ) =
,
f(t, .)I 8,%f = 0,
(1.22)
1. Harmonic Maps and the Heat Flow
245
where 7(u) = S
and
u V U6/4
1
uEU618
0
N) I distWpa (u, Kc) < b} C U. U6 = {u E C2+-'(V, 10
(1.19) is not a PDE, but after suitable reparametrization of (1.3), we are able to solve this equation. For details, cf. Theorem 7.1 in Chang [Cha101. In summary, we have the following conclusion:
Theorem 1.1. Let b = inf{E(v) I v: S2 -' N, nonconstant harmonic)
(if there is no nonconstant harmonic map from S' to N, then we define b = +oo), and let F be a component of C02+7 (19, N).
Assume that dim M = 2, 7r2(N) = 0, and that [i E C2+,, (OM, N), V E .F, with
E(W) < mf + b, where
mf = inf(E(u) u E F). Then we have (1) The heat flow, i.e., the solution of (1.3), globally exists. (2) 3 a harmonic map u` E F, and a sequence tJ T +oo such that
f (t ) (3)
in
C' (M, N).
If the infinitely dimensional manifold C, 7(M, N) is endowed with a weaker topology WP 2(M, N), p > the flow
(t,W)'-' is continuous from [0, oo) x .F - F, where f,, (t, ) denotes the flow with initial data W. (4) The set
is compact under the above topology, if c < ms + b. (5) Let K = Uc<m,+b Kc. Suppose that distw2 (f o(t, ), K) > 6 > 0
V t E R+ .
Applications to Harmonic Maps and Minimal Surfaces
246
Then we have e = E(b) > 0 such that C-
(6) For any closed neighborhood U C (M, N) of K, under the W.2-topology, where c < mf + b, 3 e > 0, a closed neighborhood V C U, and a W, - (p > °7) strong deformation retract 11
rl: [0, 1J x Ec+E -+ Ec+,, satisfying
rl(1,EcnV) c EcnU,
and
77(1, Ec+, \ V) C Ec-E,
where E. = {u E F I E(u) < a} is the level set, V a E 118+.
Remark I.I. The heat flow method was first used by J. Eells and Sampson IEeSI] in proving the existence of harmonic maps, where m is arbitrary and N has nonpositive sectional curvature. See also Hamilton 1I1am1J. Without the restriction on curvatures, but with m = 2, see M. Struwe IStr4] and K.C. Chang [Chal0J.
2. Morse Inequalities In this section, we establish Morse inequalities for harmonic maps under the assumption that all harmonic maps are isolated. As shown in Chapter I, the crucial step in the proof is to prove the following deformation lemma:
Lemma 2.1. Let 7 be a component of C, '(M, N). Suppose that there is no harmonic map with energy in the interval (c, d], where d < my + b, and that there are at most finitely many harmonic maps on the level E-1(c). Assume that a2(N) = 0. Then E. is a strong deformation retract of Ed. In order to give the proof, first we must improve conclusion (2) of Section
1, under the condition that the set of smooth harmonic maps is isolated. Namely,
Lemma 2.2. Let E(W) < ms + b, and let
c= lim E(f,(t, )). t +oo If Kc is isolated, then f,,(t, ) - is E Kc in the Wp-topology, V p > 1_°y as t -+ +00.
Proof. According to Theorem 1.1, conclusion (2), combined with a bootstrap iteration, shows that 3 u E Kc and t; T +oo such that
f ,(t,,.)
u,
(Af, N),
d 7' E (0, 7)
2. Morse Inequalities
247
If our conclusion were not correct, there would be a 6 > 0 such that the neighborhood U6 = (u E C2 N) I distw,2 (u, u) < 6} contains the single element u in K.., and a sequence tj 1 +oo such that f,(ti, ) It U6. Therefore 3 (t; , t;') satisfying
(1) ti,t;' -+00,
(2) f,,(t; , ) E 8026, f,,(ti',
E 8U6, and
E U26 \ U6 d t (3) On the one hand, we had
6 < Ilfc,(ti,)
-
C6It; - t--I'/',
provided by the embedding theorem. On the other hand, according to Theorem 1.1, conclusion (5) states
E(f,,(ti ))
- E(fv(ti
,
)) = f f I8tf (t, )I 2 dV9dt t,
M
rti' = J t, JM Iof(t, )I2dVdt > e(6)It;' - 41.
Since the left hand side of the inequality tends to zero as i -. oo, this is a contradiction. Now we return to the proof of Lemma 2.1. The basic idea is to reparametrize
the heat flow fp(t, ). Let r = p(t), where p(t) =
c)-'
J0
e
Ilof,(s, )Ili3da,
if E(W) > c, and let
Or, ) = At, ) Then we have the following relations: E( V)
(1) 8,9(r, ) = dr8ef (t, ) = II(O eT
(2)
-II) O9(r, ),
E(9(r, )) _ - fm (8,9(r, ), A9(r, ))dV9 = -(E(W) - c).
Therefore
E(g(r, )) = (1 - r)E(,p) + rc,
d r E [0, 1].
(3) The functionp: 10, oo) -+ l1F'is continuous and monotone increas-
Applications to Harmonic Maps and Minimal Surfaces
248
ing which satisfies the following properties: P(0) = 0,
p(+oo)=1 if
as t
limo E(fp(t, )) < c.
Let us define a function n: (0, 11 x Ed - Ed as follows: r1(r'cP)-
gV0(r, )
if (r, cp) E (0,1] x (Ed \ Es),
cp
if (r,go)E(0,11xE,
In order to show that EE is a deformation retract of Ed, only continuity at the following sets is needed:
(1) {1}xA,where A={VEEd\EEI f,'(oo, )EKK} (2) (0,1] x E- I (c).
Verification for case (1). V Wo E A, V E > 0, we want to find b > 0 such
that distw2 (W, cpo) < b }
r>1-b
implies distal (g"' (7, ), u') < f,
where ii = f,oo (oo, ). Choose co = fo(bs) as in conclusion (5), i.e., IIOfw(t, )IIL2 > Eo
and choose
if
disttiV2 (fw(t, ), K) > bl V t,
/ E \ 2/7 0
such that
2
Again, we choose b2 > 0 such that distwa (cp, cPo) < b2 implies distww (gwo ( 1 - 51, . ) ,
g,(1 - 5 , )) < 2
Therefore we have
dist$V2 (g,0(1 - b1, ),i) < e
V V E B6,(cpo).
We want to prove distvv (g,, (r, ), u) < e
V (r, gyp) E (1 - b,,1( x B6,(<po)
2. Morse Inequalities
249
If not, 3 T" > T' > 1 - 61 and W1 E B6i(cpo) such that 9,v, (T', ) E 8Br/2(u),
(T", .)
9v,
E 8Br(u)
and
9,o, (T, -) E B.(u)
Br/2(u) tlT E (TT').
Then we have 2
< distw; (9v, (T , ), 9,v, (T", )) = distw, (f,,, (t', ), fw, (t", .)) < Celt' - t"Iry/2.
On the other hand, Colt' -
till s
t"
llof", (t, )Il22dt
J
-
= E(ffo, (t", )) E(f E(9w, (,r", )) - E(9,v, (T', (E(TV) -
))
C)IT" - T'l
< 61(E(c0) - c),
which implies that
It - till > E(w) - c £0
61
£p >_
(£
) 2/7
(E(SP) - c) \ 2c
This is a contradiction. Verification for case (2). V ,po E E-1(c), V £ > 0, we want to find 6 > 0 such that dist(W,(o) < 6 implies dist(n(T, gyp), Vo) < C.
Similar to the above argument, let us choose
r 0
ll2/y
Find 0 < 6 < £/2 such that
E(,p)-c
V WE B6(Wo)
If our conclusion were not true, by the same procedure, we would have
(i) (ii) s < C'It' -
t"l,,/2, and
£olt' - t"I S (E(co) - c) IT - T'l < 61. This is again a contradiction.
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Applications to Harmonic Maps and Minimal Surfaces
The continuity of rl is proved, so that E0 is a strong deformation retract of Ed, d < mF + b. Suppose that Y d < mf+b there are only isolated harmonic maps. Since K n Ed is compact, they are finite. There are only isolated critical values (at most with limit mf + b)
mj=co
K, = ( u,j I? = 1,2,... ,m,).
Vd<my+b, let
Mq = E
rank C.(u,j; C)
c,
be the qth Morse type number, q = 0, 1, 2, ... , for the manifold Ed n.f and let
AQ =rankllq(EdnF,C) be the qth Betti number, q = 0, 1, 2, ... , for Ed n r. Comparing Corollary 4.1 and Theorem 4.3 of Chapter I with the above deformation, we obtain Morse inequalities for harmonic maps below a certain level.
Theorem 2.1. Let F be a component of C0'(Af , N), and let d < mf + b. Assume that ir2(N) = 0, and that in the level set Ed n.l`' there are only isolated harmonic maps. Then there exists a formal power series with nonnegative coefficients Qd(t) such that 00
00
EA1gt4 = Eg tq+(1+t)Qd(t). q=o
q=o
3. Morse Decomposition In this section, we will study the handle body decomposition of the level sets of the energy function, under the assumption that all harmonic maps in these level sets are nondegenerate. Let uo be a harmonic map from Al to N. Let E = uoTN be the pull back bundle over Al. Let 0 be a neighborhood in C' (Al, N) which contains the section uo(M). It is obvious that 0 is diffeomorphic to a neighborhood
3. Morse Decomposition
251
OF of the zero section of the tangent space T,,,(E). The diffeomorphism is realized by the exponential map o E OE
xo"°`'
0 n
n
T,, (E)
C;' (M, N).
After this we do not concern ourselves with the tangent vector or with
its exponential map exp,,,(2) o(x). We shall restrict our studies to the neighborhood OE of the vector space T,,,(E). The Taylor expansion of the energy functional at uo is as follows:
E(u) = E(uo) + I d2E(uo)(a, o) + R(o), where u(x) = expt,,(,,) a(x), and the remainder R(o) satisfies
IR(a)I = o U IvoI2) , and
1/ 2
dR(a)I I
=o
((J
M
IV UI2)
As to the Hessian d2E(uo), it is well-known (see Eells-Lemaire [ELI]) that,
`da,gEC°°(T,,,(E)), d2 E(uo) (or, 17) = JM
where
J,,,o = -A°Oo - Trace R''(duo,a)duo is the Jacobi operator. Noticing that J,,, is a linear self-adjoint elliptic differential operator, 0 with domain W2 n 1' 2 (T.. (E)), J,,, can be extended to be a continuous 0
bilinear form on the Hilbert space W2(T,,,(E)). And since d2E(uo)(a,a) ? 11011 W - Cl(uo)Ila112, 21
where C1(uo) is a constant depending on uo, the negative eigenspace of Ju, must be finitely dimensional. The dimension of the negative eigenspace of
J,,, is called the Morse index of the harmonic map uo and is denoted by ind(uo). uo is called nondegenerate if J,,, is invertible. For the self adjoint operator J,,,, it is well known that we have a spectral decomposition EA and two projections P+ and P_, which correspond the
252
Applications to Harmonic Maps and Minimal Surfaces
positive and negative eigenspaces respectively. For any a E C2 -'(uoTN), we have
of := Pfa E C,2,,(uoTN). The two square roots (P±(±J-o)P+)i/2
A+
are well defined, and we have that IIA±cIIL2 is equivalent to IIc±IIw; . In the following, we shall denote IIAfaIILa by 1(7± 1, and let Iai2 = Ia_I2. Thus, the energy function is written as follows:
1(7+ 12 +
E(u) = c+ 2 (Ia+I2 - Ia-I2) + R(a). For any given 0 < ry < 1, we choose r > 0, satisfying r
1-r<
f1---7
1+v
and 6 > 0 such that, for a Vii -hall B6 with radius 6, centered at the zero section of C2,7(uoTN), we have IR(a)I < 2ylal2
(3.1)
and
IdR(a)I < rIol. (3.2) V a E U = B6. (In the following we always denote B6 by U.) These imply
that
2(1-'Y)Ia+I2- 2(1+7)Ia-I2 < E(u) - a < 2(1+7)Ia+I2-
2(1-'Y)Ia-I2.
(3.3)
Now we are going to construct a series of deformations, which deform the level set Ec+, (for suitable e > 0) to Ec_£ attached with cells: (1) According to Lemma 2.1 we have a strong deformation retract pl, which deforms E,+, into Ec, for E > 0 small, if E-' (c, c + e) n K = 0.
(2) By conclusion (6) of Theorem 1.1, we have e > 0 and a strong deformation retract
, which deforms Ec into Ec_e U (Ec n U) and satisfies
72(1,EcnV) C EcnU, rt2 (1,Ec\V) C E,_1. (3) Let us define two conical neighborhoods:
C7=
1la_I},
C7={aEUlla+I<
1+71a-I}
l+ry
Inequality (3.3) implies that
c,.cEcnUcC..
253
3. Morse Decomposition
Lemma 3.1. There exists a strong deformation retract r3, which deforms E, U (Ec n U) into Ec_1 U C7.
Proof. Noticing that V a E, U C.r, with a E U, we have 1
la-1!5
ryb.
2 - 1 (> 0), and define a flow on U as follows:
Let K =
n(t, a) = (1 - t)a+ + (1 + tK)a_. We have (a) 77(0, a) = a.
(b) Y7(1, a) =
a_ E U if a M Cry.
(c) Letting W(t) = E(n(t, )), we have z
w(i) _
1*l+ It
- KIn_12 + (dR(n(t, )), -a+ + Ka_) z
<(1-T) f-li+It -Kln-1'+ TT (1 I t+K) In+IIn-I 1
[-ln±i (In+I-
TTIn_1) I
-Kln-1 (In-1-
rTln+I)
where n = (n+, n_). If n E (Ec n U) \ C7 C C, \ C,, then we have F1
In+I _! V
1+tIn-1?1TTln-1
and
In-1?
Vf -L 1+71n+I?
TTIn+I
It follows that
ap(t)<0
VnEC.r\Cy.
(3.4)
Combining (a) and (b) with (c), we obtain n(t, a) E (EE n U)
V(t, a) E 10,11 x ((EE\U)\(C,, U Ec_1)),
provided by the fact that Cry C E, n U. From (a) and (b), we see that if a 0 Ec_e U C.,, but a E E, n U, then
there is a unique t' E (0, t) such that n(t', a) E E-1(c - c) U BC,.. The uniqueness and the continuous dependence of t' on a are verified by the transversality n fi E-1(c - e) U 8C,., which follows from inequality (3.4).
Applications to Harmonic Maps and Minimal Surfaces
254
Let us define 173(t,a)rl(t't,a)
aE(EEnU)\(E, UCy)
if
a E Ec_e U Cy.
or
This is the deformation we need. (4) Noticing that b' a E EE_, n C.,,
-e > E(u) - c>
1
2
y la+l2
-
1 2 y la-I2,
we have 2s
Ia-I>
1+ y'
so that EE_,nC,CS:_ {aEC-yIIa_I>
1+?`
On the other hand, V a E S, Ia-I ? kola+I + 5o, where kO
1
l+y
2
1- y
and
bo = 21
2e
1+ 7.
Let us define
Tk.,b. = ( a E Cy I Ia-I ? kola+I + bo). In the following, we prove
Lemma 3.2. There is a strong deformation retract 174 which deforms Ec_e U Cy into E, U Tko,bo U {0+} x B 6k., where k = ind(uo).
Proof. We define 114(t,a)
a
il a+(1-ta+
a E EE_, U Tko,bo
l1
Ia- koc+0),a+ aECy'bo:5
aEC,n{lo_I<5o}.
(5) Choose e > 0 small enough that
E<
52(1 - y)
2
Define
0<<
1-(-y+P) 1+(y+3T
la-ISkola+I+bo,
3. Morse Decomposition
255
We consider the energy function on the conical section of the sphere 0B6: S. _
{a E 0B6 I Io+I < plo_I}. Letting a E S,,, we have
E(u)-c<
1
2ryla+I2-
c (1 27µe-
271a-I1
27/ 10-12
--1 (1-p -ry)Io-I'(1+µs). I+ µ2
J
Since 62
= to+I' + Io_I2 < (1 + p2)la-I',
then
E(u)-c<-2 (1+ µ22 - ) 1
-e,
0
i.e., S C Ec_e.
Lemma 3.3. The exit set of the Row 17(t, O) = e- k, to+ + ekata-
on the ball Bs, is the set S,,, where 1 = µs, k1, k2 > 0.
Proof. The flow , remains on the plane generated by the two vectors o+ and o_. Suppose that t meets 0B6 at time to, and let *l+ = e-k'toa+, rl_ = ekl°oa_. Choosing suitable coordinates (rl+, rl_) = 6(cos 0, sin 0), we assume that the flow rl leaves the ball B6. By comparing the tangents of the ball with the tangents of the flow, we see
-kz tg8 > -ctge, i
In+I < P117-1-
In other words, (y7+,77-) E Sµ.
0
Lemma 3.4. There is a strong deformation retract rls which deforms the set Ec_, U Tk,,6o U ({6+} x B6A:,) into EE_e u ({B+} x B6).
Proof. We use the flow rl defined in Lemma 3.3. Because S. C Ec_et if
a it Ec_e, then there must beat' E (0, oo) such that rl(t', o) E E-' (c- e).
256
Applications to Harmonic Maps and Minimal Surfaces
On the other hand, q(t, -)is transversal to the level set E-1(c-e), provided by the fact that dt E(q(t, a)) _ -(17+, kl r7+) - (q_ k2'7-) + (dR(7l), -klr1+ + k2*7-)
5 -kiln+12 - k2Iri-12 + r(177+I + I77-I)(kiln+I + k21r1-I)
= -(1- r)[k,Ivl+12 +k2In-12 - 1
T
r(kl +k2)Ii7+I In-IJ
= -(1- r)kl [111+12 +F<2Iq-12 - 1 r
(1 +µ2)1r7+I 1n-1]
< 0,
if we choose F
r
1
(3.8)
1 + F12
T Therefore, V a E Tko,6p \ E,_F, t' = t' (o) is uniquely determined and
continuous. We define our deformation retract as follows: 17±(t, a) =
o
if o E EE_F
?(t*(o)t, Cr)
if o E Tko6o \ E,,_,
ek2t'(6oo-/Ic-I)a_
if or E {9+} x BI60
.
For any two strong deformation retracts Al
X2 - Y, +o2
WI
we define their composition as follows: W, (2t, x)
tE
[O,J
P2(2t -
tE
[11,
P(t, x) =
1
1J.
This is again a strong deformation retract gyp: X1 -' Y, which is denoted by W = V2 0 (P1
Now we come to our main conclusion in this section.
Theorem 3.1. Assume that 7r2(N) = 0, and let F be a component of C2,,,(M2,N). Suppose that on the level E'1(c) fl F, c < mj + b, there are only nondegenerate harmonic maps ul,... , ut, with Morse indices ml,... , me respectively. Then the level set EE_, fl f', attached with f handles, whose dimensions correspond to these indices, is a strong deformation retract of Ec+E fl f, for suitable e > 0.
Proof. We choose -y = s , r = , and it = i Then we have 6 > 0 3 hold. Choose e > 0 small enough small enough such that (3.1) and (3.2) .
4. Existence and Multiplicity for Harmonic Maps
257
such that e < lb and that conclusion (6) holds. Inequalities (3.6), (3.7) and (3.8) are satisfied automatically. The strong deformation retract now is defined to be P= P5 0P40P3 oP2 0P1.
Combining Lemmas 3.1 and 3.2 with 3.4, we obtain our conclusion.
Corollary 3.1. Suppose that uo is a nondegenerate harmonic map with E(uo) = c and tzo E C "(M, N), ry > 0. Assume that c < ms+b, if uo ET and ir2(N) = 0. Then we have C,(uo;G) = 69k G,
where k = ind(uo).
Sections 2 and 3 are adapted from Chang [Chal1).
4. Existence and Multiplicity for Harmonic Maps We present here a few theorems about the existence and the multiplicity for harmonic maps. We follow the notations in previous sections. Theorem 4.1. (Sacks-Uhlenbeck [SaUl), Lemaire [Lemll). If 7r2(N) 0, then for any homotopy class .F of maps from M to N (with prescribed C2',, (BM, N) in the case 8M 36 0), there exists a boundary value , E harmonic map.
Proof. We choose any d E (m7, ms + b). Obviously, #0d
:= rank Ho(EdflF;C)0.
It follows Mod 96 0. Consequently, there exists a minimum u of the energy function. Therefore u E N) is a harmonic map in F.
Theorem 4.2. (Brezis-Coron [BrC11, Jost [Jos1J). Suppose that N = S2, and that 0 E C2''y(BM,52) is not a constant. Then there exist at least two homotopically different harmonic maps.
Proof. By the argument used in Theorem 4.1, we obtain a minimal energy harmonic map u among all homotopy classes E(u) = m. The second harmonic map will be obtained by constructing a map v homotopically different from u having energy E(v) < m + b.
(4.1)
The construction of the map v is as follows: Choose a small disc Do on M, take an isometric copy Dl, and identify Do and Dl along their boundary to obtain a 2-sphere S2. Take a map w: Sz -. N = Sx, which
Applications to Harmonic Maps and Minimal Surfaces
258
represents the generator of ir2(N), and coincides with u on Do, such that the map v
={
u
M\Do
wID I
on Do and identify Do with D1
satisfies (4.1).
We shall construct w explicitly. Since 10 # const., u # const., we can
choose a point xo E M, for which Du(xo) # 0. Rotating S2, one can assume that ii(xo) is the south pole. Let 7r: S2 -i (C be the stereographic map from the north pole. We choose local coordinates z E C in a neighborhood of x0, such that z(xo) = 0, IzI < e, e > 0. Thus Pr o ii(z) - V(7r o u)(0)zl = O(lzl2)
We denote V(7r o ti)(0) by a, which is a nonzero complex number, and write z = reie. Letting
we have
t(Ec'B) = it o u(ce'o)
and t((E - E2 WO) = ace 'B.
Then we define a function cp: C -+ C as follows:
w(z) =
Izl <E-E2
az t(z)
E - E2 < IZI < E
In o u(Ee'B)I2(7r o u(E2z-1)
IzI > E.
If e > 0 is small, ip is continuous and surjective. The map w is defined to be 7r-10(007x. Noticing that it is conformal, and that the energy is conformal invariant, we may compute the energy of w by the energy of 7r-1 c0. Now
E(v) = 1 2
f
M\B(zo.e)
Ivul2 +
Iow12
1
JB(xo,ee2)
+ 2
f
B.\B._,2
Since 1
2
f
lVul2 = E(ti) - O(E2)
M\B(zo.e)
1
2 JB(x,,e_e2)
IDwl2<7r=b,
(a 34 0)
IV(7r1t)12.
4. Existence and Multiplicity for Harmonic Maps
259
and 1,V (7r -'t)12
pe
:
2wr
I(7r o u)(ee'B) - aee'012E, +
/
+ia Ile1
r
o u)e(ee'B) 1 £2 -
1
_1 fe
12]
e'B
- e2r /
rdrdO
= O(e3), where C = Maxlldir-1112. Therefore, for e > 0 small enough, we have
E(v) < m + b. The remaining part of the proof is the same as that in Theorem 4.1. Theorem 4.3. Given a Riemann surface M with boundary 8M, if >G E
C2+,'(OM, S"), y > 0, n > 3, is not a constant map, then there exist at least two harmonic maps from M to S" in the (nontrivial) homotopy class
.FCC0 Proof. The proof is similar to the above case. Since ir2(S") = 0, n > 3, first, we apply Theorem 4.1 to obtain a minimum of the energy functional in F, which is a harmonic map uf. Then we shall prove that there exists d < ms + b such that A"_2 (Ed fl F) 34 0, where F is regarded as a component of C24 (M, S"); or, equivalently, we construct an essential map a E C(S"-2, Ed fl .F) such that
sup E(o(s)) < d < ms + b. SES^-2
The existence of such a map o was constructed by Benci-Coron [BeC1), in a manner very similar to the construction of v in Theorem 4.2. We are satisfied with pointing out the main idea, and we omit the details. As in the proof of Theorem 4.2, 0 76 const., ii 96 cont. Again we choose a local chart U, outside which a is defined to be u, and inside which, we choose a small disk Bo(zo) on which a(s)(z): S"-2 x B6(zo) -' S" is a homeomorphism. The map or is connected smoothly. After a careful construction, this makes E(a(s)) < mf + b.
By the fact that sr2(S") = 0, we see that o(s) E F V a E S"-2. We point out first that a is essential. Indeed, a(s)(z) = ii(z)
for
z E M \ U
260
Applications to Harmonic Maps and Minimal Surfaces
so, outside the ball Bo(zo), a(s)(z) is contractible. And inside B6(zo), 0.:Sn-2 x Bo(zo) - S" is a homeomorphism. Therefore, the image of a is homotopic to a n-topological ball. Thus or cannot shrink to a constant map, i.e., a is essential. Next, we apply the Hurewicz theorem, which implies the existence of an
integer 0 < k < n - 2 such that Ilk (Ed n 1,72) # 0. The existence of the second harmonic map in .F follows from Morse inequalities (Theorem 2.3).
Remark 4.1. Theorem 4.3 was obtained by Benci-Coron [BeCl] in case M = D2, the 2-disk. See also W.Y. Ding [Dinl] and K.C. Chang [Chal0]. For other results in this direction, readers are referred to Sacks-Uhlenbeck [SaU1,2], Schoen-Yau [ScYl) and Jost [Josl]. Readers who need to know more about harmonic maps are referred to Eells-Lemaire [EeL1-2].
5. The Plateau Problem for Minimal Surfaces The Problem. Given a Jordan curve r in 118", one asks for a surface S with minimal area spanning I'. For technical reasons, we assume that r is defined by a: S1 --+ R', which is a C2-diffeomorphism. It is known from differential geometry that such a surface S, if it exists, has mean curvature 0 (see, for example, Ossermann [Ossl]).
We introduce isothermal coordinates on S, which parametrize S by a function gyp: D -+ ll8", where
D = (z = (x, y) I x2 + y2 < 1),
with 8D = S1.
Thus, V satisfies the following nonlinear differential system:
AV =0 hP.l2 - 1WyF = (W., WY) = 0
in D, in D,
W Ieo:BD
is an oriented parametrization of r,
-+t
(5.1)
where (, ) is the scalar product on R', and I = (., Although solutions of the differential system (5.1) no longer require S having minimal area, they are also called minimal surfaces. We use complex notations,
z=x+i11, and introduce the conformal group on the disc:
G=
{g(z)=e0tZ
l
l + az I aEC,Ial < 1oER}. JJ
It is easily seen that if cp is a solution of (5.1), then cp o g, V g E G, are all solutions of (5.1), i.e., the system is G-equivariant.
5. The Plateau Problem for Minimal Surfaces
261
In order to avoid the nondetermination of solutions, we normalize the solutions as follows: Let P1, P2, P3 be an oriented triple of distinct points on r, and assume
W(eP)=PJ,
j=1,2,3.
We shall study this problem via critical point theory. Inspired by the early work of Courant [foul], the conformal condition (the second group of equations) can be solved by minimizing the Dirichlet integral. This suggests the following strategy: Define a parametrization set
M = {uECnH112(I0,21r),R')lu(37
_ 2j rr, 3
j = 0, 1, 2, 3,
and u is nondecreasing}.
This is a closed convex subset of the Banach space X = CnH1/2([0,27r), R1). (1) V u E M, solve the Dirichlet problem:
Aw=0 in
D,
VIBD=aou. The unique solution So E H'(D,R^), depending on u continuously, is denoted by cp = R(a o u). Then, we define a functional on JN:
J(u) = 2
VR(a o u)I2dx A dy. ID
(2) We find the critical point of J with respect to the closed convex subset M, and then (a) verify that the generalized critical points solve the conformal condition, and (b) the extended critical point theory provides the existence and multiplicity results.
Applications to Harmonic Maps and Minimal Surfaces
262
Before going to the minimal surface problem, we study the Banach space X = C n H1/2([0, 2ir], lR") and related Sobolev spaces. Recall that the Sobolev space H'(D, Rn) has the following norm: IIXIIH1(D) = IIxIIL2(D)+IIoxIIL2(D)
The trace space of H1(D, R") is the fractional order space H1/2(S1, Rl), x(n)I2d£dn
IIxIIi/2 = IXIi2(sI) + f f
If -
s1 Xs'
712
In particular, b' x E H1/2(S1, R"), we have the Fourier series expansion cmeime,
x(O) =Re m=-oo
where
s,r
c,n =
in = 0, ±1, ±2,...
f x(O)e-imodO, 27r
It is known (cf. Adams [Adal]) that 00
(1 + IrI)Icm[2
IIxIIi/2 = IIXIIL2 + IXI1/2 = m=-oo
is an equivalent norm, where Ixl112 = (Emlc,,,I2)1/2 is called a seminorm which induces a semi-inner product (, )1/2. We need the following preparations:
(1) Let u a o u be the Nemytski operator from the Banach linear submanifold X = {u E CnH1/2([0,2ir],R1) I u(0) = 0, u(27r) = 27r} to X.
Then a is a C1-map and J E C1(9).
Claim. The Gateaux derivative of u i-+ a o u at uo is a'(uo), so we topology. only want to verify that u " a'(u) is continuous in Since the embedding H1/2([0, 2n]) ti LP[0, 27r]
dp>1
is continuous, we have 1 /r
s,r
III 0 uIIC(TT(X),H'/3) S (10
I
a' o u(B)I''dO)
for
r1
=1-?. p
Therefore, the H1/2)-continuity of a' o u with respect to u in X follows from the Lebesgue dominance theorem.
5. The Plateau Problem for Minimal Surfaces
263
It remains to verify the £(T(2), C)-continuity of a' o u with respect to u in X. Since I(a' o u) VI 5 Ia' o uic(10,2w),i.-) ItIc,
the Cl-continuity of a implies that 0 as
Iia' o u - Q o uoIIG(T..(-k).C)
IIu - uoilC -+ 0.
This is just what we need. (2) Suppose that V = R(7), with 7 E H1/2(SI,IR"), i.e., {
Orp=O in D
`'PIOD=7'
According to the Poisson formulas, we have 00
V(r, 0) = Re E c,,,rmeime m=-00 00
E
m
z = rete
M=-00
where
00
7(0) =Re E cmeimB m=-oo
(3) V x, y E H112(S',1R"), suppose that rp, t(' are corresponding solutions
of the Dirichlet problem, with boundary data x and y respectively. Then we have
(x,U)1/z=Re E ImIcJm= fD m=-oo
02,.
(Wr, IW0.
(4) `d x, y E Hl/2 n C(SI, R"), we have (x, y) E H1/2 n C(SI, RI) and I(x, y)I1/2 5 IIxIIc
11111/2 + IIvIIC
IxII/2
Claim. We only verify that
'xS'
J
=
fI(x,y)(e) - (x, y) (1)1d n It-1712
r [
J JS'XS' <- IIxIIc 11111/2 + IIvIIC
11(17)) + (x(f) - x(17),11(17))1
It-1712 IxII/2.
d 17
Applications to Harmonic Maps and Minimal Surfaces
264
(5) If p E C1 (R2,Ilr;") and a E H'/2(S',1R2), then poa E H'/2(S',R"), and IP°aII12 5 IIVP°allr.-Ia1112 Claim.
I-
Ia(E) - a(o)IZ
7112
IIVP0a11i-I
If - 1712
a(1)0,(
)I2
1712
Integrating both sides of the inequality, we obtain the desired conclusion. Next, we need to verify that critical points of J with respect to M satisfy the conformal condition. Letting cp = R(a o u) and
we have
IF(z)I2 = (W. - i'Py.S0= - i97 )
= I9=I2 - IVYI2 - 2i(9z,wy) Therefore,
IF(z) 12 = 0
if and only if V is conformal.
However, we observe that
arlF(z)12 = 2(a1F(z), F(z)) = (oip(z), F(z)) = 0; therefore IF(z) 12 is analytic in D. In polar coordinates, IF(z)I2 = z-2[r219Pr1 - IWPOI2 - 2ir(<pr,We)].
Lemma 5.1. V U E M, letting V = R(a o u), we have (car, We) 1,=, E C' (aD)' C D', the Schwartz distribution space.
Proof. First, we assume u E C°°(aD). In this case (Tr, We)Ir_, E C(OD).
Then
(HU, a) = f Z x (W-,, cPe) . a(O)dG,
d a E C' (OD),
0
defines a linear continuous functional on C'(OD).
5. The Plateau Problem for Minimal Surfaces
265
In order to extend this functional continuously to M, V u, v E C' (OD), we let cp = R(a o u), 0 = R(a o v), and we make the following estimates:
[(cr, T e) -
J
j2w [((c
a dO
- b)r, we) + (+lOr, (co - lk)e)] o d9
= fD [(V (w - 0), V (wea)) + (Vt, V((v - O)eo)Jdxdy,
where the function o in the last integral is understood to be an extension o the same function defined on OD. The last integral is split into two terms: ID'(V(,p - 0),VWe)o + (Vu', V(W - 0)e)a]dxdy
+ ID [(V((p - t/i), r'oVa) + (Vt, (i0 - l')eVa)]dxdy.
Noticing that a (V, ID
V(w - b)e)dxdy / (V (gyp - '), aeOtl, + avtPe)dxdy,
we have JD a(V
- 0), V - O)e)dxdyl fD8BIV(W
2I
2IID IV(('-l)I2.cedxdyl
-
< 2IIcP -,Glly,(D) Ik IIC'(8D).
The remaining three terms are estimated by II'P - +I'IIH-(D)(2IIt'IIHa(D) + II9IIH1(D))IIQIICI(OD).
In summary,
(H.-Hv,a,)I < C[Ilw -'GIIH1(D) + IIp -1GIIHI(D)(2II1GIIH'(D) + II'PIIHI(D))] Ilallci(OD)
5 C(Ilu - vIIHt!2, IIuhIH./2,
vii H'/2(8D) - II17IIc'(8D)-
Applications to Harmonic Maps and Minimal Surfaces
266
Since C°°(8D) is dense in M the domain of H can be extended to M, such that d u E M, Hu E (C' (8D)) the dual of C' (8D). The lemma is proved.
We turn now to finding out the derivatives of J. V Q E C'(8D), with IQ'(O)I < 1, define p,(0) = 0 + eo (B),
for
f e j < 1.
By definition, po(O) = 0, and Pe(0) = o(B).
For any u E 0(8D), 2w
d Au o PO = 1 (wr, as o u)u'(0) c(e)de 2ff = j(corcoo)ad0 ,
= (Hu, o), Generally speaking, however, V u E M, u o p;' does not satisfy the three point condition, so we do not know if it is in M. In order to find the derivative of J with respect to M, we need more work. Note that there is a conformal mapping w,: D -+ D satisfying w, : exp
{iuo;' 2j(t)J - exp [ 2ijn 3
j =0,1,2,3.
If we define rr(6) _ -iln ww(e'e)
-
ye -
re
0u0 Pe
and
,
then
Uc E M.
Lemma 5.2. If u E M is a critical point of J with respect to M, then the distribution Hu = 0.
Proof. Choose a sequence uk E C2(OD) such that uk - u in X = C n H1/2. Observing that J is invariant tinder rr, we have
0=
d J(rr 0 uk) = _J(YC oPE), de-
5. The Plateau Problem for Minimal Surfaces
267
where y; k = rr o uk o p; 1. Therefore
(Jl(uk),
dyek 1,=0
k
0
+
a) = 0.
Since yk E M, and it is a critical point of J with respect to M, we have
/ J'( u k)
(
which implies
d
/J'(uk)
jim(
k1
,d
C-o
ddOk
0
,
o\ )
>0 ,
< 0.
Exploiting Green's formula, it follows that k
J,(uk)+ ;i
'
or)
_ (Hu.,o) =
Therefore
(Hu, o) < 0
`d a E C' (8D),
i.e., H. = 0.
Theorem 5.1. Suppose that it E M is a critical point of J with respect to M. Then V = R(a o u) is conformal, i.e., (,P=, ivy) = 0.
W=12 -
Proof. Observe the analytic function z2IF(z)12: Since Im z2IF(z)1218D = Hu = 0,
this implies Re z2IF(z)I2 = const. The constant is determined by the value at z = 0. Hence,
z2IF(z)12=0
dzED,
which implies IF(z) 12 = 0.
Therefore V is conformal.
We turn now to verify the (PS) condition of J on M. The following Courant-Lebesgue lemma is useful.
Applications to Harmonic Maps and Minimal Surfaces
268
Lemma 5.3. (Courant-Lebesgue). For each constant M > 0, the set JM12 ={ it E M I f I7R(a o u)I2d2 A dy <
D
111
MI 11
is compact in the C°-toplogy.
Proof. 1. Let C. be the circular arc centered at (x0, y0) E OD with radius p > 0. We want to show that V 6 E (0, 1), 3 p E (6, /) such that 2M
f
pln(1/6)'
Co
for each W = 1Z(a o u), with u E JM, where gyp, denotes the directional derivative along the arc CP. Claim. Define an integral
I=
f
I.
1W, 12 dadr.
Cr
6
Since
Ipr12 < Io(p12,
we have
I < f Ippl2 < M. Letting
P(r) = r
fc
r
it follows that Vb-
f6
dr p(r) r < M.
Thus, 3 p E (6, f) such that 2
f,
In - . P(p) < M,
V.12ds <
2M
pln(1/6).
2. We show that {a o u} is equicontinuous.
Claim. For the Jordan curve r, d e > 0, 3 d > 0 such that for any two points P, P' E r, r \ {P, P'} possesses one component with diameter < e, provided 0 < dist(P, P') < d.
5. The Plateau Problem for Minimal Surfaces
269
According to (1), b zo E OD, b 6 > 0, 3 p E (6, f) such that the arc length of the curve C. has the following estimate: t(V(Cv))z <
(f l'P.Ide c
J,
/z
5 (L, Ico.I2ds)
2irp
47rM ln(1/6)'
We choose 6 > 0 such that 4irM
d2
ln(1/6)
and that at least two of the following three inequalities hold:
(z -ell > f
V z E OD, and j = 0,1,2.
One may assume e < Min distj#k(pp,pk). Therefore V zo E 8D, 3 p E
(6, f) such that t(cp(C,)) < d and the arc Q. divides 8D into a large arc A' and a smaller arc A" which correspond to the two arcs A and A" on r respectively. (The large arc A' is defined to be the arc which includes at least two of the three points e- r , j = 0, 1, 2). Let P, P be the intersections of rp(C.) fl r. Then dist(P, P') < d. Consequently, diam(A') < e. Therefore, for any two points z, z' E 8D, satisfying Iz-z'I < 6, we must have dist(cp(z),ap(z')) < C. 3. Obviously, (a 0 u u E JM12 } is bounded. According to the ArzelaAscoli theorem, {aou I u E JM/2 } is compact in C(S',)!e"). Since a: M a(M) C C(S1, lR") is a homeomorphism, JM/2 is compact.
Lemma 5.4. The functional J satisfies the (PS) condition on M. Proof. Assume that fu,.) C M satisfies J(urn) < M, and I - J'(um)Iu... -+ 0.
We want to show that it is subconvergent. According to the Courant-Lebesgue lemma, there exists a subsequence {u,n) (without changing the subscripts), such that u,n -+ u* (C),
and
urn --+ u'(H1/2).
Applications to Harmonic Maps and Minimal Surfaces
270
Since M is closed under the C-topology, it suffices to prove that um u'(H1/2). Indeed, according to (3), we have
Ium -u'Ii/2 <J Iv(R(aoum)-7Z(aou'))I2dxdy rD
= fD IV(vm - cp')I2dxdy
=f
(5.2)
'P'),
wh ere tom = R(a o um) and cp' = R(a o u*). Integrating by parts, we have
(FPM-`P*)I8D=a°um(O)-aou'(9) u
u,,,(B)
(B)
= a' o Um (0)(um(0) - u'(9)) - fu
f
a"(B)dBdO' .
Introducing p: IR2 - !R', and o: S1 -' R2 by P(C '7) =
n
J J
tt
a"(9)d9d9'
and
we obtain
IPoall/2 < IIDP0UJIL- I°i1/2 from (5). By differentiation,
aP
=
- {)
It follows that IIVPOaIIL-
and
IP a11/2 < Co(IurI1/2 + 1u'11/2)
Ilum - u'IIL-,
where C. is a constant depending on a. The condition um - u' implies corn - cp', and then
I (cor,W.
J BD
- cp')d9 - 0.
5. The Plateau Problem for Minimal Surfaces
271
Again by the Courant-Lebesgue lemma and (3), we have
J8"''
P(um(O), u (e))d9I
= LI
Lm,VR(poa))dxdyl
< QW.11H. - lip -
aIIwh/2 - 0.
Moreover, returning to (5.2), 2,r
lum - u' I1/2 =
JO
((com)r, a' o um(8) - (um(8) - u' (8))dO + o(1)
_ (J'(um),ur - u') +O(1) = (-J'(um),u' - um) +o(1) < I - J'(um)Iu.,, (IlumllHh/2nC + IIU*IIH-/2nC) + o(1).
However, there is a constant C > 0 such that Iumll/2 <- Cla 0 tt nll/2
< CJ IVpml2dxdy D
= CM; therefore, Urn - u'(H1/2).
0
Now we can apply the critical point theory on the closed convex set M to obtain the following results.
Theorem 5.2. (Existence). Assume that a: SI -. r is a C2 diffeomorphism. Then the Plateau problem is solvable.
Proof. It follows directly from the results in Section 6.2 of Chapter 1, because J is bounded from below and satisfies the (PS) condition on M.
Theorem 5.3. (Morse-Tompkins-Shiffman). By the same assumption as Theorem 5.2, if, further, we assume that system (5.1) possesses two distinct solutions which associate with two local minima of the Dirichlet integral J, then system (5.1) has the third solution. Proof. It is a direct consequence of the Morse relations for J on M.
Remark 5.1. Theorem 5.2 is due to J. Douglas (Doul] (1936). In his paper, 1' is only assumed to be a Jordan curve. Theorem 5.3 was obtained by M. Morse, C.B. Tompkins (MoT1) and M. Shiffman [Shill independently in 1939. The above proof, based on the modern critical point theory is due to M. Struwe [Str2] (1984).
272
Applications to Harmonic Maps and Minimal Surfaces
Theorem 5.2 was extended by J. Jost [Jos2J, to where the disk D is replaced by a compact oriented surface of type (p, k), and IR" is replaced by a complete Riemannian manifold (N, h) with nonpositive sectional curvatures. His method is taken from hyperbolic geometry. While Theorem 5.3 is also extended to that generality by K.C. Chang and J. Eells [ChEI], an a priori assumption excluding the change of topological type is made. Recently, J. Jost and M. Struwe [JoS1J removed the a priori assumption. They successfully developed a Morse theory for minimal surfaces of varying topological type, but one in which the target space N = 1R3 is assumed.
Another related result is due to M. Ji and G.Y. Wang [JiW1]. They proved the Morse-Tomplins-Shiffman type result in the case M = D, the unit disk, and N equal to a compact Riemannian manifold which admits no minimal sphere. We also mention the work of T. Tromba [Tro2-31, in which degree theory is used to study disk-type minimal surfaces, and of Struwe [Str3] for annulus type.
APPENDIX
Witten's Proof of Morse Inequalities
0. Introduction In his paper "Supersymmetry and Morse Theory," E. Witten (Witl( presented an analytic proof of Morse inequalities. It is the purpose of this appendix to introduce his proof. According to de Rham-Hodge Theory, the Betti numbers of a differential
manifold M are related to the dimensions of harmonic forms. In the first section, we shall briefly review Hodge theory. The idea of Witten's proof is to introduce a perturbed elliptic complex for a given Morse function f as follows: AP-' ( M )
do_1
e-ti, 1 AP-I ( M )
d°-'
AP (M )
d'
An+'(M )
e-t,l
e-t! 1
AP(M)
AP(M)
-4...
with
dip = e-t.dpetf,
p = 0, 1, 2,... , n - 1,
and to compute the perturbed Laplacian Q` = de de + d=''d`-'.
= A + t(... I + t2(... I. W e range the eigenvalues of A as follows 0 < A; (t) < az(t) < ... < J1k(t) < ..
The Hodge theory implies that there are Op eigenvalues equal to 0, where )3p is the pth Betti number, p = 0,1, ... , n - 1. In local coordinates,
0f z
De =Op+t21Vfl2+tEf 8xioxj
Witten's Proof of Morse Inequalities
274
Assume that x' is a critical point of f. We approximate Di in a neighborhood of x' and obtain the approximate perturbed Laplacian: 02
P
:.1
t2µi (x')2 + tui [dx'A, id,,, j
(8xi)2
where {pi} are the eigenvalues of the Hessian d2 f (x'). If we put all these t ; together (in the product space) for all critical points {xj*} as a new operator, and range all eigenvalues as follows: 0 < tei < tez < ...
the number of zero-eigenvalues is then proved to be the number of critical points with Morse indices p. The simple version of Morse inequalities
pp < mp := #{x! I ind(f,xJ) = p}, then will be proved if we have the following asymptotics: lim Ak(t) = e' too t
k = 1, 2, ...
.
A revised elliptic complex is used to prove the final version of the Morse inequalities.
The material of this appendix is based on [Witl], F. Annik [Annl], G. Henniart [Henl] and B. Helffer [Hell].
1. A Review of Hodge Theory Let (M, g) be a compact, connected, C°°, n-Riemannian manifold without boundary. TM denotes the tangent bundle; T2M denotes the tangent space at x E M; T'M denotes the cotangent bundle; T. *M denotes the cotangent space at x E M; APT'M denotes the anti-symmetric tensor product of T'M. The section of APT' M is called a p-form over M. C°°(APT'M), the set of all C°° p-forms, is denoted by AP(M). V w E AP(M), in the local coordinates (x1, ... , xn), is expressed as follows: W=
A
A dx'o.
1
We write gi, = g (FZI a , I), (gig)-1 is positive definite.
g`j
= g(dx',dx'), i < i, j 5 n. Then (g'") _
1.
Hodge Theory
275
We may also extend g to p-forms:
g (dx'' A ... A dx", dx" A ... A
where (kl,... , kp) runs over (1, ... , p), and
if
Ek,...k = ±1
1, ... , p
is even
kl, ... , 4
is odd.
The differential operator d: AP(M) - AP+'(M) is defined to be (1) a linear operator, E 1 e- Ldxt A dx" A ... A dx'. (2) d A ... A From the definition, it is easily seen that (i) d2 = 0. (ii) d(w A 0) = dw A 0+(-1)"w A dO, V w E AP(M), V 0 E A9(M).
The Hodge star operator *: AP(M) - A^-p(M) is defined as follows: (1) *(a(x)w + b(x)O) = a(x) * w + b(x) * 0, (2) *(dx" Adx'^, where
A
1 < ji < ... < jp < n,
1 < ip+l < ... < in < n,
{ j1 i
... , jp, ip+I
in } is a permutation of { I.... n}, {kl, ... , kp} is a perIgI'"2ee,,...,t,,, IgI = det(ggj), and
mutation of (1,... ,p},
at,...t = fl
(tl,,e2,...tn)
if
its oda
Then we have
(i) *1=ri, *rl= 1 where rl=IgI1I2dx'A...Adx^, (ii) * * w = (-1)P(^-p)w V w E AP(M), (iii) g(w, 0)rl = w A (*0) V w, 0 E AP(M).
Claim. We only want to verify this identity for w = dx'' A A 0 = dxi' A ... A with 1 < it < ... < ip < n, 1 < jl < ... < jp < n.
gWr .
LHS = E
171
FLHS =
dx A
A dx" A
, .k = IgI1
2
k,k
9i
g'':., ...
, ... gi, dx" A ... A dx'^
dx
)
Witten's Proof of Morse Inequalities
276
Since {i1, ... , in} must be a permutation of 11,... , n}, and {t1 .. tP, io } is a permutation of {1,... , n} with t1 < t2 < < tp, we have
ip+1
it = t1,... i, = t,,. Therefore,
RHS = lgl1"2e,,...,. E ek,...k,g'lik, ... gioik'&," A ... A dx'^ k, ... kP
= Igl1/2
L,
Ekl...k,gi'ik, ... gi'-k. dx1 A ... A dxn
k,...k,
= LHS.
The scalar product on AP(M) is defined by
(w,9) = JM g(w'B)n
=Jo*0). M
It is real, symmetric, bilinear and positive definite. The completion of AP(M) with respect to (, ) is denoted by Ap,(M). It is a Hilbert space. The codifferential operator d': AP(M) -+ AP-1(M) is defined to be the adjoint operator of d with respect to (, ), i.e., (d* w, p) = (w, dp)
Vw E AP(M), tl p E AP-1(M). Note. The scalar products on both sides are different! Basic properties of d'. (i) d' = (-1)n(P-1)+1 * d*. Claim. (d*w, p) _ (dp, w)
_ =
(-1)P(n-P)(dp,* *w)
r
dp A (*w)
M
_(-1)P JMpA(d*w)
(ii) d'd' = 0.
_
(-1)P+(P-1)(n-P+1)
_
(-1)(P-1)(n-P)+1 (p,
A (* * d *
JM *d * w)
277
Hodge 77seory
1.
The Laplacian. AP: AP(M) - AP(M) is defined to be d'd + dd'. A pform w satisfying APw = 0 is called a p-harmonic form. Denote IIP(M) _ ker(AP). Example. p = 0. V f E C°°(M),
of = d'df =
-IgI_1/2
E I,j
; (IgI12gh1__f). 0xi
This is the Laplace-Beltrami operator on (M, g). We have (1) Let D(AP), be the space of WW -Sobolev sections of the vector bundle
APT'M. Then AP is positive and self-adjoint. Claim. V 0, w E AP(M), we have (APw, 9) = ((d'd + dd' )w, 9)
= (dw, dO) + (d'w, d'9)
_ (w,(d'd + dd')9) _ (w, APO).
Riedrich's extension provides the self adjointness. The positiveness is obvious.
(ii) AP is an elliptic operator. See (vi) In the following paragraph. (iii) AP possesses only discrete spectrum, i.e., it has only eigenvalues ), with al > 0, AP , +oo as k oo, and each o(AP) = JA < AP2 < eigenvalue has only finite multiplicity. This follows from Riesz-Schauder theory. Exterior and interior product. V w E A' (M), wA: dxl' A ... A dx"P - w A dx{' A .
A dx+,, AP (M) -' AP+1(M),
P
iW: dx" A ... A dx" -- F ,(-1)i+'g(w, dx")dx" A
..Adx" AP(M) - AP-(M), are called the exterior and interior product with respect to w respectively. These products are extended to AP(M) linearly. (i) One has d 0 E AP-(M), d to E AP(M), (w A 0, 1P) = (0, i,+G).
(ii) V f E A°(M), V 0 E AP(M),
d'(f0) = fd'O - idfO.
Witten's Proof of Morse Inequalities
278
Claim. V t/' E AP-1(M), (t0,d*(fO)) = (doo,fO)
_ (fdty,B) = (d(f i,b) - df A tb, 0) = (f 0, d* 0) - ('0, id/ B)
= (0, fd'0 - idf0). (iii) V w,, w2 E A' (M), V 0 E AP(M), i"', (w2 A 0) = 9(wi, w2)0 - w2 A i,,,, 0.
Claim. We may verify this for w1 = dx' and w2 = dx1 or dx2 in suitable coordinates.
(iv) The principal symbol of the differential operator d is oLd = it;A, where t C,dx', (£1, ... , E T'M. Therefore d
axjdxl A dx" A ... A dx".
(a;,..., dx" n ... A dx'n)
Therefore, aLdw = i> t jdx' A w, d w E AP(M). Note. For d, the symbol ad = the principal symbol aLd. (v) oLd' = i it, where C = t:idxl. Claim. Letting 9, w denote the Fourier transforms (in local coordinates) for 0 E AP(M) and 11' E AP+1(M) respectively, (B, d' p)) = (dO, t')
= (uLd B, tG) = i(C A W, 'P)
= i(0, it Therefore (B, ad' 11i) = i(B,ittl'). (vi) aLo = -ICI2. Claim.
oL(d'd + dd') = CLd'oLd + OLdaLd'
=-(t;Ait+it-CA) By choosing
e
along an axis, say£=(i1,...,1,,)=ICle1, el A it., + ie, e1 A = 1.
1.
Hodge Theory
279
Elliptic complex. Let Mn be a Riemannian manifold, and let E = {E{}"0 be a family of vector bundles over M. Let d = {di}o-1,
E{-+CEi+1),
di:
i=0,1,...,n-1,
be a family of pseudo differential operators (t/iDO) of order r, satisfying (1) di+ldi = 0, (2) V x E M, V { E T= M \ {B}, the sequence
-+ El _- ...
0 - ED
En -+ 0
is exact, where oLd(x, ) is the principal symbol of the ODO d. We say that (E, d) is an elliptic complex. Example (de Rham). We define (E, d) as follows:
E = {APT'M};=o,
d = {dp}D_o,
where dp is the differential operator. This is an elliptic complex. Claim. We only want to verify the exactness of the sequence do
- A'T'M
°
°y
Since V w E Ap(M), o,Ld(x, f )w= = ie A W. ,
Cidx and it is easy to see that
where
ker oLd(x,l:) = Im (Ld(x, along an axis, say
el,
elA(ei1A...Aei,) =
0
(Choose
el nei, A...Aei.
if i1=1 if i1>1
Therefore, ker oLd(x, l ; ) = Span {ei, n
A ei, I 1 = i1 < .
< i , , < n}
= Im elA = IM aLd(x, t).) Let (E, d) be an elliptic complex, define Di: Coo(Ei) -+ C°°(E1) as follows:
Di = di-l di-1 + di di,
i = 0,1, ... 'n - 1.
Witten's Proof of Morse Inequalities
280
We have
(i) Di is symmetric (and it has a self-adjoint extension), and positive. The proof is quite similar to those for Op. (ii) The tbDO Di is elliptic, i.e.,
aLDi = atdi_I CLds_1 + olds aLdi is invertible. Claim. Assume that for 0 E Ei, (0LDi)O = 0, then 9 (I(cLd_1 (atds-1) + (aids) - (cLd;)J9, 0) = 0, 9((aLdi)e, (c'Ldi)O) = 0, 9((atds_I )9, (aLds-1)B = (aLdi)9 = 0.
By the exactness of the sequence, (aLdi) 0 = 0 = 3 tt E Ei_I such that 9 = 7Ldi-1'', therefore 0 = (aids -1)e = (aids * 9((aLd;-1)O, (at d;-1)O) = 0,
*0=aLdi-10 =0. In the following, we use the same notations di (and d!), representing the differential operators with domains COO(E;) as well as their closed extensions in L2(Ei), i = 1, 2, ... , n - 1.
Hodge Theorem. Let (E, d) be an elliptic complex, and let Di = di-Ids_1 + dsd;,
i=0,1,2,... ,n-1.
Then we have (i) L2 (E,) = N(D;) ® R(di-I) ® R(ds);
(ii) N(d;) = R(di_1) ®N(D;); (iii) N(ds-1) = R(ds) ®N(D;); (iv) 11`(E,d) = N(d,)/R(di_1) defined to be the cohomology group of the elliptic complex. Then for each i = 0, 1, ... , n - 1, the following isomorphism holds:
H'(E,d)
N(Di),
where we denote N(D) = ker(D), R(D) = Im(D) for each linear operator.
Proof. Di has a self-adjoint extension, which is denoted by the same notation. We have L2(Ei) = N(Di) ® R(D;) (Because Di is elliptic, Di has closed range.)
2.
The Witten Complex
281
By definition,
R(D1) C R(d,) + R(di_1), however,
didi-1 = 0
implies that (d{w,d0j-1) = 0, dw E C°°(E,+1), YO E C°°(Ei_1), = R(di*) I R(di_i) = R(D4) C R(d,') ® R(di_1).
On the other hand, R(di*) C N(di)1, R(di_1) C N(d;_1)1
R(di*) ® R(di_1) C N(di)1 + N(d-1)1 C N(Di)1 = R(D1) The last inclusion follows from
N(D1) C N(di) n N(d;_1). We obtain the first conclusion: R(Di) = R(di*) ® R(di_1), and
L2(Ei) = N(Di) ® R(d;) ® R(di-1) For (ii), since N(di) C R(di )1, we have N(di) C N(D4) ® R(di_1). Conversely, N(D1) C N(di) is known, and
R(di_1) C N(di)
follows from
didi_1 = 0.
(ii) follows. (iii) is obtained in a similar manner. (iv) is a direct consequence of (ii).
Corollary. For the de Rham complex,
H'(M) = N(di)IR(di-1) is defined to be the ith cohomology group of M, which is isomorphic to
N(&),i=0,1,...,n-1. The Betti numbers
f3i = dim Hi (M)
= dim H'(M) = dim N(D'),
Witten's Proof of Morse Inequalities
282
i=0,1,...,n-1. 2. The Witten Complex Let f : Mn -+ IR' be a C°°-function. xo E M is called a critical point of f if
df(xo) = 9.
Let K be the set of critical points of f. A function f is called nondegenerate if d2 f (x) is invertible for each x E K. For a given nondegenerate function f, we define a new complex (E, de) as follows:
E = {A3(M) I p = 0,1,... ,n}.
p=0,1,...,n-1,
dP = e-tJdpeef ,
de =
{d' I p=0,1,... ,n- 1},
A"(M) -
0 - A°(M) -.... e-tf1 0
-
A° ( M
0
AP+'(M) - ...
0.
a-tf
a-ef1
) - ... -
- ...
Ap+1(M)
AP ( M)
It is easily verified that (E, de) is an elliptic complex, d t > 0. Claim.
(1) ded'-' =
e-efdpdp-reef
= 0,
(2) QL(d') = QL(dp) = i£A, so that the sequence
cc(di ')
0- A 0
ATM
ot(di)
is exact. Similarly, we define (4* w, 0) = (w, 4 0) V w E Ap+l, V 0 E AP; therefore d'*w = eefdde-ef.
Then define AP
= -"d' + dP-'dP-''.
The Witten Complex
2.
283
By the Hodge theorem for elliptic complexes,
kerAP
kerd't/Imdt-1 ker dP/Im d4_1
ker AP,
= /3P = dim ker Apt.
Claim. The second isomorphism holds, because a: w -4 a-tfw satisfies (1) a l ker ker dp -' ker d' is an isomorphism, (2) a Im dP-1 511-Im dP-1. e Next, we compute At. (1)
dtw =
e-'f d(etfw)
= e-tf (tetfdf A w + etfdw)
= tdf Aw+dw. (2)
dt w = etf de (e-tfw)
= etf (e-tf d'w - ide-,/ w)
= d'w - etfi_te-I;d, w = d'w + tidf w. (3)
Atw = dtd* w + dt dtw
= t df A d; w + d(dt w) + d' (dtw) + t idl(dtw)
= t df A (d'w + t idfw) + d(d'w + t idfw)
+d'(tdf Aw+dw)+tid/(tdf Aw+dw) = Aw+t[df Ad'w+d(idrw)+d'(df Aw)+id/dw] +t'(df Aidfw+idf (df Aw)] = Aw + t2g(df, df)w + t Pew ,
where
Pjw = idrdw + d(idfw) + d' (df A w) + df A d' w.
Let us express Pdf explicitly in local coordinates. First we observe that P,y (W) = ,pidfdw + idf(d(p A w) +,pd(idfw) + dcp A (id fw)
+,pd'(df Aw)-id,,(df Aw)+Ipdf Ad'w - df A id,,w ='pPdlw+9(df,dV)w-9(d'p,4f)w = Ip P4fw.
Witten's Proof of Morse Inequalities
284
Next, we assume that
K={x*Ij=1,2,...a). We may find coordinate charts {(U5, ip,) I j = 1, 2,... , s} such that x* E Uj, U, nUj = 0 if i 34 j, vj: Uj R", with Wf(x*) = 0, and assign a special metric gj on U, such that 94W'
=bkt,
\ayk/' V'
j = 1,2,... ,s, k,e= 1,2,... ,n, where y = Vj(x). In this case, gj on U, is flat, so
P4dxt = d(idfdx') +d*(df A dx') d
(F"a-Lf
= 1: a a 2f k't
t dxt A irk dxf + > - fk d(idlk dxl )
+E
aftd'(dxt A dx')
"2f (dx' A
_ k,1
ext dxt A dxl
xk:dyk dxr ` + d'
axkaxt
2
-
aata kid=k(dxt ndxt)
id=k - id=k dxtn)dxl
where we use the notation
dx' = dx" A ... A dx`a,
W.
I=
Let us introduce the commutator: [dxt A, id=k ] = dxt A id=k - idSk dxt A
.
We obtain
a = E 8 axt [dx'A, idxk]w, z
P'dfw
b' w E AD(M).
IC't
In summary, for a suitable metric g,, in a neighborhood of a critical point x' of f, we have O°W = Opw + t2
(a
a2f fk
w+t 2
kt
axkaxt
[dxtn, id=k Jw.
285
The Witten Complex
2.
It is important to note that neither the Betti numbers Pp, p =
0,1, ... , n, nor the Morse type numbers mp, p = 0,1, ... , n, are influenced by the changing of the Riemannian metric g, so we could choose a suitable g to simplify our computations. First, by the Morse lemma, we find neighborhoods Uj of critical points xj* of f , j = 1,... , a, as well as local charts Vj, such that U, n Uf, = 0, if j & j', co1(xi) = e,
1: 14kyk
f(x) - f(xj) =
2
y = Vj (x) for x E Uj,
k=1
where
d2f (xj) = diag(µi,... , pn). Second, let Vj be an open neighborhood of Uj, with Vj n Vj' = 0 if j # j',
j = 1,... , s, and let
V0=M\UU1. j=1
Then {Vj }p is an open covering of M. We have a C°°-partition of unity E"=o nj, where supp % C Vi and v7j = 1 on Uj, j = 1, ... '8. Define
9=7709+t77j9j j=1
This is the metric we need. Provided by the new metric g` on M, Di equals the following operator in Uj:
at
rn
:1 =k=1u -
(0)2 + t2µk2xk + tµk
(dxkA,
id,,. 1
It is an operator of separable variables. Notice that
Ht =
- (d)2 + t2p2x2
is the Hermite operator in mathematical physics (harmonic oscillation). It has eigenvalues t II
with eigenfunctions
'PN(x) = HN( tJPJx)e-
,
where HN are the hermitian polynomials, N = 0, 1, 2, ... .
Witten's Proof of Morse Inequalities
286
Denote
Hi .) _ - ( -x
+ t2 Pk2xk,
/
C7xkI and
Kk = [dxkn,id=,.],
k = 1, 2,... , n. We have n
0t=; _ E(Hi'j+tA,Kk). k=1
Since
KkdxI = ekdx', where I
if kEI
1
ek -
if k¢ I,
{- 1
obviously Kk is a scalar operator on A9(Rn). Thus the operator AP. . is self-adjoint, with eigenvalues n
t > [(1 + 2 Nk)Iµki +
ekf
µk]
k=1
and eigenvectors (orthonormal) t
11
WN1
t'/4 II HNk ( tlµkI xk J exp -2 E IµkIxk k=1
dxli,
k=1
where Nj _ (Ni , ... , Nn) runs over Fln, and 1P H with i, < < ii,, and j = 1, 2, ... , s.
i ,) runs over
We define the direct sum space H = ® AP, (ll&n),
(s-copies),
i=1
and a self-adjoint operator A'
AP(wl,...
)
P
p
We range the eigenvalues of AP as follows: 0 < tei < teZ <
< tek <
).
The Weak Morse Inequalities
3.
287
Theorem. dim ker(Ai) = mp := #{xj* E K I ind(f, xf) = p}.
Proof. By definition, Di :; t 'IJ = 0 iff n
E[(1+2Nk)Ir4I+fk'pk] =0 k=1 ajNk
0
l IµkI+jAke = 0,
k
1,...,n
Nk = 0 141 > 0
µk<0
if k $ Ij = ind(f, x!) = p. if k E P
Therefore, each xj* with Morse index p has a one-dimensional contribution to the null space, but if ind(f, x1) j4 p, there is no contribution and therefore dim ker(A') = mp.
3. The Weak Morse Inequalities We shall prove the following inequalities:
mp>pp,
p=0,1,...,n.
If we compare with the two operators AP and A= , we see that
mp = dim ker(A') and Op = dim ker(0' ). We range the eigenvalues of the operator AP as follows:
i
0 < AT(t) <_ ... < Ak(t) < ..
.
The weak Morse inequalities hold, if we can prove lim t-.+oo
AM t
= ek.
First, let us pull back the eigenvector in H
a={Nj,1jI1<j<s}, onto the differential manifold M.
We have charts (Uj, s,), where Uj is a neighborhood of xx , Wj is a coordinate such that the Morse lemma holds, and, on (Uj, Wj ), the metric g' is Euclidean, j = 1,. . . , s.
Witten's Proof of Morse Inequalities
288
Define a cut-off function p E C°°(Rn), such that 0 < p < 1, and (yI < 1
P(y) _ (
And define
1
lyl > 2.
0
a
> P(t"Scoj (x)) (cPt)i o cPj (x), j=1
where (V ')j is the j-component of cpQ, which is a vector in A'(M). For t > 0 large, the support of iPQ is concentrated in U =1 Uj. These vectors are considered to be "approximate eigenvectors" for the operator L
.
In order to prove (*), the following Rayleigh-Ritz principle is needed.
Theorem (Rayleigh-Ritz). Assume that A is a self-adjoint operator bounded below on a Hilbert space H. If A only possesses discrete spectrum, < An < consisting of eigenvalues with finite multiplicities, Al < A2 < , then (Ax, x) Sup inf An = xED(A) xEspan(W,,...
Ilx112
Proof. According to the spectral decomposition theorem,
(Ax,x) =
Aix.,
where xi = (x, e;), and e; is the ortho-normal eigenvector corresponding to
A,,i = 1,2..... Therefore, (Ax, x)
An
xED(A)
Ilx112
xEspan {e, ,... ,en _, }
<
sup
inf
xED(A) zEspan{,p,,... ,Wn-1)1
(Ax, x) Ilxll2
On the other hand, V {cp1, ... , (pn_1 } # lei,... en-1), we may choose xo = j=1 xjej such that xo 1 {cp1,... ,Vn_1} and
I > (Axo, x0) " Ilxo112 This proves the equality in the theorem. Let us make some computations: (i)
(r/,, ,V,) = bap + O(exp(-ati/s)) as t -+ +oo, where a > 0 is a
constant.
3.
The Weak Morse Inequalities
289
Claim. Noticing that (ip gyp) = b p in the space H, we have
J E* p2(t2"y)(v' )j(y)((pp)j(y)dy _ (ova 1 w) -
f [1- p2 (t21 sy)] (,o
(y) (Vtp)1(y)dy
=6o + I. Since 0 < p < 1, p(y) = 1 for jyI < 1. Letting we have
I < Js I>_t'i'o o)L('pa)I (z)(W')j(z)dz, f=i <
P(z)e-I'12dx = O(exp(-at"')),
J
where the explicit expansions for Hermitian functions are used, P(z) is a polynomial of z, and a is any positive number less than 1. Before computing (Ot , Ot i*tp), we need (ii) V h E C°°(M), [h, (h, &P]] = -2(Oh)2. Claim. V w E AP(M), [h, (h, AP)]w = (h2Op - 2hOph + &Ph2)W = h2dd'W - 2hdd'(hw) + dd'(h2w)
+ h2d'dw - 2hd'd(hw) + d'd(h2w) = h2dd'W - 2h2dd'w + h2dd'W
-2hdhAd'w+2hdhnd'w + 2hdidhw - 2hdidhW - 2dh A tdhW
+ h2d'dw - 2h2d'dw + h2d'dw + 2hidhdw - 2hidhdw - 2hd' (dh A w) + 2hd' (dh A w) - 2idh (dh A w) = -2(dh A idhW + idh(dh A w))
= -2(Vh)2w.
Witten's Proof of Morse Inequalities
290
(iii) (0',L 1/'p) = 1(ee + ep)(0.1,0') + O(exp(-at1/5)) as t - +oo, where eQ and eQ are the eigenvalues associated with W1 and V. respectively,
and a > 0 is a constant. Claim. (+G.,
Di oo) - t (eQ + ep) (io, 100')
[(P(t215y)(p)j,
_
-
Di ;P(t2/5y)(W))o(B^) )')AP(E^) - 2(P2
2(Apx;('Pa)1,P2 M
1 t((,pt )j (2Pz'z.P- 'p2 -
2OP
j=1
-2
_
)
t
((Vot)j, [p, [P,Dt' ((Pp1)j)AP(B^)
_ - Na)j, [P, JA API] (V'O)j) A(1-) j=1 e
_ E(((Pt)j, (Vp(t2/5y))2(co j=1
because AP,_,; = Op+ terms without differentials, which commute with p. Again, we see (VP(t2'5y))2
= t4/51(VP)(t2/1y)12,
which is equal to zero outside It2/5y1 < 2, and therefore
((V,)', (VP(t2/5))2(Vtt)')AP(m^) t4/5P(z)e-t Z12dz = O(exp(-atl/5)41>20/10
41>20/10 where P(z) is another polynomial of z.
Now, we turn to the first half of our conclusion: lim
k(t)
t_+oo t
< ek.
Proof. We range {?Gk I k = 1, 2.... } in such a way that Vk corresponds
to the eigenvalue ek, k = 1,2,.... By the Gram-Schmidt procedure we obtain
(_:ek k
'Pt k
- E CjkVj j=1
The Weak Morse Inequalities
3.
291
where k-1
cfk(Pt
i = 1,... ,k - 1.
1) _
j=1 Therefore,
cjk = O(exp(-atl/s))
as
t -' +oo.
It follows that
Di k) = 2 (t j + t k)ajk + O(exp(-atlas)), j, k = 1, 2, ... , and that {t14 I k = 1, 2.... )
is an orthonormal basis.
By the Rayleigh-Ritz principle, )1k(t)
-
1
AN
(tG, tAttG)
t
'ESpsn{lb,.....lbw-l)1
<
1
inf
sup
(Pop, t Di PV IN
M011=1,OE9pan{O1,... ,O1k_1) '
where V = span(t/il,... > kt), and PV is the orthogonal projection on V. Therefore
k(t) t
<
sup
inf
(PV'+G, 10Pt PV P)
t
$EV 11011=1.*E.pan{Pv#3....
sup
inf
(t/i, 10POP)
t
#ON=1.OEapan {,l'i,... ,1br _, )1 1
sup (w,t A
,OEV
t
IIkII=1
< ek + O(exp(-at1/5))
t - +oo.
as
This proves lim
k(t) t
t--,+oo
< eP.
The rest of this section is devoted to proving the second half of our conclusion, i.e., t
Ak(t) lim > tk. 4700
t
Witten's Proof of Morse Inequalities
292
On the manifold M, we define a cut-off function J; (x)
0
x51 Uj
P(t2/5cPj (x))
xEUj
t>1, j=1,2,...,s
and let (Jo)2 = 1
- (J;)2 j-1
Then we have
(iv) A = Fj=OJJAtJJ Claim. Substituting It = J,' in (ii), we obtain (Jjt)2At
- 2Jj'A'J + i
(J; )2
= [Jill [Jill Atl] = [Jj I (Ji I On)]
= -2(VJjt )2. Since
e E(J4t)2
= 1,
j=o
it follows that
e
e
AP
)2. t = Ej=0J.i At Jit - )L(VJf j=0
Lemma. Suppose that ek < r < ek+1 Then for large t > 0, there is a finite rank operator Fk(t): AP L2(M) - 42(M) with dim Im Fk(t) < k, such that
At > rt Id + Fk(t). Proof. Since e
AP
a
= JJOtJo + j JJOt JJ j=1
j=0
the operator of the second term acts as the same as the operator At together with a cut-off function. Let Pk be the orthogonal projection onto the subspace spanned by the first k eigenvectors, corresponding to the eigenvalues e°, ... , ek. Then the operator
Fk(t) = E JjtF'kAt PkJ1t j=1
3.
The Weak Morse Inequalities
293
(Pk stands for the pull back of Pk on AP(M)) is of finite rank, with dim Im Fk(t) < k. We have V 0 E AP(M), (i)
(JoD' Jo1,, +G) = (A Jo+', Job) f jot 0, jot 0)
= (APJoo, Jo+G) + t2IVf I2IIJo+GII2 + t(Pd
= T1 + T2 + T3i where T1 > 0.
As for T2, 3 co > 0 such that
for xEVo=M\ 6 U,.
IVfI2>eo,
j=1
Since IV!(x)I2 = IIJolpII2
for x E Uj, y = Vj(x), we know
. IVf(x)I2 =
0(t-011)IIJo,OII2,
for x E Uj,
and therefore T2 = t2IVfI2IIVJo+,II2 >- E1tIl/5IIJoGII2,
for some E1 > 0.
As for T3, Pqr is a bounded operator, which commutes with the multiplications of a function, and therefore
T3 > -MtIIJoeII2,
In summary, (JoD'
for some constant M > 0. tek+1 IIJo+PII2 for t large,
(Jit DiJjO, 0) = (APOt, Ot),
(ii) j=1
where IPt E H equals the element {p(t2/sy)O(Wj to the orthogonal decomposition,
And, according
(APOt, Ot) _ (Ai (I - Pk)1Gt, (I - Pk)t,ht) + (Fk(t)O, 0) > tek+1110t1I2 + (Fk(t)+G, 0)
= tek+1 >((J!)2) + (Fk(t)+G, 0), j=1
where Fk(t) = Ej'=1 JjtPk(A - tek+l)PkJ,.
Witten's Proof of Morse Inequalities
294
(iii) We know that (OJJ (x))2 = 0
if x
(VJ(x))2
((/'\)
= k>
Uj
k
1
P(t2'SAP,(x)))
= O(t4/5) k> (1 = O(t4/5)
k / (t2/5Vj (x)))
if x E Uj/,
j = 1,2,... s. And 1/2 $
I - r(JJ(x))2)
J0, (x) =
j=1
so that
if x E Vo = M \ U Uj,
(OJo(x))2 = 0
j=1 a.I0
u (axk)
(OJO(x))2 = >
aa
= t4/5 E k
/[I
2
L
(
1
k)
(x)) ' P(t2/5'Pj (x))
J
- P(t215'Pj (x))2]
= 0(t4/5) if x E Uj. Then, finally, we obtain V 0 E AP(M), e
(Did, 7P) > tek+1((JO)2W, V)) + tek+1 F((Jj )2',, 7p)
j=1
0(t4/5)IIbII2
+ (Fk(t)tp,tp) + 0(t-1/5))II II2 + (Fk(t)'+G, ). = t(ek+1 + If ek < r < ek+1+ then for large t > 0, we have
AP > t r Id + Fk(t). Now we are going to prove lim :Kt c) > ek. The proof is divided into two cases.
e moo
Morse Inequalities
4.
295
(1) ek_1 < ek. We choose c > 0 such that
ek_1 <ek-E<ek. Then we have Fk_I(t) (a bounded operator with rank < k - 1) such that A P > t(ek - E)Id + Fk_1(t)
for t > 0 large.
According to the Rayleigh-Ritz principle, --fit t)
=
i nf OED(AP)
sup ek - e
(
0 1t AN) 1
for t > 0 large,
provided we take E > 0 is arbitrary, we have
as a basis of the subspace Im Fk_1(t). Since
lim
tFOo
k(t) t
> ek.
(2) ek_I = ek. We may assume that ek > 0, and then 3 d > 1 such that ek. According to case (1), we have 4-d < ek-d+l '\k(t) too t >Z 1imoo
'\k-d+1(t)
li
t
>-d+1 = ek
This proves our conclusion.
Theorem. Suppose that M is a compact, connected, orientable C°°manifold. Then there exists a Riemannian metric g such that lim
t-.+oo
k - = ek. t
4. Morse Inequalities We have defined pp, mp, p = 0, 1, ... , n in Sections 1 and 2. Now we are going to prove the following inequalities: mo > Ro,
MI -MID >fll -00
Mn -mn-I +"'+(-I)nm0
Nn -FOn_1
++(-1)n/jo,
Witten's Proof of Morse Inequalities
296
or, in a compact form, letting PM (t) =
,Ope,
Mf (t) =
mptp,
we have
Mf(t) = P-11 (t) + (1 + t)Q(t), where Q(t) is a formal power series with nonnegative coefficients.
Let 0 < E < Min{em,+1 I p = 0,1, ... , n}. Fixing t large enough, we define a new cohomology complex as follows:
XP = Xt = jw E AP(M) it is an eigenvector of At, with eigenvalue am (t) such that '"t(t) < E}. According to the theorem in Section 3, we see that
dim X' =mp,
p=0,1,... n,
and we have
(i) dt : XP -' Xp+1, dt-1:XP-XP-1. Claim. V w E X", we have O'w = am(t)w with AMP (t) < et. Therefore
Ot+'dtw =
(dt+1'dt+1
+dtd' )dtw
= dt dt * dt w
=
dt-1dt-1-)w
= dtA w = \P (t)dtw. This implies that dtw E Xp+1 Similarly, one proves dt-1.w E XP-1, so we obtain a smaller cohomology complex,
0-+X°
do,
, X1d1'+... d"-I X"--'0.
= Qp. (ii) dim Warning. This is different from the property stated in Section 2 because the complex is different. N(dt)IR(dt-1)
Claim. We see easily that (1) N(AP) C XP n N(dt). (2) V w E XP n N(dt) n N(At)1, we have Otw = am(t)w where
AP (t) 34 0,
Morse Inequalities
4.
297
and
AN = (dp dt + dt-1dt-1*)w = dt-1dt-' w. Since dt-'*w E XP-1, we see
di-ldt am(t)
E R(dtp-1),
i.e., those p-forms in Xpf1N(dt ), which have contributions in N(dt)/R(dt are just At harmonic forms. Therefore,
N(dt)/R(dt -') °-` N(AP) in the smaller cohomology complex.
Theorem. Suppose that M is a compact, connected, orientable C°° manifold and that f: M - IR' is a nondegenerate C°° function. Then the Morse inequalities hold.
Proof. We start with the following cohomology complex:
0 - X0
` X 1 ` ...
Xn
+ 0 for large t.
We have shown that (i) dim XP = mp, and (ii) dim N(d1t)/R(dr1) = OpSince
dim XP = dim N(dt) + dim R(dt), and
dim N(dt) = dim R(dt-1) + Op, we obtain
mp = Op + dim R(dt) + dim R(dt-1),
p = 1, ... , n, where we assume di = 0. It follows that E(-1),n-p(mp
- (3p) = dim R(di)
> 0,
P=O
f o r m = 0, 1, 2, ...
, n. And for the last one, it is an equality: n
n
E(-1)n-pmp = E(-1)n-ppp. p=o
p=O
-1),
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INDEX OF NOTATION
df fa
K
K. (PS)
differential off level set of f, not above the level a critical set critical set with critical value a
Palais-Smale condition exp(-) exponential map A Laplacian operator OM Laplace-Beltrami operator A tension operator V gradient operator mes(-) measure IAA measure of A direct sum ® WT A1A OA
Fix(.) A
i,,,
Id
transpose of the matrix W loop space on A cardinal number of the set A fixed point set exterior product interior product identity operator
19
20 19 21
20 72
142
230
230
141, 229 235 175
180 182
204
216 216 277
277 99
INDEX
Arnold conjecture on fixed points, 216
on Lagrangian intersections, 217 Banach manifold, 14 Betti number, 3 bifurcation, 129, 161 blow up analysis, 232 Bott 79, 206
cap product, 9 category, 105
relative category, 109 conformal group, 360 convex set, 60 locally, 60 Courant Lebesgue lemma, 268 critical group, 32 critical manifold, 69 critical orbit, 67 critical point, 18 w.r.t. a locally convex closed set, 62
critical set, 18 critical value, 18 cuplength, 9 cup product, 9 Deformation lemma, 21 Deformation retract, 20 strong, 21 Deformation theorem first, 29 equivariant first, 67 second, 23 equivariant second, 68 degenerate critical point, 43 non, 33, 41
Euler characteristic, 6
Finsler manifold, 18 Finsler structure, 15 Fredholm operator, 47, 97 G-action, 66 G-cohomology, 75 G-critical group, 76 G-equivariant, 66 C-space, 66 Galerkin approximation, 111 general boundary condition, 55 genus, 96 cogenus,96 gradient flow, 19 Cromoll-Meyer pair, 48 Cromoll-Meyer theory, 43
Hamiltonian system, 179 handle body theorem, 38 harmonic map, 229 harmonic oscillation, 285 heat flow, 229 Hilbert Riemannian manifold, 19 Hilbert vector bundle, 70 Hodge theory, 274 homology group, 3 relative, 3 homotopy group, 12 relative, 12 Hurewicz isomorphism theorem, 13 hyperbolic operator, 41 invariant function, 111 isolated critical manifold, 69 isolated critical orbit, 74 isolated critical point, 43
Index
312
Jacobi operator, 251 jumping nonlinearity, 164 Kenneth formula, 5,
Poincare-Hopf theorem, 99 projective space real, 6, 11 complex, 6, 111 pseudo gradient vector field, 19
Landesman-Lazer condition, 153 Leray-Schauder degree, 99 link homological, 84
homotopical, 83 Ljusternik-Schnirelman theorem, 105 locally convex set, 60
Marino-Prodi theorem, 53 G-equivariant, 80 Maslov index, 183 maximum principle, 143 strong, 143 minimal surface, 260 minimax principle, 87 Morse decomposition, 250 Morse index, 33 Morse inequality, 36, 79 Morse lemma, 33 Morse-Tompkins-Shiffman theorem, 271
regular point, 18 regular set, 18 regular value, 18 saddle point reduction, 188 shifting theorem, 50 Sobolev embedding, 141 Sobolev space, 141, 231 splitting theorem, 44 strong resonance, 156 subordinate classes, 10 subsolution, 145 supersolution, 145 symplectic form, 215 symplectic matrix, 183
tangent bundle, 15 cotangent bundle, 15
Morse type number, 35 mountain pass point, 90 variational inequality, 65, 177 vector bundle, 15 Nemytcki operator, 141 normal bundle, 70
Palais-Smale condition, 20 w.r.t. a convex set, 62 (PS)*, 117 Palais theorem, 14 pendulum, 209 periodic solution, 179 perturbation on critical manifold, 131 Uhlenbeck's method, 136 Plateau problem, 260
Witten complex, 282
Kung-ching Chang Infinite Dimensional Morse Theory and Multiple Solution Problems
In this first book to discuss various critical point theorems in a unified framework, the author treats Morse theory as a tool to study multiple solutions to differential equations arising in the calculus of variations. Critical groups for isolated critical points or orbits - which provide more information than the Leray-Schauder index - are introduced. Topics covered include basic Morse theory and its various extensions; minimax principles in Morse theory; and applications of semilinear boundary value problems, periodic solutions of Hamiltonian systems, and harmonic maps. In a self-contained appendix, the author presents Witten's proof of Morse inequalities. Containing several new results, this volume will be attractive and germane to researchers and graduate students working in nonlinear analysis, nonlinear functional analysis, partial differential equations, ordinary differential equations, differential geometry, and topology.
ISBN 0-6176-345,-7
000
Birkhauser Boston Basel Berlin
111111111
9
80817 634513
