2 0. MORSE THEORY AND VARIATIONAL PROBLEMS
0.1. Compactness Conditions
It is usually necessary to assume some sort of a “compactness condition”
when seeking critical points of a functional. The following condition was
originally introduced by Palais and Smale [91]: Φ satisfies the Palais-Smale
compactness condition at the level c, or pPSqc for short, if every sequence
puj q Ă W such that
Φpuj q Ñ c,
Φ1puj
q Ñ 0,
called a pPSqc sequence, has a convergent subsequence; Φ satisfies pPSq if it
satisfies pPSqc for every c P R, or equivalently, if every sequence such that
Φpuj q is bounded and Φ1puj q Ñ 0, called a pPSq sequence, has a convergent
subsequence.
The following weaker version was introduced by Cerami [25]: Φ satisfies
the Cerami condition at the level c, or pCqc for short, if every sequence such
that
Φpuj q Ñ c,
`
1 ` }uj }
˘
Φ1puj
q Ñ 0,
called a pCqc sequence, has a convergent subsequence; Φ satisfies pCq if it
satisfies pCqc for every c, or equivalently, if every sequence such that Φpuj q is
bounded and
`
1 ` }uj }
˘
Φ1puj
q Ñ 0, called a pCq sequence, has a convergent
subsequence. This condition is weaker since a pCqc (resp. pCq) sequence is
clearly a pPSqc (resp. pPSq) sequence also.
The limit of a pPSqc (resp. pPSq) sequence is in
Kc
(resp. K) since Φ
and
Φ1
are continuous. Since any sequence in
Kc
is a pCqc sequence, it
follows that
Kc
is a compact set when pCqc holds.
0.2. Deformation Lemmas
An essential tool for locating critical points is the deformation lemmas,
which allow to lower sublevel sets of a functional, away from its critical set.
The main ingredient in their proofs is a suitable negative pseudo-gradient
flow, a notion due to Palais [93]: a pseudo-gradient vector field for Φ on
Ă
W
is a locally Lipschitz continuous mapping V :
Ă
W Ñ W satisfying
}V puq} ď

›Φ1puq›˚

, 2
Φ1puq,V
puq ě
`›
›Φ1puq›˚
˘2
@u P
Ă
W.
Such a vector field exists, and may be chosen to be odd when Φ is even.
The first deformation lemma provides a local deformation near a (pos-
sibly critical) level set of a functional.
Lemma 0.1 (First Deformation Lemma). If c P R, C is a bounded set
containing
Kc,
δ, k ą 0, and Φ satisfies pCqc, then there are an ε0 ą 0 and,
for each ε P p0, ε0q, a map η P Cpr0, 1s ˆ W, W q satisfying
piq ηp0, ¨q id
W
,
piiq ηpt, ¨q is a homeomorphism of W for all t P r0, 1s,
piiiq ηpt, ¨q is the identity outside A Φc´2εzNδ{3pCq
c`2ε
for all t P r0, 1s,
pivq }ηpt, uq ´ u} ď
`
1 ` }u}
˘
δ{k @pt, uq P r0, 1s ˆ W ,
pvq Φpηp¨,uqq is nonincreasing for all u P W ,
Previous Page Next Page