0.4. MINIMAX PRINCIPLE 7
0.4. Minimax Principle
Minimax principle originated in the work of Ljusternik and Schnirelmann
[76] and is a useful tool for finding critical points of a functional. Note that
the first deformation lemma implies that if c is a regular value and Φ satisfies
pCqc, then the family Dc,
ε
of maps 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 Φc´2ε
c`2ε
for all t P r0, 1s,
pivq Φpηp¨,uqq is nonincreasing for all u P W ,
pvq ηp1,
Φc`εq
Ă
Φc´ε
is nonempty for all sufficiently small ε ą 0. We say that a family F of
subsets of W is invariant under Dc,
ε
if
M P F, η P Dc,
ε
ùñ ηp1,M q P F.
Theorem 0.14 (Minimax Principle). If F is a family of subsets of W ,
c :“ inf
M PF
sup
uPM
Φpuq
is finite, F is invariant under Dc,
ε
for all sufficiently small ε ą 0, and Φ
satisfies pCqc, then c is a critical value of Φ.
We say that a family Γ of continuous maps γ from a topological space
X into W is invariant under Dc,
ε
if
γ P Γ, η P Dc,
ε
ùñ ηp1, ¨q ˝ γ P Γ.
Minimax principle is often applied in the following form, which follows by
taking F γpXq : γ P Γ
(
in Theorem 0.14.
Theorem 0.15. If Γ is a family of continuous maps γ from a topological
space X into W ,
c :“ inf
γPΓ
sup
uPγpXq
Φpuq
is finite, Γ is invariant under Dc,
ε
for all sufficiently small ε ą 0, and Φ
satisfies pCqc, then c is a critical value of Φ.
Some references for Theorems 0.14 and 0.15 are Palais [93], Nirenberg
[89], Rabinowitz [111], and Ghoussoub [53].
Minimax methods were introduced in Morse theory by Marino and Prodi
[79]. The following result is due to Liu [71].
Theorem 0.16. If σ P
HkpΦb, Φaq
is a nontrivial singular homology class
where ´8 ă a ă b ď `8 are regular values,
c :“ inf
zPσ
sup
uP|z|
Φpuq
where |z| denotes the support of the singular chain z, Φ satisfies pCqc, and
Kc
is a finite set, then there is a u P
Kc
with CkpΦ,uq 0.
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