October 29, 2013
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Index Theory Summary
Kelly & Alexandria
Review
Definition 1.1. A sequence um ∈ V is a Palais-Smale sequence for E if |E(um )| ≤ c uniformly
in m, while ||DE(um )|| → 0 as m → ∞. (Struwe)
Definition 1.2. E satisfies the Palais-Smale condition (P-S) if any Palais-Smale sequence has
a strongly convergent subsequence. (Struwe)
Theorem 1.3. Minimax Principle: Supoose M is a C 1,1 Finsler manifold and E ∈ C 1 (M ) satisfies P-S. Also suppose F ⊂ P(M ) is a collection of sets which is invariant with respect to any
continuous semi-flow Φ : M × [0, ∞) → M such that Φ(·, 0) = id, Φ(·, t) is a homeomorphism
of M for any t ≥ 0, and E(Φ(u, t)) is non-increasing in t ∀u ∈ M . Then if
β = inf sup E(u)
F ∈F u∈F
is finite, β is a critical value of E. (Struwe)
Theorem 1.4. Suppose M is a complete C 1,1 Finsler manifold and E ∈ C 1 (M ) satisfies PS. Let β ∈ R, > 0 be given and let N be any neighborhood of Kβ . Then ∃ ∈ (0, ) and a
continuous 1-parameter family of homeomorphisms Φ(·, t) of M , 0 ≤ t < ∞, with the properties
1. Φ(u, t) = u if t = 0, or DE(u) = 0, or |E(u) − β| ≥ .
2. E(Φ(u, t)) is non-increasing in t for any u ∈ M .
3. Φ(Eβ+ \ N, 1) ⊂ Eβ− and Φ(Eβ+ , 1) ⊂ Eβ− ∪ N .
If M admits a compact symmetry group G and E is G−invariant, then Φ can be constructed
so that Φ(g(u), t) = g(Φ(u, t)). (Struwe)
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Krasnoelskii Genus
While working with the Krasnoelskii genus, our set-up will always be:
V is a Banach space.
M is a closed symmetric C 1,1 submanifold of V .
E is an even C 1 functional on M and satisfies P-S.
A = {A ⊂ V |A closed , A = −A}.
h is any odd continuous map.
Eβ = {u ∈ M |E(u) < β}.
Kβ = {u ∈ V |E(u) = β, DE(u) = 0}.
Definition 2.1. For A ∈ A with A 6= ∅, let
(
inf{m|∃h ∈ C 0 (A; Rm \ {0}), h(−u) = −h(u)}
γ(A) =
∞, if {...} = ∅, in particular, if 0 ∈ A,
and define γ(∅) = 0. We call γ(A) the Krasnoselskii genus of A. (Struwe)
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October 29, 2013
Index Theory Summary
Kelly & Alexandria
Proposition 2.2. Let A, A1 , A2 ∈ A and h ∈ C 0 (V ; V ) odd. Then:
1.
2.
3.
4.
5.
γ(A) ≥ 0 and γ(A) = 0 iff A = ∅.
A1 ⊂ A2 iff γ(A1 ) ≤ γ(A2 ).
γ(A1 ∪ A2 ) ≤ γ(A1 ) + γ(A2 ).
γ(A) ≤ γ(h(A)).
If A ∈ A is compact and 0 6= A, then γ(A) < ∞ and ∃ a neighborhood N of A in V such
that N ∈ A and γ(A) = γ(N ). (Struwe)
Lemma 2.3. With the set-up given above, suppose for k, ` there holds
−∞ < βk = βk+1 = ... = βk+`−1 = β < ∞,
where k ≤ γ(M ). Then γ(Kβ ) ≥ `. In particular, if ` > 0, then Kβ is infinite.
Theorem 2.4. Given the same set-up above, suppose in addition that E is bounded from below
on M . Let γ̂(M ) = sup{γ(K)|K ⊂ M compact, symmetric}. Then E has at least γ̂(M ) ≤ ∞
pairs of distinct critical points. (Struwe & Costa)
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General Index Theory
For the case of general index theory, our set-up is:
M is a complete C 1,1 Finsler manifold with a compact group action G.
A = {A ⊂ V |A closed, g(A) = A for all g ∈ G}.
Γ = {h ∈ C 0 (M ; M )|h ◦ g = g ◦ h for all g ∈ G}.
Fix G = {u ∈ M |gu = u for all g ∈ G}.
Definition 3.1. An index for (G, A, Γ) is a mapping i : A → N0 ∪ {∞} such that for all
A, B ∈ A and h ∈ Γ there holds
i(A) ≥ 0 and i(A) = 0 iff A = ∅.
A ⊂ B iff i(A) ≤ i(B).
i(A ∪ B) ≤ i(A) + i(B).
i(A) ≤ i(h(A)).
If A ∈ A is compact and A ∩ Fix G = ∅, then i(A) < ∞ and ∃ a G−invariant neighborhood N of A such that i(N ) = i(A).
6. If u 6= Fix G, then i(∪g∈G gu) = 1. (Struwe)
1.
2.
3.
4.
5.
Theorem 3.2. With the set-up given above, suppose the functional E ∈ C 1 (M ) is bounded from
below and satisfies P-S. Suppose G acts on M without fixed points and let i be an index for
(G, A, Γ). Define î(M ) = sup{i(K)|K ⊂ M with K compact and G−invariant} ≤ ∞. Then E
admits at least î(M ) critical points which are distinct modulo G. (Struwe)
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References
• Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems by M. Struwe
• An Invitation to Variational Methods in Differential Equations by D. Costa.
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