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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