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J. Math. Anal. S. Palais. In a substantial part of modern analysis characterized by the tendency to avoid differentiability assumptions, this principle will likely play at least the same role as, say, the contraction mapping principle plays in “smooth” analysis. The elegance of the proofs and the natural way the principle appears in them lend much support to this belief. From a Review of [I. Ekeland, Nonconvex minimization problems. Bull. Am. Math. D. Ioffe. This principle discovered in 1972 has found a multitude of applications in different fields of Analysis.

J (u) j∈I Since the cover {Ni }i∈I is locally finite, the sum makes sense. The functions (ρi )i are Lipschitz continuous with support in Ni and are such that ˜ is a convex 0 ≤ ρi (u) ≤ 1 and i∈I ρi ≡ 1 on X˜ . Then, at each u˜ ∈ X˜ , the vector v(u) combination of pseudo-gradient vectors for at u. Moreover, the functional v is locally Lipschitz continuous. 2. When is even, that is, (u) = vector field v˜ (−u) = −˜v (u). It suffices to set (−u), it admits an odd pseudo-gradient v(u) − v(−u) , 2 where v is any pseudo-gradient vector field for on X˜ .

Notice the very important fact that it is not possible to deform the functional on critical points. When X is a Banach space, the notion of gradient does not exist. A notion of pseudo-gradient vector field was introduced by Palais [696] especially to handle this case. Moreover, it requires the functional to be only of class C 1 . Denote the set of regular points of in X by X˜ = u ∈ X ; where (x) = 0 , : X → R is supposed to be a C 1 -functional. 1. A pseudo-gradient vector v0 ∈ X for satisfies at u ∈ X˜ is a vector that (u) , 1.

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A nondegeneracy result for a nonlinear elliptic equation by Grossi M.


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