By S.Francinou, H.Gianella, S.Nicolas

ISBN-10: 2842251415

ISBN-13: 9782842251413

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**Example text**

So, we can find a better approximation of the infimum of as near as we want to our initial point with the supplementary property of minimizing the perturbation that appears in c. ) Proof. Consider the relation defined in X by u ≺ v ⇐⇒ (u) ≤ (v) − ε dist (u, v), δ where ≺ defines a partial ordering in X depending on δ. The reflexivity and antisymmetry are obvious. For the transitivity, suppose that both u ≺ v and v ≺ u hold. Since dist (u, u) ≤ dist (u, v) + dist (v, u), we conclude immediately that u ≺ u.

3 Some Criteria for Checking (PS) Checking (PS) using its definition is not always the best way to proceed. We present some prototypes of functionals that satisfy (PS). When the dimension of the space X is finite, say, X = R N , one has the following result. 1. Let ∈ C 1 (R N ; R) where X is a Banach space. If the function | |+ : RN → R is coercive, that is, it tends to +∞ as x goes to +∞, then satisfies (PS). Proof. Since X is finite dimensional, it is locally compact. So, if we suppose that | |+ is coercive, then any Palais-Smale sequence is bounded and hence contains a convergent subsequence.

A pseudo-gradient vector v0 ∈ X for satisfies at u ∈ X˜ is a vector that 1. v0 < 2 (u) , 2. v0 , (u) ≥ (u) 2 . And a pseudo-gradient vector field for is a locally Lipschitz continuous functional v : X˜ → X such that for all u ∈ X˜ , v(u) is a pseudo-gradient vector of at u. 1. Notice that 1. Any convex combination of pseudo-gradient vectors (resp. of pseudo-gradient vector fields) is a pseudo-gradient vector (resp. a pseudo-gradient vector field). Hence, such a functional may exist but is not necessarily unique.

### Exercices de mathématiques Oraux de l'ENS : Analyse 2 by S.Francinou, H.Gianella, S.Nicolas

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