# Sur les courbes intégrales du champ de gradient

Abstract : This work is devoted to the study of the trajectories of gradient vector fields of functions definable in an o-minimal structure. We focus on the behaviour of the integral curves in some neighbourhood of an atypical fiber.

The first chapter recalls some geometric properties of definable sets.

In the second chapter, we study a definable family of functions defined on open sets all contained in the same compact set. By the Cauchy-Crofton formula, we prove that the length of the gradient field trajectories of each function is bounded by some constant depending only on the dimension and the compact set. We then deduce an explicit bound in the case of a generic polynomial of given degree.

The third chapter is devoted to $C^1$ functions defined on non bounded open sets. We prove that the set of values at which the Malgrange condition fails (asymptotic critical values) is finite and contains the atypical values which are not critical values.

In the fourth chapter, we prove an embedding theorem from an arbitrary connected component of an asymptotic critical fiber in a connected component of a close typical fiber. This result, obtained from some Lojasiewicz type inequality at infinity, gives a better understanding on the changes of the topological types of the fibers of a definable function in a neighbourhood of an atypical value. In dimension two, we describe the set of points of a typical fiber at which passes a trajectory of the gradient field which does not reach the atypical level.

In the last chapter, we study some remarkable integral curves of the gradient field. We show that a curve on which the norm of the gradient is minimal on the levels is an integral curve of the gradient field if and only if it is a line. Such a result leads to ask the question of the finiteness of the gradient field separatrix in the polynomial case.
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Document type :
Theses
Domain :

https://tel.archives-ouvertes.fr/tel-00002710
Contributor : Didier d'Acunto <>
Submitted on : Monday, April 14, 2003 - 10:17:54 AM
Last modification on : Friday, November 6, 2020 - 3:30:28 AM
Long-term archiving on: : Monday, September 20, 2010 - 11:29:35 AM

### Identifiers

• HAL Id : tel-00002710, version 2

### Citation

Didier d'Acunto. Sur les courbes intégrales du champ de gradient. Mathématiques [math]. Université de Savoie, 2001. Français. ⟨tel-00002710v2⟩

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