Adaptation of the generic PDE's results to the notion of prevalence

Joseph Fourier

Description

Niveau: Supérieur, Doctorat, Bac+8
Adaptation of the generic PDE's results to the notion of prevalence Romain JOLY Institut Fourier UMR 5582, Universite Joseph Fourier, CNRS 100, rue des Maths, BP74 F-38402 St Martin d'Heres, FRANCE Abstract : Many generic results have been proved, especially concerning the qualitative behaviour of solutions of partial differential equations. Recently, a new notion of “almost always”, the prevalence, has been developped for vectorial spaces. This notion is inter- esting since, for example, prevalence sets are equivalent to the full Lebesgue measure sets in finite dimensional spaces. The purpose of this article is to adapt the generic PDE's results to the notion of prevalence. In particular, we consider the cases where Sard-Smale theorems or arguments of analytic perturbations of the parameters are used. Keywords : prevalence, genericity, Smale theorem, Sard-Smale theorem. AMS classification codes (2000) : 35B30, 35P05, 37C20, 37D05, 37D15, 47F05, 58B15. 1 Introduction Many important properties of partial differential equations are not always satisfied but seem to hold except for some particular cases. For example, they may hold except for a small set of coefficients of the equation. Since these properties may be very useful, one hopes to show that they “almost always” hold.

• infinite-dimensional spaces


• lebesgue measure


• smale theorem


• been obtained


• inter- esting since


• genericity results


• generic


• results has


• zero

Sujets

• Fourier
• Lebesgue measure
• Generic

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 9
• Langue: English

Extrait

Adaptation of the generic PDE’s results to the notion of prevalence Romain JOLY Institut Fourier UMR5582,UniversiteJosephFourier,CNRS 100, rue des Maths, BP74 F-38402StMartind’Heres,FRANCE [e-mail]

Abstract : Many generic results have been proved, especially concerning the qualitative behaviour of solutions of partial dieren tial equations. Recently, a new notion of “almost always”, the prevalence, has been developped for vectorial spaces. This notion is inter- esting since, for example, prevalence sets are equivalent to the full Lebesgue measure sets in nite dimensional spaces. The purpose of this article is to adapt the generic PDE’s results to the notion of prevalence. In particular, we consider the cases where Sard-Smale theorems or arguments of analytic perturbations of the parameters are used. Keywords : prevalence, genericity, Smale theorem, Sard-Smale theorem. AMS classication codes (2000) : 35B30, 35P05, 37C20, 37D05, 37D15, 47F05, 58B15.

1 Introduction Manyimportantpropertiesofpartialdierentialequationsarenotalwayssatisedbut seem to hold except for some particular cases. For example, they may hold except for a small set of coecien ts of the equation. Since these properties may be very useful, one hopes to show that they “almost always” hold. The problem is to give a sense to this “almost always” notion. The study of PDE’s requires innite-dimensional spaces. Thus,

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