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Comptes Rendus Mathématique
Volume 343, n° 7
pages 467-472 (octobre 2006)
Doi : 10.1016/j.crma.2006.07.014
accepted : 12 July 2006
Two remarks on liftings of maps with values into  
Deux remarques sur les relèvements dʼapplications à valeurs dans  
 

Benoît Merlet
LAGA, Institut Galilée, université Paris 13, 99, avenue J.-B. Clément, 93430 Villetaneuse, France 

Abstract

Given a map   with some regularity:  , we consider the problem of finding a lifting of u (i.e. a measurable function satisfying  ) with the same regularity and with an optimal control  . Two cases are treated here:

(i)   is a  -seminorm, with   and  . We find a lifting such that   and we show that the exponent   cannot be improved.

(ii)   is the  -seminorm where   is a smooth open set. We give a simplified proof of a previous result [J. Dàvila, R. Ignat, Lifting of BV functions with values in  , C. R. Acad. Sci. Paris, Ser. I 337 (3) (2003) 159-164]: there exists   satisfying  . To cite this article: B. Merlet, C. R. Acad. Sci. Paris, Ser. I 343 (2006).

The full text of this article is available in PDF format.
Résumé

Étant donnée une application   ayant une certaine régularité :  , nous cherchons un relèvement de u (i.e. une fonction mesurable telle que  ) ayant la même régularité et avec le meilleur contrôle possible de   en fonction de R . On traite deux cas :

(i)   est une seminorme  , avec   et  . Nous trouvons un relèvement satisfaisant   et nous montrons que lʼexposant   ne peut être amélioré.

(ii)   est la seminorme   où   est un ouvert régulier. Nous donnons une preuve simplifiée dʼun résultat préexistant [J. Dàvila, R. Ignat, Lifting of BV functions with values in  , C. R. Acad. Sci. Paris, Ser. I 337 (3) (2003) 159-164] : il existe   telle que  . Pour citer cet article : B. Merlet, C. R. Acad. Sci. Paris, Ser. I 343 (2006).

The full text of this article is available in PDF format.


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