Given a map with some regularity: , we consider the problem of finding a lifting φ 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 φ 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 .
Étant donnée une application ayant une certaine régularité : , nous cherchons un relèvement φ 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 φ satisfaisant et nous montrons que l'exposant ne peut être amélioré.
(ii) est la seminorme où est un ouvert régulier. Nous donnons une preuve simplifiée d'un 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 .
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Benoît Merlet 1
@article{CRMATH_2006__343_7_467_0, author = {Beno{\^\i}t Merlet}, title = {Two remarks on liftings of maps with values into $ {S}^{1}$}, journal = {Comptes Rendus. Math\'ematique}, pages = {467--472}, publisher = {Elsevier}, volume = {343}, number = {7}, year = {2006}, doi = {10.1016/j.crma.2006.07.014}, language = {en}, }
Benoît Merlet. Two remarks on liftings of maps with values into $ {S}^{1}$. Comptes Rendus. Mathématique, Volume 343 (2006) no. 7, pp. 467-472. doi : 10.1016/j.crma.2006.07.014. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.07.014/
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