Comptes Rendus
Mathematical Analysis
Decomposition of S1-valued maps in Sobolev spaces
[Décomposition des applications unimodulaires dans les espaces de Sobolev]
Comptes Rendus. Mathématique, Volume 348 (2010) no. 13-14, pp. 743-746.

Soient n2, s>0, p1 tels que 1sp<2. Nous montrons que, pour chaque uWs,p(Sn;S1), il existe φWs,p(Sn;R) et vWsp,1(Sn;S1) tels que u=veıφ. Ceci donne une décomposition de u comme produit d'un facteur qui se relève dans Ws,p, eıφ, et d'un facteur « plus régulier » que u mais qui ne se relève pas, à savoir v. Notre décomposition généralise un résultat antérieur de Bourgain et Brezis (qui ont traité le cas s=1/2, p=2). Une conséquence de notre résultat est une preuve intuitive de l'existence du jacobien au sens des distributions Ju pour les applications uWs,p(Sn;S1) (résultat dû, avec un argument différent, à Bourgain, Brezis et l'auteur). En complétant un résultat de Bousquet, nous caractérisons les distributions de la forme Ju.

Let n2, s>0, p1 be such that 1sp<2. We prove that for each map uWs,p(Sn;S1) one can find φWs,p(Sn;R) and vWsp,1(Sn;S1) such that u=veıφ. This yields a decomposition of u into a part that has a lifting in Ws,p, eıφ, and a map “smoother” than u but without lifting, namely v. Our result generalizes a previous one of Bourgain and Brezis (which corresponds to the case s=1/2, p=2). As a consequence, we find an intuitive proof for the existence of the distributional Jacobian Ju of maps uWs,p(Sn;S1) (originally due to Bourgain, Brezis and the author). By completing a result of Bousquet, we characterize the distributions of the form Ju.

Reçu le :
Accepté le :
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DOI : 10.1016/j.crma.2010.06.020

Petru Mironescu 1

1 Université de Lyon, CNRS, Université Lyon 1, Institut Camille-Jordan, 43, boulevard du 11 novembre 1918, 69622 Villeurbanne cedex, France
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     title = {Decomposition of $ {\mathbb{S}}^{1}$-valued maps in {Sobolev} spaces},
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Petru Mironescu. Decomposition of $ {\mathbb{S}}^{1}$-valued maps in Sobolev spaces. Comptes Rendus. Mathématique, Volume 348 (2010) no. 13-14, pp. 743-746. doi : 10.1016/j.crma.2010.06.020. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.06.020/

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