Decomposition of $ {\mathbb{S}}^{1}$-valued maps in Sobolev spaces

Mironescu, Petru

doi:10.1016/j.crma.2010.06.020

Mathematical Analysis

Decomposition of

S^{1}

-valued maps in Sobolev spaces
[Décomposition des applications unimodulaires dans les espaces de Sobolev]

Présenté par : Haïm Brezis

Mironescu, Petru ¹

¹ Université de Lyon, CNRS, Université Lyon 1, Institut Camille-Jordan, 43, boulevard du 11 novembre 1918, 69622 Villeurbanne cedex, France

Comptes Rendus. Mathématique, Tome 348 (2010) no. 13-14, pp. 743-746.

Résumé
Abstract

Soient $n ⩾ 2$ , $s > 0$ , $p ⩾ 1$ tels que $1 ⩽ sp < 2$ . Nous montrons que, pour chaque $u \in W^{s, p} (S^{n}; S^{1})$ , il existe $φ \in W^{s, p} (S^{n}; R)$ et $v \in W^{sp, 1} (S^{n}; S^{1})$ tels que $u = v e^{ı φ}$ . Ceci donne une décomposition de u comme produit d'un facteur qui se relève dans $W^{s, p}$ , $e^{ı φ}$ , 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 $u \in W^{s, p} (S^{n}; S^{1})$ (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 $n ⩾ 2$ , $s > 0$ , $p ⩾ 1$ be such that $1 ⩽ sp < 2$ . We prove that for each map $u \in W^{s, p} (S^{n}; S^{1})$ one can find $φ \in W^{s, p} (S^{n}; R)$ and $v \in W^{sp, 1} (S^{n}; S^{1})$ such that $u = v e^{ı φ}$ . This yields a decomposition of u into a part that has a lifting in $W^{s, p}$ , $e^{ı φ}$ , 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 $u \in W^{s, p} (S^{n}; S^{1})$ (originally due to Bourgain, Brezis and the author). By completing a result of Bousquet, we characterize the distributions of the form Ju.

Reçu le : 2010-06-22
Accepté le : 2010-06-23
Publié le : 2010-07-01

DOI : 10.1016/j.crma.2010.06.020

Affiliations des auteurs :

Mironescu, Petru 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},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {743--746},
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Mironescu, Petru. Decomposition of $ {\mathbb{S}}^{1}$-valued maps in Sobolev spaces. Comptes Rendus. Mathématique, Tome 348 (2010) no. 13-14, pp. 743-746. doi : 10.1016/j.crma.2010.06.020. http://www.numdam.org/articles/10.1016/j.crma.2010.06.020/

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