Lifting default for $ {\mathbb{S}}^{1}$-valued maps

Mironescu, Petru

doi:10.1016/j.crma.2008.08.001

Mathematical Analysis/Partial Differential Equations

Lifting default for

S^{1}

-valued maps
[Défaut de relèvement pour les applications à valeurs dans le cercle unité]

Présenté par : Haïm Brezis

Mironescu, Petru ¹

¹ Université de Lyon, Université Lyon1, CNRS, UMR 5208, institut Camille-Jordan, bâtiment du Doyen Jean-Braconnier, 43, boulevard du 11 novembre 1918, 69200 Villeurbanne cedex, France

Comptes Rendus. Mathématique, Tome 346 (2008) no. 19-20, pp. 1039-1044.

Résumé
Abstract

Soit $φ \in C^{\infty} ({[0, 1]}^{N}, R)$ . Si $0 < s < 1$ , $p ⩾ 1$ et $1 ⩽ s p < N$ , alors la semi-norme ${| φ |}_{W^{s, p}}$ n'est pas contrôlée par ${| g |}_{W^{s, p}}$ , où $g : = e^{ı φ}$ [J. Bourgain, H. Brezis, P. Mironescu, Lifting in Sobolev spaces, J. Anal. Math. 80 (2000) 37–86]. [Cette question est liée à l'existence, pour des g à valeurs dans $S^{1}$ , de relèvements φ aussi réguliers que g le permet.] Dans [J. Bourgain, H. Brezis, P. Mironescu, Lifting, degree, and distributional Jacobian revisited, Commun. Pure Appl. Math. 58 (2005) 529–551], il est conjecturé que ${| g |}_{W^{s, p}}$ contrôle une quantité plus faible que ${| φ |}_{W^{s, p}}$ , plus spécifiquement ${| φ |}_{W^{s, p} + W^{1, s p}}$ . L'existence d'un tel contrôle est due à J. Bourgain and H. Brezis [J. Bourgain, H. Brezis, On the equation div $Y = f$ and application to control of phases, J. Amer. Math. Soc. 16 (2003) 393–426] pour $1 < p ⩽ 2$ et $s = 1 / p$ et à H.-M. Nguyen [H.-M. Nguyen, Inequalities related to liftings and applications, C. R. Acad. Sci. Paris, Ser. I 346 (17–18) (2008) 957–962] pour $N = 1$ , $p > 1$ et $s p ⩾ 1$ ou pour $N ⩾ 2$ , $p > 1$ et $s p > 1$ . Dans cette Note, nous montrons l'existence d'un contrôle pour tout $s < 1$ , $p ⩾ 1$ et N.

Let $φ \in C^{\infty} ({[0, 1]}^{N}, R)$ . When $0 < s < 1$ , $p ⩾ 1$ and $1 ⩽ s p < N$ , the $W^{s, p}$ -semi-norm ${| φ |}_{W^{s, p}}$ of φ is not controlled by ${| g |}_{W^{s, p}}$ , where $g : = e^{ı φ}$ [J. Bourgain, H. Brezis, P. Mironescu, Lifting in Sobolev spaces, J. Anal. Math. 80 (2000) 37–86]. [This question is related to existence, for $S^{1}$ -valued maps g, of a lifting φ as smooth as allowed by g.] In [J. Bourgain, H. Brezis, P. Mironescu, Lifting, degree, and distributional Jacobian revisited, Commun. Pure Appl. Math. 58 (2005) 529–551], the authors suggested that ${| g |}_{W^{s, p}}$ does control a weaker quantity, namely ${| φ |}_{W^{s, p} + W^{1, s p}}$ . Existence of such control is due to J. Bourgain and H. Brezis [J. Bourgain, H. Brezis, On the equation div $Y = f$ and application to control of phases, J. Amer. Math. Soc. 16 (2003) 393–426] when $1 < p ⩽ 2$ , $s = 1 / p$ and to H.-M. Nguyen [H.-M. Nguyen, Inequalities related to liftings and applications, C. R. Acad. Sci. Paris, Ser. I 346 (17–18) (2008) 957–962] when $N = 1$ , $p > 1$ and $s p ⩾ 1$ or when $N ⩾ 2$ , $p > 1$ and $s p > 1$ . In this Note, we establish existence of control for all $s < 1$ , $p ⩾ 1$ and N.

Reçu le : 2008-07-30
Publié le : 2008-08-23

DOI : 10.1016/j.crma.2008.08.001

Affiliations des auteurs :

Mironescu, Petru ¹

¹ Université de Lyon, Université Lyon1, CNRS, UMR 5208, institut Camille-Jordan, bâtiment du Doyen Jean-Braconnier, 43, boulevard du 11 novembre 1918, 69200 Villeurbanne cedex, France

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     title = {Lifting default for $ {\mathbb{S}}^{1}$-valued maps},
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     pages = {1039--1044},
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Mironescu, Petru. Lifting default for $ {\mathbb{S}}^{1}$-valued maps. Comptes Rendus. Mathématique, Tome 346 (2008) no. 19-20, pp. 1039-1044. doi : 10.1016/j.crma.2008.08.001. http://www.numdam.org/articles/10.1016/j.crma.2008.08.001/

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Cité par

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[4] Bourgain, J.; Brezis, H.; Mironescu, P. Lifting, degree, and distributional Jacobian revisited, Commun. Pure Appl. Math., Volume 58 (2005), pp. 529-551

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[9] P. Mironescu, Lifting of $S^{1}$ -valued maps in sums of Sobolev spaces, J. European Math. Soc., submitted for publication

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