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

Petru Mironescu

doi:10.1016/j.crma.2008.08.001

Mathematical Analysis/Partial Differential Equations

Lifting default for

S^{1}

-valued maps

Presented by: Haïm Brezis

Petru Mironescu 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, Volume 346 (2008) no. 19-20, pp. 1039-1044.

Abstract
Résumé

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.

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.

Received: 2008-07-30
Published online: 2008-08-23

DOI: 10.1016/j.crma.2008.08.001

Author's affiliations:

Petru Mironescu 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},
     publisher = {Elsevier},
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     year = {2008},
     doi = {10.1016/j.crma.2008.08.001},
     language = {en},
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Petru Mironescu. Lifting default for $ {\mathbb{S}}^{1}$-valued maps. Comptes Rendus. Mathématique, Volume 346 (2008) no. 19-20, pp. 1039-1044. doi : 10.1016/j.crma.2008.08.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.08.001/

References
Cited by

[1] F. Bethuel; D. Chiron Some questions related to the lifting problem in Sobolev spaces (H. Berestycki; M. Bertsch; F. Browder; L. Nirenberg, eds.), Perspectives in Nonlinear Partial Differential Equations, Contemporary Mathematics, vol. 446, Amer. Math. Soc., 2007, pp. 125-152

[2] J. Bourgain; H. Brezis On the equation div $Y = f$ and application to control of phases, J. Amer. Math. Soc., Volume 16 (2003), pp. 393-426

[3] J. Bourgain; H. Brezis; P. Mironescu Lifting in Sobolev spaces, J. Anal. Math., Volume 80 (2000), pp. 37-86

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

[5] H. Brezis; P. Mironescu On some questions of topology for $S^{1}$ -valued fractional Sobolev spaces, Rev. R. Acad. Cien., Serie A Mat., Volume 95 (2001), pp. 121-143

[6] H. Brezis, P. Mironescu, H.-M. Nguyen, in preparation

[7] J.-Y. Chemin Fluides parfaits incompressibles, Astérisque, Volume 230 (1995)

[8] P. Mironescu Sobolev maps on manifolds: degree, approximation, lifting (H. Berestycki; M. Bertsch; F. Browder; L. Nirenberg; L.A. Peletier; L. Véron, eds.), Perspectives in Nonlinear Partial Differential Equations, In honor of Haïm Brezis, Contemporary Mathematics, vol. 446, Amer. Math. Society, 2007, pp. 413-436

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

[10] H.-M. Nguyen Inequalities related to liftings and applications, C. R. Acad. Sci. Paris, Ser. I, Volume 346 (2008) no. 17–18, pp. 957-962

[11] H. Triebel Theory of Function Spaces, Birkhäuser Verlag, 1983

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