Mathematical Analysis/Partial Differential Equations
Lifting default for S1-valued maps
Comptes Rendus. Mathématique, Volume 346 (2008) no. 19-20, pp. 1039-1044.

Let φC([0,1]N,R). When 0<s<1, p1 and 1sp<N, the Ws,p-semi-norm |φ|Ws,p of φ is not controlled by |g|Ws,p, where g:=eıφ [J. Bourgain, H. Brezis, P. Mironescu, Lifting in Sobolev spaces, J. Anal. Math. 80 (2000) 37–86]. [This question is related to existence, for S1-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|Ws,p does control a weaker quantity, namely |φ|Ws,p+W1,sp. 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<p2, 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 sp1 or when N2, p>1 and sp>1. In this Note, we establish existence of control for all s<1, p1 and N.

Soit φC([0,1]N,R). Si 0<s<1, p1 et 1sp<N, alors la semi-norme |φ|Ws,p n'est pas contrôlée par |g|Ws,p, où g:=eıφ [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 S1, 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|Ws,p contrôle une quantité plus faible que |φ|Ws,p, plus spécifiquement |φ|Ws,p+W1,sp. 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<p2 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 sp1 ou pour N2, p>1 et sp>1. Dans cette Note, nous montrons l'existence d'un contrôle pour tout s<1, p1 et N.

Received:
Published online:
DOI: 10.1016/j.crma.2008.08.001
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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Mironescu, Petru. 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. http://www.numdam.org/articles/10.1016/j.crma.2008.08.001/

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

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

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

[5] Brezis, H.; Mironescu, P. On some questions of topology for S1-valued fractional Sobolev spaces, Rev. R. Acad. Cien., Serie A Mat., Volume 95 (2001), pp. 121-143

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[9] P. Mironescu, Lifting of S1-valued maps in sums of Sobolev spaces, J. European Math. Soc., submitted for publication

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

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