Two remarks on liftings of maps with values into $ {S}^{1}$

Benoît Merlet

doi:10.1016/j.crma.2006.07.014

Partial Differential Equations

Two remarks on liftings of maps with values into

S^{1}

[Deux remarques sur les relèvements d'applications à valeurs dans

S^{1}

]

Présenté par : Haïm Brezis

Benoît Merlet ¹

¹ LAGA, Institut Galilée, université Paris 13, 99, avenue J.-B. Clément, 93430 Villetaneuse, France

Comptes Rendus. Mathématique, Volume 343 (2006) no. 7, pp. 467-472.

Résumé
Abstract

Étant donnée une application $u \in L_{loc}^{1} (Ω, S^{1})$ ayant une certaine régularité : ${| u |}_{X} = R < \infty$ , nous cherchons un relèvement φ de u (i.e. une fonction mesurable telle que $u = e^{i φ}$ ) ayant la même régularité et avec le meilleur contrôle possible de ${| φ |}_{X}$ en fonction de R. On traite deux cas :

(i) ${| \cdot |}_{X}$ est une seminorme $W^{s, p} (0, 1)$ , avec $0 < s < 1 < p$ et $s p > 1$ . Nous trouvons un relèvement φ satisfaisant ${| φ |}_{W^{s, p} (I)} ⩽ C (R + R^{1 / s})$ et nous montrons que l'exposant $1 / s$ ne peut être amélioré.

(ii) ${| \cdot |}_{X}$ est la seminorme $BV (Ω)$ où $Ω \subset R^{d}$ est un ouvert régulier. Nous donnons une preuve simplifiée d'un résultat préexistant [J. Dàvila, R. Ignat, Lifting of BV functions with values in $S^{1}$ , C. R. Acad. Sci. Paris, Ser. I 337 (3) (2003) 159–164] : il existe $φ \in BV (Ω)$ telle que ${| φ |}_{BV} ⩽ 2 R$ .

Given a map $u \in L_{loc}^{1} (Ω, S^{1})$ with some regularity: ${| u |}_{X} = R < \infty$ , we consider the problem of finding a lifting φ of u (i.e. a measurable function satisfying $u = e^{i φ}$ ) with the same regularity and with an optimal control ${| φ |}_{X} ⩽ g (R)$ . Two cases are treated here:

(i) ${| \cdot |}_{X}$ is a $W^{s, p} (0, 1)$ -seminorm, with $0 < s < 1 < p$ and $s p > 1$ . We find a lifting φ such that ${| φ |}_{W^{s, p} (I)} ⩽ C (R + R^{1 / s})$ and we show that the exponent $1 / s$ cannot be improved.

(ii) ${| \cdot |}_{X}$ is the $BV (Ω)$ -seminorm where $Ω \subset R^{d}$ is a smooth open set. We give a simplified proof of a previous result [J. Dàvila, R. Ignat, Lifting of BV functions with values in $S^{1}$ , C. R. Acad. Sci. Paris, Ser. I 337 (3) (2003) 159–164]: there exists $φ \in BV (Ω)$ satisfying ${| φ |}_{BV} ⩽ 2 R$ .

Accepté le : 2006-07-12
Publié le : 2006-10-01

DOI : 10.1016/j.crma.2006.07.014

Affiliations des auteurs :

Benoît Merlet ¹

¹ LAGA, Institut Galilée, université Paris 13, 99, avenue J.-B. Clément, 93430 Villetaneuse, France

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     author = {Beno{\^\i}t Merlet},
     title = {Two remarks on liftings of maps with values into $ {S}^{1}$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {467--472},
     publisher = {Elsevier},
     volume = {343},
     number = {7},
     year = {2006},
     doi = {10.1016/j.crma.2006.07.014},
     language = {en},
}

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Benoît Merlet. Two remarks on liftings of maps with values into $ {S}^{1}$. Comptes Rendus. Mathématique, Volume 343 (2006) no. 7, pp. 467-472. doi : 10.1016/j.crma.2006.07.014. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.07.014/

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[3] J. Bourgain; H. Brezis; P. Mironescu Another look at Sobolev spaces, Optimal Control and Partial Differential Equations, 2001, pp. 439-455

[4] J. Bourgain; H. Brezis; P. Mironescu $H^{1 / 2}$ maps with values into the circle: minimal connections, lifting, and the Ginzburg–Landau equation, Publ. Math. Inst. Hautes Études, Volume 99 (2004), pp. 1-115

[5] H. Brezis How to recognize constant functions. A connection with Sobolev spaces, Uspekhi Mat. Nauk, Volume 57 (2002) no. 4(346), pp. 59-74

[6] D. Chiron, On the definitions of Sobolev and BV spaces into singular spaces and the trace problem, Preprint, Laboratoire J.A. Dieudonné, Université Nice-Sophia Antipolis, 2006

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[8] J. Dávila; R. Ignat Lifting of BV functions with values in $S^{1}$ , C. R. Math. Acad. Sci. Paris, Ser. I, Volume 337 (2003) no. 3, pp. 159-164

[9] A.I. Vol'pert Spaces BV and quasilinear equations, Math. Sb. (N.S.), Volume 73 (1967) no. 115, pp. 255-302 (in Russian)

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