Comptes Rendus
Partial Differential Equations
Two remarks on liftings of maps with values into S1
Comptes Rendus. Mathématique, Volume 343 (2006) no. 7, pp. 467-472.

Given a map uLloc1(Ω,S1) with some regularity: |u|X=R<, we consider the problem of finding a lifting φ of u (i.e. a measurable function satisfying u=eiφ) with the same regularity and with an optimal control |φ|Xg(R). Two cases are treated here:

(i) ||X is a Ws,p(0,1)-seminorm, with 0<s<1<p and sp>1. We find a lifting φ such that |φ|Ws,p(I)C(R+R1/s) and we show that the exponent 1/s cannot be improved.

(ii) ||X is the BV(Ω)-seminorm where ΩRd 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 S1, C. R. Acad. Sci. Paris, Ser. I 337 (3) (2003) 159–164]: there exists φBV(Ω) satisfying |φ|BV2R.

Étant donnée une application uLloc1(Ω,S1) ayant une certaine régularité : |u|X=R<, nous cherchons un relèvement φ de u (i.e. une fonction mesurable telle que u=eiφ) ayant la même régularité et avec le meilleur contrôle possible de |φ|X en fonction de R. On traite deux cas :

(i) ||X est une seminorme Ws,p(0,1), avec 0<s<1<p et sp>1. Nous trouvons un relèvement φ satisfaisant |φ|Ws,p(I)C(R+R1/s) et nous montrons que l'exposant 1/s ne peut être amélioré.

(ii) ||X est la seminorme BV(Ω)ΩRd est un ouvert régulier. Nous donnons une preuve simplifiée d'un résultat préexistant [J. Dàvila, R. Ignat, Lifting of BV functions with values in S1, C. R. Acad. Sci. Paris, Ser. I 337 (3) (2003) 159–164] : il existe φBV(Ω) telle que |φ|BV2R.

Accepted:
Published online:
DOI: 10.1016/j.crma.2006.07.014

Benoît Merlet 1

1 LAGA, Institut Galilée, université Paris 13, 99, avenue J.-B. Clément, 93430 Villetaneuse, France
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     title = {Two remarks on liftings of maps with values into $ {S}^{1}$},
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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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