Decomposition of $ {\mathbb{S}}^{1}$-valued maps in Sobolev spaces

Petru Mironescu

doi:10.1016/j.crma.2010.06.020

Mathematical Analysis

Decomposition of

S^{1}

-valued maps in Sobolev spaces
[Décomposition des applications unimodulaires dans les espaces de Sobolev]

Présenté par : Haïm Brezis

Petru Mironescu ¹

¹ Université de Lyon, CNRS, Université Lyon 1, Institut Camille-Jordan, 43, boulevard du 11 novembre 1918, 69622 Villeurbanne cedex, France

Comptes Rendus. Mathématique, Volume 348 (2010) no. 13-14, pp. 743-746.

Résumé
Abstract

Soient $n ⩾ 2$ , $s > 0$ , $p ⩾ 1$ tels que $1 ⩽ sp < 2$ . Nous montrons que, pour chaque $u \in W^{s, p} (S^{n}; S^{1})$ , il existe $φ \in W^{s, p} (S^{n}; R)$ et $v \in W^{sp, 1} (S^{n}; S^{1})$ tels que $u = v e^{ı φ}$ . Ceci donne une décomposition de u comme produit d'un facteur qui se relève dans $W^{s, p}$ , $e^{ı φ}$ , et d'un facteur « plus régulier » que u mais qui ne se relève pas, à savoir v. Notre décomposition généralise un résultat antérieur de Bourgain et Brezis (qui ont traité le cas $s = 1 / 2$ , $p = 2$ ). Une conséquence de notre résultat est une preuve intuitive de l'existence du jacobien au sens des distributions Ju pour les applications $u \in W^{s, p} (S^{n}; S^{1})$ (résultat dû, avec un argument différent, à Bourgain, Brezis et l'auteur). En complétant un résultat de Bousquet, nous caractérisons les distributions de la forme Ju.

Let $n ⩾ 2$ , $s > 0$ , $p ⩾ 1$ be such that $1 ⩽ sp < 2$ . We prove that for each map $u \in W^{s, p} (S^{n}; S^{1})$ one can find $φ \in W^{s, p} (S^{n}; R)$ and $v \in W^{sp, 1} (S^{n}; S^{1})$ such that $u = v e^{ı φ}$ . This yields a decomposition of u into a part that has a lifting in $W^{s, p}$ , $e^{ı φ}$ , and a map “smoother” than u but without lifting, namely v. Our result generalizes a previous one of Bourgain and Brezis (which corresponds to the case $s = 1 / 2$ , $p = 2$ ). As a consequence, we find an intuitive proof for the existence of the distributional Jacobian Ju of maps $u \in W^{s, p} (S^{n}; S^{1})$ (originally due to Bourgain, Brezis and the author). By completing a result of Bousquet, we characterize the distributions of the form Ju.

Reçu le : 2010-06-22
Accepté le : 2010-06-23
Publié le : 2010-07-01

DOI : 10.1016/j.crma.2010.06.020

Affiliations des auteurs :

Petru Mironescu ¹

¹ Université de Lyon, CNRS, Université Lyon 1, Institut Camille-Jordan, 43, boulevard du 11 novembre 1918, 69622 Villeurbanne cedex, France

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Petru Mironescu. Decomposition of $ {\mathbb{S}}^{1}$-valued maps in Sobolev spaces. Comptes Rendus. Mathématique, Volume 348 (2010) no. 13-14, pp. 743-746. doi : 10.1016/j.crma.2010.06.020. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.06.020/

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Jean van Schaftingen Lifting of fractional Sobolev mappings to noncompact covering spaces, Annales de l'Institut Henri Poincaré C. Analyse Non Linéaire, Volume 42 (2025) no. 1, pp. 41-84 | DOI:10.4171/aihpc/98 | Zbl:7988204
Petru Mironescu; Emmanuel Russ; Yannick Sire Lifting in Besov spaces, Nonlinear Analysis. Theory, Methods Applications. Series A: Theory and Methods, Volume 193 (2020), p. 44 (Id/No 111489) | DOI:10.1016/j.na.2019.03.012 | Zbl:1458.46024
Antonin Monteil; Jean Van Schaftingen Uniform boundedness principles for Sobolev maps into manifolds, Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, Volume 36 (2019) no. 2, pp. 417-449 | DOI:10.1016/j.anihpc.2018.06.002 | Zbl:1418.46035
Petru Mironescu Sum-Intersection Property of Sobolev Spaces, Current Research in Nonlinear Analysis, Volume 135 (2018), p. 203 | DOI:10.1007/978-3-319-89800-1_8
Petru Mironescu; Ioana Molnar Phases of unimodular complex valued maps: optimal estimates, the factorization method, and the sum-intersection property of Sobolev spaces, Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, Volume 32 (2015) no. 5, pp. 965-1013 | DOI:10.1016/j.anihpc.2014.04.005 | Zbl:1339.46037
Haïm Brezis; Petru Mironescu Density in $W^{s, p} (Ω; N)$ , Journal of Functional Analysis, Volume 269 (2015) no. 7, pp. 2045-2109 | DOI:10.1016/j.jfa.2015.04.005 | Zbl:1516.46021
Pierre Bousquet; Petru Mironescu Prescribing the Jacobian in critical spaces, Journal d'Analyse Mathématique, Volume 122 (2014), pp. 317-373 | DOI:10.1007/s11854-014-0010-0 | Zbl:1317.46021
Fabrice Bethuel A new obstruction to the extension problem for Sobolev maps between manifolds, Journal of Fixed Point Theory and Applications, Volume 15 (2014) no. 1, pp. 155-183 | DOI:10.1007/s11784-014-0185-0 | Zbl:1321.46035
Hoai-Minh Nguyen Estimates for the topological degree and related topics, Journal of Fixed Point Theory and Applications, Volume 15 (2014) no. 1, pp. 185-215 | DOI:10.1007/s11784-014-0182-3 | Zbl:1321.46037
Radu Ignat Two-dimensional unit-length vector fields of vanishing divergence, Journal of Functional Analysis, Volume 262 (2012) no. 8, pp. 3465-3494 | DOI:10.1016/j.jfa.2012.01.014 | Zbl:1246.46031
Petru Mironescu Decomposition of $S^{1}$ -valued maps in Sobolev spaces, Comptes Rendus. Mathématique. Académie des Sciences, Paris, Volume 348 (2010) no. 13-14, pp. 743-746 | DOI:10.1016/j.crma.2010.06.020 | Zbl:1205.46017
P. Mironescu $S^{1}$ -valued Sobolev maps, Journal of Mathematical Sciences (New York), Volume 170 (2010) no. 3, pp. 340-355 | DOI:10.1007/s10958-010-0090-z | Zbl:1307.46024
Petru Mironescu Lifting default for $S^{1}$ -valued maps, Comptes Rendus. Mathématique. Académie des Sciences, Paris, Volume 346 (2008) no. 19-20, pp. 1039-1044 | DOI:10.1016/j.crma.2008.08.001 | Zbl:1168.46305

Cité par 13 documents. Sources : Crossref, zbMATH

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