Profile decomposition and phase control for circle-valued maps in one dimension

Petru Mironescu

doi:10.1016/j.crma.2015.09.030

Mathematical analysis

Profile decomposition and phase control for circle-valued maps in one dimension
[Décomposition en profils et contrôle des phases des applications unimodulaires en dimension un]

Présenté par : Haïm Brézis

Petru Mironescu ¹

¹ Université de Lyon, Université Lyon-1, CNRS UMR 5208, Institut Camille-Jordan, 43 bd du 11-Novembre-1918, 69622 Villeurbanne cedex, France

Comptes Rendus. Mathématique, Volume 353 (2015) no. 12, pp. 1087-1092.

Résumé
Abstract

Si $1 < p < \infty$ , les applications f appartenant à $W^{1 / p, p} ((0, 1); S^{1})$ ont des phases φ dans $W^{1 / p, p}$ , mais la seminorme $W^{1 / p, p}$ de φ n'est pas contrôlée par celle de f. L'absence de contrôle est illustrée par « le pli » : $f = e^{ı φ}$ , où la phase φ augmente rapidement de 0 à 2π. Pour des applications $f : S^{1} \to S^{1}$ , le même phénomène apparaît, avec les transformations de Moebius jouant le rôle des plis. Nous prouvons que cet exemple est essentiellement le seul : toute application $f : S^{1} \to S^{1}$ telle que ${| f |}_{W^{1 / p, p}}^{p} \leq M$ s'écrit $f = e^{ı ψ} \prod_{j = 1}^{K} {(M_{a_{j}})}^{\pm 1}$ , où $M_{a_{j}}$ est une transformation de Moebius s'annulant en $a_{j} \in D$ , tandis que l'entier $K = K (f)$ et la phase ψ sont contrôlés par M. En particulier, nous avons $K \leq c_{p} M$ pour une constante $c_{p}$ . Pour $p = 2$ , nous obtenons la valeur optimale de $c_{2}$ , qui est $c_{2} = 1 / (4 π^{2})$ . Comme application, nous obtenons l'existence d'une application minimale de degré un dans $W^{1 / p, p} (S^{1}; S^{1})$ avec $p \in] 2 - ε, 2 [$ .

When $1 < p < \infty$ , maps f in $W^{1 / p, p} ((0, 1); S^{1})$ have $W^{1 / p, p}$ phases φ, but the $W^{1 / p, p}$ -seminorm of φ is not controlled by the one of f. Lack of control is illustrated by “the kink”: $f = e^{ı φ}$ , where the phase φ moves quickly from 0 to 2π. A similar situation occurs for maps $f : S^{1} \to S^{1}$ , with Moebius maps playing the role of kinks. We prove that this is the only loss of control mechanism: each map $f : S^{1} \to S^{1}$ satisfying ${| f |}_{W^{1 / p, p}}^{p} \leq M$ can be written as $f = e^{ı ψ} \prod_{j = 1}^{K} {(M_{a_{j}})}^{\pm 1}$ , where $M_{a_{j}}$ is a Moebius map vanishing at $a_{j} \in D$ , while the integer $K = K (f)$ and the phase ψ are controlled by M. In particular, we have $K \leq c_{p} M$ for some $c_{p}$ . When $p = 2$ , we obtain the sharp value of $c_{2}$ , which is $c_{2} = 1 / (4 π^{2})$ . As an application, we obtain the existence of minimal maps of degree one in $W^{1 / p, p} (S^{1}; S^{1})$ with $p \in (2 - ε, 2)$ .

Reçu le : 2015-09-18
Accepté le : 2015-09-30
Publié le : 2015-10-31

DOI : 10.1016/j.crma.2015.09.030

Affiliations des auteurs :

Petru Mironescu ¹

¹ Université de Lyon, Université Lyon-1, CNRS UMR 5208, Institut Camille-Jordan, 43 bd du 11-Novembre-1918, 69622 Villeurbanne cedex, France

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     author = {Petru Mironescu},
     title = {Profile decomposition and phase control for circle-valued maps in one dimension},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1087--1092},
     publisher = {Elsevier},
     volume = {353},
     number = {12},
     year = {2015},
     doi = {10.1016/j.crma.2015.09.030},
     language = {en},
}

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Petru Mironescu. Profile decomposition and phase control for circle-valued maps in one dimension. Comptes Rendus. Mathématique, Volume 353 (2015) no. 12, pp. 1087-1092. doi : 10.1016/j.crma.2015.09.030. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.09.030/

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