Uniqueness of degree-one Ginzburg–Landau vortex in the unit ball in dimensions <i>N</i> ≥ 7

Ignat, Radu; Nguyen, Luc; Slastikov, Valeriy; Zarnescu, Arghir

doi:10.1016/j.crma.2018.07.006

Partial differential equations/Calculus of variations

Uniqueness of degree-one Ginzburg–Landau vortex in the unit ball in dimensions N ≥ 7
[Unicité du tourbillon de Ginzburg–Landau de degré un dans la boule unité en dimension N ≥ 7]

Présenté par : Haïm Brézis

Ignat, Radu ¹ ; Nguyen, Luc ² ; Slastikov, Valeriy ³ ; Zarnescu, Arghir ^4,^5,⁶

¹ Institut de mathématiques de Toulouse & Institut universitaire de France, UMR 5219, Université de Toulouse, CNRS, UPS IMT, 31062 Toulouse cedex 9, France
² Mathematical Institute and St Edmund Hall, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, United Kingdom
³ School of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW, United Kingdom
⁴ IKERBASQUE, Basque Foundation for Science, Maria Diaz de Haro 3, 48013, Bilbao, Bizkaia, Spain
⁵ BCAM, Basque Center for Applied Mathematics, Mazarredo 14, E48009 Bilbao, Bizkaia, Spain
⁶ “Simion Stoilow” Institute of the Romanian Academy, 21 Calea Griviţei, 010702 Bucharest, Romania

Comptes Rendus. Mathématique, Tome 356 (2018) no. 9, pp. 922-926.

Résumé
Abstract

Nous considérons la fonctionnelle de Ginzburg–Landau pour les applications à valeurs dans $R^{N}$ définies dans la boule unité $B^{N} \subset R^{N}$ avec la donnée de tourbillon x au bord $\partial B^{N}$ . En dimension $N \geq 7$ , nous montrons que, pour tout $ε > 0$ , il existe un unique minimiseur global $u_{ε}$ à ce problème ; de plus, $u_{ε}$ est symétrique de la forme $u_{ε} (x) = f_{ε} (| x |) \frac{x}{| x |}$ pour $x \in B^{N}$ .

For $ε > 0$ , we consider the Ginzburg–Landau functional for $R^{N}$ -valued maps defined in the unit ball $B^{N} \subset R^{N}$ with the vortex boundary data x on $\partial B^{N}$ . In dimensions $N \geq 7$ , we prove that, for every $ε > 0$ , there exists a unique global minimizer $u_{ε}$ of this problem; moreover, $u_{ε}$ is symmetric and of the form $u_{ε} (x) = f_{ε} (| x |) \frac{x}{| x |}$ for $x \in B^{N}$ .

Reçu le : 2018-07-12
Accepté le : 2018-07-13
Publié le : 2018-08-14

DOI : 10.1016/j.crma.2018.07.006

Affiliations des auteurs :

Ignat, Radu ¹ ; Nguyen, Luc ² ; Slastikov, Valeriy ³ ; Zarnescu, Arghir ^{4, 5, 6}

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     title = {Uniqueness of degree-one {Ginzburg{\textendash}Landau} vortex in the unit ball in dimensions {\protect\emph{N} \ensuremath{\geq} 7}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {922--926},
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Ignat, Radu; Nguyen, Luc; Slastikov, Valeriy; Zarnescu, Arghir. Uniqueness of degree-one Ginzburg–Landau vortex in the unit ball in dimensions N ≥ 7. Comptes Rendus. Mathématique, Tome 356 (2018) no. 9, pp. 922-926. doi : 10.1016/j.crma.2018.07.006. http://www.numdam.org/articles/10.1016/j.crma.2018.07.006/

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