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

Radu Ignat; Luc Nguyen; Valeriy Slastikov; Arghir Zarnescu

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

Presented by: Haïm Brézis

Radu Ignat ¹; Luc Nguyen ²; Valeriy Slastikov ³; Arghir Zarnescu ^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, Volume 356 (2018) no. 9, pp. 922-926.

Abstract
Résumé

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 $x \in B^{N}$ .

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 $x \in B^{N}$ .

Received: 2018-07-12
Accepted: 2018-07-13
Published online: 2018-08-14

DOI: 10.1016/j.crma.2018.07.006

Author's affiliations:

Radu Ignat ¹; Luc Nguyen ²; Valeriy Slastikov ³; Arghir Zarnescu ^{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

@article{CRMATH_2018__356_9_922_0,
     author = {Radu Ignat and Luc Nguyen and Valeriy Slastikov and Arghir Zarnescu},
     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},
     publisher = {Elsevier},
     volume = {356},
     number = {9},
     year = {2018},
     doi = {10.1016/j.crma.2018.07.006},
     language = {en},
}

TY  - JOUR
AU  - Radu Ignat
AU  - Luc Nguyen
AU  - Valeriy Slastikov
AU  - Arghir Zarnescu
TI  - Uniqueness of degree-one Ginzburg–Landau vortex in the unit ball in dimensions N ≥ 7
JO  - Comptes Rendus. Mathématique
PY  - 2018
SP  - 922
EP  - 926
VL  - 356
IS  - 9
PB  - Elsevier
DO  - 10.1016/j.crma.2018.07.006
LA  - en
ID  - CRMATH_2018__356_9_922_0
ER  -

%0 Journal Article
%A Radu Ignat
%A Luc Nguyen
%A Valeriy Slastikov
%A Arghir Zarnescu
%T Uniqueness of degree-one Ginzburg–Landau vortex in the unit ball in dimensions N ≥ 7
%J Comptes Rendus. Mathématique
%D 2018
%P 922-926
%V 356
%N 9
%I Elsevier
%R 10.1016/j.crma.2018.07.006
%G en
%F CRMATH_2018__356_9_922_0

Radu Ignat; Luc Nguyen; Valeriy Slastikov; Arghir Zarnescu. Uniqueness of degree-one Ginzburg–Landau vortex in the unit ball in dimensions N ≥ 7. Comptes Rendus. Mathématique, Volume 356 (2018) no. 9, pp. 922-926. doi : 10.1016/j.crma.2018.07.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.07.006/

References
Cited by

[1] F. Bethuel; H. Brezis; F. Hélein Ginzburg–Landau Vortices, Progress in Nonlinear Differential Equations and Their Applications, vol. 13, Birkhäuser Boston Inc., Boston, MA, USA, 1994

[2] H. Brezis Symmetry in nonlinear PDE's, La Pietra 1996 (Florence, Italy) (Proceedings of Symposia in Pure Mathematics), Volume vol. 65, American Mathematical Society, Providence, RI, USA (1999), pp. 1-12

[3] H. Brezis; J.-M. Coron; E.H. Lieb Harmonic maps with defects, Commun. Math. Phys., Volume 107 (1986) no. 4, pp. 649-705

[4] J.-M. Coron; R. Gulliver Minimizing p-harmonic maps into spheres, J. Reine Angew. Math., Volume 401 (1989), pp. 82-100

[5] R. Ignat; L. Nguyen; V. Slastikov; A. Zarnescu Uniqueness results for an ODE related to a generalized Ginzburg–Landau model for liquid crystals, SIAM J. Math. Anal., Volume 46 (2014) no. 5, pp. 3390-3425

[6] R. Ignat; L. Nguyen; V. Slastikov; A. Zarnescu Stability of the melting hedgehog in the Landau–de Gennes theory of nematic liquid crystals, Arch. Ration. Mech. Anal., Volume 215 (2015) no. 2, pp. 633-673

[7] R. Ignat; L. Nguyen; V. Slastikov; A. Zarnescu On the uniqueness of minimisers of Ginzburg–Landau functionals, 2017 (To appear in Ann. Sci. Éc. Norm. Supér) | arXiv

[8] W. Jäger; H. Kaul Rotationally symmetric harmonic maps from a ball into a sphere and the regularity problem for weak solutions of elliptic systems, J. Reine Angew. Math., Volume 343 (1983), pp. 146-161

[9] F.-H. Lin A remark on the map $x / | x |$ , C. R. Acad. Sci. Paris, Ser. I, Volume 305 (1987) no. 12, pp. 529-531

[10] V. Millot; A. Pisante Symmetry of local minimizers for the three-dimensional Ginzburg–Landau functional, J. Eur. Math. Soc., Volume 12 (2010) no. 5, pp. 1069-1096

[11] P. Mironescu Les minimiseurs locaux pour l'équation de Ginzburg–Landau sont à symétrie radiale, C. R. Acad. Sci. Paris, Ser. I, Volume 323 (1996) no. 6, pp. 593-598

[12] F. Pacard; T. Rivière Linear and Nonlinear Aspects of Vortices, Progress in Nonlinear Differential Equations and Their Applications, vol. 39, Birkhäuser Boston, Inc., Boston, MA, USA, 2000 (The Ginzburg–Landau model)

[13] A. Pisante Two results on the equivariant Ginzburg–Landau vortex in arbitrary dimension, J. Funct. Anal., Volume 260 (2011) no. 3, pp. 892-905

Cited by Sources:

Comments - Policy