Gradient vector fields with values into $ {S}^{1}$

Radu Ignat

doi:10.1016/j.crma.2011.07.024

Partial Differential Equations/Functional Analysis

Gradient vector fields with values into

S^{1}

[Champs de gradient à valeurs dans

S^{1}

]

Présenté par : Haïm Brezis

Radu Ignat ¹

¹ Laboratoire de Mathématiques, Université Paris-Sud 11, Bât. 425, 91405 Orsay cedex, France

Comptes Rendus. Mathématique, Volume 349 (2011) no. 15-16, pp. 883-887.

Abstract (VO)
Résumé

We state the following regularity result: if a two-dimensional gradient vector field $v = \nabla ψ$ with values into the unit circle $S^{1}$ belongs to $H^{1 / 2}$ (or $W^{1, 1}$ ) then v is locally Lipschitz except at a locally finite number of vortices. We also state approximation results for such vector fields.

Le résultat de régularité suivant a lieu : Si un champ de gradient $v = \nabla ψ$ est à valeurs dans le cercle unité $S^{1}$ et appartient à $H^{1 / 2}$ (ou $W^{1, 1}$ ) alors v est localement Lipschitz en dehors dʼun nombre localement fini de points singuliers. Ensuite, des résultats de densité sont énoncés pour cette classe de champs de gradient.

Reçu le : 2011-04-18
Accepté le : 2011-07-28
Publié le : 2011-08-01

DOI : 10.1016/j.crma.2011.07.024

Affiliations des auteurs :

Radu Ignat ¹

¹ Laboratoire de Mathématiques, Université Paris-Sud 11, Bât. 425, 91405 Orsay cedex, France

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Radu Ignat. Gradient vector fields with values into $ {S}^{1}$. Comptes Rendus. Mathématique, Volume 349 (2011) no. 15-16, pp. 883-887. doi : 10.1016/j.crma.2011.07.024. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.07.024/

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Pierre Bochard; Radu Ignat Kinetic formulation of vortex vector fields, Analysis PDE, Volume 10 (2017) no. 3, pp. 729-756 | DOI:10.2140/apde.2017.10.729 | Zbl:1364.35071
RADU IGNAT SINGULARITIES OF DIVERGENCE-FREE VECTOR FIELDS WITH VALUES INTO S1OR S2: APPLICATIONS TO MICROMAGNETICS, Confluentes Mathematici, Volume 04 (2012) no. 03, p. 1230001 | DOI:10.1142/s1793744212300012
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

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