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.

Résumé
Abstract

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.

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.

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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     author = {Radu Ignat},
     title = {Gradient vector fields with values into $ {S}^{1}$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {883--887},
     publisher = {Elsevier},
     volume = {349},
     number = {15-16},
     year = {2011},
     doi = {10.1016/j.crma.2011.07.024},
     language = {en},
}

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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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