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}$},
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     pages = {883--887},
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     number = {15-16},
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     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/

Bibliographie
Cité par

[1] F. Bethuel; X.M. Zheng Density of smooth functions between two manifolds in Sobolev spaces, J. Funct. Anal., Volume 80 (1988), pp. 60-75

[2] J. Bourgain; H. Brezis; P. Mironescu $H^{1 / 2}$ maps with values into the circle: minimal connections, lifting and the Ginzburg–Landau equation, Publ. Math. Inst. Hautes Études Sci., Volume 99 (2004), pp. 1-115

[3] H. Brezis; P. Mironescu; A. Ponce $W^{1, 1}$ -maps with values into $S^{1}$ , Geometric Analysis of PDE and Several Complex Variables, Contemp. Math., vol. 368, Amer. Math. Soc., Providence, RI, 2005, pp. 69-100

[4] J. Dávila; R. Ignat Lifting of BV functions with values in $S^{1}$ , C. R. Math. Acad. Sci. Paris, Volume 337 (2003), pp. 159-164

[5] A. DeSimone; H. Knüpfer; F. Otto 2-d stability of the Néel wall, Calc. Var. Partial Differential Equations, Volume 27 (2006), pp. 233-253

[6] A. DeSimone; R.V. Kohn; S. Müller; F. Otto A reduced theory for thin-film micromagnetics, Comm. Pure Appl. Math., Volume 55 (2002), pp. 1408-1460

[7] M. Giaquinta; G. Modica; J. Soucek Cartesian Currents in the Calculus of Variations, vol. II, Springer, 1998

[8] F. Golse; P.-L. Lions; B. Perthame; R. Sentis Regularity of the moments of the solution of a transport equation, J. Funct. Anal., Volume 76 (1988), pp. 110-125

[9] R. Ignat The space $BV (S^{2}, S^{1})$ : minimal connection and optimal lifting, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 22 (2005), pp. 283-302

[10] R. Ignat, Two-dimensional unit-length vector fields of vanishing divergence, submitted for publication.

[11] R. Ignat; H. Knüpfer Vortex energy and 360° Néel walls in thin-film micromagnetics, Comm. Pure Appl. Math., Volume 63 (2010), pp. 1677-1724

[12] R. Ignat; F. Otto A compactness result in thin-film micromagnetics and the optimality of the Néel wall, J. Eur. Math. Soc. (JEMS), Volume 10 (2008), pp. 909-956

[13] P.-E. Jabin; F. Otto; B. Perthame Line-energy Ginzburg–Landau models: zero-energy states, Ann. Sc. Norm. Super. Pisa Cl. Sci., Volume 1 (2002), pp. 187-202

[14] T. Rivière Dense subsets of $H^{1 / 2} (S^{2}, S^{1})$ , Ann. Global Anal. Geom., Volume 18 (2000), pp. 517-528

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