Partial Differential Equations/Functional Analysis
Gradient vector fields with values into S1
[Champs de gradient à valeurs dans S1]

Présenté par : Haïm Brezis

Ignat, Radu1
1 Laboratoire de Mathématiques, Université Paris-Sud 11, Bât. 425, 91405 Orsay cedex, France
Comptes Rendus. Mathématique, Tome 349 (2011) no. 15-16, pp. 883-887.

Le résultat de régularité suivant a lieu : Si un champ de gradient v=∇ψ est à valeurs dans le cercle unité S1 et appartient à H1/2 (ou W1,1) 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=∇ψ with values into the unit circle S1 belongs to H1/2 (or W1,1) then v is locally Lipschitz except at a locally finite number of vortices. We also state approximation results for such vector fields.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2011.07.024
Ignat, Radu 1

1 Laboratoire de Mathématiques, Université Paris-Sud 11, Bât. 425, 91405 Orsay cedex, France 
@article{CRMATH_2011__349_15-16_883_0,
     author = {Ignat, Radu},
     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},
     url = {https://www.numdam.org/articles/10.1016/j.crma.2011.07.024/}
}
TY  - JOUR
AU  - Ignat, Radu
TI  - Gradient vector fields with values into $ {S}^{1}$
JO  - Comptes Rendus. Mathématique
PY  - 2011
SP  - 883
EP  - 887
VL  - 349
IS  - 15-16
PB  - Elsevier
UR  - https://www.numdam.org/articles/10.1016/j.crma.2011.07.024/
DO  - 10.1016/j.crma.2011.07.024
LA  - en
ID  - CRMATH_2011__349_15-16_883_0
ER  - 
%0 Journal Article
%A Ignat, Radu
%T Gradient vector fields with values into $ {S}^{1}$
%J Comptes Rendus. Mathématique
%D 2011
%P 883-887
%V 349
%N 15-16
%I Elsevier
%U https://www.numdam.org/articles/10.1016/j.crma.2011.07.024/
%R 10.1016/j.crma.2011.07.024
%G en
%F CRMATH_2011__349_15-16_883_0
Ignat, Radu. Gradient vector fields with values into $ {S}^{1}$. Comptes Rendus. Mathématique, Tome 349 (2011) no. 15-16, pp. 883-887. doi : 10.1016/j.crma.2011.07.024. https://www.numdam.org/articles/10.1016/j.crma.2011.07.024/

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

[2] Bourgain, J.; Brezis, H.; Mironescu, P. H1/2 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] Brezis, H.; Mironescu, P.; Ponce, A. W1,1-maps with values into S1, Geometric Analysis of PDE and Several Complex Variables, Contemp. Math., vol. 368, Amer. Math. Soc., Providence, RI, 2005, pp. 69-100

[4] Dávila, J.; Ignat, R. Lifting of BV functions with values in S1, C. R. Math. Acad. Sci. Paris, Volume 337 (2003), pp. 159-164

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

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

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

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

[9] Ignat, R. The space BV(S2,S1): 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] Ignat, R.; Knüpfer, H. Vortex energy and 360° Néel walls in thin-film micromagnetics, Comm. Pure Appl. Math., Volume 63 (2010), pp. 1677-1724

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

[14] Rivière, T. Dense subsets of H1/2(S2,S1), Ann. Global Anal. Geom., Volume 18 (2000), pp. 517-528

Cité par Sources :