Comptes Rendus
no. 6
Partial Differential Equations
On an open problem for Jacobians raised by Bourgain, Brezis and Mironescu
[Un problème ouvert sur les Jacobiens soulevé par Bourgain, Brezis et Mironescu]

Présenté par : Haïm Brezis

Fabrice Bethuel1, 2 ; Giandomenico Orlandi3 ; Didier Smets1
1 Laboratoire Jacques-Louis Lions, Université de Paris 6, 4, place Jussieu, BC 187, 75252 Paris, France
2 Institut Universitaire de France, 103, bd Saint-Michel, 75005 Paris, France
3 Dipartimento di Informatica, Università di Verona, Strada le Grazie, 37134 Verona, Italy
Comptes Rendus. Mathématique, Volume 337 (2003) no. 6, pp. 381-385.

Nous démontrons une estimée pour des Jacobiens dans le contexte de la fonctionnelle de Ginzburg–Landau. Cela répond à une conjecture dans un travail récent de Bourgain, Brezis et Mironescu.

We establish a Jacobian estimate in the context of Ginzburg–Landau theory, which was conjectured in a recent work of Bourgain, Brezis and Mironescu.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(03)00367-4
Fabrice Bethuel 1, 2 ; Giandomenico Orlandi 3 ; Didier Smets 1

1 Laboratoire Jacques-Louis Lions, Université de Paris 6, 4, place Jussieu, BC 187, 75252 Paris, France
2 Institut Universitaire de France, 103, bd Saint-Michel, 75005 Paris, France
3 Dipartimento di Informatica, Università di Verona, Strada le Grazie, 37134 Verona, Italy 
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     journal = {Comptes Rendus. Math\'ematique},
     pages = {381--385},
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Fabrice Bethuel; Giandomenico Orlandi; Didier Smets. On an open problem for Jacobians raised by Bourgain, Brezis and Mironescu. Comptes Rendus. Mathématique, Volume 337 (2003) no. 6, pp. 381-385. doi : 10.1016/S1631-073X(03)00367-4. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00367-4/

[1] G. Alberti, S. Baldo, G. Orlandi, Variational convergence for functionals of Ginzburg–Landau type, Preprint, 2002

[2] J. Bourgain, H. Brezis, Work on the curl–div system, in preparation

[3] J. Bourgain, H. Brezis, P. Mironescu, H1/2 maps with values into the circle: minimal connections, lifting, and the Ginzburg–Landau equation, in press

[4] F. Bethuel, G. Orlandi, D. Smets, Approximation with bounded vorticity for the Ginzburg–Landau functional, in preparation

[5] F. Bethuel; H. Brezis; F. Hélein Ginzburg–Landau Vortices, Birkhäuser, Boston, 1994

[6] H. Federer Geometric Measure Theory, Springer, Berlin, 1969

[7] R. Jerrard Lower bounds for generalized Ginzburg–Landau functionals, SIAM J. Math. Anal., Volume 30 (1999), pp. 721-746

[8] R.L. Jerrard; H.M. Soner The Jacobian and the Ginzburg–Landau energy, Calc. Var. Partial Differential Equations, Volume 14 (2002), pp. 151-191

[9] E. Sandier Lower bounds for the energy of unit vector fields and applications, J. Funct. Anal., Volume 152 (1998), pp. 379-403

[10] E. Sandier; S. Serfaty A rigorous derivation of a free-boundary problem arising in superconductivity, Ann. Sci. ENS, Volume 33 (2000), pp. 561-592

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