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

Bethuel, Fabrice1, 2 ; Orlandi, Giandomenico3 ; Smets, Didier1
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, Tome 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
Bethuel, Fabrice 1, 2 ; Orlandi, Giandomenico 3 ; Smets, Didier 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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Bethuel, Fabrice; Orlandi, Giandomenico; Smets, Didier. On an open problem for Jacobians raised by Bourgain, Brezis and Mironescu. Comptes Rendus. Mathématique, Tome 337 (2003) no. 6, pp. 381-385. doi : 10.1016/S1631-073X(03)00367-4. http://archive.numdam.org/articles/10.1016/S1631-073X(03)00367-4/

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

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

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

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

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

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