no. 7
Partial Differential Equations
New estimates for the Laplacian, the div–curl, and related Hodge systems
[Nouvelles estimées pour le Laplacien, le système div–rot et autres systèmes de Hodge]

Présenté par : Haïm Brezis

Bourgain, Jean1 ; Brezis, Haïm2, 3
1 Institute for Advanced Study, Olden Lane, Princeton, NJ 08540, USA
2 Analyse numérique, Université P. et M. Curie, BC 187, 4, place Jussieu, 75252 Paris cedex 05, France
3 Department of Mathematics, Rutgers University, Hill Center, Busch Campus, 110, Frelinghuysen Road, Piscataway, NJ 08854, USA
Comptes Rendus. Mathématique, Tome 338 (2004) no. 7, pp. 539-543.

On établit de nouvelles estimées pour le Laplacien, le système div–rot et autres systèmes de Hodge en dimension quelconque. On présente une application aux minimiseurs de l'énergie de Ginzburg–Landau.

We establish new estimates for the Laplacian, the div–curl system, and more general Hodge systems in arbitrary dimension, with an application to minimizers of the Ginzburg–Landau energy.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2003.12.031
Bourgain, Jean 1 ; Brezis, Haïm 2, 3

1 Institute for Advanced Study, Olden Lane, Princeton, NJ 08540, USA
2 Analyse numérique, Université P. et M. Curie, BC 187, 4, place Jussieu, 75252 Paris cedex 05, France
3 Department of Mathematics, Rutgers University, Hill Center, Busch Campus, 110, Frelinghuysen Road, Piscataway, NJ 08854, USA 
@article{CRMATH_2004__338_7_539_0,
     author = {Bourgain, Jean and Brezis, Ha{\"\i}m},
     title = {New estimates for the {Laplacian,} the div{\textendash}curl, and related {Hodge} systems},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {539--543},
     publisher = {Elsevier},
     volume = {338},
     number = {7},
     year = {2004},
     doi = {10.1016/j.crma.2003.12.031},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.crma.2003.12.031/}
}
TY  - JOUR
AU  - Bourgain, Jean
AU  - Brezis, Haïm
TI  - New estimates for the Laplacian, the div–curl, and related Hodge systems
JO  - Comptes Rendus. Mathématique
PY  - 2004
SP  - 539
EP  - 543
VL  - 338
IS  - 7
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.crma.2003.12.031/
DO  - 10.1016/j.crma.2003.12.031
LA  - en
ID  - CRMATH_2004__338_7_539_0
ER  - 
%0 Journal Article
%A Bourgain, Jean
%A Brezis, Haïm
%T New estimates for the Laplacian, the div–curl, and related Hodge systems
%J Comptes Rendus. Mathématique
%D 2004
%P 539-543
%V 338
%N 7
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/j.crma.2003.12.031/
%R 10.1016/j.crma.2003.12.031
%G en
%F CRMATH_2004__338_7_539_0
Bourgain, Jean; Brezis, Haïm. New estimates for the Laplacian, the div–curl, and related Hodge systems. Comptes Rendus. Mathématique, Tome 338 (2004) no. 7, pp. 539-543. doi : 10.1016/j.crma.2003.12.031. http://archive.numdam.org/articles/10.1016/j.crma.2003.12.031/

[1] Bethuel, F.; Orlandi, G.; Smets, D. On an open problem for Jacobians raised by Bourgain, Brezis and Mironescu, C. R. Acad. Sci. Paris, Ser. I, Volume 337 (2003) no. 6, pp. 381-385

[2] F. Bethuel, G. Orlandi, D. Smets, Approximation with vorticity bounds for the Ginzburg–Landau functional, Comm. Contemp. Math., in press

[3] Bourgain, J.; Brezis, H. On the equation div Y=f and application to control of phases, J. Amer. Math. Soc., Volume 16 (2003), pp. 393-426 (Announced in C. R. Acad. Sci. Paris, Ser. I, 334, 2002, pp. 973-976)

[4] J. Bourgain, H. Brezis, in preparation

[5] J. Bourgain, H. Brezis, P. Mironescu, H1/2-maps into the circle: minimal connections, lifting and the Ginzburg–Landau equation, Publ. Math. IHES, in press

[6] Iwaniec, T. Integrability Theory of the Jacobians, Lecture Notes, Universität Bonn, 1995

[7] Smirnov, S.K. Decomposition of solenoidal vector charges into elementary solenoids and the structure of normal one-dimensional currents, Algebra i Analiz, Volume 5 (1993), pp. 206-238 (in Russian); English translation St. Petersburg Math. J., 5, 1994, pp. 841-867

[8] Van Schaftingen, J. A simple proof of an inequality of Bourgain, Brezis and Mironescu, C. R. Acad. Sci. Paris, Ser. I, Volume 338 (2004) no. 1, pp. 23-26

Cité par Sources :