Comptes Rendus
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

Jean Bourgain1 ; Haïm Brezis2, 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, Volume 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
Jean Bourgain 1 ; Haïm Brezis 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 
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     author = {Jean Bourgain and Ha{\"\i}m Brezis},
     title = {New estimates for the {Laplacian,} the div{\textendash}curl, and related {Hodge} systems},
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     pages = {539--543},
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     language = {en},
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Jean Bourgain; Haïm Brezis. New estimates for the Laplacian, the div–curl, and related Hodge systems. Comptes Rendus. Mathématique, Volume 338 (2004) no. 7, pp. 539-543. doi : 10.1016/j.crma.2003.12.031. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2003.12.031/

[1] F. Bethuel; G. Orlandi; D. Smets 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] J. Bourgain; H. Brezis 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] T. Iwaniec Integrability Theory of the Jacobians, Lecture Notes, Universität Bonn, 1995

[7] S.K. Smirnov 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] J. Van Schaftingen 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

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