Partial Differential Equations
On the distributions of the form ∑i(δpiδni)
Comptes Rendus. Mathématique, Volume 336 (2003) no. 7, pp. 571-576.

We present some properties of the distributions of the form T=∑i(δpiδni), with ∑id(pi,ni)<∞, which arise in the 3-d Ginzburg–Landau problem studied by Bourgain, Brezis and Mironescu (C. R. Acad. Sci. Paris, Ser. I 331 (2000) 119–124). We show that there always exists an irreducible representation of T. We also extend a result of Smets (C. R. Acad. Sci. Paris, Ser. I 334 (2002) 371–374) which says that T is a measure iff T can be written as a finite sum of dipoles.

On présente quelques propriétés sur les distributions T de la forme ∑i(δpiδni), avec ∑id(pi,ni)<∞, qui interviennent dans le problème de Ginzburg–Landau en 3-d étudié par Bourgain, Brezis et Mironescu (C. R. Acad. Sci. Paris, Ser. I 331 (2000) 119–124). Même dans un cadre plus général, ces formes linéaires ont toujours une représentation irréductible. Notre approche permet aussi de montrer que T est une mesure si et seulement si T peut être écrite comme une somme finie de masses de Dirac, ce qui généralise un résultat de Smets (C. R. Acad. Sci. Paris, Ser. I 334 (2002) 371–374).

Accepted:
Published online:
DOI: 10.1016/S1631-073X(03)00133-X
Ponce, Augusto C. 1, 2

1 Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, BC 187, 4, pl. Jussieu, 75252 Paris cedex 05, France
2 Rutgers University, Dept. of Math., Hill Center, Busch Campus, 110 Frelinghuysen Rd, Piscataway, NJ 08854, USA
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Ponce, Augusto C. On the distributions of the form ∑i(δpiδni). Comptes Rendus. Mathématique, Volume 336 (2003) no. 7, pp. 571-576. doi : 10.1016/S1631-073X(03)00133-X. http://www.numdam.org/articles/10.1016/S1631-073X(03)00133-X/

[1] Bourgain, J.; Brezis, H.; Mironescu, P. On the structure of the Sobolev space H1/2 with values into the circle, C. R. Acad. Sci. Paris, Ser. I, Volume 331 (2000), pp. 119-124

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

[3] Brezis, H. Liquid crystals and energy estimates for S2-valued maps, Theory and Applications of Liquid Crystals, Minneapolis, MN, 1985, IMA Vol. Math. Appl., 5, Springer, New York, 1987, pp. 31-52

[4] H. Brezis, The interplay between analysis and topology in some nonlinear PDE's, Bull. Amer. Math. Soc., to appear

[5] Brezis, H.; Coron, J.-M.; Lieb, E. Harmonic maps with defects, Comm. Math. Phys., Volume 107 (1986), pp. 649-705

[6] A.C. Ponce, On the distributions of the form ∑i(δpiδni), to appear

[7] Smets, D. On some infinite sums of integer valued Dirac's masses, C. R. Acad. Sci. Paris, Ser. I, Volume 334 (2002), pp. 371-374

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