On the representation as exterior differentials of closed forms with <i>L</i><sup>1</sup>-coefficients

Eduard Curcă

doi:10.1016/j.crma.2019.04.011

Partial differential equations/Harmonic analysis

On the representation as exterior differentials of closed forms with L¹-coefficients
[Sur la représentation comme différentielles extérieures des formes fermées à coefficients L¹]

Présenté par : Haïm Brézis

Eduard Curcă ¹

¹ Université de Lyon, Université Lyon-1, CNRS UMR 5208, Institut Camille-Jordan, 43, bd du 11-Novembre-1918, 69622 Villeurbanne cedex, France

Comptes Rendus. Mathématique, Volume 357 (2019) no. 4, pp. 355-359.

Résumé
Abstract

Soit $N \geq 2$ . Si $g \in L_{c}^{1} (R^{N})$ est d'intégrale nulle, alors en général il n'est pas possible de résoudre l'équation $div X = g$ avec $X \in W_{loc}^{1, 1} (R^{N}; R^{N})$ [6], ou même $X \in L_{loc}^{N / (N - 1)}$ $(R^{N}; R^{N})$ [2]. En utilisant ces résultats, nous prouvons que, pour $N \geq 3$ et $2 \leq ℓ \leq N - 1$ , il existe une ℓ-forme $f \in L_{c}^{1} (R^{N}; Λ^{ℓ})$ avec $d f = 0$ et telle que l'équation $d λ = f$ n'a pas de solution $λ \in W_{loc}^{1, 1} (R^{N}; Λ^{ℓ - 1})$ . Ceci donne une réponse négative à une question posée par Baldi, Franchi et Pansu [1].

Let $N \geq 2$ . If $g \in L_{c}^{1} (R^{N})$ has zero integral, then the equation $div X = g$ need not have a solution $X \in W_{loc}^{1, 1} (R^{N}; R^{N})$ [6] or even $X \in L_{loc}^{N / (N - 1)}$ $(R^{N}; R^{N})$ [2]. Using these results, we prove that, whenever $N \geq 3$ and $2 \leq ℓ \leq N - 1$ , there exists some ℓ-form $f \in L_{c}^{1} (R^{N}; Λ^{ℓ})$ such that $d f = 0$ and the equation $d λ = f$ has no solution $λ \in W_{loc}^{1, 1} (R^{N}; Λ^{ℓ - 1})$ . This provides a negative answer to a question raised by Baldi, Franchi, and Pansu [1].

Reçu le : 2019-04-23
Accepté le : 2019-04-28
Publié le : 2019-04-01

DOI : 10.1016/j.crma.2019.04.011

Affiliations des auteurs :

Eduard Curcă ¹

¹ Université de Lyon, Université Lyon-1, CNRS UMR 5208, Institut Camille-Jordan, 43, bd du 11-Novembre-1918, 69622 Villeurbanne cedex, France

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     author = {Eduard Curc\u{a}},
     title = {On the representation as exterior differentials of closed forms with {\protect\emph{L}\protect\textsuperscript{1}-coefficients}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {355--359},
     publisher = {Elsevier},
     volume = {357},
     number = {4},
     year = {2019},
     doi = {10.1016/j.crma.2019.04.011},
     language = {en},
}

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Eduard Curcă. On the representation as exterior differentials of closed forms with L¹-coefficients. Comptes Rendus. Mathématique, Volume 357 (2019) no. 4, pp. 355-359. doi : 10.1016/j.crma.2019.04.011. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2019.04.011/

Bibliographie
Cité par

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