Let . If has zero integral, then the equation need not have a solution [6] or even [2]. Using these results, we prove that, whenever and , there exists some ℓ-form such that and the equation has no solution . This provides a negative answer to a question raised by Baldi, Franchi, and Pansu [1].
Soit . Si est d'intégrale nulle, alors en général il n'est pas possible de résoudre l'équation avec [6], ou même [2]. En utilisant ces résultats, nous prouvons que, pour et , il existe une ℓ-forme avec et telle que l'équation n'a pas de solution . Ceci donne une réponse négative à une question posée par Baldi, Franchi et Pansu [1].
Accepted:
Published online:
Eduard Curcă 1
@article{CRMATH_2019__357_4_355_0, 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}, }
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/
[1] -Poincaré inequalities for differential forms on Euclidean spaces and Heisenberg groups, 2019 | HAL
[2] On the equation and application to control of phases, J. Amer. Math. Soc., Volume 16 (2003) no. 2, pp. 393-426
[3] New estimates for eliptic equations and Hodge type systems, J. Eur. Math. Soc., Volume 9 (2007) no. 2, pp. 277-315
[4] On Bogovskii and regularized Poincaré integral operators for de Rham complexes on Lipschitz domains, Math. Z., Volume 265 (2010) no. 2, pp. 297-320
[5] Lipschitz maps and nets in Euclidean space, Geom. Funct. Anal., Volume 8 (1998), pp. 304-314
[6] On the representation of functions as a sum of derivatives, C. R. Acad. Sci. Paris, Ser. I, Volume 328 (1999) no. 4, pp. 303-306
[7] Limiting Bourgain–Brezis estimates for systems of linear differential equations: theme and variations, J. Fixed Point Theory Appl., Volume 15 (2014) no. 2, pp. 273-297
Cited by Sources:
Comments - Policy