Retractions onto the Space of Continuous Divergence-free Vector Fields
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 20 (2011) no. 4, p. 767-779

We prove that there does not exist a uniformly continuous retraction from the space of continuous vector fields onto the subspace of vector fields whose divergence vanishes in the distributional sense. We then generalise this result using the concept of m-charges, introduced by De Pauw, Moonens, and Pfeffer: on any subset X n satisfying a mild geometric condition, there is no uniformly continuous representation operator for m-charges in X.

On prouve qu’il n’existe pas de rétraction uniformément continue de l’espace des champs de vecteurs continus à valeurs dans le sous-espace des champs dont la divergence est nulle au sens des distributions. Les m-charges, telles que définies par De Pauw, Moonens, et Pfeffer permettent de généraliser le résultat précédent. On prouve qu’il n’y a pas d’opérateur de représentation uniformément continu des m-charges dans une partie X n vérifiant une hypothèse faible.

@article{AFST_2011_6_20_4_767_0,
     author = {Bouafia, Philippe},
     title = {Retractions onto the Space of Continuous Divergence-free Vector Fields},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 20},
     number = {4},
     year = {2011},
     pages = {767-779},
     doi = {10.5802/afst.1323},
     mrnumber = {2918213},
     zbl = {1234.47058},
     language = {en},
     url = {http://www.numdam.org/item/AFST_2011_6_20_4_767_0}
}
Bouafia, Philippe. Retractions onto the Space of Continuous Divergence-free Vector Fields. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 20 (2011) no. 4, pp. 767-779. doi : 10.5802/afst.1323. http://www.numdam.org/item/AFST_2011_6_20_4_767_0/

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