Weak approximation by bounded Sobolev maps with values into complete manifolds

Pierre Bousquet; Augusto C. Ponce; Jean Van Schaftingen

doi:10.1016/j.crma.2018.01.017

Mathematical analysis

Weak approximation by bounded Sobolev maps with values into complete manifolds

Presented by: Haïm Brézis

Pierre Bousquet ¹; Augusto C. Ponce ²; Jean Van Schaftingen ²

¹ Université de Toulouse, Institut de mathématiques de Toulouse, UMR CNRS 5219, Université Paul-Sabatier Toulouse 3, 118 route de Narbonne, 31062 Toulouse Cedex 9, France
² Université catholique de Louvain, Institut de recherche en mathématique et physique, chemin du cyclotron 2, bte L7.01.02, 1348 Louvain-la-Neuve, Belgium

Comptes Rendus. Mathématique, Volume 356 (2018) no. 3, pp. 264-271.

Abstract
Résumé

We have recently introduced the trimming property for a complete Riemannian manifold $N^{n}$ as a necessary and sufficient condition for bounded maps to be strongly dense in $W^{1, p} (B^{m}; N^{n})$ when $p \in {1, \dots, m}$ . We prove in this note that, even under a weaker notion of approximation, namely the weak sequential convergence, the trimming property remains necessary for the approximation in terms of bounded maps. The argument involves the construction of a Sobolev map having infinitely many analytical singularities going to infinity.

Nous avons récemment introduit la propriété dite trimming property pour une variété riemanienne complète $N^{n}$ : il s'agit d'une condition nécessaire et suffisante pour que les applications bornées soient fortement denses dans $W^{1, p} (B^{m}; N^{n})$ pour $p \in {1, \dots, m}$ . Nous prouvons dans cette note que, même pour une notion de convergence plus faible, à savoir la convergence séquentielle faible, la trimming property reste nécessaire pour l'approximation en termes de fonctions bornées. La preuve repose sur la construction d'une application de Sobolev qui a une infinité de singularités hors de tout compact.

Received: 2017-10-09
Accepted: 2018-01-29
Published online: 2018-02-21

DOI: 10.1016/j.crma.2018.01.017

Author's affiliations:

Pierre Bousquet ¹; Augusto C. Ponce ²; Jean Van Schaftingen ²

¹ Université de Toulouse, Institut de mathématiques de Toulouse, UMR CNRS 5219, Université Paul-Sabatier Toulouse 3, 118 route de Narbonne, 31062 Toulouse Cedex 9, France
² Université catholique de Louvain, Institut de recherche en mathématique et physique, chemin du cyclotron 2, bte L7.01.02, 1348 Louvain-la-Neuve, Belgium

@article{CRMATH_2018__356_3_264_0,
     author = {Pierre Bousquet and Augusto C. Ponce and Jean Van Schaftingen},
     title = {Weak approximation by bounded {Sobolev} maps with values into complete manifolds},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {264--271},
     publisher = {Elsevier},
     volume = {356},
     number = {3},
     year = {2018},
     doi = {10.1016/j.crma.2018.01.017},
     language = {en},
}

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Pierre Bousquet; Augusto C. Ponce; Jean Van Schaftingen. Weak approximation by bounded Sobolev maps with values into complete manifolds. Comptes Rendus. Mathématique, Volume 356 (2018) no. 3, pp. 264-271. doi : 10.1016/j.crma.2018.01.017. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.01.017/

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