Mathematical analysis
Weak approximation by bounded Sobolev maps with values into complete manifolds
Comptes Rendus. Mathématique, Volume 356 (2018) no. 3, pp. 264-271.

We have recently introduced the trimming property for a complete Riemannian manifold Nn as a necessary and sufficient condition for bounded maps to be strongly dense in W1,p(Bm;Nn) when p{1,,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 Nn : il s'agit d'une condition nécessaire et suffisante pour que les applications bornées soient fortement denses dans W1,p(Bm;Nn) pour p{1,,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.

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DOI: 10.1016/j.crma.2018.01.017
Bousquet, Pierre 1; Ponce, Augusto C. 2; Van Schaftingen, Jean 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
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Bousquet, Pierre; Ponce, Augusto C.; Van Schaftingen, Jean. 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. http://www.numdam.org/articles/10.1016/j.crma.2018.01.017/

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