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
[Densité faible des fonctions de Sobolev bornées à valeurs dans des variétés complètes]

Présenté par : 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.

Résumé
Abstract

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.

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.

Reçu le : 2017-10-09
Accepté le : 2018-01-29
Publié le : 2018-02-21

DOI : 10.1016/j.crma.2018.01.017

Affiliations des auteurs :

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

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     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/

Bibliographie
Cité par

[1] F. Bethuel A characterization of maps in $H^{1} (B^{3}, S^{2})$ which can be approximated by smooth maps, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 167 (1990), pp. 269-286

[2] F. Bethuel The approximation problem for Sobolev maps between two manifolds, Acta Math., Volume 167 (1991), pp. 153-206

[3] F. Bethuel A counterexample to the weak density of smooth maps between manifolds in Sobolev spaces (available at) | arXiv

[4] P. Bousquet; A.C. Ponce; J. Van Schaftingen Strong density for higher order Sobolev spaces into compact manifolds, J. Eur. Math. Soc. (JEMS), Volume 17 (2015), pp. 763-817

[5] P. Bousquet; A.C. Ponce; J. Van Schaftingen Density of bounded maps in Sobolev spaces into complete manifolds, Ann. Mat. Pura Appl. (4), Volume 196 (2017), pp. 2261-2301

[6] H. Brezis; J.-M. Coron Large solutions for harmonic maps in two dimensions, Commun. Math. Phys., Volume 92 (1983), pp. 203-215

[7] H. Brezis; J.-M. Coron; E.H. Lieb Harmonic maps with defects, Commun. Math. Phys., Volume 107 (1986), pp. 649-705

[8] H. Brezis; Y. Li Topology and Sobolev spaces, J. Funct. Anal., Volume 183 (2001), pp. 321-369

[9] P. Hajłasz Approximation of Sobolev mappings, Nonlinear Anal., Volume 22 (1994), pp. 1579-1591

[10] P. Hajłasz; A. Schikorra Lipschitz homotopy and density of Lipschitz mappings in Sobolev spaces, Ann. Acad. Sci. Fenn., Math., Volume 39 (2014), pp. 593-604

[11] F. Hang Density problems for $W^{1, 1} (M, N)$ , Commun. Pure Appl. Math., Volume 55 (2002), pp. 937-947

[12] F. Hang; F. Lin Topology of Sobolev mappings. II, Acta Math., Volume 191 (2003), pp. 55-107

[13] M.R. Pakzad; T. Rivière Weak density of smooth maps for the Dirichlet energy between manifolds, Geom. Funct. Anal., Volume 13 (2003), pp. 223-257

[14] R. Schoen; K. Uhlenbeck Boundary regularity and the Dirichlet problem for harmonic maps, J. Differ. Geom., Volume 18 (1983), pp. 253-268

[15] B. White Homotopy classes in Sobolev spaces and the existence of energy minimizing maps, Acta Math., Volume 160 (1988), pp. 1-17

[16] W. Ziemer Weakly Differentiable Functions, Graduate Texts in Mathematics, Springer, New York, 1989

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