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Strong density results for manifold valued fractional Sobolev maps
Domenico Mucci1
1 Università di Parma DSMFI Parco Area delle Scienze 53/A I-43124 Parma (Italy)
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 33 (2024) no. 3, pp. 581-610.

We consider fractional Sobolev classes W s,p of maps defined in high dimensional domains and with values into compact smooth manifolds. The problem of strong density of smooth maps for s lower than one is discussed. An equivalent energy convergence defined through extensions in suitable weighted Sobolev spaces is exploited to obtain a new proof of the density of maps with “small” singular set. Moreover, a homotopy-type property is analyzed, yielding to a characterization of approximable maps through topological arguments. We then focus on maps taking values into high dimensional spheres, where homological tools allow to describe the singular set. For suitable values of the product sp, in fact, strong density of smooth maps is characterized by the triviality of the current of the singularities.

On considère des classes de Sobolev fractionnaires W s,p d’applications définies dans des domaines de grande dimension et à valeurs dans des variétés régulières compactes. Le problème de la forte densité des applications régulières pour s inférieur à un est discuté. Une convergence d’énergie équivalente définie grâce à des extensions dans des espaces de Sobolev à poids appropriés est exploitée pour obtenir une nouvelle preuve de la densité des applications avec un « petit » ensemble singulier. De plus, une propriété de type homotopie est analysée, aboutissant à une caractérisation des applications approximables par des arguments topologiques. Nous nous concentrons ensuite sur des applications prenant des valeurs dans des sphères de grande dimension, où les outils homologiques permettent de décrire l’ensemble singulier. Pour des valeurs appropriées du produit sp, la forte densité des applications régulières se caractérise en effet par la trivialité du courant des singularités.

Received:
Accepted:
Published online:
DOI: 10.5802/afst.1782
Classification: 49Q20, 46E35, 28A75, 58D15
Keywords: fractional Sobolev spaces, weighted Sobolev spaces, maps between manifolds, singularities
Mot clés : espaces de Sobolev fractionnaires, espaces de Sobolev a poids, applications entre variétés, singularités

Domenico Mucci 1

1 Università di Parma DSMFI Parco Area delle Scienze 53/A I-43124 Parma (Italy)
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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     title = {Strong density results for manifold valued  fractional {Sobolev} maps},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {581--610},
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Domenico Mucci. Strong density results for manifold valued  fractional Sobolev maps. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 33 (2024) no. 3, pp. 581-610. doi : 10.5802/afst.1782. https://afst.centre-mersenne.org/articles/10.5802/afst.1782/

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