Trace theory for Sobolev mappings into a manifold

Petru Mironescu; Jean Van Schaftingen

doi:10.5802/afst.1675

Petru Mironescu ¹; Jean Van Schaftingen ²

¹ Université Lyon 1, CNRS UMR 5208 Institut Camille Jordan, 43 Boulevard du 11 Novembre 1918, 69622 Villeurbanne Cedex, France, and, Simion Stoilow Institute of Mathematics of the Romanian Academy, Calea Griviţei 21, 010702 Bucureşti, România
² Université catholique de Louvain, Institut de Recherche en Mathématique et Physique, Chemin du Cyclotron 2 bte L7.01.01, 1348 Louvain-la-Neuve, Belgium

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 30 (2021) no. 2, pp. 281-299.

Abstract
Abstract

We review the current state of the art concerning the characterization of traces of the spaces $W^{1, p} (𝔹^{m - 1} \times (0, 1), 𝒩)$ of Sobolev mappings with values into a compact manifold $𝒩$ . In particular, we exhibit a new analytical obstruction to the extension, which occurs when $p < m$ is an integer and the homotopy group $π_{p} (𝒩)$ is non trivial. On the positive side, we prove the surjectivity of the trace operator when the fundamental group $π_{1} (𝒩)$ is finite and $π_{2} (𝒩) ≃ \dots ≃ π_{⌊ p - 1 ⌋} (𝒩) ≃ {0}$ . We present several open problems connected to the extension problem.

Nous examinons l’état de l’art de la caractérisation des traces des espaces $W^{1, p} (𝔹^{m - 1} \times (0, 1), 𝒩)$ d’applications Sobolev à valeurs valeurs dans une variété compacte $𝒩$ . En particulier, nous mettons en évidence une nouvelle obstruction analytique à l’extension, qui se produit lorsque $p < m$ est un entier et que le groupe d’homotopie $π_{p} (𝒩)$ n’est pas trivial. Du côté positif, nous démontrons la surjectivité de l’opérateur de trace lorsque le groupe fondamental $π_{1} (𝒩)$ est fini et que $π_{2} (𝒩) ≃ \dots ≃ π_{⌊ p - 1 ⌋} (𝒩) ≃ {0}$ . Nous présentons plusieurs problèmes ouverts liés au problème d’extension.

Published online: 2021-07-09

DOI: 10.5802/afst.1675

Classification: 46T10, 46E35, 58D15
Keywords: Trace spaces, fractional Sobolev spaces, homotopy groups, lifting of Sobolev mappings.

Author's affiliations:

Petru Mironescu ¹; Jean Van Schaftingen ²

¹ Université Lyon 1, CNRS UMR 5208 Institut Camille Jordan, 43 Boulevard du 11 Novembre 1918, 69622 Villeurbanne Cedex, France, and, Simion Stoilow Institute of Mathematics of the Romanian Academy, Calea Griviţei 21, 010702 Bucureşti, România
² Université catholique de Louvain, Institut de Recherche en Mathématique et Physique, Chemin du Cyclotron 2 bte L7.01.01, 1348 Louvain-la-Neuve, Belgium

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Petru Mironescu and Jean Van Schaftingen},
     title = {Trace theory for {Sobolev} mappings into a manifold},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {281--299},
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Petru Mironescu; Jean Van Schaftingen. Trace theory for Sobolev mappings into a manifold. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 30 (2021) no. 2, pp. 281-299. doi : 10.5802/afst.1675. https://afst.centre-mersenne.org/articles/10.5802/afst.1675/

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