Density of smooth maps for fractional Sobolev spaces W s,p into simply connected manifolds when s1
Confluentes Mathematici, Volume 5 (2013) no. 2, pp. 3-24.

Given a compact manifold N n ν and real numbers s1 and 1p<, we prove that the class C (Q ¯ m ;N n ) of smooth maps on the cube with values into N n is strongly dense in the fractional Sobolev space W s,p (Q m ;N n ) when N n is sp simply connected. For sp integer, we prove weak sequential density of C (Q ¯ m ;N n ) when N n is sp-1 simply connected. The proofs are based on the existence of a retraction of ν onto N n except for a small subset of N n and on a pointwise estimate of fractional derivatives of composition of maps in W s,p W 1,sp .

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DOI: 10.5802/cml.5
Classification: 58D15, 46E35, 46T20
Keywords: Strong density; weak sequential density; Sobolev maps; fractional Sobolev spaces; simply connectedness
Pierre Bousquet 1; Augusto C. Ponce 2; Jean Van Schaftingen 2

1 Aix-Marseille Université, Laboratoire d’analyse, topologie, probabilités UMR7353, CMI 39, Rue Frédéric Joliot Curie, 13453 Marseille Cedex 13, 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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Pierre Bousquet; Augusto C. Ponce; Jean Van Schaftingen. Density of smooth maps for fractional Sobolev spaces $W^{s, p}$ into $\ell $ simply connected manifolds when $s \ge 1$. Confluentes Mathematici, Volume 5 (2013) no. 2, pp. 3-24. doi : 10.5802/cml.5. https://cml.centre-mersenne.org/articles/10.5802/cml.5/

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