# CONFLUENTES MATHEMATICI

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.

Given a compact manifold ${N}^{n}\subset {ℝ}^{\nu }$ and real numbers $s\ge 1$ and $1\le p<\infty$, we prove that the class ${C}^{\infty }\left({\overline{Q}}^{m};{N}^{n}\right)$ of smooth maps on the cube with values into ${N}^{n}$ is strongly dense in the fractional Sobolev space ${W}^{s,p}\left({Q}^{m};{N}^{n}\right)$ when ${N}^{n}$ is $⌊sp⌋$ simply connected. For $sp$ integer, we prove weak sequential density of ${C}^{\infty }\left({\overline{Q}}^{m};{N}^{n}\right)$ when ${N}^{n}$ is $sp-1$ simply connected. The proofs are based on the existence of a retraction of ${ℝ}^{\nu }$ onto ${N}^{n}$ except for a small subset of ${N}^{n}$ and on a pointwise estimate of fractional derivatives of composition of maps in ${W}^{s,p}\cap {W}^{1,sp}$.

Revised:
Accepted:
Published online:
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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