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, p. 3-22

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 ${W}^{s,p}\cap {W}^{1,sp}$.

DOI : https://doi.org/10.5802/cml.5
Classification:  58D15,  46E35,  46T20
Keywords: Strong density; weak sequential density; Sobolev maps; fractional Sobolev spaces; simply connectedness
@article{CML_2013__5_2_3_0,
author = {Bousquet, Pierre and Ponce, Augusto C. and Van Schaftingen, Jean},
title = {Density of smooth maps for fractional Sobolev spaces $W^{s, p}$ into $\ell$ simply connected manifolds when $s \ge 1$},
journal = {Confluentes Mathematici},
publisher = {Institut Camille Jordan},
volume = {5},
number = {2},
year = {2013},
pages = {3-22},
doi = {10.5802/cml.5},
mrnumber = {3145030},
language = {en},
url = {http://www.numdam.org/item/CML_2013__5_2_3_0}
}

Bousquet, Pierre; Ponce, Augusto C.; Van Schaftingen, Jean. 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. 3-22. doi : 10.5802/cml.5. http://www.numdam.org/item/CML_2013__5_2_3_0/

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