Density of smooth maps for fractional Sobolev spaces $W^{s, p}$ into $\ell $ simply connected manifolds when $s \ge 1$

Bousquet, Pierre; Ponce, Augusto C.; Van Schaftingen, Jean

Density of smooth maps for fractional Sobolev spaces

W^{s, p}

into

ℓ

simply connected manifolds when

s \geq 1

Bousquet, Pierre ; Ponce, Augusto C. ; Van Schaftingen, Jean

Confluentes Mathematici, Tome 5 (2013), p. 3-22 / Harvested from Numdam

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Résumé

Given a compact manifold $N^{n} \subset ℝ^{ν}$ and real numbers $s \geq 1$ and $1 \leq p < \infty$ , we prove that the class $C^{\infty} ({\bar{Q}}^{m}; N^{n})$ 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 $⌊ s p ⌋$ simply connected. For $s p$ integer, we prove weak sequential density of $C^{\infty} ({\bar{Q}}^{m}; N^{n})$ when $N^{n}$ is $s p - 1$ simply connected. The proofs are based on the existence of a retraction of $ℝ^{ν}$ onto $N^{n}$ except for a small subset of $N^{n}$ and on a pointwise estimate of fractional derivatives of composition of maps in $W^{s, p} \cap W^{1, s p}$ .

Publié le : 2013-01-01

MR 3145030

DOI : https://doi.org/10.5802/cml.5
Classification: 58D15, 46E35, 46T20

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     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},
     volume = {5},
     year = {2013},
     pages = {3-22},
     doi = {10.5802/cml.5},
     mrnumber = {3145030},
     language = {en},
     url = {http://dml.mathdoc.fr/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, Tome 5 (2013) pp. 3-22. doi : 10.5802/cml.5. http://gdmltest.u-ga.fr/item/CML_2013__5_2_3_0/

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