Density of smooth maps for fractional Sobolev spaces W s,p into simply connected manifolds when s1
Confluentes Mathematici, Volume 5 (2013) no. 2, p. 3-22

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 .

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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