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

Confluentes Mathematici, Tome 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 $C^{\infty }({\overline{Q}}^m;N^n)$ 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 $\lfloor sp\rfloor$ simply connected. For $sp$ integer, we prove weak sequential density of $C^{\infty }({\overline{Q}}^m;N^n)$ 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 $W^{s,p}\cap W^{1,sp}$.

Classification : 58D15, 46E35, 46T20
Mots-clés : Strong density; weak sequential density; Sobolev maps; fractional Sobolev spaces; simply connectedness

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