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

Abstract

Given a compact manifold $N^n subset mathbb{R}^u$ 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 $lfloor sp floor$ 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 $mathbb{^ u$ onto $N^n$ except for a small subset of $N^n$ and on a pointwise estimate of fractional derivatives of composition of maps in $W^{s, p} cap W^{1, sp}$.