Strong approximation of fractional Sobolev maps

Brezis and Mironescu have announced several years ago that for a compact manifold $N^n\subset {\mathbb{R}}^{\nu }$ and for real numbers $0<s<1$ and $1 \le p < \infty$theclass$C^\infty(\overline{Q}^m; N^n)$ of smooth maps on the cube with values into $N^n$ is dense with respect to the strong topology in the Sobolev space $W^{s,p}(Q^m;N^n)$ when the homotopy group $\pi_{\lfloor sp \rfloor}(N^n)$oforder$\lfloor sp \rfloor$ is trivial. The proof of this beautiful result is long and rather involved. Under the additional assumption that $N^n$ is $\lfloor sp\rfloor$ simply connected, we give a shorter proof of their result. Our proof for $sp\ge 1$ is based on the existence of a retraction of ${\mathbb{R}}^{\nu }$ onto $N^n$ except for a small subset in the complement of $N^n$ and on the Gagliardo-Nirenberg interpolation inequality for maps in $W^{1,q}\cap L^{\mathrm{\infty }}$. In contrast, the case $sp<1$ relies on the density of step functions on cubes in $W^{s,p}$.