A complete answer to the strong density problem in Sobolev spaces with values in compact manifolds

Abstract

We consider the problem of the strong density of smooth maps in the Sobolev space $W^{s,p}(Q^m;\mathcal{N})$, where $0<s<+\infty$, $1\le p<+\infty$, $Q^m$ is the unit cube in ${\mathbb{R}}^m$, and $\mathcal{N}$ is a smooth compact connected Riemannian manifold without boundary. Our main result fully answers the strong density problem in the whole range $0<s<+\infty$: the space ${\mathcal{C}}^{\infty }({\overline{Q}}^m;\mathcal{N})$ is dense in $W^{s,p}(Q^m;\mathcal{N})$ if and only if ${\pi }_{[sp]}(\mathcal{N})=\{0\}$. This completes the results of Bethuel ($s=1$), Brezis and Mironescu ($0<s<1$), and Bousquet, Ponce, and Van Schaftingen ($s=2$, $3$, …). We also consider the case of more general domains $\Omega$, in the setting studied by Hang and Lin when $s=1$.