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

Abstract

We consider the problem of 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 $s=1$.

arXiv commentRevised version; accepted for publication at J. Eur. Math. Soc. (JEMS) Minor typo fixes and corrections