Density in $W^{s,p}(\mathrm{\Omega };N)$

Let $\mathrm{\Omega }$ be a smooth bounded domain in ${\mathbb{R}}^n$, $0\text{textless}s\text{textless}\mathrm{\infty }$ and $1\le p\text{textless}\mathrm{\infty }$. We prove that $C^{\mathrm{\infty }}(\bar{\mathrm{\Omega }};{\mathbb{S}}^1)$ is dense in $W^{s,p}(\Omega ; {\mathbb S}^1)$exceptwhen$1\le sp\textless{}2$and$n\ge 2$. The main ingredient is a new approximation method for $W^{s,p}$-maps when $s\text{textless}1$. With $0\text{textless}s\text{textless}1$, $1\le p\text{textless}\mathrm{\infty }$ and $sp\text{textless}n$, $\mathrm{\Omega }$ a ball, and $N$ a general compact connected manifold, we prove that $C^{\mathrm{\infty }}(\bar{\mathrm{\Omega }};N)$ is dense in $W^{s,p}(\mathrm{\Omega };N)$ if and only if $\pi \mathrm{\_}[sp](N)=0$. This supplements analogous results obtained by Bethuel when $s=1$, and by Bousquet, Ponce and Van Schaftingen when $s=2,3,\dots$ [General domains $\mathrm{\Omega }$ have been treated by Hang and Lin when $s=1$; our approach allows to extend their result to $s\text{textless}1$.] The case where $s\text{textgreater}1$, $s\in ̸\mathbb{N}$, is still open.