Density in $W^{s,p}(\Omega ; N)$
Résumé
Let $\Omega$ be a smooth bounded domain in ${\mathbb R}^n$, $s\in (0,\infty)$ and $p\in [1,\infty)$. We prove that $C^\infty(\overline\Omega ; {\mathbb S}^1)$ is dense in $W^{s,p}(\Omega ; {\mathbb S}^1)$ except when $sp\in [1,2)$ and $n\ge 2$. The main ingredient is a new approximation method for $W^{s,p}$-maps when $s\in (0,1)$. With $s\in (0,1)$, $p\in [1,\infty)$ and $sp\in (0,n)$, $\Omega$ a ball, and $N$ a general compact connected manifold, we prove that $C^\infty(\overline\Omega ; N)$ is dense in $W^{s,p}(\Omega ; N)$ if and only if $\pi_{[sp]}(N)=0$. This supplements analogous results obtained by Bethuel when $s=1$, and by Bousquet, Ponce and Van Schaftingen when $s=2,3,\ldots$ (General domains $\Omega$ have been treated by Hang and Lin when $s=1$; our approach allows to extend their result to $s\in (0,1)$.) The case where $s\in (1,\infty)$, $s\not\in{\mathbb N}$, is still open.
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