Density

Abstract

We investigate here density questions. For real-valued Sobolev spaces, $C^\infty (\overline{\varOmega }; \mathbb R)$ is dense in $W^{s,p}(\varOmega ; \mathbb R)$, for any $s>0$ and $1\le p<\infty$. This need not be true for the Sobolev spaces $W^{s,p}(\varOmega ; {\mathscr {N}})$, where $\mathscr {N}$ is a manifold. In particular, this is not always the case when ${\mathscr {N}}={\mathbb S}^1$. We present the optimal conditions on s and p so that $C^{\infty }(\overline{\varOmega }; {\mathbb S}^1)$ is dense in $W^{s,p}(\varOmega ; {\mathbb S}^1)$.