  We prove that for a bounded domain $\Omega\subset \mathbb R^n$ which is
Gromov hyperbolic with respect to the quasihyperbolic metric, especially when
$\Omega$ is a finitely connected planar domain, the Sobolev space
$W^{1,\,\infty}(\Omega)$ is dense in $W^{1,\,p}(\Omega)$ for any $1\le
p<\infty$. Moreover if $\Omega$ is also Jordan or quasiconvex, then
$C^{\infty}(\mathbb R^n)$ is dense in $W^{1,\,p}(\Omega)$ for $1\le p<\infty$.
