A density problem for Sobolev spaces on Gromov hyperbolic domains

Abstract: 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 $1\le p<\infty$.