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 $1\le p<\infty$.

22 pages, 6 figures