A density problem for Sobolev spaces on Gromov hyperbolic domains

We prove that for a bounded domain Ω⊂Rn which is Gromov hyperbolic with respect to the quasihyperbolic metric, especially when Ω is a finitely connected planar domain, the Sobolev space W1,∞(Ω) is dense in W1,p(Ω) for any 1≤p<∞. Moreover if Ω is also Jordan or quasiconvex, then C∞(Rn) is dense in W1,p(Ω) for 1≤p<∞.