A density problem for Sobolev spaces on Gromov hyperbolic domains

Abstract

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 < ∞.