A density result for homogeneous Sobolev spaces on planar domains

We show that in a bounded simply connected planar domain $\mathrm{\Omega }$ the smooth Sobolev functions $W^{k,\mathrm{\infty }}(\mathrm{\Omega })\cap C^{\mathrm{\infty }}(\mathrm{\Omega })$ are dense in the homogeneous Sobolev spaces $L^{k,p}(\mathrm{\Omega })$.