Smooth Approximation of Sobolev Functions on Planar Domains

Abstract

We examine two related problems concerning a planar domain Ω. The first is whether Sobolev functions on Ω can be approximated by global C∞ functions, and the second is whether approximation can be done by functions in C∞(Ω) which, together with all derivatives, are bounded on Ω. We find necessary and sufficient conditions for certain types of domains, such as starshaped domains, and we construct several examples which show that the general problem is quite difficult, even in the simply connected case.