SMOOTH APPROXIMATION OF SOBOLEV FUNCTIONS ON PLANAR DOMAINS

We examine two related problems concerning a planar domain OMEGA. The first is whether Sobolev functions on OMEGA can be approximated by glo bal C(infinity) functions, and the second is whether approximation can be done by functions in C(infinity)(OMEGA) which, together with all d erivatives, are bounded on OMEGA. We find necessary and sufficient con ditions for certain types of domains, such as starshaped domains, and we construct several examples which show that the general problem is q uite difficult, even in the simply connected case.