Smooth Approximation of Sobolev Functions on Planar Domains

Abstract

We examine two related problems concerning a planar domain Cl. The first is whether Sobolev functions on Q can be approximated by global C 00 functions, and the second is whether approximation can be done by functions in C°°(n) which, together with all derivatives, are bounded on Q. 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.

Extension theorems for external cusps with minimal regularity

“…The definition of uniform domain used in [Smith et al 1994] (for the proof of this lemma) is slightly different than the one used here. For the equivalence between the two see [Väisälä 1988] and [Martio 1980].…”

“…We use the following result stated by Smith, Stanoyevitch and Stegenga: Lemma 8.3 [Smith et al 1994]. Let 1 and 2 be uniform domains with finite diameters.…”

A Density Problem for Sobolev Spaces on Planar Domains

“…Their interior segment property is weaker than the usual segment property that actually implies that the boundary is locally the graph of a continuous function. In [11] it was also inquired if the measure density together with lack of two-sided boundary points would suffice for the density of C ∞ (Ω), but C.J. Bishop [4] gave a counterexample to this statement.…”

“…He proved that C ∞ (R 2 ) is dense in W 1,p (Ω) for every 1 < p < ∞ when Ω is a Jordan domain: the bounded component of R 2 \ γ, where γ is a Jordan curve. Lewis A. Steganga showed in [11] that domains which satisfy their interior segment property allow approximation of functions in W 1,p (Ω) for 1 ≤ p < ∞ by bounded smooth functions with bounded derivatives and with global smooth functions if the boundary of Ω satisfies a suitable additional exterior density condition. Their interior segment property is weaker than the usual segment property that actually implies that the boundary is locally the graph of a continuous function.…”

A counterexample concerning smooth approximation

“…In their paper [2] Smith, Stanoyevitch and Stegenga prove several interesting theorems describing when C ∞ (Ω) is dense in W k,p (Ω) and give several examples where it is not dense. Based on their results they asked the following.…”