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Smooth Approximation of Sobolev Functions on Planar Domains
Author(s) -
Smith Wayne,
Stanoyevitch Alexander,
Stegenga David A.
Publication year - 1994
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/49.2.309
Subject(s) - sobolev space , planar , bounded function , mathematics , domain (mathematical analysis) , pure mathematics , construct (python library) , mathematical analysis , computer science , computer graphics (images) , programming language
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.