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Journal of the London Mathematical Society 1994 49(2):309-330; doi:10.1112/jlms/49.2.309
© 1994 by London Mathematical Society
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© The London Mathematical Society

Smooth Approximation of Sobolev Functions on Planar Domains

Wayne Smith, Alexander Stanoyevitch and David A. Stegenga

Department of Mathematics, University of Hawaii Honolulu, Hawaii 96822, USA

Received 19 March 1992. Revision received 27 July 1992.

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

E-mail: waynes{at}math.hawaii.edu

alex{at}math.hawaii.edu

davids{at}math.hawaii.edu

The first author is supported in part by a grant from the National Science Foundation.


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