Vol. 32, No. 1, 1970

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 306: 1
Vol. 305: 1  2
Vol. 304: 1  2
Vol. 303: 1  2
Vol. 302: 1  2
Vol. 301: 1  2
Vol. 300: 1  2
Vol. 299: 1  2
Online Archive
The Journal
Editorial Board
Special Issues
Submission Guidelines
Submission Form
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Author Index
To Appear
Other MSP Journals
Compact Sobolev imbeddings for unbounded domains

Robert Alexander Adams

Vol. 32 (1970), No. 1, 1–7

A condition on an open set G En which is both necessary and sufficient for the compactness of the (Sobolev) imbedding H0ru+1(G) H0m(G) is not yet known. C. Clark has given a necessary condition (quasiboundedness) and a much stronger sufficient condition. We show here that (unless n = 1) quasiboundedness is not sufficient, and answer in the negative a question raised by Clark on whether the imbedding can be compact if ∂G consists of isolated points. We also substantially weaken Clark’s sufficient condition so as to include a wide class of domains with null exterior. The gap between necessary and sufficient conditions is thus considerably narrowed.

Mathematical Subject Classification
Primary: 46.38
Received: 8 January 1968
Published: 1 January 1970
Robert Alexander Adams