Vol. 32, No. 1, 1970

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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