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.
