Very loosely speaking, noncompactness of a Sobolev embedding on a domain
can be caused by:
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a) some bad behavior of the boundary of the domain;
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b) the norm of the domain or target function space is too weak or too
strong, respectively;
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c) the domain is too big (let's say, of infinite measure).
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Unlike in the other cases, surprisingly little attention has been paid
to quantitative questions as to the lack of compactness (e.g., "how much
is my embedding noncompact?") when the lack is caused by too big a domain.
<br>
In my recent joint paper with D.E. Edmunds and J. Lang, we studied such
questions for the Sobolev embedding W^{1,p}_0(S) \to L^p(S), in which
p\in(1,\infty) and S is an infinite strip in R^n, answering an open
question raised by D.E.E. and W.D. Evans almost four decades ago. The
old question and some related ones (and answers to them) are what my
talk is going to be about.