In this entry, we continue to talk about the Concentration-Compactness Principle discovered by P.L. Lions [here]. In the previous entry, we already discussed two forms of non-compactness due to unbounded domains. Here we discuss what happens when passing to the limit on those functionals along weakly convergent subsequences.
Theorem (Lions). Let
be sequence in
weakly convergent to
and such that
converges weak* to a nonnegative measure
,
converges weak* to a nonnegative measure
.
Then there exists an at most countable index set
, sequence
,
,
,
, such that
and
and
where
is the best Sobolev constant and
are Dirac measures assigned to
. If
and
then
is a singleton and
for some
.
Apparently, the theorem does not provide any information about possible loss of mass at infinity of a weakly convergent minimizing sequence. We shall consider that case in the forthcoming topic.
See also:
- Concentration-Compactness principle, I.
- Concentration-Compactness Principle: The loss of mass at infinity in the subcritical case.
- Jan Chabrowski, Variational methods for potential operator equations, Walter de Gruyter, 1997.