Â. Convergence of Averages
Before proceeding further it may be useful to pause and develop some
further insight into the meaning of weak convergence. So assume now 1
q oo and
(1.2) fk-f i n L\U).
Then if Å c U is a bounded measurable set, we deduce upon setting g = ÷
in (1.1) that
(1.3) /"/***- f fax.
This implies that the averages of the functions {fk}T_ over the set Å con-
verge to the average of / over Å. (Conversely, it is straightforward to check
oo Q
that if {fk} is bounded in L (U) and verifies (1.3) for each bounded,
measurable set Å c U, then (1.2) holds.) Á problem we will continually
confront in PDE applications is that this convergence of averages does not
imply norm or even a.e. convergence. It may very well be that the sequence
{fk} °°_ effects its weak convergence to / by virtue of perhaps unbounded,
very high frequency and quite irregulär oscillations. Such behavior utterly ex-
cludes any simple analysis of nonlinear functionals of the sequence {fk } °°_ .
In particular we note that (1.2) does not imply F(fk) » F(f) for any
nonlinear, real-valued function F. To see this, select real numbers á b
and 0 ë 1 so that
F{Xa + (1 - X)b) ö ÄF{a) + (1 - X)F{b).
Assuming U = (0, 1) c R, we set
/*(*) - {
Then fk -^ / = ëá + (1 - X)b in L , whereas
F(fk) -1 F = XF(a) + (1 - X)F(b) ø F(f).
C. Compactness in Sobolev Spaces
1. Embeddings. For later reference we record here the Gagliardo-Nirenberg-
Sobolev inequality, which asserts that if 1 q ç and q* = qn/(n - q) is
the Sobolev conjugate of q, then
(1-4) ||/H
V )
for all functions f e Cc (R ), the optimal constant Cq depending only on
q and ç . Invoking usual approximations, we see that this estimate is also
valid provided f £ L , Df e L .
á i f £ x ^ , V = 0 , . . . , / c - l
b otherwise.
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