6 WEAK CONVERGENCE

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.

JE JE

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

L

,

V )

Cq\\Df\\Lq^X)

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.