WEAK CONVERGENCE 7
If U denotes as usual a bounded smooth open subset of R , it follows
from (1.4) and Standard extension theorems that
(1.5) u\\LplV) c\\nwl.,(ü)
for each 1 ñ q* and / e
Wl,q(U),
the constant C depending only on
q , ç , and U.
2. Compactness theorems. Our next assertion sharpens the preceding Ob-
servation, demonstrating that the embedding W
,q(U)
c L (U) is in fact
compact if 1 q n , 1 p q*:
THEOREM
5
(RELLICH'S COMPACTNESS THEOREM).
Assume that the se-
quence {fk}™ is bounded in
Wl,q(U).
Then {fk}°?_ is precompact in
L (U) for each 1 ñ q*.
Á proof of this assertion is available in Gilbarg-Trudinger [68].
We will later devote considerable effort to understanding just how com-
pactness fails for the critical case ñ = q*: see §D below and Chapter 4,
§A.
THEOREM
6
(COMPACTNESS FOR MEASURES).
Assume thesequence {ßk}T_
is bounded in Jt(U). Then {ßk}°° is precompact in W~
,q{U)
for each
1 ú 1 *.
oo
PROOF.
1. Using Theorem 4 we may extract a subsequence {/^ } . c
CO
{ì^ } so that ßk * ì in J?{U), for some measure ì .
2. Set q =q/(q-\) and denote by  the closed unit ball in
W^q
(U).
Since 1 q 1*, we have q n\ and so  is compact in CQ(U). Thus
Í(å)
given å 0 , there exist functions {/,} _ c CQ(U) so that
min I I ö - ö. I I ^ _ å
liN(e)
l
C(U)
for each ö e Â.
3. Thusif 0 €# ,
\ öÜì. - öÜì\ sup , |(ß/) + / 0. Üì, - É ö.Üì
\Ju
J
Ju É j
j
\Ju
J
Ju
for some index 1 / Í (å). Consequently,
' = 0 ,
lim sup
and so ì
ßò
- ì in
W~Uq{U)
/ öÜì]^- \ öÜì
Ju
j
Ju
Previous Page Next Page