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 Â the closed unit ball in

W^q

(U).

Since 1 q 1*, we have q n\ and so Â 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\ 2å sup |ì , |(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