1286 CHAPTER 37. BASIC THEORY OF SOBOLEV SPACES

C(||φu(·+v)−φu(·)||Lp(Gv)

,mn (Gv))(∫

Gv|φu(x+v)−φu(x)|dx

)θq

≤C(

2 ||φu(·)||Lp(U) ,mn (U))(∫

Gv|φu(x+v)−φu(x)|dx

)θq

≤ C (φ ,M,mn (U))

(∫Gv|φu(x+v)−φu(x)|dx

)θq

= C (φ ,M,mn (U))

(∫Rn

∣∣∣φ̃u(x+v)− φ̃u(x)∣∣∣dx)θq

. (37.1.23)

Now by Lemma 37.1.15,∫Rn

∣∣∣φ̃u(x+v)− φ̃u(x)∣∣∣dx≤C

(φ , ||u||1,1,U

)|v| (37.1.24)

and so from 37.1.23 and 37.1.24, and adjusting the constants∫Rn

∣∣∣φ̃u(x+v)− φ̃u(x)∣∣∣q dx ≤ C (φ ,M,mn (U))

(C(

φ , ||u||1,1,U)|v|)θq

= C (φ ,M,mn (U)) |v|θq

which verifies 37.1.21 whenever |v| is sufficiently small. This proves the lemma becausethe conditions of Theorem 37.1.14 are satisfied.

Theorem 37.1.17 Let U be a bounded open set and define for p > 1

S≡{

u ∈W 1,1 (U)∩Lp (U) : ||u||1,1,U + ||u||Lp(U) ≤M}

(37.1.25)

Then S is precompact in Lq (U) where 1≤ q < p.

Proof: If suffices to show that every sequence, {uk}∞

k=1 ⊆ S has a subsequence whichconverges in Lq (U) . Let {Km}∞

m=1 denote a sequence of compact subsets of U with theproperty that Km ⊆ Km+1 for all m and ∪∞

m=1Km = U. Now let φ m ∈ C∞c (U) such that

φ m (x) ∈ [0,1] and φ m (x) = 1 for all x ∈ Km. Let Sm ≡ {φ mu : u ∈ S}. By Lemma 37.1.16there exists a subsequence of {uk}∞

k=1 , denoted here by{

u1,k}∞

k=1 such that{

φ 1u1,k}∞

k=1converges in Lq (U) . Now S2 is also precompact in Lq (U) and so there exists a subse-quence of

{u1,k}∞

k=1 , denoted by{

u2,k}∞

k=1 such that{

φ 2u2,k}∞

k=1 converges in L2 (U) .

Thus it is also the case that{

φ 1u2,k}∞

k=1 converges in Lq (U) . Continue taking subse-quences in this manner such that for all l ≤ m,

{φ lum,k

}∞

k=1 converges in Lq (U). Let{wm}∞

m=1 = {um,m}∞

m=1 so that {wk}∞

k=m is a subsequence of{

um,k}∞

k=1 . Then it followsfor all k, {φ kwm}∞

m=1 must converge in Lq (U) . For u ∈ S,

||u−φ ku||qLq(U)=

∫U|u|q (1−φ k)

q dx

≤(∫

U|u|p dx

)q/p(∫U(1−φ k)

qr dx)1/r

≤ M(∫

U(1−φ k)

qr dx)1/r