1268 CHAPTER 37. BASIC THEORY OF SOBOLEV SPACES

{ηk}∞

k=1 such that each spt(ηk) ∈Cc (Vk) . Let ψm denote the sum of all the ηk which arenon zero at some point of Vm. Thus

spt(ψm)⊆Um+2 \Um−2,ψm ∈C∞c (U) ,

∞

∑m=1

ψm (x) = 1 (37.0.3)

for all x ∈U, and ψmu ∈W m,p (Um+2) .

Now let φ l be a mollifier and consider

J ≡∞

∑m=0

uψm ∗φ lm (37.0.4)

where lm is chosen large enough that the following two conditions hold:

spt(uψm ∗φ lm

)⊆Um+3 \Um−3, (37.0.5)

∣∣∣∣(uψm)∗φ lm −uψm

∣∣∣∣m,p,Um+3

=∣∣∣∣(uψm)∗φ lm −uψm

∣∣∣∣m,p,U <

δ

2m+5 , (37.0.6)

where 37.0.6 is obtained from Theorem 37.0.5. Because of 37.0.3 only finitely many termsof the series in 37.0.4 are nonzero and therefore, J ∈C∞ (U) . Now let N > 10, some largevalue.

||J−u||m,p,UN−3=

∣∣∣∣∣∣∣∣∣∣ N

∑k=0

(uψk ∗φ lk −uψk

)∣∣∣∣∣∣∣∣∣∣m,p,UN−3

≤N

∑k=0

∣∣∣∣∣∣uψk ∗φ lk −uψk

∣∣∣∣∣∣m,p,UN−3

≤N

∑k=0

δ

2m+5 < δ .

Now apply the monotone convergence theorem to conclude that ||J−u||m,p,U ≤ δ . Thisproves the theorem.

Note that J = 0 on ∂U. Later on, you will see that this is pathological.In the study of partial differential equations it is the space W m,p (U) which is of the

most use, not the space Xm,p (U) . This is because of the density of C∞(U). Nevertheless,

for reasonable open sets, U, the two spaces coincide.

Definition 37.0.9 An open set, U ⊆ Rn is said to satisfy the segment condition if for allz ∈U , there exists an open set Uz containing z and a vector a such that

U ∩Uz + ta⊆U

for all t ∈ (0,1) .