1328 CHAPTER 38. SOBOLEV SPACES BASED ON L2

same properties. To see this, let GW (u) = (u′,φW (u′)−un) . Then HW = R−1W ◦GW and

G−1W =(u′,φW (u′)−un) and so H−1

W =G−1W ◦RW . Note also that as indicated in the picture,

RW (W ∩Ω) ={

u ∈ Rn : u′ ∈U ′ and 0 < un < φW(u′)}

.

Since Γ = ∂Ω is compact, there exist finitely many of these open sets, W, denoted by{Wi}q

i=1 such that Γ⊆ ∪qi=1Wi. Let the corresponding sets, U ′ be denoted by U ′i and let the

functions, φ be denoted by φ i. Also let hi = hWi etc. Now let {ψ i}qi=1 be a C∞ partition of

unity subordinate to the {Wi}qi=1. If u ∈ Ht (Ω) , then by Corollary 38.3.5 on Page 1314 it

follows that uψ i ∈ Ht (Wi∩Ω) . Now

Hi : Ui ≡ {u ∈ Rn : u′ ∈U ′i , 0 < un < φ i(u′)}→Wi∩Ω

and Hi and its inverse are defined on Rn and are in Cm,1 (Rn) . Therefore, by Lemma 38.3.3on Page 1312,

H∗i ∈L(Ht (Wi∩Ω) ,Ht (Ui)

).

Provide t = m+ s where s > 0.Now it is possible to define the trace on Γ≡ ∂Ω. For u ∈ Ht (Ω) ,

γu≡q

∑i=1

g∗i (γH∗i (uψ i)) . (38.6.25)

I must show it satisfies what it should. Recall the definition of what it means for a functionto be in Ht−1/2 (Γ) where t = m+ s.

Definition 38.6.8 Let s ∈ (0,1) and m is a nonnegative integer. Also let µ denote thesurface measure for Γ. A µ measurable function, u is in Hm+s (Γ) if whenever

{Wi,ψ i,Γi,Ui,hi,gi}∞

i=1

is described above, h∗i (uψ i) ∈ Hm+s (Ui) and

||u||Hm+s(Γ) ≡

(∞

∑i=1||h∗i (uψ i)||

2Hm+s(Ui)

)1/2

< ∞.

Recall that all these norms which are obtained from various partitions of unity andfunctions, hi and gi, are equivalent. Here there are only finitely many Wi so the sum is afinite sum. The theorem is the following.

Theorem 38.6.9 Let Ω be a bounded open set having Cm,1 boundary as discussed abovein Definition 38.6.7. Then for t ≤ m+1, there exists a unique

γ ∈L(

Ht (Ω) ,Ht−1/2 (Γ))

which has the property that for µ the measure on the boundary,

γu(x) = u(x) for µ a.e. x ∈ Γwhenever u ∈S|Ω. (38.6.26)