38.6. SOBOLEV SPACES ON MANIFOLDS 1323

It only remains to show γ is continuous. Let u ∈ Ht (U) . Thus there exists v ∈ Ht (Rn)such that u = v|U . Then

||γu||Ht−1/2(U ′) ≤ ||γv||Ht−1/2(Rn−1) ≤C ||v||Ht (Rn)

for C independent of v. Then taking the inf for all such v ∈ Ht (Rn) which are equal to ua.e. on U, it follows

||γu||Ht−1/2(U ′) ≤C ||v||Ht (Rn)

and this proves γ is continuous.

38.6 Sobolev Spaces On Manifolds38.6.1 General TheoryThe type of manifold, Γ for which Sobolev spaces will be defined on is:

Definition 38.6.1 1. Γ is a closed subset of Rp where p≥ n.

2. Γ = ∪∞i=1Γi where Γi = Γ∩Wi for Wi a bounded open set.

3. {Wi}∞

i=1 is locally finite.

4. There are open bounded sets, Ui and functions hi : Ui → Γi which are one to one,onto, and in Cm,1 (Ui) . There exists a constant, C, such that C ≥ Liphr for all r.

5. There exist functions, gi : Wi →Ui such that gi is Cm,1 (Wi) , and gi ◦hi = id on Uiwhile hi ◦gi = id on Γi.

This will be referred to as a Cm,1 manifold.

Lemma 38.6.2 Let gi, hi,Ui,Wi, and Γi be as defined above. Then

gi ◦hk : Uk ∩h−1k (Γi)→Ui∩h−1

i (Γk)

is Cm,1. Furthermore, the inverse of this map is gk ◦hi.

Proof: First it is well to show it does indeed map the given open sets. Let x ∈Uk ∩h−1

k (Γi) . Then hk (x) ∈ Γk ∩ Γi and so gi (hk (x)) ∈ Ui because hk (x) ∈ Γi. Now sincehk (x) ∈ Γk, gi (hk (x)) ∈ h−1

i (Γk) also and this proves the mappings do what they shouldin terms of mapping the two open sets. That gi ◦hk is Cm,1 follows immediately from thechain rule and the assumptions that the functions gi and hk are Cm,1. The claim about theinverse follows immediately from the definitions of the functions.

Let {ψ i}∞

i=1 be a partition of unity subordinate to the open cover {Wi} satisfying ψ i ∈C∞

c (Wi) . Then the following definition provides a norm for Hm+s (Γ) .

Definition 38.6.3 Let s ∈ (0,1) and m is a nonnegative integer. Also let µ denote thesurface measure for Γ defined in the last section. 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

< ∞.