Chapter 46

Sobolev Spaces On Manifolds46.1 Basic Definitions

Consider the following situation. There exists a set, Γ ⊆ Rm where m > n, mappings,hi : Ui→ Γi = Γ∩Wi for Wi an open set in Rmwith Γ⊆ ∪l

i=1Wi and Ui is an open subset ofRn which hi one to one and onto. Assume hi is of the form

hi (x) = Hi (x,0) (46.1.1)

where for some open set, Oi, Hi : Ui×Oi→Wi is bilipschitz having bilipschitz inverse suchthat for G = Hi or H−1

i ,Dα G is Lipschitz for |α| ≤ k.For example, let m = n+1 and let

Hi (x,y) =(

xφ (x)+ y

)where φ is a Lipschitz function having Dα φ Lipschitz for all |α| ≤ k. This is an example ofthe sort of thing just described if x ∈Ui ⊆Rn and Oi =R, because it is obvious the inverseof Hi is given by

H−1i (x,y) =

(x

y−φ (x)

).

Also let {ψ i}li=1 be a partition of unity subordinate to the open cover {Wi} satisfying ψ i ∈

C∞c (Wi) . Then the definition of W s,p (Γ) follows.

Definition 46.1.1 u ∈ W s,p (Γ) if whenever {Wi,ψ i,Γi,Ui,hi,Hi}li=1 is described above

with hi ∈Ck,1, h∗i (uψ i) ∈W s,p (Ui) . The norm is given by

||u||s,p,Γ ≡l

∑i=1||h∗i (uψ i)||s,p,Ui

It is not at all obvious this norm is well defined. What if{W ′i ,φ i,Γi,Vi,gi,Gi

}ri=1

is as described above. Would the two norms be equivalent? To begin with consider thefollowing lemma which involves a particular choice for {Wi,ψ i,Γi,Ui,hi,Hi}l

i=1 .

Lemma 46.1.2 W s,p (Γ) as just described, is a Banach space. If p > 1 then it is reflexive.

Proof: Let L : W s,p (Γ)→∏li=1 W s,p (Ui) be defined by (Lu)i ≡ h∗i (uψ i) . Let

{u j}∞

j=1

be a Cauchy sequence in W s,p (Γ) . Then{

h∗i (u jψ i)}∞

j=1 is a Cauchy sequence in W s,p (Ui)

for each i. Therefore, for each i, there exists wi ∈W s,p (Ui) such that

limj→∞

h∗i (u jψ i) = wi in W s,p (Ui) .

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