1516 CHAPTER 46. SOBOLEV SPACES ON MANIFOLDS

Ω∩Wi

Ui Ω

x̂k +φ i(x̂k)ekWi

Assume Ω is Cm−1,1. Define

hi (x̂k) = x̂k +φ i (x̂k)ek,Hi (x)≡ x̂k +(φ i (x̂k)+ xk)ek,

and let ψ i ∈C∞c (Wi) with ∑

li=0 ψ i (x) = 1 on Ω. Thus

{Wi,ψ i,∂Ω∩Wi,Ui,hi,Hi}li=1

satisfies all the conditions for defining W s,p (∂Ω) for s≤m. Let u∈C∞(Ω)

and let hi be asjust described. The trace, denoted by γ is that operator which evaluates functions in C∞

(Ω)

on ∂Ω. Thus for u ∈C∞(Ω), and y ∈ ∂Ω,

u(y) =l

∑i=1

(uψ i)(y)

and so using the notation to suppress the reference to y,

γu =l

∑i=1

γ (uψ i)

It is necessary to show this is a continuous map. Letting u ∈W m,p (Ω) , it follows fromTheorem 45.4.3, and Theorem 37.0.14,

||γu||m− 1p ,p,∂Ω

=l

∑i=1||h∗i (γ (ψ iu))||m− 1

p ,p,Ui

=l

∑i=1||h∗i γ (ψ iu)||m− 1

p ,p,Rn−1k≤C

l

∑i=1||H∗i (ψ iu)||m,p,Rn

+

≤Cl

∑i=1||H∗i (ψ iu)||m,p,Ui×(ai,0) ≤C

l

∑i=1||(ψ iu)||m,p,Wi∩Ω