41.2. THE CASE OF BOUNDED OPEN SETS 1359

RW

W

Ω⋂

W RW (Ω⋂

W )

0

b

u′ ∈U ′

For the situation described in the above definition, let hW : U ′→ Γ∩W be defined by

hW(u′)≡ R−1

W(u′,φW

(u′))

, gW (x)≡ (RW x)′ , HW (u)≡ R−1W(u′,φW

(u′)−un

).

where x′ ≡ (x1, · · · ,xn−1) for x = (x1, · · · ,xn). Thus gW ◦hW = id on U ′ and hW ◦gW = idon Γ∩W. Also note that HW is defined on all of Rn is Cm,1, and has an inverse with thesame properties. To see this, let GW (u) = (u′,φW (u′)−un) . Then HW = R−1

W ◦GW andG−1

W =(u′,φW (u′)−un) and so H−1W =G−1

W ◦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 ,GWi = Gi etc. Now let

i : GiRi (Ω∩W )≡Vi→Ω∩Wi

be defined by

i (y)≡ R−1i ◦G−1

i (y) .

Thus i,−1i ∈Cm,1 (Rn). The following picture might be helpful.