1288 CHAPTER 37. BASIC THEORY OF SOBOLEV SPACES

Proof: For x ∈ ∂U, simply look at a single open set, Qx described in the above whichcontains x. Then consider an open set whose intersection with U is of the form

RT ({y :ŷ ∈ B,g(ŷ)− ε < yn < g(ŷ)})

and a vector of the form εRT (−en) where ε is chosen smaller than min{

g(x) : x ∈ B}−a.

There is nothing to prove for points of U.

One way to extend many of the above theorems to more general open sets than Rn isthrough the use of an appropriate extension theorem. In this section, a fairly general onewill be presented.

Lemma 37.2.3 Let B× (a,b) be as described in Definition 37.2.1 and let

V− ≡ {(ŷ,yn) : yn < g(ŷ)} , V+ ≡ {(ŷ,yn) : yn > g(ŷ)},

for g a Lipschitz function of the sort described in this definition. Suppose u+ and u− areLipschitz functions defined on V+ and V− respectively and suppose that u+ (ŷ,g(ŷ)) =u− (ŷ,g(ŷ)) for all ŷ ∈ B. Let

u(ŷ,yn)≡{

u+ (ŷ,yn) if (ŷ,yn) ∈V+

u− (ŷ,yn) if (ŷ,yn) ∈V−

and suppose spt(u)⊆ B× (a,b). Then extending u to be 0 off of B× (a,b), u is continuousand the weak partial derivatives, u,i, are all in L∞ (Rn)∩Lp (Rn) for all p > 1 and u,i =(u+),i on V+ and u,i = (u−),i on V−.

Proof: Consider the following picture which is descriptive of the situation.

a

b

spt(u)

B

Note first that u is Lipschitz continuous. To see this, consider |u(y1)−u(y2)| where(ŷi,yi

n)= yi. There are various cases to consider depending on whether yi

n is above g(ŷi) .

Suppose y1n < g(ŷ1) and y2

n > g(ŷ2) . Then letting K ≥max(Lip(u+) ,Lip(u−) ,Lip(g)) ,∣∣u(ŷ1,y1n)−u(ŷ2,y2

n)∣∣≤