41.2. THE CASE OF BOUNDED OPEN SETS 1363

Then D1,W2, · · · ,Wl cover ∂Ω. Let C2 = ∂Ω\(D1∪

(∪l

i=3Wi))

. Then C2 is a closed subsetof W2. Choose an open set, D2 such that

C2 ⊆ D2 ⊆ D2 ⊆W2.

Thus D1,D2,W3 · · · ,Wl covers ∂Ω. Continue in this way to get Di ⊆Wi, and ∂Ω⊆∪li=1Di,

and Di is an open set. Now letD0 ≡Ω\∪l

i=1Di.

Also, let Di ⊆ Vi ⊆ Vi ⊆Wi. Therefore, D0,V1, · · · ,Vl covers Ω. Then the same estimationprocess used above yields

||u||H2(D0)≤C

(|| f ||2L2(Ω)+ ||u||

2H1(Ω)+∑

k||hk||2H1(Ω)

).

From Lemma 41.2.2

||u||H2(Vi∩Ω) ≤C

(|| f ||2L2(Ω)+ ||u||

2H1(Ω)+∑

k||hk||2H1(Ω)

)

also. This proves the theorem since

||u||H2(Ω) ≤l

∑i=1||u||H2(Vi∩Ω)+ ||u||H2(D0)

.

What about the Dirichlet problem? The same differencing procedure as above yieldsthe following.

Theorem 41.2.4 Let Ω be a bounded open set with C1,1 boundarybrownianmotiontheoremas in Definition 41.2.1, let f ∈ L2 (Ω) ,hk ∈ H1 (Ω), and suppose that for all x ∈Ω,

ai j (x)viv j ≥ δ |v|2 .

Suppose also that u ∈ H10 (Ω) and∫

Ω

ai j (x)u,i (x)v, j (x)dx+∫

Ω

hk (x)v,k (x)dx =∫

Ω

f (x)v(x)dx

for all v ∈ H10 (Ω) . Then u ∈ H2 (Ω) and for some C independent of f ,g, and u,

||u||2H2(Ω) ≤C

(|| f ||2L2(Ω)+ ||u||

2H1(Ω)+∑

k||hk||2H1(Ω)

).

What about higher regularity?