1364 CHAPTER 41. ELLIPTIC REGULARITY

Lemma 41.2.5 Let W be one of the sets described in Definition 41.2.1 and let m ≥ k. LetW1 ⊆W1 ⊆W where W1 is an open set. Suppose also that

u ∈ Hk (Ω) ,

αrs ∈ Ck−1,1 (

Ω),

f ∈ Hk−1 (Ω) ,

hs ∈ Hk (Ω) ,

and that for all v ∈ H1 (Ω∩W ) such that spt(v)⊆Ω∩W,∫Ω

ai j (x)u,i (x)v, j (x)dx+∫

Ω

hs (x)v,s (x)dx =∫

Ω

f (x)v(x)dx. (41.2.40)

Then there exists a constant, C, independent of f ,u, and g such that

||u||2Hk+1(Ω∩W1)≤C

(|| f ||2Hk−1(Ω)+ ||u||

2Hk(Ω)+∑

s||hs||2Hk(Ω)

). (41.2.41)

Proof: LetE ≡

{v ∈ Hk (Ω∩W ) : spt(v)⊆W

}u restricted to W ∩Ω is in Hk (Ω∩W ) and∫

Ω∩Wai j (x)u,iv, jdx+

∫Ω

hs (x)v,s (x)dx

=∫

Ω

f (x)v(x)dx for all v ∈ E. (41.2.42)

Now let i (y) = x. For this particular W, denote Φi more simply by Φ, Ui ≡Φi (Ω∩Wi)by U, and Vi by V. Denoting the coordinates of V by y, and letting u(x)≡ w(y) and v(x)≡z(y) , it follows that in terms of the new coordinates, 41.2.35 takes the form∫

Uai j (Φ(y))

∂w∂yr

∂yr

∂xi∂ z∂ys

∂ys

∂x j |detDΦ(y)|dy

+∫

Uhk (Φ(y))

∂ z∂yl

∂yl

∂xk |detDΦ(y)|dx

=∫

Uf (Φ(y))z(y) |detDΦ(y)|dy

Let

αrs (y)≡ ai j (Φ(y))

∂yr

∂xi∂ys

∂x j |detDΦ(y)| , (41.2.43)

h̃l (y)≡ hk (Φ(y))∂yl

∂xk |detDΦ(y)| , (41.2.44)

andf̃ (y)≡Φ

∗ f |detDΦ|(y)≡ f (Φ(y)) |detDΦ(y)| . (41.2.45)