1358 CHAPTER 41. ELLIPTIC REGULARITY

As noted earlier, the condition, 41.1.2 implies αnn (y)≥ δ > 0 and so

D2nw = − 1

αnn

(∂αrs

∂ys∂w∂yr + ∑

r≤n−1∑

s≤n−1α

rs ∂ 2w∂ys∂yr +

∑s

αns ∂ 2w

∂ys∂yn +∑r

αrn ∂ 2w

∂yn∂yr +∂hs

∂ys + f).

It follows from Dα = Dβ D2n that

Dα w = Dβ

[− 1

αnn

(∂αrs

∂ys∂w∂yr + ∑

r≤n−1∑

s≤n−1α

rs ∂ 2w∂ys∂yr +

∑s

αns ∂ 2w

∂ys∂yn +∑r

αrn ∂ 2w

∂yn∂yr +∂hs

∂ys + f)]

.

Now you note that terms like Dβ

(∂ 2w

∂ys∂yn

)have αn = j− 1 and so, from the induction

hypothesis along with the assumptions on the given functions,

||Dα w||L2(U1)≤C

(|| f ||Hk−1(U)+ ||w||Hk(U)+∑

s||hs||Hk(U)

).

This proves the corollary.

41.2 The Case Of Bounded Open SetsThe main interest in all this is in the application to bounded open sets. Recall the followingdefinition.

Definition 41.2.1 A bounded open subset, Ω, of Rn has a Cm,1boundary if it satisfies thefollowing conditions. For each p∈ Γ≡Ω\Ω, there exists an open set, W , containing p, anopen interval (0,b), a bounded open box U ′ ⊆Rn−1, and an affine orthogonal transforma-tion, RW consisting of a distance preserving linear transformation followed by a translationsuch that

RWW =U ′× (0,b), (41.2.30)

RW (W ∩Ω) = {u ∈ Rn : u′ ∈U ′, 0 < un < φW(u′)} (41.2.31)

where φW ∈ Cm,1(U ′)

meaning φW is the restriction to U ′ of a function, still denoted byφW which is in Cm,1

(Rn−1

)and

inf{

φW(u′)

: u′ ∈U ′}> 0

The following picture depicts the situation.