1354 CHAPTER 41. ELLIPTIC REGULARITY

Now from the Lipschitz assumption on αrs, it follows

F ≡ ∑r,s≤n−1

∂

∂ys

(α

rs ∂w∂yr

)+ ∑

s≤n−1

∂

∂ys

(α

ns ∂w∂yn

)−∑

s

∂hs

∂ys + f

∈ L2 (U1)

and

||F ||L2(U1)≤C

(||w||2H1(U)+ || f ||

2L2(U)+Cε ∑

s||hs||2H1(U)

). (41.1.22)

Therefore, from density of C∞c (U1) in L2 (U1) ,

− ∂

∂yn

(α

nn (y)∂w∂yn

)= F, no sum on n

and so

−∂αnn

∂yn∂w∂yn −α

nn ∂ 2w

∂ (yn)2 = F

By 41.1.2 αnn (y)≥ δ and so it follows from 41.1.22 that there exists a constant,C depend-ing on δ such that ∣∣∣∣∣ ∂ 2w

∂ (yn)2

∣∣∣∣∣L2(U1)

≤C(|F |L2(U1)

+ ||w||H1(U)

)which with 41.1.21 and 41.1.22 implies the existence of a constant, C depending on δ suchthat

||w||2H2(U1)≤C

(||w||2H1(U)+ || f ||

2L2(U)+Cε ∑

s||hs||2H1(U)

),

proving the lemma.What if more regularity is known for f , hs,α

rs and w? Could more be said about theregularity of the solution? The answer is yes and is the content of the next corollary.

First here is some notation. For α a multi-index with |α| = k− 1, α = (α1, · · · ,αn)define

Dhα l (y)≡

n

∏k=1

(Dh

k

)αkl (y) .

Also, for α and τ multi indices, τ < α means τ i < α i for each i.

Corollary 41.1.2 Suppose in the context of Lemma 41.1.1 the following for k ≥ 1.

w ∈ Hk (U) ,

αrs ∈ Ck−1,1 (U) ,hs ∈ Hk (U) ,

f ∈ Hk−1 (U) ,