1350 CHAPTER 41. ELLIPTIC REGULARITY

for all z ∈ H1 (U) having the property that spt(z) ⊆ V. Then w ∈ H2 (U1) and for someconstant C, independent of f ,w, and g, the following estimate holds.

||w||2H2(U1)≤C

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

2L2(U)+∑

s||hs||2H1(U)

). (41.1.8)

Proof: Define for small real h,

Dhk l (y)≡ 1

h(l (y+hek)− l (y)) .

Let U1 ⊆U1 ⊆W ⊆W ⊆ V and let η ∈ C∞c (W ) with η (y) ∈ [0,1] , and η = 1 on U1 as

shown in the following picture.

R

Rn−1

U

V

Γ

U1

W

For h small (3h < dist(W ,VC

)), let

z(y)≡ 1h

{η

2 (y−hek)

[w(y)−w(y−hek)

h

]

−η2 (y)

[w(y+hek)−w(y)

h

]}(41.1.9)

≡ −D−hk

(η

2Dhkw), (41.1.10)

where here k < n. Thus z can be used in equation 41.1.7. Begin by estimating the left sideof 41.1.7. ∫

Uα

rs (y)∂w∂yr

∂ z∂ys dy

=1h

∫U

αrs (y+hek)

∂w∂yr (y+hek)

∂(η2Dh

kw)

∂ys dy

−1h

∫U

αrs (y)

∂w∂yr

∂(η2Dh

kw)

∂ys dy