41.1. THE CASE OF A HALF SPACE 1355
and ∫U
αrs (y)
∂w∂yr
∂ z∂ys dy+
∫U
hs (y)∂ z∂ys dy =
∫U
f zdy (41.1.23)
for all z ∈ H1 (U) or H10 (U) such that spt(z) ⊆ V. Then there exists C independent of w
such that
||w||Hk+1(U1)≤C
(|| f ||Hk−1(U)+∑
s||hs||Hk(U)+ ||w||Hk(U)
). (41.1.24)
Proof: The proof involves the following claim which is proved using the conclusion ofLemma 41.1.1 on Page 1349.
Claim : If α = (α ′,0) where |α ′| ≤ k− 1, then there exists a constant independent ofw such that
||Dα w||H2(U1)≤C
(|| f ||Hk−1(U)+∑
s||hs||Hk(U)+ ||w||Hk(U)
). (41.1.25)
Proof of claim: First note that if |α|= 0, then 41.1.25 follows from Lemma 41.1.1 onPage 1349. Now suppose the conclusion of the claim holds for all |α| ≤ j−1 where j < k.Let |α| = j and α = (α ′,0) . Then for z ∈ H1 (U) having compact support in V, it followsthat for h small enough,
D−hα z ∈ H1 (U) , spt
(Dh
α z)⊆V.
Therefore, you can replace z in 41.1.23 with D−hα z. Now note that you can apply the fol-
lowing manipulation. ∫U
p(y)D−hα z(y)dy =
∫U
Dhα p(y)z(y)dy
and obtain∫U
(Dh
α
(α
rs ∂w∂yr
)∂ z∂ys +Dh
α (hs)∂ z∂ys
)dy =
∫U
((Dh
α f)
z)
dy. (41.1.26)
Letting h→ 0, this gives∫U
(Dα
(α
rs ∂w∂yr
)∂ z∂ys +Dα (hs)
∂ z∂ys
)dy =
∫U((Dα f )z)dy.
Now
Dα
(α
rs ∂w∂yr
)= α
rs ∂ (Dα w)∂yr + ∑
τ<α
C (τ)Dα−τ (αrs)∂ (Dτ w)
∂yr
where C (τ) is some coefficient. Therefore, from 41.1.26,
∫U
αrs ∂ (Dα w)
∂yr∂ z∂ys dy+
∫U
(∑
τ<α
C (τ)Dα−τ (αrs)∂ (Dτ w)
∂yr +Dα (hs)
)∂ z∂ys dy