1352 CHAPTER 41. ELLIPTIC REGULARITY
≤
(∫ h
0
(∫W|∇w(y+ tek) · ek|2 dy
)1/2 dth
)≤ ||∇w||L2(U ;Rn)
so by density of such functions in H1 (U) , 41.1.17 holds. Therefore, changing ε, yields
B.≤Cε (η ,Lip(α) ,α) ||∇w||2L2(U ;Rn)+ ε
∣∣∣∣∣∣η∇Dhkw∣∣∣∣∣∣2
L2(W∩U ;Rn). (41.1.18)
With 41.1.14 and 41.1.18 established, consider the other terms of 41.1.7.∣∣∣∣∫Uf zdy
∣∣∣∣≤
∣∣∣∣∫Uf(−D−h
k η2Dh
kw)
dy∣∣∣∣
≤(∫
U| f |2 dy
)1/2(∫U
∣∣∣D−hk
(η
2Dhkw)∣∣∣2 dy
)1/2
≤ || f ||L2(U)
∣∣∣∣∣∣∇(η2Dh
kw)∣∣∣∣∣∣
L2(U ;Rn)
≤ || f ||L2(U)
(∣∣∣∣∣∣2η∇ηDhkw∣∣∣∣∣∣
L2(U ;Rn)+∣∣∣∣∣∣η2
∇Dhkw∣∣∣∣∣∣
L2(U ;Rn)
)≤ C || f ||L2(U) ||∇w||L2(U ;Rn)+ || f ||L2(U)
∣∣∣∣∣∣η∇Dhkw∣∣∣∣∣∣
L2(U ;Rn)
≤ Cε
(|| f ||2L2(U)+ ||∇w||2L2(U ;Rn)
)+ ε
∣∣∣∣∣∣η∇Dhkw∣∣∣∣∣∣2
L2(U ;Rn)(41.1.19)
∣∣∣∣∫Uhs (y)
∂ z∂ys dy
∣∣∣∣≤
∣∣∣∣∣∫
Uhs (y)
∂(−D−h
k
(η2Dh
kw))
∂ys dy
∣∣∣∣∣≤
∣∣∣∣∣∫
UDh
khs (y)∂((
η2Dhkw))
∂ys
∣∣∣∣∣≤
∫U
∣∣∣∣Dhkhs2η
∂η
∂ys Dhkw∣∣∣∣dy+
∫U
∣∣∣∣∣(ηDhkhs
)(η
∂(Dh
kw)
∂ys
)∣∣∣∣∣dy
≤ C∑s||hs||H1(U)
(||w||H1(U)+
∣∣∣∣∣∣η∇Dhkw∣∣∣∣∣∣
L2(U ;Rn)
)≤ Cε ∑
s||hs||2H1(U)+ ||w||
2H1(U)+ ε
∣∣∣∣∣∣η∇Dhkw∣∣∣∣∣∣2
L2(U ;Rn). (41.1.20)
The following inequalities in 41.1.14,41.1.18, 41.1.19and 41.1.20 are summarized here.
A.≥ δ
∫U
η2∣∣∣∇Dh
kw∣∣∣2 dy,