41.1. THE CASE OF A HALF SPACE 1353

B.≤Cε (η ,Lip(α) ,α) ||∇w||2L2(U ;Rn)+ ε

∣∣∣∣∣∣η∇Dhkw∣∣∣∣∣∣2

L2(W∩U ;Rn),

∣∣∣∣∫Uf zdy

∣∣∣∣≤Cε

(|| f ||2L2(U)+ ||∇w||2L2(U ;Rn)

)+ ε

∣∣∣∣∣∣η∇Dhkw∣∣∣∣∣∣2

L2(U ;Rn)

∣∣∣∣∫Uhs (y)

∂ z∂ys dy

∣∣∣∣ ≤ Cε ∑s||hs||2H1(U)

+ ||w||2H1(U)+ ε

∣∣∣∣∣∣η∇Dhkw∣∣∣∣∣∣2

L2(U ;Rn).

Therefore,

δ

∣∣∣∣∣∣η∇Dhkw∣∣∣∣∣∣2

L2(U ;Rn)

≤ Cε (η ,Lip(α) ,α) ||∇w||2L2(U ;Rn)+ ε

∣∣∣∣∣∣η∇Dhkw∣∣∣∣∣∣2

L2(U ;Rn)

+Cε ∑s||hs||2H1(U)+ ||w||

2H1(U)+ ε

∣∣∣∣∣∣η∇Dhkw∣∣∣∣∣∣2

L2(U ;Rn)

+Cε

(|| f ||2L2(U)+ ||∇w||2L2(U ;Rn)

)+ ε

∣∣∣∣∣∣η∇Dhkw∣∣∣∣∣∣2

L2(U ;Rn).

Letting ε be small enough and adjusting constants yields∣∣∣∣∣∣∇Dhkw∣∣∣∣∣∣2

L2(U1;Rn)≤∣∣∣∣∣∣η∇Dh

kw∣∣∣∣∣∣2

L2(U ;Rn)≤

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

2L2(U)+Cε ∑

s||hs||2H1(U)

)where the constant, C, depends on η ,Lip(α) ,α,δ . Since this holds for all h small enough,it follows ∂w

∂yk ∈ H1 (U1) and ∣∣∣∣∣∣∣∣∇ ∂w∂yk

∣∣∣∣∣∣∣∣2L2(U1;Rn)

≤

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

2L2(U)+Cε ∑

s||hs||2H1(U)

)(41.1.21)

for each k < n. It remains to estimate∣∣∣∣∣∣ ∂ 2w

∂y2n

∣∣∣∣∣∣2L2(U1)

. To do this return to 41.1.7 which must

hold for all z ∈C∞c (U1) . Therefore, using 41.1.7 it follows that for all z ∈C∞

c (U1) ,∫U

αrs (y)

∂w∂yr

∂ z∂ys dy =−

∫U

∂hs

∂ys zdy+∫

Uf zdy.