1342 CHAPTER 40. KORN’S INEQUALITY

Consider the first integral on the right in 40.1.1. Changing the variables, letting

yn = 2g(x)− xn

in the first term of the integrand and 3g(x)−2xn in the next, it equals

−3∫

U−

∂φ

∂xn(x,2g(x)− yn) f (x,yn)dyndx

+2∫

U−

∂φ

∂xn

(x,

32

g(x)− yn

2

)f (x,yn)dyndx.

For (x,yn) ∈U−, and defining

ψ (x,yn)≡ φ (x,yn)+3φ (x,2g(x)− yn)−4φ

(x,

32

g(x)− yn

2

),

it follows ψ = 0 when yn = g(x) and so∫Rn

f∂φ

∂xndx =

∫U−

∂ψ

∂ynf (x,yn)dxdyn.

Now from the definition of ψ given above,

||ψ||1,2,U− ≤Cg ||φ ||1,2,U− ≤Cg ||φ ||1,2,Rn

and so ∣∣∣∣∣∣∣∣ ∂ f∂xn

∣∣∣∣∣∣∣∣−1,2,Rn

≡

sup{∫

Rnf

∂φ

∂xndx : φ ∈C∞

c (Rn) , ||φ ||1,2,Rn ≤ 1}≤

sup{∣∣∣∣∫U−

f∂ψ

∂xndxdyn

∣∣∣∣ : ψ ∈ H10(U−), ||ψ||1,2,U− ≤Cg

}=Cg

∣∣∣∣∣∣∣∣ ∂ f∂xn

∣∣∣∣∣∣∣∣−1,2,U−

(40.1.2)

It remains to establish a similar inequality for the case where the derivatives are takenwith respect to xi for i < n. Let φ ∈C∞

c (Rn) . Then∫Rn

f∂φ

∂xidx =

∫U−

f∂φ

∂xidx

∫U+

∂φ

∂xi[−3 f (x,g(x)− xn)+4 f (x,3g(x)−2xn)]dx.

Changing the variables as before, this last integral equals

−3∫

U−Diφ (x,2g(x)− yn) f (x,yn)dyndx