1346 CHAPTER 40. KORN’S INEQUALITY

Now if ||φ ||1,2 ≤ 1, then for a suitable constant, Cψ ,

||ψφ ||1,2 ≤Cψ ||φ ||1,2 ≤Cψ ,

∣∣∣∣∣∣∣∣φ ∂ψ

∂xi

∣∣∣∣∣∣∣∣1,2≤Cψ .

Therefore, ∣∣∣∣∣∣∣∣∂ ( f ψ)

∂xi

∣∣∣∣∣∣∣∣−1,2,Ω

≤ sup||η ||1,2≤Cψ

∣∣∣∣∫Ω

f∂η

∂xidx∣∣∣∣+ sup||η ||1,2≤Cψ

∣∣∣∣∫Ω

f ηdx∣∣∣∣

≤Cψ

(∣∣∣∣∣∣∣∣ ∂ f∂xi

∣∣∣∣∣∣∣∣−1,2,Ω

+ || f ||−1,2,Ω

). (40.1.7)

Now using 40.1.7 and 40.1.6

∣∣∣∣∣∣ f ψ j

∣∣∣∣∣∣0,2,Ω≤Cg

∣∣∣∣∣∣ f ψ j

∣∣∣∣∣∣−1,2,Ω

+n

∑i=1

∣∣∣∣∣∣∣∣∣∣∣∣∂

(f ψ j

)∂xi

∣∣∣∣∣∣∣∣∣∣∣∣−1,2,Ω

≤Cψ jCg

(|| f ||−1,2,Ω +

n

∑i=1

∣∣∣∣∣∣∣∣ ∂ f∂xi

∣∣∣∣∣∣∣∣−1,2,Ω

).

Therefore, letting C = ∑pj=1 Cψ jCg,

|| f ||0,2,Ω ≤p

∑j=1

∣∣∣∣∣∣ f ψ j

∣∣∣∣∣∣0,2,Ω≤C

(|| f ||−1,2,Ω +

n

∑i=1

∣∣∣∣∣∣∣∣ ∂ f∂xi

∣∣∣∣∣∣∣∣−1,2,Ω

). (40.1.8)

This proves the hard half of the inequality of Theorem 40.1.1.To complete the proof, let f denote the zero extension of f off Ω. Then

|| f ||−1,2,Ω +n

∑i=1

∣∣∣∣∣∣∣∣ ∂ f∂xi

∣∣∣∣∣∣∣∣−1,2,Ω

≤∣∣∣∣ f ∣∣∣∣−1,2,Rn +

n

∑i=1

∣∣∣∣∣∣∣∣ ∂ f∂xi

∣∣∣∣∣∣∣∣−1,2,Rn

≤Cn∣∣∣∣ f ∣∣∣∣0,2,Rn =Cn || f ||0,2,Ω .

This along with 40.1.8 proves Theorem 40.1.1.

40.2 Korn’s InequalityThe inequality in this section is known as Korn’s second inequality. It is also known ascoercivity of strains. For u a vector valued function in Rn, define

ε i j (u)≡12(ui, j +u j,i)

This is known as the strain or small strain. Korn’s inequality says that the norm given by,

|||u||| ≡

(n

∑i=1||ui||20,2,Ω +

n

∑i=1

n

∑j=1

∣∣∣∣ε i j (u)∣∣∣∣2

0,2,Ω

)1/2

(40.2.9)