40.2. KORN’S INEQUALITY 1347

is equivalent to the norm,

||u|| ≡

(n

∑i=1||ui||20,2,Ω +

n

∑i=1

n

∑j=1

∣∣∣∣∣∣∣∣ ∂ui

∂x j

∣∣∣∣∣∣∣∣20,2,Ω

)1/2

(40.2.10)

It is very significant because it is the strain as just defined which occurs in many of thephysical models proposed in continuum mechanics. The inequality is far from obviousbecause the strains only involve certain combinations of partial derivatives.

Theorem 40.2.1 (Korn’s second inequality) Let Ω be any domain for which the conclusionof Theorem 40.1.1 holds. Then the two norms in 40.2.9 and 40.2.10 are equivalent.

Proof: Let u be such that ui ∈ H1 (Ω) for each i = 1, · · · ,n. Note that

∂ 2ui

∂x j,∂xk=

∂

∂x j(ε ik (u))+

∂

∂xk(ε i j (u))−

∂

∂xi

(ε jk (u)

).

Therefore, by Theorem 40.1.1,∣∣∣∣∣∣∣∣ ∂ui

∂x j

∣∣∣∣∣∣∣∣0,2,Ω≤C

[∣∣∣∣∣∣∣∣ ∂ui

∂x j

∣∣∣∣∣∣∣∣−1,2,Ω

+n

∑k=1

∣∣∣∣∣∣∣∣ ∂ 2ui

∂x j,∂xk

∣∣∣∣∣∣∣∣−1,2,Ω

]

≤C

[∣∣∣∣∣∣∣∣ ∂ui

∂x j

∣∣∣∣∣∣∣∣−1,2,Ω

+ ∑r,s,p

∣∣∣∣∣∣∣∣∂εrs (u)∂xp

∣∣∣∣∣∣∣∣−1,2,Ω

]

≤C

[∣∣∣∣∣∣∣∣ ∂ui

∂x j

∣∣∣∣∣∣∣∣−1,2,Ω

+∑r,s||εrs (u)||0,2,Ω

].

But also by this theorem,

||ui||−1,2,Ω +∑p

∣∣∣∣∣∣∣∣ ∂ui

∂xp

∣∣∣∣∣∣∣∣−1,2,Ω

≤C ||ui||0,2,Ω

and so ∣∣∣∣∣∣∣∣ ∂ui

∂x j

∣∣∣∣∣∣∣∣0,2,Ω≤C

[||ui||0,2,Ω +∑

r,s||εrs (u)||0,2,Ω

]This proves the theorem.

Note that Ω did not need to be bounded. It suffices to be able to conclude the resultof Theorem 40.1.1 which would hold whenever the boundary of Ω can be covered withfinitely many boxes of the sort to which Lemma 40.1.2 can be applied.