Let Ω be an open, bounded set in Rn (n ≥ 2) with Lipschitz boundary ∂Ω. Assume that Γ ⊆ ∂Ω has a positive Hn-1 measure and let R : Ω → SO(n) be given. We construct counterexamples to the following version of Korn’s inequality in the following two cases: (a) n = 2 and Γ ≠ ∂Ω; (b) n = 3 and Γ = ∂Ω. It is known that the above inequality holds in any dimension n and for any Γ ⊆ ∂Ω, provided that R ∈ C(Ω). However, the inequality is true for n = 2 and Γ = ∂Ω without any regularity assumptions on R.
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