Korn's inequality
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In mathematics, Korn's inequality is a result about the derivatives of Sobolev functions. Korn's inequality plays an important rôle in linear elasticity theory.
[edit] Statement of the inequality
Let Ω be an open, connected domain in n-dimensional Euclidean space Rn, n ≥ 2. Let H1(Ω) be the Sobolev space of all vector fields v = (v1, ..., vn) on Ω that, along with their weak derivatives, lie in the Lebesgue space L2(Ω). Denoting the partial derivative with respect to the ith component by ∂i, the norm in H1(Ω) is given by
Then there is a constant C ≥ 0, known as the Korn constant of Ω, such that, for all v ∈ H1(Ω),
where e denotes the symmetrized gradient given by
Inequality (1) is known as Korn's inequality.
