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 i^th component by ∂_i, the norm in H¹(Ω) is given by

$\| v \|_{H^{1} (\Omega)} := \left( \int_{\Omega} \sum_{i = 1}^{n} | v^{i} (x) |^{2} \, \mathrm{d} x \right)^{1/2} + \left( \int_{\Omega} \sum_{i, j = 1}^{n} | \partial_{j} v^{i} (x) |^{2} \, \mathrm{d} x \right)^{1/2}.$

Then there is a constant C ≥ 0, known as the Korn constant of Ω, such that, for all v ∈ H¹(Ω),

$\| v \|_{H^{1} (\Omega)}^{2} \leq C \int_{\Omega} \sum_{i, j = 1}^{n} \left( | v^{i} (x) |^{2} + | (e_{ij} v) (x) |^{2} \right) \, \mathrm{d} x, \quad (1)$