Korn's inequality

In mathematical analysis, Korn's inequality is an inequality concerning the gradient of a vector field that generalizes the following classical theorem: if the gradient of a vector field is skew-symmetric at every point, then the gradient must be equal to a constant skew-symmetric matrix. Korn's theorem is a quantitative version of this statement, which intuitively says that if the gradient of a vector field is on average not far from the space of skew-symmetric matrices, then the gradient must not be far from a particular skew-symmetric matrix. The statement that Korn's inequality generalizes thus arises as a special case of rigidity.

In (linear) elasticity theory, the symmetric part of the gradient is a measure of the strain that an elastic body experiences when it is deformed by a given vector-valued function. The inequality is therefore an important tool as an a priori estimate in linear elasticity theory.

Statement of the inequality
Let $Ω$ be an open, connected domain in $n$-dimensional Euclidean space $R^{n}$, $n ≥ 2$. Let $H^{1}(Ω)$ be the Sobolev space of all vector fields $v = (v^{1}, ..., v^{n})$ on $Ω$ that, along with their (first) weak derivatives, lie in the Lebesgue space $L^{2}(Ω)$. Denoting the partial derivative with respect to the ith component by $∂_{i}$, the norm in $H^{1}(Ω)$ is given by


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

Then there is a (minimal) constant $C ≥ 0$, known as the Korn constant of $Ω$, such that, for all $v ∈ H^{1}(Ω)$,

where $e$ denotes the symmetrized gradient given by


 * $$e_{ij} v = \frac1{2} ( \partial_{i} v^{j} + \partial_{j} v^{i} ).$$

Inequality $$ is known as Korn's inequality.