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Korn's inequality

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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 ≥ 2. Let H(Ω) be the Sobolev space of all vector fields v = (v, ..., v) on Ω that, along with their (first) weak derivatives, lie in the Lebesgue space L(Ω). Denoting the partial derivative with respect to the i component by ∂i, the norm in H(Ω) is given by

‖ v ‖ H 1 ( Ω ) := ( ∫ Ω ∑ i = 1 n | v i ( x ) | 2 d x + ∫ Ω ∑ i , j = 1 n | ∂ j v i ( x ) | 2 d x ) 1 / 2 . {\displaystyle \|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(Ω),

‖ v ‖ H 1 ( Ω ) 2 ≤ C ∫ Ω ∑ i , j = 1 n ( | v i ( x ) | 2 + | ( e i j v ) ( x ) | 2 ) d x {\displaystyle \|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} 1

where e denotes the symmetrized gradient given by

e i j v = 1 2 ( ∂ i v j + ∂ j v i ) . {\displaystyle e_{ij}v={\frac {1}{2}}(\partial _{i}v^{j}+\partial _{j}v^{i}).}

Inequality (1) is known as Korn's inequality.

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