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 Rⁿ, n ≥ 2. Let H¹(Ω) be the Sobolev space of all vector fields v = (v¹, ..., vⁿ) on Ω that, along with their (first) 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

$\Vert v{\Vert }_{H^1(\mathrm{\Omega })}:={(\underset{\mathrm{\Omega }}{\int }\sum _{i=1}^n|v^i(x){|}^2\mathrm{d}x+\underset{\mathrm{\Omega }}{\int }\sum _{i,j=1}^n|{\partial }_j{v}^i(x){|}^2\mathrm{d}x)}^{1/2}.$

Then there is a (minimal) 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

$e_{ij}v=\frac{1}{2}({\partial }_i{v}^j+{\partial }_j{v}^i).$

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

See also

• Hardy inequality
• Poincaré inequality

References

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