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 $\mathrm{\Omega }$ be an open, connected domain in Template:Math-dimensional Euclidean space ${ℝ}^n$, $n\ge 2$. Let $H^1(\mathrm{\Omega })$ be the Sobolev space of all vector fields $v=(v^1,\dots ,v^n)$ on $\mathrm{\Omega }$ that, together with their first weak derivatives, lie in the Lebesgue space $L^2(\mathrm{\Omega })$. Denoting the partial derivative with respect to the $i$-th coordinate by ${\partial }_i$, the norm in $H^1(\mathrm{\Omega })$ is given by

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

Then there is a (minimal) constant $C\ge 0$, called the Korn constant of $\mathrm{\Omega }$, such that for all $v\in H^1(\mathrm{\Omega })$ the following inequality holds:

where $e$ denotes the symmetrized gradient given by

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

Inequality Template:EquationNote is known as Korn's inequality.

See also

• Hardy inequality
• Poincaré inequality

References

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External links

• Template:SpringerEOM