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.

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

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

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

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

1

where ${\displaystyle e}$ denotes the symmetrized gradient given by

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

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

• Hardy inequality
• Poincaré inequality

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