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 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 be an open, connected domain in -dimensional Euclidean space,. Let be the Sobolev space of all vector fields on that, together with their first weak derivatives, lie in the Lebesgue space. Denoting the partial derivative with respect to the -th coordinate by, the norm in is given by
Then there is a constant, called the Korn constant of, such that for all the following inequality holds:
where denotes the symmetrized gradient given by
Inequality is known as Korn's inequality.