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 Ω be an open, connected domain in n-dimensional Euclidean space Rn, n ≥ 2.