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 be an open, connected domain in -dimensional Euclidean space , . Let be the Sobolev space of all vector fields on that, along with their (first) weak derivatives, lie in the Lebesgue space . Denoting the partial derivative with respect to the ith component by , the norm in is given by
Then there is a constant , known as the Korn constant of , such that, for all ,
where denotes the symmetrized gradient given by
Inequality is known as Korn's inequality.
See also
Linear elasticity
Hardy inequality
Poincaré inequality
References
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