Sobolev spaces | Inequalities

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. (Wikipedia).

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Related pages

Poincaré inequality | Norm (mathematics) | Rigidity (mathematics) | Dimension | Connected space | Lp space | Partial derivative | Skew-symmetric matrix | Gradient | Sobolev space | Euclidean space | Vector field | A priori estimate | Mathematical analysis | Open set