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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 (mathematics)|rigidity]].

In (linear) [[elasticity theory]], the symmetric part of the gradient is a measure of the [[deformation (mechanics)|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 Ω be an [[open set|open]], [[connected space|connected]] domain in ''n''-[[dimension]]al [[Euclidean space]] '''R'''<sup>''n''</sup>, ''n''&nbsp;≥&nbsp;2.  Let ''H''<sup>1</sup>(Ω) be the Sobolev space of all [[vector field]]s ''v''&nbsp;=&nbsp;(''v''<sup>1</sup>,&nbsp;...,&nbsp;''v''<sup>''n''</sup>) on Ω that, along with their weak derivatives, lie in the [[Lp space|Lebesgue space]] ''L''<sup>2</sup>(Ω).  Denoting the [[partial derivative]] with respect to the ''i''<sup>th</sup> component by ∂<sub>''i''</sub>, the [[norm (mathematics)|norm]] in ''H''<sup>1</sup>(Ω) is given by

:<math>\| 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}.</math>

Then there is a constant ''C''&nbsp;≥&nbsp;0, known as the '''Korn constant''' of Ω, such that, for all ''v''&nbsp;∈&nbsp;''H''<sup>1</sup>(Ω),

:<math>\| 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, \quad (1)</math>

where ''e'' denotes the symmetrized gradient given by

:<math>e_{ij} v = \frac1{2} ( \partial_{i} v^{j} + \partial_{j} v^{i} ).</math>

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

==References==

* {{cite journal
| last = Horgan
| first = Cornelius O.
| title = Korn's inequalities and their applications in continuum mechanics
| journal = SIAM Rev.
| volume = 37
| year = 1995
| pages = 491&ndash;511
| issn = 0036-1445
| doi = 10.1137/1037123
| issue = 4
}} {{MathSciNet|id=1368384}}

==External links==
* [http://www.encyclopediaofmath.org/index.php/Korn_inequality Korn inequality] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]


[[Category:Inequalities]]
[[Category:Sobolev spaces]]
[[Category:Solid mechanics]]