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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]]~~.		\ո
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	~~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'' ≥ 2~~. ~~Let ''H''<sup>1</sup>(Ω)~~ be the ~~Sobolev space of all [[vector field]]s ''v'' = (''v''~~<~~sup~~>1<~~/sup~~>,~~ ~~...~~, ''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'' ≥ 0~~, ~~known as~~ the '~~''Korn constant'''~~ of ~~Ω, such that, for all ''v'' ∈ ''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 = 491–511~~
	~~\| 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]]~~

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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'''''n'', ''n'' ≥ 2. Let ''H''1(Ω) be the Sobolev space of all [[vector field]]s ''v'' = (''v''1, ..., ''v''''n'') on Ω that, along with their weak derivatives, lie in the [[Lp space|Lebesgue space]] ''L''2(Ω). Denoting the [[partial derivative]] with respect to the ''i''th component by ∂''i'', the [[norm (mathematics)|norm]] in ''H''1(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'' ≥ 0, known as the '''Korn constant''' of Ω, such that, for all ''v'' ∈ ''H''1(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 = 491–511
| 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]]

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