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.

Statement of the inequality

Let $Ω$ be an open, connected domain in $n$ -dimensional Euclidean space $R n$ , $n \geq 2$ . Let $H 1 (Ω)$ be the Sobolev space of all vector fields $v = (v 1, ..., v n)$ on $Ω$ that, along with their weak derivatives, lie in the Lebesgue space $L 2 (Ω)$ . Denoting the partial derivative with respect to the i^th component by $\partial i$ , the norm in $H 1 (Ω)$ is given by

\|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}.

Then there is a constant $C \geq 0$ , known as the Korn constant of $Ω$ , such that, for all $v \in H 1 (Ω)$ ,

$\|v\|_{H^{1}(\Omega )}^{2}\leq C\int _{\Omega }\sum _{i,j=1}^{n}\left(|v^{i}(x)|^{2}+|(e_{ij}v)(x)|^{2}\right)\,\mathrm {d} x$

(1)

where $e$ denotes the symmetrized gradient given by

e_{ij}v={\frac {1}{2}}(\partial _{i}v^{j}+\partial _{j}v^{i}).

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

References

Cioranescu, Doina; Oleinik, Olga Arsenievna; Tronel, Gérard (1989), "On Korn's inequalities for frame type structures and junctions", Comptes rendus hebdomadaires des séances de l'Académie des Sciences, Série I: Mathématiques, 309 (9): 591–596, MR 1053284, Zbl 0937.35502 .
Horgan, Cornelius O. (1995), "Korn's inequalities and their applications in continuum mechanics", SIAM Review, 37 (4): 491–511, doi:10.1137/1037123, ISSN 0036-1445, MR 1368384, Zbl 0840.73010 .
Oleinik, Olga Arsenievna; Kondratiev, Vladimir Alexandrovitch (1989), "On Korn's inequalities", Comptes rendus hebdomadaires des séances de l'Académie des Sciences, Série I: Mathématiques, 308 (16): 483–487, MR 0995908, Zbl 0698.35067 .
Oleinik, Olga A. (1992), "Korn's Type inequalities and applications to elasticity", in Amaldi, E.; Amerio, L.; Fichera, G.; Gregory, T.; Grioli, G.; Martinelli, E.; Montalenti, G.; Pignedoli, A.; Salvini, Giorgio; Scorza Dragoni, Giuseppe, Convegno internazionale in memoria di Vito Volterra (8–11 ottobre 1990), Atti dei Convegni Lincei (in Italian), 92, Roma: Accademia Nazionale dei Lincei, pp. 183–209, ISSN 0391-805X, MR 1783034, Zbl 0972.35013. .

External links

Voitsekhovskii, M. I. (2001), "Korn inequality", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

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Korn's inequality

Statement of the inequality

See also

References

External links