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 Rⁿ, n ≥ 2. Let H¹(Ω) be the Sobolev space of all vector fields v = (v¹, ..., vⁿ) on Ω that, along with their weak derivatives, lie in the Lebesgue space L²(Ω). Denoting the partial derivative with respect to the i^th component by ∂_i, the norm in H¹(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 ≥ 0, known as the Korn constant of Ω, such that, for all v ∈ H¹(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

$e_{ij} v = \frac1{2} ( \partial_{i} v^{j} + \partial_{j} v^{i} ).$

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

See also

• Linear elasticity
• Hardy inequality
• Poincaré 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