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 (first) 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

${\displaystyle \|v\|_{H^{1}(\Omega )}:=\left(\int _{\Omega }\sum _{i=1}^{n}|v^{i}(x)|^{2}\,\mathrm {d} x+\int _{\Omega }\sum _{i,j=1}^{n}|\partial _{j}v^{i}(x)|^{2}\,\mathrm {d} x\right)^{1/2}.}$

Then there is a constant C ≥ 0, known as the Korn constant of Ω, such that, for all v ∈ H¹(Ω),

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

(1)

where e denotes the symmetrized gradient given by

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

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 (eds.), Convegno internazionale in memoria di Vito Volterra (8–11 ottobre 1990), Atti dei Convegni Lincei (in Italian), vol. 92, Roma: Accademia Nazionale dei Lincei, pp. 183–209, ISSN   0391-805X, MR   1783034, Zbl   0972.35013, archived from the original on 2017-01-07, retrieved 2014-07-27.

External links

• Voitsekhovskii, M. I. (2001) [1994], "Korn inequality", Encyclopedia of Mathematics , EMS Press