Jump to content

Korn's inequality

From Wikipedia, the free encyclopedia

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

[edit]

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 ith component by i, the norm in H1(Ω) is given by

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

(1)

where e denotes the symmetrized gradient given by

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

See also

[edit]

References

[edit]
  • 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.
[edit]