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 {\displaystyle \Omega } be an open, connected domain in n-dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} , n ≥ 2 {\displaystyle n\geq 2} . Let H 1 ( Ω ) {\displaystyle H^{1}(\Omega )} be the Sobolev space of all vector fields v = ( v 1 , … , v n ) {\displaystyle v=(v^{1},\dots ,v^{n})} on Ω {\displaystyle \Omega } that, together with their first weak derivatives, lie in the Lebesgue space L 2 ( Ω ) {\displaystyle L^{2}(\Omega )} . Denoting the partial derivative with respect to the i {\displaystyle i} -th coordinate by ∂ i {\displaystyle \partial _{i}} , the norm in H 1 ( Ω ) {\displaystyle H^{1}(\Omega )} is given by
‖ v ‖ H 1 ( Ω ) := ( ∫ Ω ∑ i = 1 n | v i ( x ) | 2 d x + ∫ Ω ∑ i , j = 1 n | ∂ j v i ( x ) | 2 d x ) 1 / 2 . {\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 (minimal) constant C ≥ 0 {\displaystyle C\geq 0} , called the Korn constant of Ω {\displaystyle \Omega } , such that for all v ∈ H 1 ( Ω ) {\displaystyle v\in H^{1}(\Omega )} the following inequality holds:
where e {\displaystyle e} denotes the symmetrized gradient given by
e i j v = 1 2 ( ∂ i v j + ∂ j v i ) . {\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 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