# Korn inequality

2010 Mathematics Subject Classification: Primary: 74B05 Secondary: 74B20 [MSN][ZBL]

An inequality concerning the derivatives of vector functions $f:\mathbb R^n\to \mathbb R^n$. Assuming that $f$ is continuosly differentiable, we denote by $Df$ the Jacobian matrix of its differential and by $D^s f$ its symmetric part, namely the matrix with entries $\frac{1}{2} \left(\frac{\partial f_j}{\partial x_i} + \frac{\partial f_i}{\partial x_j}\right)\, .$ Denoting by $|Df|$ and $|D^s f|$ the corresponding Hilbert-Schmidt norms, the original inequality of Korn (see [K2]) states that, if $f\in C^1_c (\mathbb R^n)$, then $\int |Df|^2 \leq 2 \int |D^s f|^2\, .$ In fact, when $f$ is $C^2$ a simple integration by parts yields the identity $\int |D^s f|^2 = \frac{1}{2} \int |D f|^2 + \frac{1}{2} \int ({\rm div}\, f)^2$ from which Korn's inequality is obvious. A standard approximation procedure yields then the general statement: in fact for the same reason the inequality holds for functions in the Sobolev class $H^1_0$. The Korn's inequality can also be concluded easily using the Fourier Transform.

The inequality has been subsequently generalized to

• $f\in W^{1,2} (\Omega)$, under the assumption that $\Omega$ is bounded and $\partial \Omega$ sufficiently regular (Lipschitz is sufficient);
• $f\in W^{1,p}_0 (\mathbb R^n)$ and $f\in W^{1,p} (\Omega)$ (again under the assumption that $\Omega$ is bounded and the boundary sufficiently regular) for $p\in ]1, \infty[$, in which case the inequality takes the form

$\|Du\|_{L^p} \leq C \|D^s u\|_{L^p}\, ,$ where the constant $C$ depends, additionally, upon $p$.

The latter generalization uses the Calderon-Zygmund estimates for singular integral operators, see for instance [C]. The cases $p = 1, \infty$ of the inequality are false, as implied by a more general theorem of Ornstein about the failure of $L^1$ estimates for general singular integral operators, see [O]. For a modern proof the reader might consult [CFM].

The Korn inequality has several applications in the theory of nonlinear elasticity (and was in fact originally derived by Korn in linear elasticity, see [K]); cf. [C2], [F].

How to Cite This Entry:
Korn inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Korn_inequality&oldid=30665
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article