View source for Korn inequality

{{MSC|74B05|74B20}}
[[Category:Partial differential equations]]
{{TEX|done}}

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 {{Cite|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 {{Cite|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 {{Cite|O}}. For a modern proof the reader might consult {{Cite|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 {{Cite|K}}); cf. {{Cite|C2}}, {{Cite|F}}.

====References====
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|valign="top"|{{Ref|C}}|| P. G. Ciarlet, "On Korn's inequality", ''Chinese Ann. Math., Ser B'' '''31''' (2010),  pp. 607-618.
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|valign="top"|{{Ref|C2}}|| P. G. Ciarlet, "Mathematical Elasticity", Vol. I : Three-Dimensional Elasticity, Series “Studies in Mathematics and its Applications”, North-Holland, Amsterdam, 1988.
|-
|valign="top"|{{Ref|CFM}}|| S. Conti, D. Faraco, F. Maggi, "A new approach to counterexamples to $L^1$ estimates: Korn’s inequality, geometric rigidity, and regularity for gradients of separately convex functions", ''Arch. Rat. Mech. Anal.'' '''175''', (2005), pp. 287-300.
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|valign="top"|{{Ref|K}}|| A. Korn, "Solution générale du problème d’équilibre dans la théorie de l'élasticité", ''Annales de la Faculté des Sciences de Toulouse'', '''10''', (1908), pp. 705-724
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|valign="top"|{{Ref|K2}}|| A. Korn, "Ueber einige Ungleichungen, welche in der Theorie der elastischen und elektrischen Schwingungen eine Rolle spielen", ''Bulletin internationale de l'Academie de Sciences de Cracovie'', '''9''', (1909), pp. 705-724 
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|valign="top"|{{Ref|O}}|| D.  Ornstein, "A non-inequality for differential operators in the $L^1$ norm", ''Arch. Rational Mech. Anal.'','''11''', (1962), pp. 40–49
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|valign="top"|{{Ref|F}}|| G. Fichera, "Existence theorems in elasticity theory", ''Handbuch der Physik'', '''VIa/2''', Springer  (1972)  pp. 347–389  
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