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Difference between revisions of "Korn inequality"

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An inequality for vector functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055790/k0557901.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055790/k0557902.png" />, and their derivatives, defined in some bounded domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055790/k0557903.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055790/k0557904.png" />:
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{{MSC|74B05|74B20}}
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[[Category:Partial differential equations]]
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055790/k0557905.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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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
 +
\[
 +
\int |Df|^2 \leq 2 \int |D^s f|^2\, .
 +
\]
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In fact, when $f$ is $C^2$ a simple integration by parts yields the identity
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\[
 +
\int |D^s f|^2 = \frac{1}{2} \int |D f|^2 + \frac{1}{2} \int ({\rm div}\, f)^2
 +
\]
 +
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.
  
where
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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 $p$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055790/k0557906.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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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 is also valid for vector functions in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055790/k0557907.png" /> obtained by completing the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055790/k0557908.png" /> with respect to the norm (2). The inequality (1) is sometimes called the second Korn inequality; the first Korn inequality being inequality (1) without the second term on the left.
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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}}.
 
 
The inequality was proposed by A. Korn (1908) in order to obtain an a priori estimate for the solution of non-homogeneous equations of elasticity theory.
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> G. Fichera,   "Existence theorems in elasticity theory" , ''Handbuch der Physik'' , '''VIa/2''' , Springer  (1972)  pp. 347–389  {{MR|}} {{ZBL|}} </TD></TR></table>
+
{|
 +
|-
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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.
 +
|-
 +
|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.
 +
|-
 +
|valign="top"|{{Ref|K}}|| A. Korn, "Solution general du probleme d'equilibre dans la theorie de l'elasticite", ''Annales de la Faculte de Sciences de Toulouse'', '''10''', (1908), pp. 705-724
 +
|-
 +
|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
 +
|-
 +
|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
 +
|-
 +
|valign="top"|{{Ref|F}}|| G. Fichera, "Existence theorems in elasticity theory", ''Handbuch der Physik'', '''VIa/2''', Springer  (1972)  pp. 347–389   
 +
|-
 +
|}

Latest revision as of 13:59, 3 November 2013

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].

References

[C] P. G. Ciarlet, "On Korn's inequality", Chinese Ann. Math., Ser B 31 (2010), pp. 607-618.
[C2] P. G. Ciarlet, "Mathematical Elasticity", Vol. I : Three-Dimensional Elasticity, Series “Studies in Mathematics and its Applications”, North-Holland, Amsterdam, 1988.
[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.
[K] A. Korn, "Solution general du probleme d'equilibre dans la theorie de l'elasticite", Annales de la Faculte de Sciences de Toulouse, 10, (1908), pp. 705-724
[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
[O] D. Ornstein, "A non-inequality for differential operators in the $L^1$ norm", Arch. Rational Mech. Anal.,11, (1962), pp. 40–49
[F] G. Fichera, "Existence theorems in elasticity theory", Handbuch der Physik, VIa/2, Springer (1972) pp. 347–389
How to Cite This Entry:
Korn inequality. Encyclopedia of Mathematics. URL: http://www.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