Korn inequality

2020 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 générale du problème d’équilibre dans la théorie de l'élasticité", Annales de la Faculté des 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