# Article

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Keywords:
Korn's Inequality; coercive inequalities
Summary:
If $\Omega \subset \Bbb R^n$ is a bounded domain with Lipschitz boundary $\partial \Omega$ and $\Gamma$ is an open subset of $\partial \Omega$, we prove that the following inequality $$\biggl(\int_\Omega |A(x)\nabla u(x)|^p\,dx\biggr)^{1/p} + \biggl(\int_\Gamma |u(x)|^p\,d\Cal H^{n-1}(x)\biggr)^{1/p} \geq c\, \|u\|_{W^{1,p}{(\Omega )}}$$ holds for all $u\in W^{1,p}(\Omega ;\Bbb R^m)$ and $1<p<\infty$, where $$(A(x)\nabla u(x))_k=\sum_{i=1}^m\sum_{j=1}^n\, a_k^{ij}(x)\,\frac{\partial u_i}{\partial x_j}(x) \quad (k=1,2,\ldots,r; r\geq m)$$ defines an elliptic differential operator of first order with continuous coefficients on $\overline{\Omega }$. As a special case we obtain $$\int_{\Omega }\bigl|\nabla u(x)F(x)+(\nabla u(x)F(x))^T\bigr|^p\,dx\geq c\int_{\Omega }|\nabla u(x)|^p\,dx\,, \leqno{(*)}$$ for all $u\in W^{1,p}(\Omega ;\Bbb R^n)$ vanishing on $\Gamma$, where $F:\overline{\Omega }\rightarrow M^{n\times n}(\Bbb R)$ is a continuous mapping with $\operatorname{det} F(x)\geq \mu >0$. Next we show that $(*)$ is not valid if $n\geq 3$, $F\in L^\infty(\Omega )$ and $\operatorname{det} F(x)=1$, but does hold if $p=2$, $\Gamma =\partial \Omega$ and $F(x)$ is symmetric and positive definite in $\Omega$.
References:
[1] Besov O.V.: On coercivity in nonisotropic Sobolev spaces. Math. USSR-Sbornik, vol. 2 (1967), no. 4, 521-534. Zbl 0169.47101
[2] Calderón A.P., Zygmund A.: On singular integrals. Amer. J. Math. 78 (1956), 289-309. MR 0084633
[3] Chen W., Jost J.: A Riemann version of Korn's Inequality. Calc. Var. (2001).
[4] Ciarlet P.G.: Mathematical Elasticity, Volume I: Three-dimensional Elasticity. North-Holland, 1988. MR 0936420 | Zbl 0648.73014
[5] Kałamajska A.: Coercive inequalities on weighted Sobolev spaces. Coll. Math. LXVI (1994), 309-318. MR 1268073
[6] Maz'ya V.G.: Sobolev Spaces. Springer, 1985. MR 0817985 | Zbl 1152.46002
[7] Mikhlin S.G.: Multidimensional singular integrals and integral equations. Pergamon Press, 1965. MR 0185399 | Zbl 0129.07701
[8] Nečas J.: Sur les normes équivalentes dans $W^{(k)}_p(Ømega)$ et sur la coercivité des formes formellement positives. Séminaire de mathématiques supérieures, 1965. Fasc. 19: Équations aux dérivées partielles (1966), 101-128.
[9] Nečas J., Hlaváček I.: Mathematical Theory of Elastic and Elasto-plastic Bodies: An Introduction. Elsevier Scientific Publishing Company, 1981. MR 0600655
[10] Neff P.: On Korn's First Inequality with nonconstant coefficients. Proc. Roy. Soc. Edinburgh 132A (2002), 221-243. MR 1884478
[11] Neff P.: A Korn's First Inequality with $W^{1,4}(Ømega)$-coefficients. preprint.
[12] Neff P.: Local existence and uniqueness for quasistatic finite plasticity with grain boundary relaxation. preprint. MR 2126571 | Zbl 1072.74013

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