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Korn's Inequality; coercive inequalities
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 $.
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