Korn's First Inequality with variable coefficients and its generalization

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 $10$. 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 $