Waldemar Pompe
Korn's First Inequality with variable coefficients and its generalization

Comment.Math.Univ.Carolinae 44,1 (2003) 57-70.

Abstract: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 $$ (A(x)\nabla u(x))_k=\sum _{i=1}^m\sum _{j=1}^n a_k^{ij}(x) \frac {\partial u_i}{\partial x_j}(x) \hskip 1em\relax (k=1,2,\ldots ,r; r\geq m) $$ 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 $\mathop {\fam \z@ det}\nlimits@ F(x)\geq \mu >0$. Next we show that $(*)$ is not valid if $n\geq 3$, $F\in L^\infty (\Omega )$ and $\mathop {\fam \z@ det}\nlimits@ F(x)=1$, but does hold if $p=2$, $\Gamma =\partial \Omega $ and $F(x)$ is symmetric and positive definite in $\Omega $.

Keywords: Korn's Inequality, coercive inequalities
AMS Subject Classification: 35F15, 35J55