Korn's First Inequality with variable coefficients and its generalization

@article{Pompe2003,
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 $$ (A(x)\nabla u(x))\_k=\sum\_\{i=1\}^m\sum\_\{j=1\}^n\, a\_k^\{ij\}(x)\,\frac\{\partial u\_i\}\{\partial x\_j\}(x) \quad (k=1,2,\ldots,r; r\geq m) $$ 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 $.},
author = {Pompe, Waldemar},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {generalized Korn inequality; continuous coefficients},
language = {eng},
number = {1},
pages = {57-70},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Korn's First Inequality with variable coefficients and its generalization},
url = {http://eudml.org/doc/249158},
volume = {44},
year = {2003},
}

TY - JOUR
AU - Pompe, Waldemar
TI - Korn's First Inequality with variable coefficients and its generalization
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2003
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 44
IS - 1
SP - 57
EP - 70
AB - 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 $.
LA - eng
KW - generalized Korn inequality; continuous coefficients
UR - http://eudml.org/doc/249158
ER -