Korn's First Inequality with variable coefficients and its generalization

Pompe, Waldemar. "Korn's First Inequality with variable coefficients and its generalization." Commentationes Mathematicae Universitatis Carolinae 44.1 (2003): 57-70. <http://eudml.org/doc/249158>.

@article{Pompe2003,
abstract = {If $\Omega \subset \mathbb \{R\}^n$ is a bounded domain with Lipschitz boundary $\partial \Omega $ and $\Gamma $ is an open subset of $\partial \Omega $, we prove that the following inequality \[ \biggl (\int \_\Omega |A(x)\nabla u(x)|^p\,dx\biggr )^\{1/p\} + \biggl (\int \_\Gamma |u(x)|^p\,d\mathcal \{H\}^\{n-1\}(x)\biggr )^\{1/p\} \ge c\, \Vert u\Vert \_\{W^\{1,p\}\{(\Omega )\}\} \] holds for all $u\in W^\{1,p\}(\Omega ;\mathbb \{R\}^m)$ and $1<p<\infty $, where \[ (A(x)\nabla u(x))\_k=\sum \_\{i=1\}^m\sum \_\{j=1\}^n\, a\_k^\{ij\}(x)\,\frac\{\partial u\_i\}\{\partial x\_j\}(x) \quad (k=1,2,\ldots ,r; r\ge m) \] defines an elliptic differential operator of first order with continuous coefficients on $\overline\{\Omega \}$. As a special case we obtain \[ \int \_\{\Omega \}\bigl |\nabla u(x)F(x)+(\nabla u(x)F(x))^T\bigr |^p\,dx\ge c\int \_\{\Omega \}|\nabla u(x)|^p\,dx\,, \qquad \{(*)\} \] for all $u\in W^\{1,p\}(\Omega ;\mathbb \{R\}^n)$ vanishing on $\Gamma $, where $F:\overline\{\Omega \}\rightarrow M^\{n\times n\}(\mathbb \{R\})$ is a continuous mapping with $\operatorname\{det\} F(x)\ge \mu >0$. Next we show that $(*)$ is not valid if $n\ge 3$, $F\in L^\infty (\Omega )$ 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 = {Korn's Inequality; coercive inequalities; 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 $\Omega \subset \mathbb {R}^n$ is a bounded domain with Lipschitz boundary $\partial \Omega $ and $\Gamma $ is an open subset of $\partial \Omega $, we prove that the following inequality \[ \biggl (\int _\Omega |A(x)\nabla u(x)|^p\,dx\biggr )^{1/p} + \biggl (\int _\Gamma |u(x)|^p\,d\mathcal {H}^{n-1}(x)\biggr )^{1/p} \ge c\, \Vert u\Vert _{W^{1,p}{(\Omega )}} \] holds for all $u\in W^{1,p}(\Omega ;\mathbb {R}^m)$ and $1<p<\infty $, where \[ (A(x)\nabla u(x))_k=\sum _{i=1}^m\sum _{j=1}^n\, a_k^{ij}(x)\,\frac{\partial u_i}{\partial x_j}(x) \quad (k=1,2,\ldots ,r; r\ge m) \] defines an elliptic differential operator of first order with continuous coefficients on $\overline{\Omega }$. As a special case we obtain \[ \int _{\Omega }\bigl |\nabla u(x)F(x)+(\nabla u(x)F(x))^T\bigr |^p\,dx\ge c\int _{\Omega }|\nabla u(x)|^p\,dx\,, \qquad {(*)} \] for all $u\in W^{1,p}(\Omega ;\mathbb {R}^n)$ vanishing on $\Gamma $, where $F:\overline{\Omega }\rightarrow M^{n\times n}(\mathbb {R})$ is a continuous mapping with $\operatorname{det} F(x)\ge \mu >0$. Next we show that $(*)$ is not valid if $n\ge 3$, $F\in L^\infty (\Omega )$ 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 - Korn's Inequality; coercive inequalities; generalized Korn inequality; continuous coefficients
UR - http://eudml.org/doc/249158
ER -