# Korn's First Inequality with variable coefficients and its generalization

• Volume: 44, Issue: 1, page 57-70
• ISSN: 0010-2628

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## Abstract

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If $\Omega \subset {ℝ}^{n}$ is a bounded domain with Lipschitz boundary $\partial \Omega$ and $\Gamma$ is an open subset of $\partial \Omega$, we prove that the following inequality ${\left({\int }_{\Omega }{|A\left(x\right)\nabla u\left(x\right)|}^{p}\phantom{\rule{0.166667em}{0ex}}dx\right)}^{1/p}+{\left({\int }_{\Gamma }{|u\left(x\right)|}^{p}\phantom{\rule{0.166667em}{0ex}}d{ℋ}^{n-1}\left(x\right)\right)}^{1/p}\ge c\phantom{\rule{0.166667em}{0ex}}{\parallel u\parallel }_{{W}^{1,p}\left(\Omega \right)}$ holds for all $u\in {W}^{1,p}\left(\Omega ;{ℝ}^{m}\right)$ and $1, where ${\left(A\left(x\right)\nabla u\left(x\right)\right)}_{k}=\sum _{i=1}^{m}\sum _{j=1}^{n}\phantom{\rule{0.166667em}{0ex}}{a}_{k}^{ij}\left(x\right)\phantom{\rule{0.166667em}{0ex}}\frac{\partial {u}_{i}}{\partial {x}_{j}}\left(x\right)\phantom{\rule{1.0em}{0ex}}\left(k=1,2,...,r;r\ge m\right)$ defines an elliptic differential operator of first order with continuous coefficients on $\overline{\Omega }$. As a special case we obtain ${\int }_{\Omega }{\left|\nabla u\left(x\right)F\left(x\right)+{\left(\nabla u\left(x\right)F\left(x\right)\right)}^{T}\right|}^{p}\phantom{\rule{0.166667em}{0ex}}dx\ge c{\int }_{\Omega }{|\nabla u\left(x\right)|}^{p}\phantom{\rule{0.166667em}{0ex}}dx\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{2.0em}{0ex}}\left(*\right)$ for all $u\in {W}^{1,p}\left(\Omega ;{ℝ}^{n}\right)$ vanishing on $\Gamma$, where $F:\overline{\Omega }\to {M}^{n×n}\left(ℝ\right)$ is a continuous mapping with $detF\left(x\right)\ge \mu >0$. Next we show that $\left(*\right)$ is not valid if $n\ge 3$, $F\in {L}^{\infty }\left(\Omega \right)$ and $detF\left(x\right)=1$, but does hold if $p=2$, $\Gamma =\partial \Omega$ and $F\left(x\right)$ is symmetric and positive definite in $\Omega$.

## How to cite

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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 -

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