Geodesic
Korn's First Inequality with variable coefficients and its generalization
Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003) no. 1, pp. 57-70 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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 $.
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 $1$, 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 $.
Classification : 26D10, 26D15, 35F15, 35J55
Keywords: Korn's Inequality; coercive inequalities 
@article{CMUC_2003_44_1_a4,
     author = {Pompe, Waldemar},
     title = {Korn's {First} {Inequality} with variable coefficients and its generalization},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {57--70},
     year = {2003},
     volume = {44},
     number = {1},
     mrnumber = {2045845},
     zbl = {1098.35042},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_2003_44_1_a4/}
}
TY  - JOUR
AU  - Pompe, Waldemar
TI  - Korn's First Inequality with variable coefficients and its generalization
JO  - Commentationes Mathematicae Universitatis Carolinae
PY  - 2003
SP  - 57
EP  - 70
VL  - 44
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/CMUC_2003_44_1_a4/
LA  - en
ID  - CMUC_2003_44_1_a4
ER  - 
%0 Journal Article
%A Pompe, Waldemar
%T Korn's First Inequality with variable coefficients and its generalization
%J Commentationes Mathematicae Universitatis Carolinae
%D 2003
%P 57-70
%V 44
%N 1
%U http://geodesic.mathdoc.fr/item/CMUC_2003_44_1_a4/
%G en
%F CMUC_2003_44_1_a4
Pompe, Waldemar. Korn's First Inequality with variable coefficients and its generalization. Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003) no. 1, pp. 57-70. http://geodesic.mathdoc.fr/item/CMUC_2003_44_1_a4/