| Title:
|
Korn's First Inequality with variable coefficients and its generalization (English) |
| Author:
|
Pompe, Waldemar |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
44 |
| Issue:
|
1 |
| Year:
|
2003 |
| Pages:
|
57-70 |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $. (English) |
| Keyword:
|
Korn's Inequality |
| Keyword:
|
coercive inequalities |
| MSC:
|
26D10 |
| MSC:
|
26D15 |
| MSC:
|
35F15 |
| MSC:
|
35J55 |
| idZBL:
|
Zbl 1098.35042 |
| idMR:
|
MR2045845 |
| . |
| Date available:
|
2009-01-08T19:27:22Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119367 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| . |
