Korn's First Inequality with variable coefficients and its generalization
Pompe, Waldemar
Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003), p. 57-70 / Harvested from Czech Digital Mathematics Library
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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 $.

Publié le : 2003-01-01
Classification:  26D10,  26D15,  35F15,  35J55 
@article{119367,
     author = {Waldemar Pompe},
     title = {Korn's First Inequality with variable coefficients and its generalization},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     volume = {44},
     year = {2003},
     pages = {57-70},
     zbl = {1098.35042},
     mrnumber = {2045845},
     language = {en},
     url = {http://dml.mathdoc.fr/item/119367}
}

Pompe, Waldemar. Korn's First Inequality with variable coefficients and its generalization. Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003) pp. 57-70. http://gdmltest.u-ga.fr/item/119367/

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