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Korn’s first inequality with variable coefficients and its generalization. (English) Zbl 1098.35042
For a bounded domain $$\Omega \subset {\mathbb R}^n$$ with a Lipschitz boundary $$\Gamma$$, it is shown that the functional $$f\: u\to \int _\Omega | A(x)\nabla u(x)| ^p\,\text dx +\int _\Gamma | u(x)| ^p\,\text d{\mathcal H}^{n-1}(x)$$ dominates the usual Sobolev norm on $$W^{1,p}(\Omega ;{\mathbb R}^m)$$ in the sense $$f(u)\geq c\| u\| _{W^{1,p}(\Omega ;{\mathbb R}^m)}^p$$ for some $$c>0$$, provided $$1<p<+\infty$$ and the tensor $$A(\cdot )$$ defines, through the formula $$(A\nabla u)_k:=\sum _{i=1}^m\sum _{j=1}^na_k^{ij}\partial u_i/\partial x_j$$, a suitable elliptic operator with continuous coefficients $$a_k^{ij}$$ on $$\bar \Omega$$. A particular case $$n=m$$ and $$A(x)\nabla u(x) =\nabla u(x)F(x)+(\nabla u(x)F(x))^\top$$ covers the classical Korn’s inequality if $$F =\text{the identity matrix}$$. A consequence of general results is that the mentioned estimate holds if $$F(\cdot )$$ is continuous with det$$\,F(x)\geq \mu >0$$. This is further shown to be a very fine result because it need not hold if $$F$$ is only an $$L^\infty$$-function unless $$p=2=n$$ and $$\Gamma =\partial \Omega$$.

##### MSC:
 35B45 A priori estimates in context of PDEs 35J55 Systems of elliptic equations, boundary value problems (MSC2000) 26D10 Inequalities involving derivatives and differential and integral operators
##### Keywords:
generalized Korn inequality; continuous coefficients
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