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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
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