Best constants in Korn‐Poincaré's inequalities on a slab

We establish the best constants in the Poincaré‐type and the trace‐type inequalities for the quadratic form \documentclass{article}\pagestyle{empty}\begin{document}$ \lambda ||\,{\rm div}\,{\rm u}\,||_{L^2 }^2 \, + \,2\,\mu \,||\,\,(\nabla {\rm u}\, + \,\nabla {\rm u}^{\rm T})/2\,||_{L^2 }^2 $\end{document} which is fundamental in elasticity theory, on the space of H 1 vector fields u on a slab vanishing on one or both of its sides. We similarly calculate those constants for the case of H 1 divergence‐free vector fields. Our method, which is fairly general, has another practical application to the quadratic form ∑ j , k ( a jk ∂ k u , ∂ j u ) L 2 with coefficients a jk = a kj ε C in H 1 scalar functions u on a slab.