On the optimal constants in Korn's and geometric rigidity estimates, in bounded and unbounded domains, under Neumann boundary conditions

We are concerned with the optimal constants: in the Korn inequality under tangential boundary conditions on bounded sets $\mathrm{\Omega }\subset {\mathbb{R}}^n$, and in the geometric rigidity estimate on the whole ${\mathbb{R}}^2$. We prove that the latter constant equals $\sqrt{2}$, and we discuss the relation of the former constants with the optimal Korn’s constants under Dirichlet boundary conditions, and in the whole ${\mathbb{R}}^n$, which are well known to equal $\sqrt{2}$. We also discuss the attainability of these constants and the structure of deformations/displacement fields in the optimal sets.