On the Optimal Constants in Korn's and Geometric Rigidity Estimates, in Bounded and Unbounded Domains, under Neumann Boundary Conditions

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