Optimal Korn's inequalities for divergence-free vector fields with application to incompressible linear elastodynamics

The best constants in two types of Korn's inequality for divergence-free vector fields on the half space are established. Moreover, one of those, which is equal to the square of the Rayleigh speed for a certain incompressible elastic material, proves to be important to the solvability of a certain mixed problem of incompressible linear elastodynamics with a time-dependent displacement-traction interface on the boundary.