  The inf-sup constant for the divergence, or LBB constant, is related to the
Cosserat spectrum. It has been known for a long time that on non-smooth domains
the Cosserat operator has a non-trivial essential spectrum, which can be used
to bound the LBB constant from above. We prove that the essential spectrum on a
plane polygon consists of an interval related to the corner angles and that on
three-dimensional domains with edges, the essential spectrum contains such an
interval. We obtain some numerical evidence for the extent of the essential
spectrum on domains with axisymmetric conical points by computing the roots of
explicitly given holomorphic functions related to the corner Mellin symbol.
Using finite element discretizations of the Stokes problem, we present
numerical results pertaining to the question of the existence of eigenvalues
below the essential spectrum on rectangles and cuboids.
