On variational representations of the constant in the inf sup condition for the Stokes problem

We deduce variational representations of the constant $c_\Omega$ in the inf sup condition for the Stokes problem in a bounded Lipschitz domain in $\mathbb R^d$, $d\geq2$. For any pair of admissible functions the respective variational functional provides an upper bound of $c_\Omega$ and the exact infimum of it is equal to $c_\Omega$. Minimization of the functionals over suitable finite dimensional subspaces generates monotonically decreasing sequences of numbers converging to $c_\Omega$ and, therefore, they can be used for numerical evaluation of the constant.