On Variational Representations of the Constant in the Inf-Sup Condition for the Stokes Problem

Resumo

Variational representations of the constant c_Ω in the inf-sup condition for the Stokes problem in a bounded Lipschitz domain in ℝ^d, d ≥ 2, are deduced. For any pair of admissible functions, the respective variational functional provides an upper bound of c_Ω and the exact infimum of it is equal to c_Ω. Minimization of the functionals over suitable finite dimensional subspaces generates monotonically decreasing sequences of numbers converging to c_Ω and, therefore, they can be used for numerical evaluation of the constant.