  We study the Stokes problem in a bounded planar domain $\Omega$ with a
friction type boundary condition that switches between a slip and no-slip
stage. Unlike our previous work [6], in the present paper the threshold value
may depend on the velocity field. Besides the usual velocity-pressure
formulation, we introduce an alternative formulation with three Lagrange
multipliers which allows a more flexible treatment of the impermeability
condition as well as optimum design problems with cost functions depending on
the shear and/or normal stress. Our main goal is to determine under which
conditions concerning smoothness of $\Omega$, solutions to the Stokes system
depend continuously on variations of $\Omega$. Having this result at our
disposal, we easily prove the existence of a solution to optimal shape design
problems for a large class of cost functionals.
