Stokes problem with a solution dependent slip bound: Stability of solutions with respect to domains

Abstract

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.