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

We study the Stokes problem in a bounded planar domain Ω with a friction type boundary condition that switches between a slip and no‐slip stage. Unlike our previous work , 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 the boundary of Ω, solutions to the Stokes system depend continuously on variations of Ω. 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.