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

TL;DR

This paper investigates the stability of solutions to the Stokes problem with a velocity-dependent slip boundary condition, establishing conditions for continuous dependence on domain variations and applying results to optimal shape design.

Contribution

It introduces a new formulation with Lagrange multipliers for the Stokes problem with slip boundary conditions, allowing better analysis of domain dependence and shape optimization.

Findings

Solutions depend continuously on domain variations under certain smoothness conditions.

Existence of solutions to optimal shape design problems is proven for various cost functionals.

The new formulation facilitates analysis of slip boundary conditions with velocity-dependent thresholds.

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.