We study the Stokes problems in a bounded planar domain Ω with a friction type boundary condition that switches between a slip and no-slip stage. Our main goal is to determine under which conditions concerning the smoothness of Ω solutions to the Stokes system with the slip boundary conditions 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. In order to release the impermeability condition, whose numerical treatment could be troublesome, we use a penalty approach. We introduce a family of shape optimization problems with the penalized state relations. Finally we establish convergence properties between solutions to the original and modified shape optimization problems when the penalty parameter tends to zero.
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The research of the first author was supported by the grant P201/12/0671 of GAČR. The second author acknowledges the support of the grant 201/09/0917 of GAČR and RVO: 67985840. Finally a part of this paper was done in co-operation of the first and the third author in the frame of the ERASMUS project.
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Haslinger, J., Stebel, J. & Sassi, T. Shape optimization for Stokes problem with threshold slip. Appl Math 59, 631–652 (2014). https://doi.org/10.1007/s10492-014-0077-z
- Stokes problem
- friction boundary condition
- shape optimization