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A bilevel shape optimization problem with the exterior Bernoulli free boundary problem as lower-level problem and the control of the free boundary as the upper-level problem is considered. Using the shape of the inner boundary as the control, we aim at reaching a specific shape for the free boundary. A rigorous sensitivity analysis of the bilevel shape optimization in the infinite-dimensional setting is performed. The numerical realization using two different cost functionals presented in this paper demonstrate the efficiency of the approach.

A bilevel shape optimization problem for the exterior Bernoulli free boundary value problem