  We present a general numerical approach to shape optimization with state
constraints for 2-dimensional geometries, without relaxing the constraints. To
do this we reformulate the problem on a fixed reference domain using a
conformal pull-back. The shape dependence is then hidden in a conformal
parameter, which appears as a coefficient in the differential operators. The
problem on the reference domain can be discretized, leading to an NLP which can
be handled using existing solvers. Furthermore, we deal with the question how
constraints on the conformal parameter can be used to preserve characteristic
features of the geometry. We introduce this approach with the help of a Stokes
flow, where the task is finding a shape such that the wall shear stress is
uniformly close to some given target.
