  This paper is concerned with a shape optimization problem governed by a
non-smooth PDE, i.e., the nonlinearity in the state equation is not necessarily
differentiable. We follow the functional variational approach of [36] where the
set of admissible shapes is parametrized by a large class of continuous
mappings. This methodology allows for both boundary and topological variations.
It has the advantage that one can rewrite the shape optimization problem as a
control problem in a function space. To overcome the lack of convexity of the
set of admissible controls, we provide an essential density property. This
permits us to show that each parametrization associated to the optimal shape is
the limit of global optima of non-smooth distributed optimal control problems.
The admissible set of the approximating minimization problems is a convex
subset of a Hilbert space of functions. Moreover, its structure is such that
one can derive strong stationary optimality conditions [5]. This opens the door
to future research concerning sharp first-order necessary optimality conditions
in form of a qualified optimality system.
