  This paper continues the investigations from [7] and it is concerned with the
derivation of first-order conditions for a control constrained optimization
problem governed by a nonsmooth elliptic PDE. The control appears in the state
equation as the argument of a regularization of the Heaviside function, while
the nonsmoothness is locally Lipschitz-continuous and directionally
differentiable. Because of the lack of G\^ateaux-differentiability, the
application of standard adjoint calculus is excluded. We derive conditions
under which a strong stationary optimality system can be established, i.e., a
system that is equivalent to the purely primal optimality condition saying that
the directional derivative of the reduced objective in feasible directions is
nonnegative. For this, two assumptions are made on the unknown optimizer. These
are fulfilled if the non-smoothness is locally convex around its
non-differentiable points and if an estimate involving only the given data is
true. The presented findings open the door to future research regarding limit
optimality systems for non-smooth shape optimization problems [7].
