  We prove that solution operators of elliptic obstacle-type variational
inequalities (or, more generally, locally Lipschitz continuous functions
possessing certain pointwise-a.e. convexity properties) are Newton
differentiable when considered as maps between suitable Lebesgue spaces and
equipped with the strong-weak Bouligand differential as a generalized
set-valued derivative. It is shown that this Newton differentiability allows to
solve optimal control problems with H1-cost terms and one-sided pointwise
control constraints by means of a semismooth Newton method. The superlinear
convergence of the resulting algorithm is proved in the infinite-dimensional
setting and its mesh independence is demonstrated in numerical experiments. We
expect that the findings of this paper are also helpful for the design of
numerical solution procedures for quasi-variational inequalities and the
optimal control of obstacle-type variational problems.
