  A nonlocal Cahn-Hilliard model with a nonsmooth potential of double-well
obstacle type that promotes sharp interfaces in the solution is presented. To
capture long-range interactions between particles, a nonlocal Ginzburg-Landau
energy functional is defined which recovers the classical (local) model for
vanishing nonlocal interactions. In contrast to the local Cahn-Hilliard problem
that always leads to diffuse interfaces, the proposed nonlocal model can lead
to a strict separation into pure phases of the substance. Here, the lack of
smoothness of the potential is essential to guarantee the aforementioned
sharp-interface property. Mathematically, this introduces additional inequality
constraints that, in a weak form, lead to a coupled system of variational
inequalities which at each time instance can be restated as a constrained
optimization problem. We prove the well-posedness and regularity of the
semi-discrete and continuous in time weak solutions, and derive the conditions
under which pure phases are admitted. Moreover, we develop discretizations of
the problem based on finite elements and implicit-explicit time stepping
methods that can be realized efficiently. Finally, we illustrate our
theoretical findings through several numerical experiments in one and two
spatial dimensions that highlight the differences in features of local and
nonlocal solutions and also the sharp interface properties of the nonlocal
model.
