  We introduce a fractional variant of the Cahn-Hilliard equation settled in a
bounded domain $\Omega$ of $R^N$ and complemented with homogeneous Dirichlet
boundary conditions of solid type (i.e., imposed in the entire complement of
$\Omega$). After setting a proper functional framework, we prove existence and
uniqueness of weak solutions to the related initial-boundary value problem.
Then, we investigate some significant singular limits obtained as the order of
either of the fractional Laplacians appearing in the equation is let tend to 0.
In particular, we can rigorously prove that the fractional Allen-Cahn,
fractional porous medium, and fractional fast-diffusion equations can be
obtained in the limit. Finally, in the last part of the paper, we discuss
existence and qualitative properties of stationary solutions of our problem and
of its singular limits.
