  We construct weak solutions of the anisotropic inverse mean curvature flow
(A-IMCF) under very mild assumptions both on the anisotropy (which is simply a
norm in $\mathbb R^N$ with no ellip\-ticity nor smoothness requirements, in
order to include the crystalline case) and on the initial data. By means of an
approximation procedure introduced by Moser, our solutions are limits of
anisotropic $p$-harmonic functions or $p$-capacitary functions (after a change
of variable), and we get uniqueness both for the approximating solutions (i.e.,
uniqueness of $p$-capacitary functions) and the limiting ones. Our notion of
weak solution still recovers variational and geometric definitions similar to
those introduced by Huisken-Ilmanen, but requires to work within the broader
setting of $BV$-functions. Despite of this, we still reach classical results
like the continuity and exponential growth of perimeter, as well as outward
minimizing properties of the sublevel sets. Moreover, by assuming the extra
regularity given by an interior rolling ball condition (where a sliding Wulff
shape plays the role of a ball), the solutions are shown to be continuous and
satisfy Harnack inequalities. Finally, examples of explicit solutions are
built.
