  The Cahn--Hilliard equation with anisotropic energy contributions frequently
appears in many physical systems. Systematic analytical results for the case
with the relevant logarithmic free energy have been missing so far. We close
this gap and show existence, uniqueness, regularity, and separation properties
of weak solutions to the anisotropic Cahn--Hilliard equation with logarithmic
free energy. Since firstly, the equation becomes highly non-linear, and
secondly, the relevant anisotropies are non-smooth, the analysis becomes quite
involved. In particular, new regularity results for quasilinear elliptic
equations of second order need to be shown.
