  In the Allen-Cahn theory of phase transitions, minimizers partition the
domain in subregions, the sets where a minimizer is near to one or to another
of the zeros of the potential. These subregions that model the phases are
separated by a tiny Diffuse Interface. Understanding the shape of this diffuse
interface is an important step toward the description of the structure of
minimizers. We assume Dirichlet data and present general conditions on the
domain and on the boundary datum ensuring the connectivity of the diffuse
interface. Then we restrict to the case of two dimensions and show that the
phases can be separated, in a certain optimal way, by a connected network with
a well defined structure. This network is contained in the diffuse interface
and is a priori unknown. Under general assumption on the potential and on the
Dirichlet datum, we show that, if we assume that the phase are connected, then
we can obtain precise information on the shape of the network and in turn a
detailed description of the fine structure of minimizers. In particular we can
characterize the shape and the size of the various phases and also how they
depend on the surface tensions.
