  We continue the analysis, started in Morini&Slastikov (2012), of a
two-dimensional non-convex variational problem, motivated by studies on
magnetic domain walls trapped by thin necks. The main focus is on the impact of
extreme geometry on the structure of local minimizers representing the
transition between two different constant phases. We address here the case of
general non-symmetric dumbbell-shaped domains with a small constriction and
general multi-well potentials. Our main results concern the existence and
uniqueness of non-constant local minimizers, their full classification in the
case of convex bulks, and the complete description of their asymptotic
behavior, as the size of the constriction tends to zero.
