Geometrically induced phase transitions in two-dimensional dumbbell-shaped domains

We continue the analysis, started in [23], 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.