Even existence requires triangle inequalities

Regularity theorems for compound bubbles or immiscible fluids require that the weights satisfy triangle inequalities. For example, if the interface between two regions of prescribed planar areas has high cost, two tangent disks minimizes weighted perimeter.

Here we note that even existence requires triangle inequalities, even for three fluids. Consider weight 3 for the 01, 02, and 12 interfaces (and weight 1 fo the rest), with say unit areas for 1 and 2 and small area for 3. Now the minimizer should be some kind of 12 double bubble inside a 3 bubble, with 13 and 23 interfaces coinciding with 30 interfaces. At a singularity, 13 and 30 interfaces meeting tangentially and then coincide, conjecturally the only new type of singularity.

Inspired by a question from Gary Lawlor.