Sticky-disk limit of planar N-bubbles

Abstract

Abstract We study planar N-bubbles that minimize, under an area constraint, a weighted perimeter P ε {P_{\varepsilon}} depending on a small parameter ε > 0 {\varepsilon>0} . Specifically, we weight 2 - ε {2-\varepsilon} the boundary between the bubbles and 1 the boundary between a bubble and the exterior. We prove that as ε → 0 {\varepsilon\to 0} , minimizers of P ε {P_{\varepsilon}} converge to configurations of disjoint disks that maximize the number of tangencies, each weighted by the harmonic mean of the radii of the two tangent disks. We also obtain some information on the structure of minimizers for small ε.