Sticky-disk limit of planar $N$-bubbles

TL;DR

This paper investigates the limiting behavior of planar N-bubbles minimizing a weighted perimeter as a small parameter approaches zero, revealing convergence to configurations of tangent disks with maximal tangencies.

Contribution

It establishes the limit of minimizers for a weighted perimeter problem, showing convergence to tangent disk configurations and analyzing their structure for small epsilon.

Findings

Minimizers converge to tangent disk configurations as epsilon approaches zero.

Configurations maximize the number of tangencies weighted by harmonic means of radii.

Structural properties of minimizers are characterized for small epsilon.

Abstract

We study planar $N$-clusters that minimize, under an area constraint, a weighted perimeter $P_\varepsilon$ depending on a small parameter $\varepsilon>0$. Specifically we weight $2-\varepsilon$ the boundary between the interior chambers and $1$ the boundary between an interior chamber and the exterior one. We prove that as $\varepsilon\to 0$ minimizers of $P_\varepsilon$ 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 $\varepsilon$.