The Optimal Double Bubble for Density $r^p$

TL;DR

This paper investigates the optimal double bubble configuration in a plane with density r^p, extending classical Euclidean results and providing new insights into density-influenced minimal perimeter partitions.

Contribution

It introduces the conjecture that the Euclidean double bubble with a vertex at the origin is optimal in the plane with density r^p and proves a key monotonicity property for the Euclidean case.

Findings

Verified equilibrium condition for the conjectured optimal double bubble

Provided the first direct proof of perimeter monotonicity in Euclidean case

Connected the density problem to classical minimal perimeter partition results

Abstract

In 1993 Foisy et al. proved that the optimal Euclidean planar double bubble---the least-perimeter way to enclose and separate two given areas---is three circular arcs meeting at 120 degrees. We consider the plane with density $r^p$, joining the surge of research on manifolds with density after their appearance in Perelman's 2006 proof of the Poincar\'e Conjecture. Dahlberg et al. proved that the best single bubble in the plane with density $r^p$ is a circle through the origin. We conjecture that the best double bubble is the Euclidean solution with one of the vertices at the origin, for which we have verified equilibrium (first variation or "first derivative" zero). To prove the exterior of the minimizer connected, it would suffice to show that least perimeter is increasing as a function of the prescribed areas. We give the first direct proof of such monotonicity for the Euclidean case. Such arguments were important in the 2002 Annals proof of the double bubble in Euclidean 3-space