Double Bubbles on the Real Line with Log-Convex Density

TL;DR

This paper investigates the optimal double bubble configurations in the real line and higher dimensions under strictly log-convex density, revealing diverse solutions including intervals and spheres.

Contribution

It characterizes the structure of minimal perimeter double bubbles with log-convex density in one and higher dimensions, extending classical results.

Findings

In 1D, solutions are either two or three contiguous intervals.

In higher dimensions, solutions include standard double bubbles and concentric spheres.

The shape depends on the prescribed volumes and density properties.

Abstract

The classic double bubble theorem says that the least-perimeter way to enclose and separate two prescribed volumes in $\mathbb{R}^N$ is the standard double bubble. We seek the optimal double bubble in $\mathbb{R}^N$ with density, which we assume to be strictly log-convex. For $N=1$ we show that the solution is sometimes two contiguous intervals and sometimes three contiguous intervals. In higher dimensions, we think that the solution is sometimes a standard double bubble and sometimes concentric spheres (e.g. for one volume small and the other large).