Double Bubbles on the Real Line with Log-Convex Density

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).

47 pages, 10 figures