Double Bubbles on the Real Line with Log-Convex Density

Eliot Bongiovanni, Leonardo Di Giosia, Alejandro Diaz, Jahangir Habib, Arjun Kakkar, Lea Kenigsberg, Dylanger Pittman, Nat Sothanaphan, Weitao Zhu

Published 2017-08-10Version 1

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).