Double Bubbles on the Line with Log-convex Density $f$ with $(\log f)'$ Bounded

TL;DR

This paper extends the understanding of double bubbles on the line with log-convex density by analyzing cases where the derivative of the log-density is bounded, revealing new blowup phenomena and providing a specific density example.

Contribution

It generalizes previous results to bounded derivative cases and introduces a density with finite positive blowup time for the tie function.

Findings

Tie function exists but may blow up in finite time

First example of density with positive finite blowup time

Extended results to a broader class of densities

Abstract

We extend results of Bongiovanni et al. on double bubbles on the line with log-convex density to the case where the derivative of the log of the density is bounded. We show that the tie function between the double interval and the triple interval still exists but may blow up to infinity in finite time. For the first time, a density is presented for which the blowup time is positive and finite.