Isoperimetric 3- and 4-bubble results on $\mathbb{R}$ with density $|x|$

TL;DR

This paper investigates the structure of optimal partitions (bubbles) in one-dimensional space with density |x|, revealing regular patterns for 3- and 4-bubble configurations and extending known results from simpler cases.

Contribution

It provides new insights into the structure of multi-bubble isoperimetric solutions on the real line with density |x|, specifically for three and four regions.

Findings

Optimal 3- and 4-bubble regions alternate across the origin.

Smaller regions are closer to the origin.

Results extend previous single- and double-bubble findings.

Abstract

We study the isoperimetric problem on $\mathbb{R}^1$ with a prescribed density function $f(x) = |x|$. Under these conditions, we find that isoperimetric $3$-bubble and $4$-bubble results satisfy a regular structure. As our regions increase in size, the intervals that form them alternate back-and-forth across the origin, with the smaller regions closer to the origin. This expands on previously known observations about the single- and double-bubble results on $\mathbb{R}$ with density $|x|^p$.