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arXiv · 2201.02197

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

Evan Alexander·Emily Burns·John Ross·Jesse Stovall·Zariah Whyte
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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 $|x|^p$.

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Evan Alexander, Emily Burns, John Ross, Jesse Stovall, Zariah Whyte. 2022-01-05. Isoperimetric 3- and 4-bubble results on $\mathbb{R}$ with density $|x|$. https://arxiv.org/abs/2201.02197

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