Solution to the isoperimetric $n$-bubble problem on $\mathbb{R}^1$ with log-concave density
Year: 2022
Type: preprint
Abstract: We study the isoperimetric problem on $\mathbb{R}^1$ with a prescribed density function $f$ that affects how area and perimeter are measured. We examine density functions that are symmetric, radially increasing, and satisfy two additional conditions: they have a point of zero density (at the origin), and they satisfy a "log-concavity" requirement $\left[ \log f \right]'' \leq 0$. Under these conditions, we find that isoperimetric $n$-bubbles satisfy a regular structure and can be identified for arbitrary $n$. This generalizes recent work done on the density function $|x|^p$, and stands in contrast to log-convex density functions which have no such regular structure.
Source: arXiv (Cornell University)
Author: Ross, John
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4221162586