Related papers: 1D Triple Bubble Problem with Log-Convex Density

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…

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…

We characterize the perimeter-minimizing double bubbles on all flat two-tori and, as corollaries, on the flat infinite cylinder and the flat infinite strip with free boundary. Specifically, we show that there are five distinct types of…

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…

In 1993 Foisy et al. proved that the optimal Euclidean planar double bubble---the least-perimeter way to enclose and separate two given areas---is three circular arcs meeting at 120 degrees. We consider the plane with density $r^p$, joining…

The generalized soap bubble problem seeks the least perimeter way to enclose and separate n given volumes in R^m. We study the possible configurations for perimeter minimizing bubble complexes enclosing more than two regions. We prove that…

We characterize the critical points of the double bubble problem in $\mathbb{R}^n$ and the triple bubble problem in $\mathbb{R}^3$, in the case the bubbles are convex.

We study the double bubble problem where the perimeter is taken with respect to the hexagonal norm, i.e. the norm whose unit circle in $\mathbb{R}^2$ is the regular hexagon. We provide an elementary proof for the existence of minimizing…

We study the isoperimetric problem for Euclidean space endowed with a continuous density. In dimension one, we characterize isoperimetric regions for a unimodal density. In higher dimensions, we prove existence results and we derive…

Using Brakke's Evolver, we numerically verify conjectured optimal planar double bubbles for density $r^p$ and provide conjectures for triple and quadruple bubbles.

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…

The isoperimetric problem with a density or weighting seeks to enclose prescribed weighted area with minimum weighted perimeter. According to Chambers' recent proof of the Log Convex Density Conjecture, for many densities on $\mathbb{R}^n$…

We consider three-dimensional clusters of identical bubbles packed around a central bubble and calculate their energy and optimal shape. We obtain the surface area and bubble pressures to improve on existing growth laws for…

We present a conjecture, based on computational results, on the area minimizing way to enclose and separate two arbitrary volumes in the flat cubic 3-torus. For comparable small volumes, we prove that an area minimizing double bubble in the…

We study the double bubble problem with perimeter taken with respect to the $\ell_1$ norm on $\mathbb{R}^2$. We give an elementary proof for the existence of minimizing sets for any volume ratio parameter $0<\alpha\le1$ by direct comparison…

Which convex 3D polyhedra can be obtained by gluing several regular hexagons edge-to-edge? It turns out that there are only 15 possible types of shapes, 5 of which are doubly-covered 2D polygons. We give examples for most of them, including…

Using Brakke's Evolver, we numerically verify previous conjectures for optimal double bubbles for density $r^p$ in $R^3$ and our own new conjectures for triple bubbles.

In this paper we consider the isoperimetric problem with double density in an Euclidean space, that is, we study the minimisation of the perimeter among subsets of $\mathbb{R}^n$ with fixed volume, where volume and perimeter are relative to…

We investigate minimal-perimeter configurations of two finite sets of points on the square lattice. This corresponds to a lattice version of the classical double-bubble problem. We give a detailed description of the fine geometry of…

A computer study of clusters of up to 200,000 equal-area bubbles shows for the first time that rounding conjectured optimal hexagonal planar soap bubble clusters reduces perimeter.