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Isoperimetric bubbles in spaces with density $r^p + a$

Martyn Gwynne, Simon Cox

TL;DR

This paper analyzes the isoperimetric problem under the radial density ρ(r)=r^p+a across dimensions 1 to 3, showing how the additive parameter a induces a transition from off-center to centered optimal regions as p>1. A key tool is the log-convex density theorem, which yields explicit a_crit thresholds above which symmetric balls (in 2D and 3D) or intervals (in 1D) centered at the origin minimize weighted boundary length for a given mass. In 1D, the authors derive closed-form optimal endpoints for special cases (notably p=2, p=1, and 0<p<1) and demonstrate a transition at a_crit; in 2D and 3D they provide exact results for p=2 and use numerical methods (Surface Evolver) to explore p≠2, revealing that symmetry often emerges above a_crit but higher-dimensional shapes may become non-circular/ovoidal when p≠2. The findings connect density-driven symmetry with mass constraints, offering analytic and numerical validation and potential experimental analogues in capillary-like settings. Overall, the work extends isoperimetric theory in spaces with density by detailing how additive density shifts influence geometry and optimal configurations.

Abstract

Least perimeter solutions for a region with fixed mass are sought in ${\mathbb{R}^d}$ on which a density function $ρ(r) = r^p+a$, with $p>0, a>0$, weights both perimeter and mass. On the real line ($d=1$) this is a single interval that includes the origin. For $p \le 1$ the isoperimetric interval has one end at the origin; for larger $p$ there is a critical value of $a$ above which the interval is symmetric about the origin. In the case $p=2$, for $d=2$ and $3$, the isoperimetric region is a circle or sphere, respectively, that includes the origin; the centre moves towards the origin as $a$ increases, with constant radius, and then remains centred on the origin for $a$ above the critical value as the radius decreases.

Isoperimetric bubbles in spaces with density $r^p + a$

TL;DR

This paper analyzes the isoperimetric problem under the radial density ρ(r)=r^p+a across dimensions 1 to 3, showing how the additive parameter a induces a transition from off-center to centered optimal regions as p>1. A key tool is the log-convex density theorem, which yields explicit a_crit thresholds above which symmetric balls (in 2D and 3D) or intervals (in 1D) centered at the origin minimize weighted boundary length for a given mass. In 1D, the authors derive closed-form optimal endpoints for special cases (notably p=2, p=1, and 0<p<1) and demonstrate a transition at a_crit; in 2D and 3D they provide exact results for p=2 and use numerical methods (Surface Evolver) to explore p≠2, revealing that symmetry often emerges above a_crit but higher-dimensional shapes may become non-circular/ovoidal when p≠2. The findings connect density-driven symmetry with mass constraints, offering analytic and numerical validation and potential experimental analogues in capillary-like settings. Overall, the work extends isoperimetric theory in spaces with density by detailing how additive density shifts influence geometry and optimal configurations.

Abstract

Least perimeter solutions for a region with fixed mass are sought in on which a density function , with , weights both perimeter and mass. On the real line () this is a single interval that includes the origin. For the isoperimetric interval has one end at the origin; for larger there is a critical value of above which the interval is symmetric about the origin. In the case , for and , the isoperimetric region is a circle or sphere, respectively, that includes the origin; the centre moves towards the origin as increases, with constant radius, and then remains centred on the origin for above the critical value as the radius decreases.

Paper Structure

This paper contains 14 sections, 8 theorems, 71 equations, 6 figures.

Table of Contents

  1. Introduction
  2. When the density is log-convex
  3. Isoperimetric intervals on the real line (1D)
  4. The case $p=2$ in 1D
  5. The case $p = 1$ in 1D
  6. The case $0< p < 1$ in 1D
  7. Other values of $p$ in 1D
  8. Isoperimetric solutions in the plane (2D)
  9. The case $p=2$ in 2D
  10. The cases $p \ne 2$ in 2D
  11. Isoperimetric solutions in 3D
  12. The case $p=2$ in 3D
  13. Conclusions
  14. Reduction of many intervals to one interval that includes the origin: proof of Lemma \ref{['lem:joinintervals']}

Key Result

Lemma 1

Let the density function be $\rho(r)=r^p+a$, with $p>1$ and $a>0$. A symmetric region of mass $M$ in $d$ dimensions, centred on the origin, is isoperimetric if $a$ is greater than where $k_d$ is the surface area of a $d$-dimensional unit sphere, $k_1= 1$, $k_2 = 2\pi$, $k_3 = 4\pi$.

Figures (6)

  • Figure 1: Examples of the 1D density function $\rho(x)=| x | ^p+a$ considered here and different least-perimeter intervals $[\alpha,\beta]$. (a) If $p=1$, the least perimeter interval always has one end at the origin. (b) For $p=1.5$, a symmetric interval is the least perimeter solution for large enough $a$, (c) For $p=2$, the isoperimetric solution is an asymmetric interval if $a$ is less than a critical value.
  • Figure 2: Contours of weighted perimeter $P(\alpha,\beta)$ for various values of $p$ and $a$, with the thick line indicating the constraint for an interval of unit weighted mass, in terms of the position of the ends of the interval.
  • Figure 3: (a) Minimum perimeter for different values of the exponent $p$ as the parameter $a$ varies, for an interval of unit mass $M_0=1$. For $p=2$, $a_{crit}\approx 0.52$. (b) The positions of the end-points of the interval with least perimeter for different values of the exponent $p$ as the parameter $a$ varies in the case $M_0=1$. In both panels the data for $p=4$ is obtained numerically, while other curves are from the equations in the text.
  • Figure 4: (a) Perimeter, radius and centre position of the conjectured isoperimetric circle for $p=2$, with a mass of $M=1$. The predictions from eqs. (\ref{['eq:2D_p2_perimeter']}) and (\ref{['eq:2D_p2_radius']}) are shown as thin black lines and compared with the result of Surface Evolver simulations, shown as thick coloured lines. The agreement is excellent, suggesting that the isoperimetric solution is indeed a circle. (b) The isoperimetric quotient, formed from the unweighted perimeter and area of the simulated region, $P_u/\sqrt{4A_u\pi}$versus the offset $a$ normalised by its critical value for each $p$, providing further evidence that only for $p=2$ is the isoperimetric region circular.
  • Figure 5: Isoperimetric regions in 2D calculated wth Surface Evolver for several values of $p$. The offset is $a=0.1$, and the mass is $M=1$. The axis of symmetry, and the centre of the region, is constrained to lie on the $x$-axis, without loss of generality.
  • ...and 1 more figures

Theorems & Definitions (20)

  • Definition 1
  • Lemma 1
  • proof
  • Definition 2
  • Definition 3
  • Lemma 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • ...and 10 more