Isoperimetric bubbles in spaces with density r^p + a

2025/03/21 by Martyn Gwynne, Gwynne, Martyn, Simon Cox +1
Earth and Planetary Sciences · Materials Science · #28A75 #52B60 #FOS: Mathematics #Ocean Waves and Remote Sensing #Optimization and Control (math.OC) #Pickering emulsions and particle stabilization #Ultrasound and Cavitation Phenomena

paper · pdf · doi:10.48550/arxiv.2503.17177

openalex publication_date 2025/03/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Least perimeter solutions for a region with fixed mass are sought in 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 ≤ 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.

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