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

Abstract

Least perimeter solutions for a region with fixed mass are sought in ${\mathbb{R}}^d$ on which a density function $\ddot{I}(r)=r^p+a$, 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.

22 pages