Calculus of Variations and Geometric Measure Theory

A. Figalli - F. Maggi

On the isoperimetric problem for radial log-convex densities

created by maggi on 23 Feb 2012
modified by figalli on 29 Sep 2012


Accepted Paper

Inserted: 23 feb 2012
Last Updated: 29 sep 2012

Journal: Calc. Var. Partial Differential Equations
Year: 2012


Given a smooth, radial, uniformly log-convex density $e^V$ on $\mathbb{R}^n$, $n\ge 2$, we characterize isoperimetric sets $E$ with respect to weighted perimeter $\int_{\partial E}e^Vd\mathcal{H}^{n-1}$ and weighted volume $m=\int_Ee^V$ as balls centered at the origin, provided $m \in [0,m_0)$ for some (potentially computable) $m_0>0$; this affirmatively answers a conjecture by Rosales, Canete, Bayle, and Morgan, "On the isoperimetric problem in the Euclidean space with density", Calc. Var. PDE 31 (2008), for such values of the weighted volume parameter. We also prove that the set of weighted volumes such that this characterization holds true is open, thus reducing the proof of the full conjecture to excluding the possibility of bifurcation values of the weighted volume parameter. Finally, we show the validity of the conjecture when $V$ belongs to a $C^2$-neighborhood of $c
^2$ ($c>0$).