*Accepted Paper*

**Inserted:** 23 feb 2012

**Last Updated:** 29 sep 2012

**Journal:** Calc. Var. Partial Differential Equations

**Year:** 2012

**Abstract:**

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

x

^2$ ($c>0$).

**Download:**