On the isoperimetric problem in Euclidean space with density

Abstract

We study the isoperimetric problemfor Euclidean space endowedwith a continuous density. In dimension one, we characterize isoperimetric regions for a unimodal density. In higher dimensions, we prove existence results and we derive stability conditions, which lead to the conjecture that for a radial log-convex density, balls about the origin are isoperimetric regions. Finally, we prove this conjecture and the uniqueness of minimizers for the density exp(|x|2) by using symmetrization techniques.