On the isoperimetric problem in Euclidean space with density

Details On the isoperimetric problem in Euclidean space with density On the isoperimetric problem in Euclidean space with density

2006-02-07

arxiv.org/abs/math/0602135v1

Mathematics

Differential Geometry

19 pages, 3 figures

Scientific paper

We study the isoperimetric problem for Euclidean space endowed with 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.

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