On the isoperimetric problem in Euclidean space with density

César Rosales; Antonio Cañete; Vincent Bayle; Frank Morgan

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On the isoperimetric problem in Euclidean space with density

César Rosales·Antonio Cañete·Vincent Bayle·Frank Morgan

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Abstract

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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César Rosales, Antonio Cañete, Vincent Bayle, Frank Morgan. 2006-02-07. On the isoperimetric problem in Euclidean space with density. https://arxiv.org/abs/math/0602135

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