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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First and second authors are partially supported by MCyT-Feder research project MTM2004-01387, fourth author by the National Science Foundation.
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Rosales, C., Cañete, A., Bayle, V. et al. On the isoperimetric problem in Euclidean space with density. Calc. Var. 31, 27–46 (2008). https://doi.org/10.1007/s00526-007-0104-y
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DOI: https://doi.org/10.1007/s00526-007-0104-y