On the isoperimetric problem in Euclidean space with density

Rosales, César; Cañete, Antonio; Bayle, Vincent; Morgan, Frank
January 2008
Calculus of Variations & Partial Differential Equations;Jan2008, Vol. 31 Issue 1, p27
Academic Journal
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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