December 2024
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We study the isoperimetric problem with a potential energy g in weighted by a radial density f and analyze the geometric properties of minimizers. Notably, we construct two counterexamples demonstrating that, in contrast to the classical isoperimetric case , the condition does not generally guarantee the global optimality of centered spheres. However, we demonstrate that centered spheres are globally optimal when both f and g are monotone. Additionally, we strengthen this result by deriving a sharp quantitative stability inequality.