Free energy minimizers with radial densities: classification and quantitative stability

We study the isoperimetric problem with a potential energy \(g\) in \(\mathbb{R}^n\) 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 \(g = 0\), the con...

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Published in:arXiv.org 2024-12
Main Authors: Shrey Aryan, Silini, Lauro
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Free energy
Isoperimetric problem
Optimization
Potential energy
Stability
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Silini, Lauro
description We study the isoperimetric problem with a potential energy \(g\) in \(\mathbb{R}^n\) 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 \(g = 0\), the condition \(\ln(f)'' + g' \geq 0\) 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.
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subjects Free energy
Isoperimetric problem
Optimization
Potential energy
Stability
title Free energy minimizers with radial densities: classification and quantitative stability
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