Free energy minimizers with radial densities: classification and quantitative stability

TL;DR

This paper investigates the geometric properties of minimizers in a weighted isoperimetric problem with potential energy, revealing conditions for optimality and stability, and providing counterexamples to classical assumptions.

Contribution

It introduces new counterexamples showing classical conditions are insufficient for global optimality and establishes sharp quantitative stability results for centered spheres.

Findings

Counterexamples show classical conditions do not guarantee global optimality.

Centered spheres are globally optimal when both density and potential are monotone.

A sharp quantitative stability inequality for centered spheres is derived.

Abstract

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.