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Free energy minimizers with radial densities: classification and quantitative stability

Shrey Aryan, Lauro Silini

TL;DR

The paper analyzes the weighted isoperimetric problem in $\mathbb{R}^n$ with radial density $f=e^\psi$ and radial potential $g$, introducing the energy $\mathcal{E}(F)=\mathcal{P}_f(F)+\mathcal{G}_f(F)$ under a fixed weighted volume. It provides a complete 1D classification, shows large-volume optimality by centered balls under $\kappa$-uniform admissibility via calibration, and reveals counterexamples demonstrating that $\psi''+g'>0$ is not universally sufficient for global optimality in higher dimensions. The authors develop a detailed geometric analysis of optimal sets via spherical symmetrization, deriving a constant weighted mean curvature condition and a two-curve (upper/lower) ODE structure for the generating profile, leading to a full uniqueness result for minimizers and an isoperimetric-type inequality. Finally, they establish sharp quantitative stability: a quadratic bound $\mathcal{E}(E)-\mathcal{E}(B_R) \ge c \lvert E\triangle B_R\rvert_f^2$ for volume-constrained near-minimizers, leveraging a Fuglede-type near-spherical estimate and De Giorgi regularity theory to extend to the general setting. These results advance the understanding of weighted isoperimetry in Euclidean/Riemannian contexts and provide precise stability controls for radial densities and exterior potentials.

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.

Free energy minimizers with radial densities: classification and quantitative stability

TL;DR

The paper analyzes the weighted isoperimetric problem in with radial density and radial potential , introducing the energy under a fixed weighted volume. It provides a complete 1D classification, shows large-volume optimality by centered balls under -uniform admissibility via calibration, and reveals counterexamples demonstrating that is not universally sufficient for global optimality in higher dimensions. The authors develop a detailed geometric analysis of optimal sets via spherical symmetrization, deriving a constant weighted mean curvature condition and a two-curve (upper/lower) ODE structure for the generating profile, leading to a full uniqueness result for minimizers and an isoperimetric-type inequality. Finally, they establish sharp quantitative stability: a quadratic bound for volume-constrained near-minimizers, leveraging a Fuglede-type near-spherical estimate and De Giorgi regularity theory to extend to the general setting. These results advance the understanding of weighted isoperimetry in Euclidean/Riemannian contexts and provide precise stability controls for radial densities and exterior potentials.

Abstract

We study the isoperimetric problem with a potential energy in weighted by a radial density 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 and are monotone. Additionally, we strengthen this result by deriving a sharp quantitative stability inequality.

Paper Structure

This paper contains 17 sections, 43 theorems, 218 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Structure of the paper
  4. Necessary and sufficient conditions on the weights
  5. Existence, regularity, first and second variations, and optimal profile
  6. In one dimension
  7. Large volume regime
  8. Counter-examples in higher dimensions
  9. Characterization of optimal sets
  10. Spherical symmetrization and terminology
  11. Some useful lemmas
  12. Upper Curve
  13. Lower Curve
  14. Proof of Theorem \ref{['thm:uniqueness']} and Corollary \ref{['cor:isop_ineq']}
  15. Stability
  16. ...and 2 more sections

Key Result

Theorem 1.2

Let $n=1$ and $\psi,g$ be radial admissible weights. Then, centered intervals minimize eq:optimization_problem for any volume if and only if $\min_{r\geq 0}\psi(r)=\psi(0)$. Furthermore, if $\psi,g$ are strictly admissible and $\min_{r\geq 0}\psi(r)=\psi(0)$, then centered intervals uniquely solve e

Figures (2)

  • Figure 1: First construction of $\psi$ and $g$.
  • Figure 2: Construction of $\psi$.

Theorems & Definitions (103)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Remark 1.7
  • Theorem 1.8
  • Theorem 2.1
  • proof
  • ...and 93 more