Uniqueness and shape of simplex ground states for aggregation--diffusion energies with mild repulsion
Abstract
We study global minimizers in ℝd of the free energy ℰκ=𝒲+κ𝒰, where 𝒲 is a power-law interaction energy with mild repulsion and 𝒰 is the porous-medium internal energy with exponent m>1. In a range of exponents that we call the strict simplex regime, results of Lim--McCann and Davies--Lim--McCann show that 𝒲 is minimized, uniquely up to rigid motions, by equal point masses at the vertices of a regular unit d-simplex. Diffusion excludes such atomic minimizers for every κ>0. We prove that uniqueness nevertheless persists: for every fixed m>1 and all sufficiently small κ>0, the free energy has a unique global minimizer up to rigid motions. After a suitable rigid motion, this minimizer is invariant under every symmetry of the simplex, and its support splits into d+1 clusters near the vertices, each of mass exactly 1/(d+1); in particular, it is not radially symmetric when d≥2. The clusters have width of order κ1/(d(m−1)+2), which is also the exact order of the quadratic Wasserstein distance from the minimizer to the atomic simplex measure. Rescaled at this scale, the clusters converge in L1 and L∞ to explicit Barenblatt-type profiles supported on ellipsoids determined by the Hessian of the simplex potential. For d≥2 these ellipsoids are spheroids flattened along the directions of the vertices; when the repulsive exponent exceeds two, their aspect ratio equals (d+1)−1/2, independently of all other parameters. At the endpoint (4,2) of this regime in dimensions d≥2, where the zero-diffusion minimizer is not unique, we show instead that vanishing diffusion selects the uniform spherical shell.
Related
BibTeX
@misc{lim2026uniqueness,
title = {{Uniqueness and shape of simplex ground states for aggregation--diffusion energies with mild repulsion}},
author = {Tongseok Lim and Geuntaek Seo},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02890},
eprint = {2611.02890},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02890},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Uniqueness and shape of simplex ground states for aggregation--diffusion energies with mild repulsion
AU - Tongseok Lim
AU - Geuntaek Seo
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02890
UR - https://arxiv.org/abs/2611.02890
AB - We study global minimizers in ℝ^d of the free energy ℰ_κ=𝒲+κ𝒰, where 𝒲 is a power-law interaction energy with mild repulsion and 𝒰 is the porous-medium internal energy with exponent m>1. In a range of exponents that we call the strict simplex regime, results of Lim--McCann and Davies--Lim--McCann show that 𝒲 is minimized, uniquely up to rigid motions, by equal point masses at the vertices of a regular unit d-simplex. Diffusion excludes such atomic minimizers for every κ>0. We prove that uniqueness nevertheless persists: for every fixed m>1 and all sufficiently small κ>0, the free energy has a unique global minimizer up to rigid motions. After a suitable rigid motion, this minimizer is invariant under every symmetry of the simplex, and its support splits into d+1 clusters near the vertices, each of mass exactly 1/(d+1); in particular, it is not radially symmetric when d≥2. The clusters have width of order κ^1/(d(m−1)+2), which is also the exact order of the quadratic Wasserstein distance from the minimizer to the atomic simplex measure. Rescaled at this scale, the clusters converge in L^1 and L^∞ to explicit Barenblatt-type profiles supported on ellipsoids determined by the Hessian of the simplex potential. For d≥2 these ellipsoids are spheroids flattened along the directions of the vertices; when the repulsive exponent exceeds two, their aspect ratio equals (d+1)^−1/2, independently of all other parameters. At the endpoint (4,2) of this regime in dimensions d≥2, where the zero-diffusion minimizer is not unique, we show instead that vanishing diffusion selects the uniform spherical shell.
ER -