Uniqueness and shape of simplex ground states for aggregation--diffusion energies with mild repulsion

Abstract

We study global minimizers in ${\mathbb{R}}^d$ of the free energy ${\mathcal{E}}_{\kappa }=\mathcal{W}+\kappa \mathcal{U}$, where $\mathcal{W}$ is a power-law interaction energy with mild repulsion and $\mathcal{U}$ 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 $\mathcal{W}$ 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 $\kappa >0$. We prove that uniqueness nevertheless persists: for every fixed $m>1$ and all sufficiently small $\kappa >0$, 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 $d\ge 2$. The clusters have width of order ${\kappa }^{1/(d(m-1)+2)}$, 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 $L^{\infty }$ to explicit Barenblatt-type profiles supported on ellipsoids determined by the Hessian of the simplex potential. For $d\ge 2$ 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 $d\ge 2$, where the zero-diffusion minimizer is not unique, we show instead that vanishing diffusion selects the uniform spherical shell.