Uniqueness and shape of simplex ground states for aggregation--diffusion energies with mild repulsion
We study global minimizers in $\mathbb{R}^d$ of the free energy $\mathcal{E}_κ=\mathcal{W}+κ\mathcal{U}$, 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 $κ>0$. 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 $d\geq2$. 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 $L^\infty$ to explicit Barenblatt-type profiles supported on ellipsoids determined by the Hessian of the simplex potential. For $d\geq2$ 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 $d\geq2$, where the zero-diffusion minimizer is not unique, we show instead that vanishing diffusion selects the uniform spherical shell.