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We study a large family of axisymmetric Riesz-type singular interaction potentials with anisotropy in three dimensions. We generalize some of the results of the recent work [J. A. Carrillo and R. Shu, Global minimizers of a large class of anisotropic attractive-repulsive interaction energies in 2D,  Comm. Pure Appl. Math.  (2023), 10.1002/cpa.22162] in two dimensions to the present setting. For potentials with linear interpolation convexity, their associated global energy minimizers are given by explicit formulas whose supports are ellipsoids. We show that, for less singular anisotropic Riesz potentials, the global minimizer may collapse into one or two-dimensional concentrated measures which minimize restricted isotropic Riesz interaction energies. Some partial aspects of these questions are also tackled in the intermediate range of singularities in which one-dimensional vertical collapse is not allowed. Collapse to lower-dimensional structures is proved at the critical value of the convexity but not necessarily to vertically or horizontally concentrated measures, leading to interesting open problems.

Minimizers of 3D anisotropic interaction energies