Nonlocal Anisotropic Riesz Interactions with a Physical Confinement

In this work, we fully characterize, in any space dimension, the minimizer of a class of nonlocal and anisotropic Riesz energies defined over probability measures supported on ellipsoids. In the super-Coulombic and Coulombic regimes, we prove that the minimizer is independent of the anisotropy. In contrast, in the sub-Coulombic regime, we show that this property fails: we exhibit an example of anisotropy for which the isotropic minimizer is not optimal. In order to prove our main result, we provide a formula for the potential inside an ellipsoid, valid in any space dimension and involving the hypergeometric function.