The equilibrium measure for an anisotropic nonlocal energy

Abstract

In this paper we characterise the minimisers of a one-parameter family of nonlocal and anisotropic energies $I_α$ defined on probability measures in $\R^n$, with $n\geq 3$. The energy $I_α$ consists of a purely nonlocal term of convolution type, whose interaction kernel reduces to the Coulomb potential for $α=0$ and is anisotropic otherwise, and a quadratic confinement. The two-dimensional case arises in the study of defects in metals and has been solved by the authors by means of complex-analysis techniques. We prove that for $α\in (-1, n-2]$, the minimiser of $I_α$ is unique and is the (normalised) characteristic function of a spheroid. This result is a paradigmatic example of the role of the anisotropy of the kernel on the shape of minimisers. In particular, the phenomenon of loss of dimensionality, observed in dimension $n=2$, does not occur in higher dimension at the value $α=n-2$ corresponding to the sign change of the Fourier transform of the interaction potential.