The equilibrium measure for an anisotropic nonlocal energy

In this paper we characterise the minimisers of a one-parameter family of nonlocal and anisotropic energies $$I_\alpha $$ I α defined on probability measures in $${\mathbb {R}}^n$$ R n , with $$n\ge 3$$ n ≥ 3 . The energy $$I_\alpha $$ I α consists of a purely nonlocal term of convolution type, whose interaction kernel reduces to the Coulomb potential for $$\alpha =0$$ 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 $$\alpha \in (-1, n-2]$$ α ∈ ( - 1 , n - 2 ] , the minimiser of $$I_\alpha $$ 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 $$n=2$$ n = 2 , does not occur in higher dimension at the value $$\alpha =n-2$$ α = n - 2 corresponding to the sign change of the Fourier transform of the interaction potential.