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_α defined on probability measures in Rⁿ, with n≥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∈(−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.