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 defined on probability measures in n dimensions, with n greater than or equal to 3. The energy depends on a parameter a and consists of a purely nonlocal term of convolution type, whose interaction kernel reduces to the Coulomb potential for a=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 a belonging to (-1,n-2], the minimiser 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 a=n-2 corresponding to the sign change of the Fourier transform of the interaction potential.