The equilibrium measure for an anisotropic nonlocal energy - Carrillo, JA; Mateu, J; Mora, MG; Rondi, L; Scardia, L; Verdera, J (Springer, 2021)
In this paper we characterise the minimisers of a one-parameter family of nonlocal and anisotropic energies 𝐼𝛼 defined on probability measures in ℝ𝑛, with 𝑛≥3. The energy 𝐼𝛼 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 𝐼𝛼 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.