The equilibrium measure for an anisotropic nonlocal energy

• Published in: Calculus of variations and partial differential equations Vol. 60; no. 3
• Main Authors: Carrillo, J. A., Mateu, J., Mora, M. G., Rondi, L., Scardia, L., Verdera, J.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.06.2021; Springer Nature B.V
• Subjects: Analysis; Anisotropy; Calculus of Variations and Optimal Control; Optimization; Characteristic functions; Control; Convolution; Coulomb potential; Fourier transforms; Kernels; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Systems Theory; Theoretical; Secondary 49K20; Primary 31A15
• ISSN: 0944-2669; 1432-0835
• DOI: 10.1007/s00526-021-01928-4

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 ≥ 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.