Calculus of Variations and Geometric Measure Theory

J. A. Carrillo - J. Mateu - M. G. Mora - L. Rondi - L. Scardia - J. Verdera

The equilibrium measure for an anisotropic nonlocal energy

created by mora on 28 Jun 2019
modified by scardia on 26 Nov 2020



Inserted: 28 jun 2019
Last Updated: 26 nov 2020

Journal: Calc. Var. and PDEs
Year: 2019


In this paper we characterise the minimisers of a one-parameter family of nonlocal and anisotropic energies $I_\alpha$ defined on probability measures in $\mathbb R^n$, with $n\geq 3$. The energy $I_\alpha$ consists of a purely nonlocal term of convolution type, whose interaction kernel reduces to the Coulomb potential for $\alpha=0$ 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 $\alpha\in (-1, n-2]$, the minimiser of $I_\alpha$ 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 $\alpha=n-2$ corresponding to the sign change of the Fourier transform of the interaction potential.

Keywords: Potential theory, nonlocal interactions, global minimisers