The equilibrium measure for an anisotropic nonlocal energy

Abstract

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 $n\ge 3$. 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.

References in corpus (2)

• Existence of Ground States of Nonlocal-Interaction Energies
• An Energy Estimate for Dislocation Configurations and the Emergence of Cosserat-Type Structures in Metal Plasticity