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.

在本文中，我们描述一个单参数家庭非局部和各向异性能量的minimisers \（I_ \阿尔法\）在概率测度定义\（{\ mathbb {R}} ^ N \） ，用\（N \ GE 3 \）。能量\（I_ \ alpha \）由卷积类型的一个纯粹的非局部项组成，其相互作用核降低为\（\ alpha = 0 \）的库仑势，否则为各向异性，并且具有二次约束。二维情况出现在金属缺陷研究中，作者已经通过复杂分析技术解决了这种情况。我们证明对于\（\ alpha \ in（-1，n-2] \），\（I_ \ alpha \）的最小值是唯一的，并且是球体的（归一化）特征函数。该结果是核的各向异性对极小形状的作用的一个典型例子。特别是，在维\（n = 2 \）中观察到的维数损失现象不会在较高维中发生，其值\（\ alpha = n-2 \）对应于傅立叶变换的符号变化。互动潜力。