In this paper we consider non-local energies defined on probability measures in the plane, given by a convolution interaction term plus a quadratic confinement. The interaction kernel is , , with . This kernel is anisotropic except for the Coulomb case . We present a short compact proof of the known surprising fact that the unique minimizer of the energy is the normalized characteristic function of the domain enclosed by an ellipse with horizontal semi-axis and vertical semi-axis . Letting , we find that the semicircle law on the vertical axis is the unique minimizer of the corresponding energy, a result related to interacting dislocations, and previously obtained by some of the authors. We devote the first sections of this paper to presenting some well-known background material in the simplest way possible, so that readers unfamiliar with the subject find the proofs accessible.

在本文中，我们考虑定义在平面概率度量上的非局部能量，由卷积相互作用项加上二次限制给出。交互核是，，。除了库仑情况外，该内核是各向异性的。我们提出了一个简短的紧凑证明，证明了已知的令人惊讶的事实，即能量的唯一最小值是由具有水平半轴和垂直半轴 的椭圆包围的域的归一化特征函数。出租，我们发现垂直轴上的半圆定律是相应能量的唯一极小值，这是与相互作用位错相关的结果，之前由一些作者获得。我们在本文的第一部分以尽可能最简单的方式介绍了一些众所周知的背景材料，以便不熟悉该主题的读者找到易于理解的证明。