论文标题
某些非局部各向异性能量的显式最小化器：一个简短的证明
Explicit minimisers of some nonlocal anisotropic energies: a short proof
论文作者
论文摘要
在本文中，我们考虑了通过卷积相互作用项和二次限制给出的基于平面概率度量定义的非局部能量。相互作用内核为$ - \ log | z |+α\，x^2/| z |^2，\; z = x+iy，$ a $ -1 <α<1。$ $此内核是各向异性的，除了库仑的情况$α=0。$我们提供了一个简短的紧凑型证明，表明能量的独特最小值是eLlipse nips nipse nipse nipse nipse nipse nips nips n e e e e e e e e e e emanipse n e Ellipse semiain semiain n emimigatizan $ \ semi-axpen $ \ ssqrt}的归一化特征的功能。 $ \ sqrt {1+α}。$让$α\至1^ - $我们发现垂直轴上的半圆定律是相应能量的独特最小化器，这是与相互作用的错位相关的结果，并且先前由一些作者获得。我们将本文的第一部分用于以最简单的方式展示一些著名的背景材料，以便读者不熟悉该主题
In this paper we consider nonlocal energies defined on probability measures in the plane, given by a convolution interaction term plus a quadratic confinement. The interaction kernel is $-\log|z|+α\, x^2/|z|^2, \; z=x+iy,$ with $-1 < α< 1.$ This kernel is anisotropic except for the Coulombic case $α=0.$ We present a short compact proof of the known surprising fact that the unique minimiser of the energy is the normalised characteristic function of the domain enclosed by an ellipse with horizontal semi-axis $\sqrt{1-α}$ and vertical semi-axis $\sqrt{1+α}.$ Letting $α\to 1^-$ we find that the semicircle law on the vertical axis is the unique minimiser 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
