We study a large family of Riesz-type singular interaction potentials with anisotropy in two dimensions. Their associated global energy minimizers are given by explicit formulas whose supports are determined by ellipses under certain assumptions. More precisely, by parameterizing the strength of the anisotropic part we characterize the sharp range in which these explicit ellipse-supported configurations are the global minimizers based on linear convexity arguments. Moreover, for certain anisotropic parts, we prove that for large values of the parameter the global minimizer is only given by vertically concentrated measures corresponding to one dimensional minimizers. We also show that these ellipse-supported configurations generically do not collapse to a vertically concentrated measure at the critical value for convexity, leading to an interesting gap of the parameters in between. In this intermediate range, we conclude by infinitesimal concavity that any superlevel set of any local minimizer in a suitable sense does not have interior points. Furthermore, for certain anisotropic parts, their support cannot contain any vertical segment for a restricted range of parameters, and moreover the global minimizers are expected to exhibit a zigzag behavior. All these results hold for the limiting case of the logarithmic repulsive potential, extending and generalizing previous results in the literature. Various examples of anisotropic parts leading to even more complex behavior are numerically explored.

中文翻译：

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二维中一大类各向异性吸引-排斥相互作用能的全局最小化

我们研究了一大类二维各向异性的里斯型奇异相互作用势。它们相关的全局能量最小化器由显式公式给出，其支持由某些假设下的椭圆确定。更准确地说，通过参数化各向异性部分的强度，我们描述了这些显式椭圆支持配置是基于线性凸性参数的全局最小化的尖锐范围。此外，对于某些各向异性部分，我们证明，对于较大的参数值，全局最小化器仅由与一维最小化器相对应的垂直集中度量给出。我们还表明，这些椭圆支持的配置通常不会在凸性临界值处塌陷为垂直集中的度量，从而导致之间的参数存在有趣的差距。在这个中间范围内，我们通过无穷小凹性得出结论，在适当的意义上，任何局部最小化器的任何超级集合都不具有内点。此外，对于某些各向异性零件，它们的支撑不能包含参数范围有限的任何垂直段，而且全局最小化器预计会表现出锯齿形行为。所有这些结果都适用于对数排斥势的极限情况，扩展并概括了文献中先前的结果。对导致更复杂行为的各向异性零件的各种示例进行了数值探索。