二维中一大类各向异性吸引-排斥相互作用能的全局最小值

IF 4.3 3区 材料科学 Q1 ENGINEERING, ELECTRICAL & ELECTRONIC
José A. Carrillo, Ruiwen Shu
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引用次数: 6

摘要

研究了二维上具有各向异性的riesz型奇异相互作用势。它们相关的全局能量最小值由明确的公式给出,这些公式的支持在某些假设下由椭圆决定。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构型是基于线性凸性参数的全局最小值的尖锐范围。此外,对于某些各向异性部件,我们证明了当参数值较大时,全局最小值只能由与一维最小值相对应的垂直集中测度给出。我们还表明,这些椭圆支持的构型通常不会在凸度临界值处坍缩为垂直集中的度量,导致两者之间的参数有一个有趣的间隙。在这个中间范围内,我们由无穷小凹性得出:任何局部极小器的任何超水平集在适当意义上都不存在内点。此外,对于某些各向异性部件,它们的支撑在有限的参数范围内不能包含任何垂直段,而且全局最小值预计会表现出锯齿状行为。所有这些结果都适用于对数排斥势的极限情况,扩展和推广了以往文献中的结果。各向异性部件导致更复杂的行为的各种例子进行了数值探索。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

Global minimizers of a large class of anisotropic attractive-repulsive interaction energies in 2D

Global minimizers of a large class of anisotropic attractive-repulsive interaction energies in 2D

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.

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来源期刊
CiteScore
7.20
自引率
4.30%
发文量
567
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