论文标题
3D各向异性相互作用能的最小化器
Minimizers of 3D anisotropic interaction energies
论文作者
论文摘要
我们研究了一系列的轴对称riesz型奇异相互作用势与各向异性的三个维度。我们将最近工作的一些结果概括为当前环境。对于线性插值凸电的电势，其相关的全球能量最小化是由椭圆形的支撑的显式公式给出的。我们表明，对于不太奇异的各向异性Riesz电位，全局最小化器可能会崩溃成一个或两个维的浓缩度量，从而最大程度地减少了各向同性Riesz的相互作用能量。这些问题的某些部分方面也在不允许一维垂直崩溃的奇异范围内解决。倒塌到较低的尺寸结构以凸度的临界值证明，但不一定是垂直或水平浓缩的措施，从而导致有趣的开放问题。
We study a large family of axisymmetric Riesz-type singular interaction potentials with anisotropy in three dimensions. We generalize some of the results of our recent work in two dimensions to the present setting. For potentials with linear interpolation convexity, their associated global energy minimizers are given by explicit formulas whose supports are ellipsoids. We show that for less singular anisotropic Riesz potentials, the global minimizer may collapse into one or two dimensional concentrated measures which minimize restricted isotropic Riesz interaction energies. Some partial aspects of these questions are also tackled in the intermediate range of singularities in which one dimensional vertical collapse is not allowed. Collapse to lower dimensional structures is proved at the critical value of the convexity but not necessarily to vertically or horizontally concentrated measures, leading to interesting open problems.
