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.

Automated summary

We study the characteristics of a large family of Riesz-type singular interaction potentials with anisotropy in two dimensions. We provide explicit formulas for their associated global energy minimizers, whose supports are determined by ellipses under certain assumptions. By parameterizing the anisotropic part, we characterize the range in which these configurations are the global minimizers based on linear convexity arguments. Moreover, we observe that for certain anisotropic parts, the global minimizer is only given by vertically concentrated measures for large parameter values. We also discover an interesting gap of parameters between the collapse of the ellipse-supported configurations and the critical value for convexity. Additionally, we analyze the properties of superlevel sets and the behavior of global minimizers for certain anisotropic parts. These results extend and generalize previous findings in the literature, and numerical examples further explore even more complex behavior of anisotropic parts.