Equilibrium of charges and differential equations solved by polynomials

电荷平衡与多项式求解的微分方程 II

Equilibrium of Charges and Differential Equations Solved by Polynomials II

本文解决了环量比为-2的两类二维库仑电荷平衡问题，证明需用三阶微分算子的Darboux型变换替代二阶Schrodinger算子，并发现该变换可生成含第三类电荷的平衡构型。

Logarithmic Equilibrium on the Sphere in the Presence of Multiple Point Charges

With the sphere $\mathbb{S}^2 \subset \mathbb{R}^3$ as a conductor holding a unit charge with logarithmic interactions, we consider the problem of determining the support of the equilibrium measure in the presence of an external field consisting of finitely many point charges on the surface of the sphere. We determine that for any such configuration, the complement of the equilibrium support is the stereographic preimage from the plane of a union of classical quadrature domains, whose orders sum to the number of point charges.

A vector equilibrium problem for symmetrically located point charges on a sphere

We study the equilibrium measure on the two dimensional sphere in the presence of an external field generated by r+1 equal point charges that are symmetrically located around the north pole. The support of the equilibrium measure is known as the droplet. The droplet has a motherbody which we characterize by means of a vector equilibrium problem (VEP) for r measures in the complex plane. The model undergoes two transitions which is reflected in the support of the first component of the minimizer of the VEP, namely the support can be a finite interval containing 0, the union of two intervals, or…

On equilibrium shapes of charged flat drops

Equilibrium shapes of two-dimensional charged, perfectly conducting liquid drops are governed by a geometric variational problem that involves a perimeter term modeling line tension and a capacitary term modeling Coulombic repulsion. Here we give a complete explicit solution to this variational problem. Namely, we show that at fixed total charge a ball of a particular radius is the unique global minimizer among all sufficiently regular sets in the plane. For sets whose area is also fixed, we show that balls are the only minimizers if the charge is less than or equal to a critical charge, while…

Planar equilibrium measure problem in the quadratic fields with a point charge

We consider a two-dimensional equilibrium measure problem under the presence of quadratic potentials with a point charge and derive the explicit shape of the associated droplets. This particularly shows that the topology of the droplets reveals a phase transition: (i) in the post-critical case, the droplets are doubly connected domain; (ii) in the critical case, they contain two merging type singular boundary points; (iii) in the pre-critical case, they consist of two disconnected components. From the random matrix theory point of view, our results provide the limiting spectral distribution of…

关于导电针中电荷分布的研究

On the distribution of charges in a conducting needle

本文研究导电针上 n 个点电荷的平衡与运动，证明平衡态唯一但动力学不收敛，并解释电荷端点聚集与概率分布函数趋于均匀之间的悖论。