Coulomb equilibrium in the external field of an attractive-repellent pair of charges

TL;DR

The paper addresses the Coulomb equilibrium problem in $\mathbb{R}^{d}$ with the kernel $1/|x|^{d-2}$ under a non-convex external field generated by an attractive-repellent pair of charges in $\mathbb{R}^{d+1}$. It develops a framework based on signed equilibrium measures and balayage to obtain explicit descriptions of the equilibrium measure $\mu_Q$ under both admissible and weakly admissible external fields, identifying ball and shell supports and providing exact densities. It derives explicit formulas for the ball case density $\frac{d\mu_Q}{dx}$ and for the shell geometry with inner radius $R_0$ and outer radius $R_s$, including the conditions for shell vs ball, with the outer radius determined by $\frac{\gamma_2}{(1+(h_2/R_s)^2)^{d/2}}-\frac{\gamma_1}{(1+(h_1/R_s)^2)^{d/2}}=1$ and the inner radius by $\left(\frac{R_0^{2}+h_1^{2}}{R_0^{2}+h_2^{2}}\right)^{d}=\left(\frac{\gamma_1}{\gamma_2}\right)^{2}$. The results extend prior work on Riesz kernels and radial external fields, and show that the external field can yield fully explicit solutions with ball, shell, or complement-of-ball supports, depending on parameter regimes.

Abstract

The aim of this paper is to provide a complete analysis of the Coulomb equilibrium problem in the euclidean space $\mathbb{R}^d$, $d\geq2$, associated to the kernel $1/|x|^{d-2}$, with a non-convex external field created by an attractive-repellent pair of charges placed in $\mathbb{R}^{d+1} \setminus \mathbb{R}^d$. We consider the admissible setting, where the equilibrium measure is compactly supported, as well as the limiting weakly admissible setting, with a weaker external field at infinity, where the existence of the equilibrium measure still holds but possibly with an unbounded support. The main tools for our analysis are the notions of signed equilibrium and balayage of measures. We note that for certain configurations of charges and distances to the conductor, the support of the equilibrium measure is a shell (multidimensional annulus).

Figures (4)

• Figure 1: The different types of supports for the equilibrium measure according to the heights $h_{1}$ and $h_{2}$, the admissible case on the left, the weakly admissible case on the right. The line $L_{1}$ has equation $h_{2}=(\gamma_{1}/\gamma_{2})^{1/2}h_{1}$ and the line $L_{2}$ has equation $h_{2}=(\gamma_{2}/\gamma_{1})^{1/d}h_{1}$.
• Figure 2: The different types of supports for the equilibrium measure according to the values of $\gamma_{1}$ and $\gamma_{2}$, above the line $\gamma_{2}=\gamma_{1}+1$. Below that line, the external field $Q$ is not admissible and the equilibrium problem has no solution. The circled letters $A,B,C$ refer to the different cases defined in Section \ref{['Proofs']}, see (\ref{['Cases']}).
• Figure 3: Examples in ${\mathbb R}^{3}$ of radial densities of $\mu_{Q}$ when the support is a ball, a ball with a density vanishing at the origin (transition between a ball and a shell), a shell, and the complement of a ball. The external field $Q$ is represented by the dashed curve (up to an additive constant to fit the graph). Note that the densities do not vanish on the boundaries of the supports.
• Figure 4: Behavior of the densities $g_{c}$ (solid line) and $g_{s}$ (dashed line) in the four cases $A_{1},A_{2},B,C$ as defined in (\ref{['Cases']}), near the root $R_{s}$ of $g_{s}$, where it changes sign. The density $g_{c}$ has a root $r_{c}$ in cases $A_{1}$ and $B$, but no root in cases $A_{2}$ and $C$.

Theorems & Definitions (24)

• Theorem 1
• Theorem 2
• Theorem 3
• Remark 4
• Definition 5
• Proposition 6
• proof
• Lemma 7
• Lemma 8
• proof