Riesz equilibrium on a ball in the external field of a point charge

TL;DR

The paper studies the weighted Riesz energy minimization on the unit ball $\overline{\mathbb B}$ in $\mathbb R^{d}$ under an external field generated by a point charge, exploring both attractive and repulsive regimes for a broad range of the Riesz parameter $s$ (including Coulomb $s=d-2$ and logarithmic $s=0$ in low dimensions). It develops a rigorous framework based on signed equilibrium measures, Riesz balayage, Frostman inequalities, and analytic density results, and it uses iterative balayage and complex-analytic methods to characterize the support and density of minimizers in various geometries (ball and segment). A central outcome is the shell conjecture: for repulsive charges the equilibrium support is either the full ball or a shell, with the 1D case fully resolved in Section 6 through an explicit iterative scheme. The paper also provides explicit radial densities in several regimes (Coulomb and logarithmic in 1D), and analyzes how mass transfers between interior and boundary as the external field strength varies. These results deepen understanding of external-field effects on Riesz equilibria and yield precise descriptions of when the minimizer concentrates on the boundary, resides on a shell, or fills the interior, with potential implications for potential theory and approximation theory.

Abstract

We investigate the Riesz energy minimization problem on a $d$-dimensional ball in the presence of an external field created by a point charge above the ball in $\R^{d+1}$, $d\geq1$. Both cases of an attractive charge and a repulsive charge are considered. The notion of a signed equilibrium measure is one of the main tools in the present study. For the case of a positive (repulsive) charge, the determination of the support of the equilibrium measure is a nontrivial question. We solve it in the one-dimensional case by making use of iterated balayage, a method already applied in logarithmic potential theory. Here we use a modified version of it, in order to handle the phenomenon of mass loss, characteristic of the Riesz balayage of positive measures. Moreover, we also consider minimization of Coulomb energy on the ball in dimension $d\geq2$, and of logarithmic energy on the segment in dimension 1. Different techniques are used for these two cases.

Figures (4)

• Figure 1: Radial densities of the weighted Riesz equilibrium measures for the ball in ${\mathbb R}^{3}$ and for negative charges $\gamma\in\{-20,-15,\gamma_{-}=-8.612,0\}$ (resp. in blue, orange, green, red) when $d=3$ and $s=1$. The charge $\gamma$ is located at height $y_{4}=1$ above the ball.
• Figure 2: Radial densities of the Riesz signed equilibrium measures for the ball in ${\mathbb R}^{3}$ and for positive charges $\gamma\in\{0,\gamma_{+}=1.302,5,10\}$ (resp. in blue, orange, green, red) when $d=3$ and $s=1$. The charge $\gamma$ is located at height $y_{4}=1$ above the ball.
• Figure 3: Evolution of the equilibrium measure $\mu_{Q,{\mathbb B}}$ as described in Theorem \ref{['thm:equildisk2']}. For $\gamma<\widetilde{\gamma}$, a large attractive charge, $\mu_{Q,{\mathbb B}}$ is supported on a ball of radius $<1$, which grows with $\gamma$. When $\gamma=\widetilde{\gamma}$, the radius becomes equal to 1. For $\widetilde{\gamma}<\gamma<0$, $\mu_{Q,{\mathbb B}}$ is a combination of a measure on ${\mathbb B}$ and a multiple of the uniform measure on the sphere $\mathbb{S}$. When $\gamma=0$, all of the volume measure has moved to the sphere, and $\mu_{Q,{\mathbb B}}$ is just the normalized uniform measure on $\mathbb{S}$. Then, for a positive charge $\gamma>0$, $\mu_{Q,{\mathbb B}}$ remains unchanged.
• Figure 4: Radial densities of the logarithmic equilibrium measures for negative charges $\gamma\in\{-1,\gamma_{-}=\sqrt{2}/(1-\sqrt{2}),-7\}$ (in blue, orange, green) on the left, and for positive charges $\gamma\in\{0,\gamma_{+}=\sqrt{2}+1,5\}$ (in blue, orange, green) on the right. The charge is located at height $y_{2}=1$ above the segment. On the right, the dashed curve is the density of the positive component of the signed equilibrium measure for $\gamma=5$.

Theorems & Definitions (51)

• Theorem 2.1: W
• Remark 2.2
• Proposition 2.3: Land
• Lemma 2.4
• proof
• Lemma 2.5: Superposition principle, Land
• Proposition 2.6
• proof
• Lemma 2.7
• proof