Minimum Riesz Energy Problem on the Hyperdisk

TL;DR

This paper solves the minimum Riesz energy problem on a hyperdisk in Euclidean space with external fields, providing explicit formulas for extremal measures and analyzing support structures including ring formations.

Contribution

It introduces explicit solutions for extremal measures on the hyperdisk under external fields and explores support structures and integral equation reductions.

Findings

Explicit extremal measure formulas derived

Support can form a ring structure under certain conditions

Reduction to Fredholm integral equations for measure recovery

Abstract

We consider the minimum Riesz $s$-energy problem on the unit disk $\mathbb D:=\{(x_1,\ldots,x_d)\in\mathbb R^d: x_1=0, x_2^2+x_3^2+\ldots+x_d^2\leq 1\}$ in the Euclidean space $\mathbb R^d$, $d\geq 3$, immersed into a smooth rotationally invariant external field $Q$. The charges are assumed to interact via the Riesz potential $1/r^s$, with $d-3 < s < d-1$, where $r$ denotes the Euclidean distance. We solve the problem by finding an explicit expression for the extremal measure. We then consider applications to a monomial external field and an external field generated by a positive point charge, located at some distance above the disk on the polar axis. We obtain an equation describing the critical height for the location of the point charge, which guarantees that the support of the extremal measure occupies the whole disk $\mathbb D$. We also show that under some mild restrictions on a general external field the support of the extremal measure will have a ring structure. Furthermore, we demonstrate how to reduce the problem of recovery of the extremal measure in this case to a Fredholm integral equation of the second kind.