Weighted energy problem on the unit sphere

TL;DR

This paper solves the minimal energy problem on the unit sphere with external fields by explicitly determining the equilibrium measure and density, applying the results to specific external fields like point charges and quadratic fields.

Contribution

It provides explicit solutions for the equilibrium measure and density on the sphere under external fields, advancing understanding of energy minimization in potential theory.

Findings

Explicit support of extremal measure determined

Closed-form expression for equilibrium density derived

Applications to point charge and quadratic external fields

Abstract

We consider the minimal energy problem on the unit sphere $\mathbb S^2$ in the Euclidean space $\mathbb R^3$ immersed in an external field $Q$, where the charges are assumed to interact via Newtonian potential $1/r$, $r$ being the Euclidean distance. The problem is solved by finding the support of the extremal measure, and obtaining an explicit expression for the equilibrium density. We then apply our results to the external field generated by a point charge, and a quadratic external field.