Non-standard Green energy problems in the complex plane

TL;DR

This paper investigates non-standard Green energy problems in the complex plane, analyzing their asymptotic solutions, and introduces algorithms and conditions for optimal particle arrangements and measures.

Contribution

It provides a detailed study of discrete and continuous Green energy problems with variable masses and positions, including continuity results and convergence algorithms.

Findings

Equilibrium measure and constant vary continuously with parameter R.

A greedy algorithm converges to the equilibrium measure and constant.

Optimal particle masses are uniquely determined under certain conditions.

Abstract

We consider several non-standard discrete and continuous Green energy problems in the complex plane and study the asymptotic relations between their solutions. In the discrete setting, we consider two problems; one with variable particle positions (within a given compact set) and variable particle masses, the other one with variable masses but prescribed positions. The mass of a particle is allowed to take any value in the range $0\leq m\leq R$, where $R>0$ is a fixed parameter in the problem. The corresponding continuous energy problems are defined on the space of positive measures $\mu$ with mass $\|\mu\|\leq R$ and supported on the given compact set, with an additional upper constraint that appears as a consequence of the prescribed positions condition. It is proved that the equilibrium constant and equilibrium measure vary continuously as functions of the parameter $R$ (the latter in the weak-star topology). In the unconstrained energy problem we present a greedy algorithm that converges to the equilibrium constant and equilibrium measure. In the discrete energy problems, it is shown that under certain conditions, the optimal values of the particle masses are uniquely determined by the optimal positions or prescribed positions of the particles, depending on the type of problem considered.