Optimal distribution of oppositely charged phases: perfect screening and other properties

TL;DR

This paper investigates the optimal arrangement of negative charges around a fixed positive charge, proving properties like charge neutrality and perfect screening, and analyzing the limiting behavior when negative charge dominates.

Contribution

It establishes existence, uniqueness, and regularity of the optimal charge configuration, and characterizes the limit model via $ ext{Gamma}$-convergence for high negative charge density.

Findings

Proves charge neutrality in optimal configurations

Demonstrates complete Coulomb screening by positive charge

Analyzes the limit behavior with dominant negative charge

Abstract

We study the minimum energy configuration of a uniform distribution of negative charge subject to Coulomb repulsive self-interaction and attractive interaction with a fixed positively charged domain. After having established existence and uniqueness of a minimizing configuration, we prove charge neutrality and the complete screening of the Coulomb potential exerted by the positive charge, and we discuss the regularity properties of the solution. We also determine, in the variational sense of $\Gamma$-convergence, the limit model when the charge density of the negative phase is much higher than the positive one.