Droplet phases in non-local Ginzburg-Landau models with Coulomb repulsion in two dimensions

TL;DR

This paper analyzes the formation and distribution of droplet patterns in a two-dimensional non-local Ginzburg-Landau model with Coulomb repulsion, revealing asymptotic behaviors near pattern onset.

Contribution

It provides the first detailed asymptotic analysis of droplet patterns and their thresholds in non-local Ginzburg-Landau models with Coulomb interactions.

Findings

Droplets are nearly identical and circular at onset.

Droplets become uniformly distributed in the domain.

Asymptotic thresholds and droplet sizes are precisely characterized.

Abstract

We establish the behavior of the energy of minimizers of non-local Ginzburg-Landau energies with Coulomb repulsion in two space dimensions near the onset of multi-droplet patterns. Under suitable scaling of the background charge density with vanishing surface tension the non-local Ginzburg-Landau energy becomes asymptotically equivalent to a sharp interface energy with screened Coulomb interaction. Near the onset the minimizers of the sharp interface energy consist of nearly identical circular droplets of small size separated by large distances. In the limit the droplets become uniformly distributed throughout the domain. The precise asymptotic limits of the bifurcation threshold, the minimal energy, the droplet radii, and the droplet density are obtained.