Connected Coulomb Columns: Analysis and Numerics

TL;DR

This paper investigates a modified Gamow's liquid drop model with Coulomb interaction, focusing on columnar shapes constrained by a connected perimeter, establishing existence of minimizers and analyzing their shapes through theoretical and numerical methods.

Contribution

It introduces a connected perimeter approach to ensure existence of minimizers in a Coulomb-interacting liquid drop model constrained to columnar shapes, with theoretical and numerical analysis.

Findings

Existence of minimizers for the connected isoperimetric problem with Coulomb interaction.

Characterization of minimizer shapes in small and large cross section regimes.

Numerical study of minimizers using an Ohta-Kawasaki phase field model.

Abstract

We consider a version of Gamow's liquid drop model with a short range attractive perimeter-penalizing potential and a long-range Coulomb interaction of a uniformly charged mass in $\R^3$. Here we constrain ourselves to minimizing among the class of shapes that are columnar, i.e., constant in one spatial direction. Using the standard perimeter in the energy would lead to non-existence for any prescribed cross-sectional area due to the infinite mass in the constant spatial direction. In order to heal this defect we use a connected perimeter instead. We prove existence of minimizers for this connected isoperimetric problem with long-range interaction and study the shapes of minimizers in the small and large cross section regimes. For an intermediate regime we use an Ohta-Kawasaki phase field model with connectedness constraint to study the shapes of minimizers numerically.