Droplet breakup in the liquid drop model with background potential

TL;DR

This paper studies a modified liquid drop model with background potential, analyzing how minimizers behave as the background potential's strength approaches zero, revealing the structure of optimal configurations.

Contribution

It introduces a regularized version of the Gamow liquid drop model with a background potential and characterizes minimizers in the small potential limit.

Findings

Existence of minimizers for all masses due to background potential.

Asymptotic expansion of energy as background potential strength approaches zero.

Characterization of all minimizing sequences via generalized minimizers.

Abstract

We consider a variant of Gamow's liquid drop model, with a general repulsive Riesz kernel and a long-range attractive background potential with weight $Z$. The addition of the background potential acts as a regularization for the liquid drop model in that it restores the existence of minimizers for arbitrary mass. We consider the regime of small $Z$ and characterize the structure of minimizers in the limit $Z\to 0$ by means of a sharp asymptotic expansion of the energy. In the process of studying this limit we characterize all minimizing sequences for the Gamow model in terms of "generalized minimizers".