Droplet Minimizers of an Isoperimetric Problem with long-range interactions

TL;DR

This paper analyzes the geometry of droplet patterns in a nonlocal isoperimetric problem, focusing on the sharp interface limit of the Ohta-Kawasaki energy, and extends previous results using minimal surface regularity theory.

Contribution

It provides a detailed geometric description of droplet patterns and introduces robust tools for analyzing complex patterns in nonlocal isoperimetric problems.

Findings

Detailed geometric characterization of droplet patterns.

Extension of regularity results for minimal surfaces.

Tools for analyzing complex nonlocal patterns.

Abstract

We give a detailed description of the geometry of single droplet patterns in a nonlocal isoperimetric problem. In particular we focus on the sharp interface limit of the Ohta-Kawasaki free energy for diblock copolymers, regarded as a paradigm for those energies modeling physical systems characterized by a competition between short and a long-range interactions. Exploiting fine properties of the regularity theory for minimal surfaces, we extend previous partial results in different directions and give robust tools for the geometric analysis of more complex patterns.