Droplet minimizers of an isoperimetric problem with long-range interactions

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 energies with short- and 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.