Volume constrained minimizers of the fractional perimeter with a potential energy

TL;DR

This paper studies the existence, regularity, and shape of volume-constrained minimizers of fractional perimeter with potential energy, showing they are close to balls in small volume cases.

Contribution

It introduces a new variational problem combining fractional perimeter and potential energy, proving existence, regularity, and quantitative closeness to spheres.

Findings

Existence and regularity of minimizers established.

Minimizers are close to spherical shapes in small volume regime.

Results apply to periodic potential energy cases.

Abstract

We consider volume-constrained minimizers of the fractional perimeter with the addition of a potential energy in the form of a volume inte- gral. Such minimizers are solutions of the prescribed fractional curvature problem. We prove existence and regularity of minimizers under suitable assumptions on the potential energy, which cover the periodic case. In the small volume regime we show that minimizers are close to balls, with a quantitative estimate.