Delaunay cylinders with constant non-local mean curvature

Abstract

The aim of this master's thesis is to obtain an alternative proof, using variational techniques, of an existence result for periodic sets in $\mathbb{R}^2$ that minimize a non-local version of the classical perimeter functional adapted to periodic sets. This functional was first introduced by Dávila, Del Pino, Dipierro and Valdinoci to study periodic sets of codimension 1 in $\mathbb{R}^n$ that are decreasing and cylindrically symmetric in a given direction. Our minimizers are periodic sets of $\mathbb{R}^2$ having constant non-local mean curvature. We call them non-local Delaunay cylinders (or bands) in reference to the classical Delaunay surfaces with constant mean curvature.