Graph properties for nonlocal minimal surfaces

TL;DR

This paper proves that nonlocal minimal surfaces that are graphs outside a cylinder are globally graphs and, in three dimensions, are smooth, using convolution and integral estimates to establish the Euler-Lagrange equation for nonlocal mean curvature.

Contribution

It establishes the global graphical nature and smoothness of nonlocal minimal surfaces in three dimensions, extending local properties to the nonlocal setting.

Findings

Nonlocal minimal surfaces are globally graphs if they are graphs outside a cylinder.

In three dimensions, such surfaces are proven to be smooth.

The methods rely on convolution techniques and integral estimates for the Euler-Lagrange equation.

Abstract

In this paper we show that a nonlocal minimal surface which is a graph outside a cylinder is in fact a graph in the whole of the space. As a consequence, in dimension~$3$, we show that the graph is smooth. The proofs rely on convolution techniques and appropriate integral estimates which show the pointwise validity of an Euler-Lagrange equation related to the nonlocal mean curvature.