Regularity properties of nonlocal minimal surfaces via limiting arguments

TL;DR

This paper establishes flatness and smoothness properties of nonlocal minimal surfaces as the fractional parameter approaches 1, showing all such cones are flat and surfaces are smooth in low dimensions.

Contribution

It provides an s-independent improvement of flatness for nonlocal minimal surfaces and proves all nonlocal minimal cones are flat near s=1.

Findings

All nonlocal minimal cones are flat.

Nonlocal minimal surfaces are smooth in dimensions ≤ 7 for s close to 1.

Flatness improvement is independent of s as s→1^-.

Abstract

We prove an improvement of flatness result for nonlocal minimal surfaces which is independent of the fractional parameter $s$ when $s\rightarrow 1^-$. As a consequence, we obtain that all the nonlocal minimal cones are flat and that all the nonlocal minimal surfaces are smooth when the dimension of the ambient space is less or equal than 7 and $s$ is close to 1.