Regularity and Bernstein-type results for nonlocal minimal surfaces

TL;DR

This paper proves that Lipschitz nonlocal minimal surfaces are smooth in all dimensions and extends Bernstein's theorem to the nonlocal setting, linking surface regularity to the nonexistence of certain singular cones.

Contribution

It establishes regularity results for nonlocal minimal surfaces and generalizes Bernstein's theorem to the nonlocal context, connecting surface smoothness with cone nonexistence.

Findings

Lipschitz nonlocal minimal surfaces are smooth in all dimensions

Extension of Bernstein's theorem to nonlocal minimal surfaces

Connection between surface regularity and nonexistence of singular cones

Abstract

We prove that, in every dimension, Lipschitz nonlocal minimal surfaces are smooth. Also, we extend to the nonlocal setting a famous theorem of De Giorgi stating that the validity of Bernstein's theorem in dimension $n+1$ is a consequence of the nonexistence of $n$-dimensional singular minimal cones in $\R^n$.