Flatness results for nonlocal minimal cones and subgraphs

TL;DR

This paper proves that nonlocal minimal cones that are non-singular subgraphs outside the origin must be halfspaces, simplifying existing proofs and establishing new classification results for nonlocal minimal surfaces.

Contribution

It introduces new proofs for Bernstein-type results and presents a novel nonlocal Bernstein-Moser-type classification for Lipschitz minimal subgraphs.

Findings

Nonlocal minimal cones outside the origin are halfspaces.

New simpler proofs of Bernstein-type results.

Classification of Lipschitz nonlocal minimal subgraphs outside a ball.

Abstract

We show that nonlocal minimal cones which are non-singular subgraphs outside the origin are necessarily halfspaces. The proof is based on classical ideas of~\cite{DG1} and on the computation of the linearized nonlocal mean curvature operator, which is proved to satisfy a suitable maximum principle. With this, we obtain new, and somehow simpler, proofs of the Bernstein-type results for nonlocal minimal surfaces which have been recently established in~\cite{FV}. In addition, we establish a new nonlocal Bernstein-Moser-type result which classifies Lipschitz nonlocal minimal subgraphs outside a ball.