Regularity of nonlocal minimal cones in dimension 2

TL;DR

This paper proves that in two dimensions, the only nonlocal minimal cones are trivial, leading to a bound on the Hausdorff dimension of singular sets in nonlocal minimal surfaces.

Contribution

It establishes the classification of nonlocal minimal cones in 2D and derives implications for the structure of singular sets in nonlocal minimal surfaces.

Findings

Only trivial nonlocal minimal cones exist in 2

Singular sets have Hausdorff dimension at most 0 in 2

Results hold for all s in (0,1)

Abstract

We show that the only nonlocal $s$-minimal cones in $\R^2$ are the trivial ones for all $s \in (0,1)$. As a consequence we obtain that the singular set of a nonlocal minimal surface has at most $n-3$ Hausdorff dimension.