Removable singularities for nonlocal minimal graphs

TL;DR

This paper proves that nonlocal minimal graphs can be extended across certain small singular sets, establishing a removable singularity theorem in the nonlocal minimal surface context.

Contribution

It introduces a removable singularity theorem for nonlocal minimal graphs, showing extension across compact sets of zero capacity.

Findings

Nonlocal minimal graphs extend across sets of zero capacity.

Removable singularity theorem established for nonlocal minimal graphs.

Extension results hold in general open sets.

Abstract

We prove the removable singularity theorem for nonlocal minimal graphs. Specifically, we show that any nonlocal minimal graph in $\Omega \setminus K$, where $\Omega \subset \mathbb{R}^n$ is an open set and $K \subset \Omega$ is a compact set of $(s, 1)$-capacity zero, is indeed a nonlocal minimal graph in all of $\Omega$.