Removable singularities for nonlocal minimal graphs

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

💡 Research Summary

The paper establishes a removable singularity theorem for nonlocal minimal graphs, extending the classical result for minimal surfaces to the fractional setting. The author first recalls the definition of an s‑minimal set and its s‑perimeter, introduced by Caffarelli, Roquejoffre and Savin, and explains that the boundary of an s‑minimal set that can be written as a subgraph of a function u is called a nonlocal minimal graph. The associated Euler–Lagrange equation is the nonlocal mean curvature equation H_s u = 0, which converges to the classical mean curvature equation as s → 1.

The main object of study is a more general nonlocal equation
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