Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Removable singularities for nonlocal minimal graphs

Minhyun Kim

TL;DR

The paper proves a removable singularity theorem for nonlocal minimal graphs, showing that any solution of the nonlocal mean curvature equation in a domain with a compact set $K$ of $(s,1)$-capacity zero actually extends to a solution on the whole domain. The authors develop a three-part strategy: (1) obtain $W^{s,1}_{loc}$-regularity under a mild integrability assumption, (2) demonstrate that this integrability holds via localization and nonlocal tail techniques, and (3) finalize the result by a standard approximation argument. The framework is formulated for a broad class of nonlocal equations with Carathéodory structure on the principal part and lower-order term, and includes a removable-singularity result for weak solutions when $\mathscr{B}$ is suitably controlled and independent of the solution. This work extends classical removable-singularity results to the nonlocal setting and provides tools (Caccioppoli-type estimates, double truncations, and tail analysis) applicable to prescribed nonlocal mean curvature and capillarity-type problems.

Abstract

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 $Ω$.

Removable singularities for nonlocal minimal graphs

TL;DR

The paper proves a removable singularity theorem for nonlocal minimal graphs, showing that any solution of the nonlocal mean curvature equation in a domain with a compact set of -capacity zero actually extends to a solution on the whole domain. The authors develop a three-part strategy: (1) obtain -regularity under a mild integrability assumption, (2) demonstrate that this integrability holds via localization and nonlocal tail techniques, and (3) finalize the result by a standard approximation argument. The framework is formulated for a broad class of nonlocal equations with Carathéodory structure on the principal part and lower-order term, and includes a removable-singularity result for weak solutions when is suitably controlled and independent of the solution. This work extends classical removable-singularity results to the nonlocal setting and provides tools (Caccioppoli-type estimates, double truncations, and tail analysis) applicable to prescribed nonlocal mean curvature and capillarity-type problems.

Abstract

We prove the removable singularity theorem for nonlocal minimal graphs. Specifically, we show that any nonlocal minimal graph in , where is an open set and is a compact set of -capacity zero, is indeed a nonlocal minimal graph in all of .
Paper Structure (5 sections, 11 theorems, 70 equations)

This paper contains 5 sections, 11 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. preliminaries
  3. Caccioppoli-type estimates
  4. Removable singularity theorem
  5. Removable singularity theorem for weak solutions

Key Result

Theorem 1.3

Let $\Omega \subset \mathbb{R}^n$ be open and let $K \subset \Omega$ be a compact set of $(s, 1)$-capacity zero. If $u$ is a solution of eq-NMSE in $\Omega \setminus K$, then $u$ has a representative that is a solution of eq-NMSE in $\Omega$.

Theorems & Definitions (25)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Example 1.4
  • Definition 1.5
  • Theorem 1.6
  • Proposition 2.1
  • proof
  • Theorem 2.2
  • Definition 2.3
  • ...and 15 more