Removable singularities for nonlocal minimal graphs

Technical Summary: Removable Singularities for Nonlocal Minimal Graphs

Problem Statement
This paper addresses the removable singularity problem for nonlocal minimal graphs. Specifically, it investigates whether a solution to the nonlocal minimal surface equation (and a broader class of nonlocal equations) defined on an open set $\Omega \setminus K$, where $K$ is a compact set of $(s, 1)$-capacity zero, can be extended to a solution on the entire domain $\Omega$.

The study is motivated by the classical theory of minimal surfaces, where Bers, Nitsche, and De Giorgi–Stampacchia established that singularities of $(n-1)$-dimensional Hausdorff measure zero (or 1-capacity zero) are removable for solutions to the minimal surface equation. The author seeks to generalize these results to the nonlocal setting, where the governing equation involves the nonlocal mean curvature $H_s$ or, more generally, a nonlocal operator of the form:
$$-\int_{\mathbb{R}^n} A\left(x, y, \frac{u(x)-u(y)}{|x-y|}\right) \frac{dy}{|x-y|^{n+s}} = B(x, u)$$
Here, $s \in (0, 1)$, and $A$ and $B$ satisfy specific structural conditions (monotonicity, boundedness, and growth constraints).

Methodology
The proof of the main theorem proceeds in three distinct steps, adapting techniques from the local theory (specifically Serrin's approach) to the nonlocal framework while addressing the challenges posed by nonlocality.

1. Regularity under Integrability Assumptions:
   The first step establishes that if a solution $u$ is already known to possess a certain level of local integrability (specifically $u \in L^\varepsilon_{\text{loc}}(\Omega)$ for some $\varepsilon > 0$) near the singular set $K$, then $u$ attains $W^{s,1}_{\text{loc}}(\Omega)$ regularity.

   • Technique: The author employs Caccioppoli-type estimates involving "double truncations" of the solution ($\bar{u}$ and $u$). These estimates are derived using test functions constructed via mollification and a sequence of cut-off functions $\bar{\eta}_j$ that vanish near $K$ but converge to 1 almost everywhere.
   • Tools: The proof utilizes the fractional Sobolev inequality and a Moser iteration technique to improve the integrability of $u$ from $L^\varepsilon$ to $L^\infty_{\text{loc}}$, which subsequently ensures $W^{s,1}_{\text{loc}}$ regularity.

2. Verification of Integrability:
   The second step proves that the additional integrability assumption required in Step 1 is automatically satisfied by any solution defined on $\Omega \setminus K$.

   • Challenge: Classical techniques (e.g., Serrin's double truncation with levels exceeding the boundary maximum) fail in the nonlocal setting because the "boundary" values of a nonlocal problem depend on the function's behavior on the entire complement of the domain, which may be unbounded.
   • Solution: The author introduces a localization technique. By utilizing the specific structure of the equation (corresponding to the case $p=1$ in fractional $p$-Laplacian type equations), the method avoids the need for finite supremums on the complement. Instead, it leverages the fact that the equation involves a form of "weighted $L^0$-norm" (a constant term) to adapt Serrin's proof using double truncations. This step crucially relies on the compactness of $K$.

3. Extension of the Solution:
   The final step uses a standard approximation argument to show that the extended function satisfies the weak formulation of the equation on the entire domain $\Omega$.

   • Technique: A sequence of test functions $\phi_j = \phi \bar{\eta}_j$ is used, where $\bar{\eta}_j$ vanishes near $K$. The author demonstrates that the nonlocal integral terms and the lower-order terms converge to the required limits as $j \to \infty$, utilizing the measure-zero property of sets with $(s, 1)$-capacity zero and the dominated convergence theorem.

Key Contributions and Results

• Main Theorem (Theorem 1.4): The paper proves that if $K \subset \Omega$ is a compact set of $(s, 1)$-capacity zero, any solution to the general nonlocal equation (1.3) in $\Omega \setminus K$ has a representative that is a solution in all of $\Omega$.
• Application to Nonlocal Minimal Graphs: As a direct corollary, the paper establishes that any nonlocal minimal graph defined in $\Omega \setminus K$ (where $K$ has $(s, 1)$-capacity zero) is indeed a nonlocal minimal graph in $\Omega$.
• Generalization of Equation Class: The result extends beyond the specific nonlocal minimal surface equation to a broad class of equations with prescribed nonlocal mean curvature, including cases where the right-hand side $B(x, u)$ satisfies specific growth conditions (e.g., capillarity-type equations).
• Distinction of Solution Concepts: The paper carefully distinguishes between "weak solutions" (measurable functions satisfying the integral formulation) and "solutions" (weak solutions that also belong to $W^{s,1}_{\text{loc}}$). The main theorem applies to the latter, while a separate result (Theorem 5.1) addresses weak solutions under slightly stronger assumptions on the lower-order term $B$.

Significance and Claims
The author claims that this work provides the first removable singularity theorem for nonlocal minimal graphs and a broad class of nonlocal equations. A notable feature of the result is that, unlike removable singularity theorems for nonlocal nonlinear $p$-Laplacian type equations with $p > 1$, the theorem for the $p=1$ case (which includes nonlocal minimal graphs) requires no a priori assumptions on the behavior of the solution $u$ near the singular set $K$.

The paper acknowledges that while the regularity theory for nonlocal minimal graphs implies that solutions are $C^\infty$, the regularity theory for the more general equation (1.3) is not yet fully established. Consequently, the paper does not claim that the extended solution is necessarily continuous or smooth for the general case, only that it is a valid solution in the defined sense.

The work also notes that the case where the singular set $K$ approaches the boundary of $\Omega$ remains an open question, though the paper provides a partial result for weak solutions in Theorem 5.1 where the set $E$ is relatively closed (allowing it to approach the boundary) under specific conditions on the lower-order term.