Volumetric density estimates for nonlocal minimal surfaces

Technical Summary: Volumetric Density Estimates for Nonlocal Minimal Surfaces

Problem Statement
The paper addresses the regularity theory of nonlocal minimal surfaces, specifically focusing on the validity of volumetric density estimates for viscosity subsolutions to nonlocal mean curvature equations. In the classical theory of minimal surfaces, density estimates (quantifying how "close" portions of a surface can be) are fundamental tools. For classical minimal surfaces, volumetric density estimates (V) and surface density estimates from above (SA) hold for minimizers but fail for general stationary surfaces (e.g., the catenoid or periodic slabs).

In the nonlocal setting, defined by kernels comparable to the fractional Laplacian, previous results established that (V), (SA), and (SB) hold for minimizers of the nonlocal perimeter. However, for general non-minimizing sets (stationary solutions or viscosity subsolutions), the validity of the volumetric estimate (V) remained an open question. Furthermore, for general kernels (not just the fractional Laplacian), the validity of these estimates for stationary solutions was largely unknown, particularly because existing proofs for the fractional case relied on monotonicity formulas that do not extend to general kernels.

Methodology
The authors employ a contradiction argument rooted in the properties of viscosity subsolutions and the geometry of the nonlocal mean curvature operator. The core methodology involves:

1. Viscosity Formulation: The authors work with the definition of viscosity subsolutions, where the nonlocal mean curvature $H_{K,E}$ is required to be bounded above at points where the set $E$ admits an exterior touching ball. This allows the analysis of potentially irregular sets (Borel sets) without requiring classical differentiability.
2. Hardy-Littlewood Maximal Inequality: The proof utilizes the maximal function to identify a set of points $D_\alpha$ within a ball where the set $E$ is "sparse" (i.e., has low density) at all scales.
3. Rearrangement Inequalities: A key technical tool is a rearrangement inequality (Lemma 3.2) which demonstrates that if a set $E$ has small measure in a region, the nonlocal interaction energy (and thus the curvature) at a boundary point touching an exterior ball must be large.
4. Contradiction via Density: The authors assume the volumetric density estimate fails (i.e., $E$ has arbitrarily small measure in a ball). They show that this sparsity forces the nonlocal mean curvature to exceed the bound imposed by the viscosity subsolution condition at specific points.
5. Handling Boundary Measure: A critical technical distinction is made regarding the Lebesgue measure of the boundary $\partial E$. The authors prove that if the density is too low, the set of points where the nonlocal curvature is infinite must have positive Lebesgue measure. Since the viscosity definition only constrains points with exterior touching balls, and points with infinite curvature in this context necessarily lack such balls, the authors derive a contradiction with the assumption that $\partial E$ has zero Lebesgue measure.

Key Contributions and Results

• Universal Volumetric Estimates for Subsolutions (Theorem 1.2): The primary result establishes that for any Borel set $E$ with a boundary of zero Lebesgue measure, if $E$ is a viscosity subsolution to a nonlocal mean curvature equation with a kernel comparable to the fractional Laplacian, then $E$ satisfies a universal volumetric density estimate. Specifically, $|E \cap B_R| \geq \delta |B_R|$ for all $R > 0$, where $\delta$ depends only on dimension, the fractional order $s$, and the ellipticity constants of the kernel.
• Extension to General Kernels (Corollary 1.4): Unlike previous results for $s$-minimal surfaces which relied on specific monotonicity formulas, these estimates hold for general symmetric kernels satisfying uniform ellipticity bounds (comparable to $|y|^{-(d+s)}$).
• Surface Density from Below (Corollary 1.5): By combining the new volumetric estimates with the relative isoperimetric inequality, the authors prove that general nonlocal minimal surfaces (stationary solutions) satisfy a universal surface density estimate from below (SB). This extends the validity of (SB) beyond the fractional Laplacian case.
• The "Fat Boundary" Phenomenon (Theorem 1.6): The paper proves a striking converse: if a viscosity subsolution has "too low" density (below a universal threshold), its topological boundary must have positive Lebesgue measure. Specifically, the set of points on the boundary where the nonlocal mean curvature is $+\infty$ has positive Lebesgue measure. This highlights a rigidity in the viscosity solution framework: one cannot have a "thin" set with a "thin" boundary satisfying the subsolution condition; if the set is too sparse, the boundary becomes "fat" (positive measure) and the curvature blows up on a set of positive measure.

Significance and Claims
The paper claims to resolve the validity of volumetric density estimates for non-minimizing nonlocal minimal surfaces with general kernels. The authors emphasize that their results hold for viscosity subsolutions, a class broader than minimizers or stationary points in the classical sense.

The significance lies in:

1. Universality: Establishing that volumetric density estimates are a robust property of nonlocal minimal surfaces, independent of whether the set is a global minimizer, provided the boundary is sufficiently regular (zero Lebesgue measure).
2. General Kernels: Moving beyond the specific fractional Laplacian kernel to a broad class of kernels, thereby broadening the applicability of nonlocal minimal surface theory to models with different interaction ranges.
3. Structural Insight: Theorem 1.6 provides a novel structural insight into the nature of viscosity solutions, linking low density directly to the existence of a "fat boundary" and infinite curvature, a phenomenon not present in the classical local theory in the same form.

The authors note that their results do not require stability assumptions (unlike previous results for the surface density estimate from above in the nonlocal setting) and apply to the entire space at all scales. The work relies on the interplay between the nonlocal operator's singularity and the geometric constraints imposed by the viscosity solution definition.