Volumetric density estimates for nonlocal minimal surfaces

TL;DR

This paper establishes universal volumetric estimates for viscosity subsolutions of nonlocal mean curvature equations, showing that low-density subsolutions have boundaries with positive measure.

Contribution

It provides the first universal volumetric bounds for nonlocal minimal surfaces with general symmetric kernels comparable to the fractional Laplacian.

Findings

Viscosity subsolutions satisfy volumetric estimates at all scales.

Low-density subsolutions have boundaries with positive Lebesgue measure.

Results apply to general symmetric kernels comparable to fractional Laplacian.

Abstract

In this article, we prove that viscosity subsolutions to nonlocal mean curvature-type equations satisfy universal volumetric estimates at all scales. Our results hold for general symmetric kernels that are comparable to the fractional Laplacian. Furthermore, we prove that subsolutions with low density (with respect to a universal constant) necessarily have `fat boundary', that is, have topological boundary with positive Lebesgue measure.