Volumetric density estimates for nonlocal minimal surfaces

Abstract

In this article, we prove that viscosity subsolutions to nonlocal mean curvature-type equations satisfy universal volumetric estimates at all scales provided their boundary has zero Lebesgue measure. Our results hold for a general class of symmetric kernels including 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.