Nonlocal perimeter, curvature and minimal surfaces for measurable sets

Abstract

In this paper, we study the nonlocal perimeter associated with a nonnegative radial kernel J: ℝ^N → ℝ, verifying ∫_ℝ^NJ(z)dz = 1. The nonlocal perimeter studied here is given by the interactions (measured in terms of the kernel J) of particles from the outside of a measurable set E with particles from the inside, that is,

$${P_J}(E): = \int_E {\left( {\int_{{\mathbb{R}^N}\backslash E} {J(x - y)dy} } \right)dx} .$$

As a consequence of a result by Bourgain, Brezis and Mironescu in 2001 and Dávila in 2002, when the kernel J is appropriately rescaled, this nonlocal perimeter converges to the classical local perimeter. We prove that an isoperimetric inequality also holds for it. Associated with the kernel J and the previous definition of perimeter we can consider minimal surfaces. In connexion with minimal surfaces we introduce the concept of J-mean curvature at a point x, and we show that again under rescaling we can recover the usual notion of mean curvature. In addition, we study the situation analogous to a Cheeger set in this nonlocal context and show that a set Ω is J-calibrable (Ω is a J-Cheeger set of itself) if and only if there exists τ such that τ(x)= 1 if x ∊ Ω satisfying $-{\lambda }_\Omega^J\tau\in{\Delta_1^J}\chi_\Omega$, here ${\lambda }_\Omega^J$ is the J-Cheeger constant $\lambda _{\rm{\Omega }}^J = {{{P_J}({\rm{\Omega }})} \over {{\rm{|\Omega |}}}}$ and Δ ^J₁ is given, formally, by

$${\rm{\Delta }}_1^Ju(x) = \int_{{\mathbb{R}^N}} {J(x - y){{u(y) - u(x)} \over {{\rm{|}}u(y) - u(x){\rm{|}}}}dy} .$$

Moreover, we also provide a result on J-calibrable sets and the nonlocal J-mean curvature that says that a J-calibrable set cannot include points with large curvature. Concerning examples, we show that balls are J-calibrable for kernels J that are radially nonincresing, while stadiums are J-calibrable when they are small but not when they are large.