Nonlocal s-minimal surfaces and Lawson cones

The nonlocal s-fractional minimal surface equation for Σ = ∂ E where E is an open set in RN is given by \[ H_Σ^ s (p) := ∫R^N {χ_E(x) - χE^c(x)} {{ x-p }N+s}\, dx \ =\ 0 \; for all \; p∈Σ . \] Here 0 < s < 1 , χ designates characteristic function, and the integral is understood in the principal value sense. The classical notion of minimal surface is recovered by letting s → 1. In this paper we exhibit the first concrete examples (beyond the plane) of nonlocal s-minimal surfaces. When s is close to $1$, we first construct a connected embedded s-minimal surface of revolution in R³, the nonlocal catenoid, an analog of the standard catenoid x₃ = log(r+ √r² - 1). Rather than eventual logarithmic growth, this surface becomes asymptotic to the cone x₃ = r √1 - s. We also find a two-sheet embedded s-minimal surface asymptotic to the same cone, an analog to the simple union of two parallel planes. On the other hand, for any 0 < s < 1 , n , m ≥ 1, s-minimal Lawson cones v = α u , (u, v) ∈ Rⁿ × Rᵐ, are found to exist. In sharp contrast with the classical case, we prove their stability for small s and $n + m = 7$, which suggests that unlike the classical theory (or the case s close to $1$), the regularity of s-area minimizing surfaces may not hold true in dimension $7$.