Nonlocal s-Minimal Surfaces and Lawson Cones

The nonlocal s-fractional minimal surface equation for Σ = ∂E where E is an open set in R ^N is given by H _Σ ^s (p):= _RN ^χE(x) − χEc(x ⁾ dx = 0 for all p ∈ Σ. |x − p| ^N+s 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 R ³, the nonlocal catenoid, an analog of the standard catenoid |x ₃| = log(r+ r ² − 1). Rather than eventual logarithmic growth, this surface becomes asymptotic to the cone |x ₃| = r1 − 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) ∈ R ⁿ × R ^m, 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.