Nonlocal s-minimal surfaces and Lawson cones

Abstract

The nonlocal s-fractional minimal surface equation for Sigma = partial derivative E where E is an open set in R-N is given by H-Sigma(s)(p) := integral(RN) chi E(x) - chi E-c(x)/vertical bar x - p vertical bar N + s dx = 0 for all p is an element of Sigma Here 0 < s < 1, chi 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 fi rst concrete examples (beyond the plane) of nonlocal s minimal surfaces. When s is close to 1, we fi rst construct a connected embedded s-minimal surface of revolution in R-3, the nonlocal catenoid, an analog of the standard catenoid vertical bar x(3)vertical bar = log(r + root r(2) - 1). Rather than eventual logarithmic growth, this surface becomes asymptotic to the cone vertical bar x(3)vertical bar = r root 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 vertical bar v vertical bar = alpha vertical bar u vertical bar, (u, v), is an element of R-n x 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.