Complete stickiness of nonlocal minimal surfaces for small values of the fractional parameter

Abstract

In this paper, we consider the asymptotic behavior of the fractional mean curvature when $s\to 0^{+}$. Moreover, we deal with the behavior of s-minimal surfaces when the fractional parameter $s\in (0,1)$ is small, in a bounded and connected open set with $C^2$ boundary $\mathrm{\Omega }\subset {\mathbb{R}}^n$. We classify the behavior of s-minimal surfaces with respect to the fixed exterior data (i.e. the s-minimal set fixed outside of Ω). So, for s small and depending on the data at infinity, the s-minimal set can be either empty in Ω, fill all Ω, or possibly develop a wildly oscillating boundary.

Also, we prove the continuity of the fractional mean curvature in all variables, for $s\in [0,1]$. Using this, we see that as the parameter s varies, the fractional mean curvature may change sign.