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

TL;DR

This paper investigates the behavior of nonlocal minimal surfaces as the fractional parameter approaches zero, revealing their complete stickiness, boundary oscillations, and the continuity of fractional mean curvature.

Contribution

It classifies the asymptotic behavior of s-minimal surfaces for small s and proves the continuity of fractional mean curvature across all variables.

Findings

s-minimal sets can be empty, fill the domain, or oscillate wildly for small s

Fractional mean curvature is continuous in all variables for s in (0,1]

The fractional mean curvature can change sign as s varies

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 $\Omega\subset \mathbb{R}^n$. We classify the behavior of $s$-minimal surfaces with respect to the fixed exterior data (i.e. the $s$-minimal set fixed outside of $\Omega$). So, for $s$ small and depending on the data at infinity, the $s$-minimal set can be either empty in $\Omega$, fill all $\Omega$, 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.