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

• ### Enrico Valdinoci

Department of Mathematics and Statistics, University of Western Australia, 35 Stirling Hwy, Crawley, WA 6009, Australia, Dipartimento di Matematica, Università degli Studi di Milano, Via Cesare Saldini 50, 20133 Milano, Italy, Istituto di Matematica Applicata e Tecnologie Informatiche, Consiglio Nazionale delle Ricerche, Via Ferrata 1, 27100 Pavia, Italy
• ### Claudia Bucur

School of Mathematics and Statistics, The University of Melbourne, 813 Swanston Street, Parkville, VIC 3010, Australia
• ### Luca Lombardini

Dipartimento di Matematica, Università degli Studi di Milano, Via Cesare Saldini 50, 20133 Milano, Italy, Faculté des Sciences, Université de Picardie Jules Verne, 33 Rue Saint Leu, 80039 Amiens CEDEX 1, France, Department of Mathematics and Statistics, University of Western Australia, 35 Stirling Hwy, Crawley, WA 6009, Australia

## Abstract

In this paper, we consider the asymptotic behavior of the fractional mean curvature when $s\rightarrow 0^{ + }$. 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.