In this paper, we consider the asymptotic behavior of the fractional mean curvature when . Moreover, we deal with the behavior of s-minimal surfaces when the fractional parameter is small, in a bounded and connected open set with boundary . 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 . Using this, we see that as the parameter s varies, the fractional mean curvature may change sign.
Cite this article
Enrico Valdinoci, Claudia Bucur, Luca Lombardini, Complete stickiness of nonlocal minimal surfaces for small values of the fractional parameter. Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), no. 3, pp. 655–703DOI 10.1016/J.ANIHPC.2018.08.003