(Dis)connectedness of nonlocal minimal surfaces in a cylinder and a stickiness property

TL;DR

This paper investigates how the connectivity and stickiness properties of nonlocal minimal surfaces in a cylinder depend on the slab width, revealing differences from classical minimal surface behavior.

Contribution

It demonstrates the dependence of connectedness and stickiness phenomena of nonlocal minimal surfaces on slab width, highlighting differences from classical minimal surfaces.

Findings

Large slab width leads to disconnected minimizers.

Small slab width results in connected, stickily adhering minimizers.

Quantitative bounds on stickiness are established in dimension 2.

Abstract

We consider nonlocal minimal surfaces in a cylinder with prescribed datum given by the complement of a slab. We show that when the width of the slab is large the minimizers are disconnected and when the width of the slab is small the minimizers are connected. This feature is in agreement with the classical case of the minimal surfaces. Nevertheless, we show that when the width of the slab is large the minimizers are not flat discs, as it happens in the classical setting, and, in particular, in dimension $2$ we provide a quantitative bound on the stickiness property exhibited by the minimizers. Moreover, differently from the classical case, we show that when the width of the slab is small then the minimizers completely adhere to the side of the cylinder, thus providing a further example of stickiness phenomenon.