Boundary behavior of nonlocal minimal surfaces

TL;DR

This paper investigates the boundary behavior of nonlocal minimal surfaces, revealing phenomena like boundary sticking and instability that contrast with classical minimal surfaces, supported by detailed examples and estimates.

Contribution

It provides the first detailed analysis of boundary stickiness and instability phenomena in nonlocal minimal surfaces, including explicit examples and barrier constructions.

Findings

Nonlocal minimal surfaces can stick at smooth convex boundaries.

Stickiness phenomena increase as the fractional parameter decreases.

Lines in the plane are unstable at the boundary, causing stickiness under perturbations.

Abstract

We consider the behavior of the nonlocal minimal surfaces in the vicinity of the boundary. By a series of detailed examples, we show that nonlocal minimal surfaces may stick at the boundary of the domain, even when the domain is smooth and convex. This is a purely nonlocal phenomenon, and it is in sharp contrast with the boundary properties of the classical minimal surfaces. In particular, we show stickiness phenomena to half-balls when the datum outside the ball is a small half-ring and to the side of a two-dimensional box when the oscillation between the datum on the right and on the left is large enough. When the fractional parameter is small, the sticking effects may become more and more evident. Moreover, we show that lines in the plane are unstable at the boundary: namely, small compactly supported perturbations of lines cause the minimizers in a slab to stick at the boundary, by a quantity that is proportional to a power of the perturbation. In all the examples, we present concrete estimates on the stickiness phenomena. Also, we construct a family of compactly supported barriers which can have independent interest.