The stickiness phenomena of nonlocal minimal surfaces: new results and a comparison with the classical case

TL;DR

This paper explores the stickiness phenomena of nonlocal minimal surfaces, highlighting their generic nature in non-convex domains and presenting examples, especially in highly nonlocal regimes, contrasting with classical minimal surfaces.

Contribution

It provides new insights into the stickiness behavior of nonlocal minimal surfaces and compares these phenomena with classical minimal surfaces, emphasizing the role of nonlocality.

Findings

Stickiness is generic in nonlocal minimal surfaces.

Classical minimal surfaces do not stick in convex domains.

Complete stickiness occurs in highly nonlocal regimes.

Abstract

We discuss in this note the stickiness phenomena for nonlocal minimal surfaces. Classical minimal surfaces in convex domains do not stick to the boundary of the domain, hence examples of stickiness can be obtained only by removing the assumption of convexity. On the other hand, in the nonlocal framework, stickiness is "generic". We provide various examples from the literature, and focus on the case of complete stickiness in highly nonlocal regimes.