# Nonlocal minimal graphs in the plane are generically sticky

## TL;DR

This paper demonstrates that nonlocal minimal graphs in the plane typically exhibit boundary stickiness and discontinuities, with a sharp transition between continuous and discontinuous boundary behaviors.

## Contribution

It establishes a boundary regularity dichotomy for nonlocal minimal graphs, showing the boundary is either discontinuous or smoothly regular, with no intermediate states.

## Key findings

- Boundary discontinuities occur under small perturbations.
- Nonlocal minimal graphs are either discontinuous or $C^{1,eta}$ at the boundary.
- The boundary regularity jumps from discontinuous to smooth without intermediate forms.

## Abstract

We prove that nonlocal minimal graphs in the plane exhibit generically stickiness effects and boundary discontinuities. More precisely, we show that if a nonlocal minimal graph in a slab is continuous up to the boundary, then arbitrarily small perturbations of the far-away data produce boundary discontinuities. Hence, either a nonlocal minimal graph is discontinuous at the boundary, or a small perturbation of the prescribed conditions produces boundary discontinuities. The proof relies on a sliding method combined with a fine boundary regularity analysis, based on a discontinuity/smoothness alternative. Namely, we establish that nonlocal minimal graphs are either discontinuous at the boundary or their derivative is H\"older continuous up to the boundary. In this spirit, we prove that the boundary regularity of nonlocal minimal graphs in the plane "jumps" from discontinuous to $C^{1,\gamma}$, with no intermediate possibilities allowed. In particular, we deduce that the nonlocal curvature equation is always satisfied up to the boundary. As an interesting byproduct of our analysis, one obtains a detailed understanding of the "switch" between the regime of continuous (and hence differentiable) nonlocal minimal graphs to that of discontinuous (and hence with differentiable inverse) ones.