# Boundary properties of fractional objects: flexibility of linear equations and rigidity of minimal graphs

## TL;DR

This paper investigates the boundary behavior of nonlocal minimal graphs in three dimensions, revealing a rigidity property that contrasts with the flexibility observed in solutions to related linear fractional Laplacian equations.

## Contribution

It establishes the rigidity of boundary tangent planes for nonlocal minimal graphs and highlights the fundamental differences from linear fractional Laplacian problems.

## Key findings

- Boundary tangent planes of nonlocal minimal graphs match exterior data.
- Linear fractional Laplacian solutions allow arbitrary fractional normal derivatives.
- Nonlinear fractional mean curvature equations exhibit greater rigidity than linear counterparts.

## Abstract

The main goal of this article is to understand the trace properties of nonlocal minimal graphs in~$\R^3$, i.e. nonlocal minimal surfaces with a graphical structure. We establish that at any boundary points at which the trace from inside happens to coincide with the exterior datum, also the tangent planes of the traces necessarily coincide with those of the exterior datum. This very rigid geometric constraint is in sharp contrast with the case of the solutions of the linear equations driven by the fractional Laplacian, since we also show that, in this case, the fractional normal derivative can be prescribed arbitrarily, up to a small error. We remark that, at a formal level, the linearization of the trace of a nonlocal minimal graph is given by the fractional normal derivative of a fractional Laplace problem, therefore the two problems are formally related. Nevertheless, the nonlinear equations of fractional mean curvature type present very specific properties which are strikingly different from those of other problems of fractional type which are apparently similar, but diverse in structure, and the nonlinear case given by the nonlocal minimal graphs turns out to be significantly more rigid than its linear counterpart.