has that $\bar a$ is exactly the fractional normal derivative of the solution of a linear equation, which, in view of Theorem \[FLFL\], can be prescribed in an essentially arbitrary way.

In this spirit, if the linearization procedure produced a “good approximation” of the nonlinear geometric problem, one would expect that the original nonlocal minimal graph $u$ is well approximated near the origin by a term of the form ${\varepsilon}\,\bar a \,x_2^{\sigma}$, with no prescription whatsoever on $\bar a$. Quite surprisingly, formula  (applied here with $\zeta_0:=0$) tells us that this is not the case, and the correct behavior of a nonlocal minimal graph at the boundary [*cannot be simply understood “by linearization”*]{}.

![*[[How we expect the graph of the function $f$ defined in  when $n=2$ and the exterior datum is $$u(x):=\chi_{(-2,0)\times(-1,-3)}(x)-
    \chi_{(0,2)\times(-1,-3)}$$ for all $x\in{\ensuremath{\mathbb{R}}}^2\setminus\big( (-2,2)\times(0,4)\big)$. Notice the horizontal tangency at the origin.]{}]{}*[]{data-label="DOMA-2"}](traccia-2.pdf){width="8.9cm"}

Theorem \[NLMS\] also reveals a [*structural difference*]{} of the boundary regularity theory of fractional minimal surfaces embedded in ${\ensuremath{\mathbb{R}}}^{n+1}$ when $n=2$ with respect to the case in which $n=1$. Indeed, when $n=1$, an $s$-minimal graph in which the exterior datum is attained continuously at a boundary point is necessarily $C^{1,\frac{1+s}2}$ in a neighborhood of such a point 