discuss now the case of fractional linear equations, showing that the fractional boundary derivative of the solutions can be essentially arbitrarily prescribed, up to a small error. For this, we denote by $B'_r$ the $(n-1)$-dimensional ball of radius $r$ centered at the origin, namely $$B'_r := \{ x'\in{\ensuremath{\mathbb{R}}}^{n-1} {\mbox{ s.t. }} |x'|<r\}.$$ As customary, given $\sigma\in(0,1)$, we define the fractional Laplacian as $$(-\Delta)^\sigma u(x):=\int_{{\ensuremath{\mathbb{R}}}^n}\frac{2u(x)-u(x+y)-u(x-y)}{|y|^{n+\sigma}}
\,dy.$$ Then, we have:

\[FLFL\] Let $n{\geqslant}2$, $\sigma\in(0,1)$, $k\in{\ensuremath{\mathbb{N}}}$ and $f\in C^k(\overline{B_1'})$. Then, for every ${\varepsilon}>0$ there exist $f_{\varepsilon}\in C^k({\ensuremath{\mathbb{R}}}^{n-1})$ and $u_{\varepsilon}\in C({\ensuremath{\mathbb{R}}}^n)$ such that $$\label{56:01} \begin{cases}
(-\Delta)^\sigma u_{\varepsilon}=0 & {\mbox{ in }} B_1\cap \{x_n>0\},\\
u_{\varepsilon}=0 & {\mbox{ in }} \{x_n<0\},\end{cases}$$ $$\label{56:02}
\displaystyle\lim_{x_n\searrow0} \frac{u_{\varepsilon}(x',x_n)}{x_n^\sigma}=f_{\varepsilon}(x')\qquad{\mbox{ for all }}
x'\in B_1',$$ and $$\label{56:03}
\| f_{\varepsilon}-f\|_{C^k(\overline{B_1'})}{\leqslant}{\varepsilon}.$$

On the one hand, Theorem \[FLFL\] falls in the research line opened in [@MR3626547] according to which “all functions are $\sigma$-harmonic up to a small error”, namely it states an interesting flexibility offered by solutions of fractional equations which can adapt themselves in order to capture essentially any prescribed behavior. This flexible feature has been recently studied in several fractional contexts, including time-fractional derivatives, non-elliptic operators, and higher order operators, see [@MR3716924; @MR3935264; @KRYL; @CAR; @CARBOO]. In addition, the flexible properties of fractional 