equations can be effectively exploited to construct interesting counterexamples, see [@MR3783214], and they have consequences in concrete scenarios involving also mathematical biology and inverse problems, see [@MR3579567; @MR3774704]. Differently from the previous literature, Theorem \[FLFL\] aims at detecting a “boundary” flexibility of fractional equations, rather than an “interior” one.

On the other hand, differently from all the other fractional flexibility results in the literature, which have no counterpart for the case of the classical Laplacian, Theorem \[FLFL\] shares a common treat with the Laplace equation and possesses a full classical analogue (we present its classical counterpart in Appendix \[APPA\]).

Boundary rigidity of fractional minimal surfaces
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We now discuss the boundary behavior of $s$-minimal surfaces and we will show its striking differences with respect to the linear fractional equations. To this end, we recall the setting introduced in [@MR2675483]. Given $s\in(0,1)$, we consider the interaction of two disjoint (measurable) sets $F$, $G\subseteq{\ensuremath{\mathbb{R}}}^N$ defined by $$\label{nNdi}
{\mathcal{I}}_s(F,G):=\iint_{F\times G}\frac{dx\,dy}{|x-y|^{N+s}}.$$ Given a bounded reference domain $\Omega$ with Lipschitz boundary, we define the $s$-perimeter of a set $E\subseteq{\ensuremath{\mathbb{R}}}^N$ in $\Omega$ by $${\rm Per}_s(E;\Omega):=
{\mathcal{I}}_s(E\cap\Omega,E^c\cap\Omega)+
{\mathcal{I}}_s(E\cap\Omega,E^c\cap\Omega^c)+
{\mathcal{I}}_s(E\cap\Omega^c,E^c\cap\Omega).$$ As customary, we have used here the complementary set notation $E^c:={\ensuremath{\mathbb{R}}}^N\setminus E$.

Let $\Omega\subset{\ensuremath{\mathbb{R}}}^N$ be bounded and with 