Lipschitz boundary. Let $E\subseteq{\ensuremath{\mathbb{R}}}^N$. We say that $E$ is $s$-minimal in $\Omega$ if ${\rm Per}_s(E;\Omega)<+\infty$ and $$\label{MINIMIZZ} {\rm Per}_s(E;\Omega){\leqslant}{\rm Per}_s(E';\Omega)$$ for every $E'\subseteq{\ensuremath{\mathbb{R}}}^N$ such that $E'\cap\Omega^c=E\cap\Omega^c$.

Moreover, given $U\subset{\ensuremath{\mathbb{R}}}^N$, we say that $E$ is locally $s$-minimal in $U$ if it $s$-minimal in $
\Omega$, for every $\Omega$ which is bounded, with Lipschitz boundary and strictly contained in $U$.

We remark that one can make sense of the minimization procedure also in unbounded domains by saying that $E$ is locally $s$-minimal in a (possibly unbounded) domain $\Omega$ if $E$ is $s$-minimal in every bounded and Lipschitz domain $\Omega'\Subset\Omega$ (see Section 1.3 in [@MR3827804] for additional details on these minimality notions).

The regularity theory of nonlocal minimal surfaces is a fascinating topic of investigation, still possessing a number of fundamental open problems. We refer to [@MR3090533; @MR3107529; @MR3331523; @BV] for interior regularity results, [@MR3798717; @CCC] for a precise discussion on stable nonlocal cones, and [@MR3588123; @MR3824212] for recent surveys containing the state of the art of this problem.

A particularly important case of locally $s$-minimal sets is given by the ones which have a graph structure. Namely, given $\Omega_0\subseteq{\ensuremath{\mathbb{R}}}^n$ and $u:\Omega_0\to{\ensuremath{\mathbb{R}}}$, we let $$\label{LEYUDEF} E_u:=\{ X=(x,x_{n+1}) {\mbox{ s.t. $x\in\Omega_0$ and $ x_{n+1}<u(x)$}}\}.$$ 