  Given~$s,\sigma\in(0,1)$ and a bounded domain~$\Omega\subset\R^n$, we
consider the following minimization problem of $s$-Dirichlet plus
$\sigma$-perimeter type $$ [u]_{ H^s(\R^{2n}\setminus(\Omega^c)^2) }
  + \Per_\sigma\left(\{u>0\},\Omega\right), $$ where~$[ \cdot]_{H^s}$ is the
fractional Gagliardo seminorm and $\Per_\sigma$ is the fractional perimeter.
  Among other results, we prove a monotonicity formula for the minimizers,
glueing lemmata, uniform energy bounds, convergence results, a regularity
theory for the planar cones and a trivialization result for the flat case.
  Several classical free boundary problems are limit cases of the one that we
consider in this paper, as $s\nearrow1$, $\sigma\nearrow1$
or~$\sigma\searrow0$.
