A nonlocal free boundary problem

Given~$s,\sigma \in (0,1)$ and a bounded domain~$\mathrm{\Omega }\subset {\mathbb{R}}^n$, 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 ${\text{Per}}_{\sigma }$ 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 $s\nearrow 1$, $\sigma \nearrow 1$ or~$\sigma \searrow 0$.