A nonlocal free boundary problem

Given s,σ ∈ (0,1) and a bounded domain Ω ⊂ ℝn, we consider the following minimization problem of s-Dirichlet-plus-σ-perimeter-type | u|Hsℝ2n\(Ωc)2) + Perσ(u > 0,Ω), where [·]hs is the fractional Gagliardo seminorm and Perσ 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. The classical free boundary problems are limit cases of the one that we consider in this paper, as s ↗ 1, σ ↗ 1, or σ 0.