Two-phase almost minimizers for a fractional free boundary problem

In this paper, we study almost minimizers to a fractional Alt–Caffarelli–Friedman type functional. Our main results concern the optimal \(C^{0,s}\) $C^{0,s}$ regularity of almost minimizers as well as the structure of the free boundary. We first prove that the two free boundaries \(F^+(u)=\partial \{u(\cdot ,0)>0\}\) $F^{+}(u)=\partial \{u(·,0)>0\}$ and \(F^-(u)=\partial \{u(\cdot ,0)<0\}\) $F^{-}(u)=\partial \{u(·,0)<0\}$ cannot touch, that is, \(F^+(u)\cap F^-(u)=\emptyset \) $F^{+}(u)\cap F^{-}(u)=\mathrm{\varnothing }$ . Lastly, we prove a flatness implies \(C^{1,\gamma }\) $C^{1,\gamma }$ result for the free boundary.