A nonlocal free boundary problem
  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~$\sigma\searrow0$. 
Serena Dipierro, Ovidiu Savin and Enrico Valdinoci
math.AP