A nonlocal free boundary problem

Abstract: 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~$\sigma\searrow0$.