A nonlocal free boundary problem

Given~$s,σ\in(0,1)$ and a bounded domain~$Ω\subset\R^n$, we consider the following minimization problem of $s$-Dirichlet plus $σ$-perimeter type $$ [u]_{ H^s(\R^{2n}\setminus(Ω^c)^2) } + \Per_σ\left(\{u>0\},Ω\right), $$ where~$[ \cdot]_{H^s}$ is the fractional Gagliardo seminorm and $\Per_σ$ 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 $s\nearrow1$, $σ\nearrow1$ or~$σ\searrow0$.

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