A Nonlocal Free Boundary Problem

Source abstract

Given $s,\sigma\in(0,1)$ and a bounded domain $\Omega\subset\mathbb{R}^n$, we consider the following minimization problem of $s$-Dirichlet--plus--$\sigma$-perimeter-type $[u]_{ H^s(\mathbb{R}^{2n}\setminus(\Omega^c)^2) } + {\rm Per}_\sigma (\{u>0\},\Omega),$ where $[ \cdot]_{H^s}$ is the fractional Gagliardo seminorm and ${\rm Per}_\sigma$ 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. The classical free boundary problems are limit cases of the one that we consider in this paper, as $s\nearrow1$, $\sigma\nearrow1$, or $\sigma\searrow0$.