Density estimates for a variational model driven by the Gagliardo norm

We prove density estimates for level sets of minimizers of the energy $$\eps^{2s}\|u\|_{H^s(\Omega)}^2+\int_\Omega W(u)\,dx,$$ with $s \in (0,1)$, where $\|u\|_{H^s(\Omega)}$ denotes the total contribution from $\Omega$ in the $H^s$ norm of $u$, and $W$ is a double-well potential. As a consequence we obtain, as $\eps \to 0$, the uniform convergence of the level sets of $u$ to either a $H^s$-nonlocal minimal surface if $s\in(0,\frac 1 2)$, or to a classical minimal surface if $s \in[\frac 1 2,1)$.