Density estimates for a variational model driven by the Gagliardo norm

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