Density estimates for a nonlocal variational model via the Sobolev inequality

We consider the minimizers of the energy $$\mathrm{\parallel }u{\mathrm{\parallel }}_{H^s(\mathrm{\Omega })}^2+{\int }_{\mathrm{\Omega }}W(u)dx,$$ with $s\in (0,1\mathrm{/}2)$, 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. By using a fractional Sobolev inequality, we give a new proof of the fact that the sublevel sets of a minimizer $u$ in a large ball $B_R$ occupy a volume comparable with the volume of $B_R$.