Density estimates for a nonlocal variational model via the Sobolev inequality

We consider the minimizers of the energy $\|u\|_{H^s(Ω)}^2+\int_ΩW(u)\,dx,$ with $s \in (0,1/2)$, where $\|u\|_{H^s(Ω)}$ denotes the total contribution from $Ω$ 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 $B_R$.

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