Density estimates for a variational model driven by the Gagliardo norm

We prove density estimates for level sets of minimizers of the energy $$\eps^{2s}\|u\|_{H^s(Ω)}^2+\int_ΩW(u)\,dx,$$ with $s \in (0,1)$, 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. 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 $s \in[\frac 1 2,1)$.

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