Density estimates for a variational model driven by the Gagliardo norm

We prove density estimates for level sets of minimizers of the energyε2s{norm of matrix}u{norm of matrix}Hs(Ω)2+∫ΩW(u)dx, with s∈(0, 1), where {norm of matrix}u{norm of matrix}Hs(Ω) denotes the total contribution from Ω in the Hs norm of u, and W is a double-well potential.As a consequence we obtain, as ε→0+, the uniform convergence of the level sets of u to either an Hs-nonlocal minimal surface if s∈(0,12), or to a classical minimal surface if s∈[12,1).