Density Estimates for a Variational Model Driven by the Gagliardo Norm

We prove density estimates for level sets of minimizers of the energy ε‖u‖Hs(Ω) + ∫ Ω W (u) dx, with s ∈ (0, 1), where ‖u‖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 a Hs-nonlocal minimal surface if s ∈ (0, 1 2 ), or to a classical minimal surface if s ∈ [ 1 2 , 1).