Γ-convergence for Nonlocal Phase Transitions

We discuss the Γ-convergence, under the appropriate scaling, of the energy functional ‖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. When s ∈ [1/2, 1), we show that the energy Γ-converges to the classical minimal surface functional – while, when s ∈ (0, 1/2), it is easy to see that the functional Γ-converges to the nonlocal minimal surface functional.