Γ-convergence for nonlocal phase transitions

Abstract

We discuss the Γ-convergence, under the appropriate scaling, of the energy functional‖u‖Hs(Ω)2+∫ΩW(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.