$Γ$-convergence for nonlocal phase transitions

Abstract

We discuss the $Γ$-convergence, under the appropriate scaling, of the energy functional $$ \|u\|_{H^s(Ω)}^2+\int_ΩW(u)dx,$$ 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. When $s\in [1/2,\,1)$, we show that the energy $Γ$-converges to the classical minimal surface functional -- while, when $s\in(0,\,1/2)$, it is easy to see that the functional $Γ$-converges to the nonlocal minimal surface functional.