$\Gamma$-convergence for nonlocal phase transitions

Abstract

We discuss the $\Gamma$-convergence, under the appropriate scaling, of the energy functional $$\Vert u{\Vert }_{H^s(\Omega )}^2+{\int }_{\Omega }W(u)dx,$$ with $s\in (0,1)$, where $\Vert u{\Vert }_{H^s(\Omega )}$ denotes the total contribution from $\Omega$ 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 $\Gamma$-converges to the classical minimal surface functional -- while, when $s\in (0,1/2)$, it is easy to see that the functional $\Gamma$-converges to the nonlocal minimal surface functional.