$\mathrm{\Gamma }$-convergence for nonlocal phase transitions

We discuss the $\mathrm{\Gamma }$-convergence, under the appropriate scaling, of the energy functional $$\mathrm{\parallel }u{\mathrm{\parallel }}_{H^s(\mathrm{\Omega })}^2+{\int }_{\mathrm{\Omega }}W(u)dx,$$ with $s \in (0,1)$,where$|u|_{H^s(\Omega)}$denotesthetotalcontributionfrom$\Omega$inthe$H^s$normof$u$,and$W$ is a double-well potential. When $s\in [1\mathrm{/}2,1)$, we show that the energy $\mathrm{\Gamma }$-converges to the classical minimal surface functional – while, when $s\in (0,1\mathrm{/}2)$, it is easy to see that the functional $\mathrm{\Gamma }$-converges to the nonlocal minimal surface functional.