*Published Paper*

**Inserted:** 31 jul 2023

**Last Updated:** 31 jul 2023

**Journal:** Le Matematiche

**Volume:** LXXV

**Number:** 1

**Pages:** 195--220

**Year:** 2020

**Doi:** 10.4418/2020.75.1.10

**Links:**
https://lematematiche.dmi.unict.it/index.php/lematematiche/article/view/1982/1096

**Abstract:**

We study the asymptotic behavior as $\varepsilon$ goes to 0 of an appropriate scaling of the following nonlocal Allen-Cahn energy,
\[
E^s_\varepsilon(u)=\varepsilon^{2s} \int\int_{I\times I}\frac{

u(x)-u(y)

^2}{

x-y

^{1+2s}}\, {\rm d}x{\rm d}y+ \int_I W(u)\, {\rm d}x,
\]
where $I$ is an interval in $\mathbb{R}$, and $W$ is a double-well potential. We provide a $\Gamma$-convergence result for any $s \in (0,1)$, by extending the case when $s=1/2$ studied by Alberti, BouchittÃ© and Seppecher (C.R. 1994, ARMA 1998).
We also investigate the convergence as $s \nearrow 1$ of the related optimal profile problem to the local counterpart.

**Keywords:**
phase transitions, singular perturbations, fractional Laplacian, nonlocal energies, Gagliardo norms