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, Bouchitté 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