Asymptotic expansion of a nonlocal phase transition energy

We study the asymptotic behavior of the fractional Allen--Cahn energy functional in bounded domains with prescribed Dirichlet boundary conditions. When the fractional power $s \in (0,\frac12)$, we establish establish the first-order asymptotic development up to the boundary in the sense of $\Gamma$-convergence. In particular, we prove that the first-order term is the nonlocal minimal surface functional. Also, we show that, in general, the second-order term is not properly defined and intermediate orders may have to be taken into account. For $s \in [\frac12,1)$, we focus on the one-dimensional case and we prove that the first order term is the classical perimeter functional plus a penalization on the boundary. Towards this end, we establish existence of minimizers to a corresponding fractional energy in a half-line, which provides itself a new feature with respect to the existing literature.