  This article is mainly devoted to the asymptotic analysis of a fractional
version of the (elliptic) Allen-Cahn equation in a bounded domain
$\Omega\subset\mathbb{R}^n$, with or without a source term in the right hand
side of the equation (commonly called chemical potential). Compare to the usual
Allen-Cahn equation, the Laplace operator is here replaced by the fractional
Laplacian $(-\Delta)^s$ with $s\in(0,1/2)$, as defined in Fourier space. In the
singular limit $\varepsilon\to 0$, we show that arbitrary solutions with
uniformly bounded energy converge both in the energetic and geometric sense to
surfaces of prescribed nonlocal mean curvature in $\Omega$ whenever the
chemical potential remains bounded in suitable Sobolev spaces. With no chemical
potential, the notion of surface of prescribed nonlocal mean curvature reduces
to the stationary version of the nonlocal minimal surfaces introduced by L.A.
Caffarelli, J.M. Roquejoffre, and O. Savin. Under the same Sobolev regularity
assumption on the chemical potential, we also prove that surfaces of prescribed
nonlocal mean curvature have a Minkowski codimension equal to one, and that the
associated sets have a locally finite fractional $2s^\prime$-perimeter in
$\Omega$ for every $s^\prime\in(0,1/2)$.
