  We study existence, unicity and other geometric properties of the minimizers
of the energy functional $$ \|u\|^2_{H^s(\Omega)}+\int_\Omega W(u)\,dx, $$
where $\|u\|_{H^s(\Omega)}$ denotes the total contribution from $\Omega$ in the
$H^s$ norm of $u$ and $W$ is a double-well potential. We also deal with the
solutions of the related fractional elliptic Allen-Cahn equation on the entire
space $\mathbb{R}^n$.
  The results collected here will also be useful for forthcoming papers, where
the second and the third author will study the $\Gamma$-convergence and the
density estimates for level sets of minimizers.
