Local and global minimizers for a variational energy involving a fractional norm

We study existence, unicity and other geometric properties of the minimizers of the energy functional $$ \|u\|^2_{H^s(\Omega)} + \int W(u) dx, $$ where $\|u\|^2_{H^s(\Omega)}$ 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.