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

We study existence, uniqueness, and other geometric properties of the minimizers of the energy functional where {double pipe}u{double pipe} H_s(Ω) denotes the total contribution from Ω 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 ℝ_n. The results collected here will also be useful for forthcoming papers, where the second and the third author will study the Γ-convergence and the density estimates for level sets of minimizers.