Stable solutions to the fractional Allen-Cahn equation in the nonlocal perimeter regime

We study stable solutions to the fractional Allen-Cahn equation (-Δ)s/2u=u-u³, $|u|<1$ in Rⁿ. For every s∈(0,1) and dimension n≥ 2, we establish sharp energy estimates, density estimates, and the convergence of blow-downs to stable nonlocal s-minimal cones. As a consequence, we obtain a new classification result: if for some pair $(n,s)$, with n≥ 3, hyperplanes are the only stable nonlocal s-minimal cones in Rⁿ\0\, then every stable solution to the fractional Allen-Cahn equation in Rⁿ is 1D, namely, its level sets are parallel hyperplanes. Combining this result with the classification of stable s-minimal cones in R³\0\ for s~ 1 obtained by the authors in a recent paper, we give positive answers to the ``stability conjecture'' in R³ and to the ``De Giorgi conjecture'' in R⁴ for the fractional Allen-Cahn equation when the order s∈ (0,1) of the operator is sufficiently close to $1$.