STABLE SOLUTIONS TO THE FRACTIONAL ALLEN-CAHN EQUATION IN THE NONLOCAL PERIMETER REGIME

We study stable solutions to the fractional Allen-Cahn equation (−∆)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.