Submitted Paper

Inserted: 12 nov 2021
Last Updated: 12 nov 2021

Year: 2021

Abstract:

We study stable solutions to the fractional Allen-Cahn equation $(−Δ)^{s/2}u=u−u^3$, $
u
<1$ in $\mathbb R^n$. For every $s\in(0,1)$ and dimension $n\ge 2$, 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 $\mathbb R^n\setminus\{0\}$, then every stable solution to the fractional Allen-Cahn equation in $\mathbb R^n$ is 1D, namely, its level sets are parallel hyperplanes. Combining this result with the classification of stable s-minimal cones in $\mathbb R^3\setminus\{0\}$ for s∼1 obtained by the authors in a recent paper, we give positive answers to the "stability conjecture" in $\mathbb R^3$ and to the "De Giorgi conjecture" in $\mathbb R^4$ for the fractional Allen-Cahn equation when the order $s\in(0,1)$ of the operator is sufficiently close to 1.