The strongly nonlocal Allen-Cahn problem

We study the sharp interface limit of the fractional Allen-Cahn equation $$\epsilon {\mathrm{\partial }}_t{u}^{\epsilon }={\mathcal{I}}_n^s[u^{\epsilon }]-\frac{1}{{\epsilon }^{2s}}{W}^{\mathrm{\prime }}(u^{\epsilon })\text{in}(0,\mathrm{\infty })\times {\mathbb{R}}^n,n\ge 2,$$ where $\epsilon >0$, ${\mathcal{I}}_n^s=-c_{n,s}(-\mathrm{\Delta }{)}^s$ is the fractional Laplacian of order $2s\in (0,1)$ in ${\mathbb{R}}^n$, and $W$ is a smooth double-well potential with minima at 0 and 1. We focus on the singular regime $s\in (0,\frac{1}{2})$, corresponding to strongly nonlocal diffusion. For suitably prepared initial data, we prove that the solution $u^{\epsilon }$ converges, as $\epsilon \to 0$, to the minima of $W$ with the interface evolving by fractional mean curvature flow. This establishes the first rigorous convergence result in this regime, complementing and completing previous work for $s\ge \frac{1}{2}$.