The strongly nonlocal Allen-Cahn problem
We study the sharp interface limit of the fractional Allen-Cahn equation $$ \varepsilon \partial_t u^{\varepsilon} = \mathcal{I}^s_n [u^{\varepsilon}] -\frac{1}{\varepsilon ^{2s}} W'(u^\varepsilon) \quad \hbox{in}~(0,\infty)\times\mathbb{R}^n, ~n \geq 2, $$ where $\varepsilon >0$, $\mathcal{I}^s_n=-c_{n,s}(-\Delta )^s$ 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^\varepsilon $ converges, as $\varepsilon\to0$, 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 $s\geq \frac{1}{2}$.