The strongly nonlocal Allen-Cahn problem

TL;DR

This paper rigorously analyzes the sharp interface limit of the fractional Allen-Cahn equation in the strongly nonlocal regime, showing convergence to fractional mean curvature flow, thus completing the understanding across all fractional orders.

Contribution

It provides the first rigorous proof of convergence to fractional mean curvature flow for the Allen-Cahn equation when the fractional order is less than 1/2.

Findings

Convergence of solutions to minima of W as epsilon approaches zero.

Interface evolution follows fractional mean curvature flow.

Completes the theoretical framework for all fractional orders.

Abstract

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 $s\geq \frac{1}{2}$.