Convergence of nonlocal threshold dynamics approximations to front propagation

TL;DR

This paper proves that scaled threshold dynamics algorithms for the fractional Laplacian converge to front propagation, with interface motion governed by either weighted mean curvature or nonlocal fractional velocity depending on the order.

Contribution

It establishes convergence of nonlocal threshold dynamics algorithms to front motion, extending to anisotropic schemes and characterizing the interface velocity for different fractional orders.

Findings

For α ≥ 1, interfaces move by weighted mean curvature.

For α < 1, interface velocity is nonlocal and fractional.

Results extend to general anisotropic threshold schemes.

Abstract

In this note we prove that appropriately scaled threshold dynamics-type algorithms corresponding to the fractional Laplacian of order $\alpha \in (0,2)$ converge to moving fronts. When $\alpha \geqq 1$ the resulting interface moves by weighted mean curvature, while for $\alpha <1$ the normal velocity is nonlocal of ``fractional-type.'' The results easily extend to general nonlocal anisotropic threshold dynamics schemes.