Convergence of Nonlocal Threshold Dynamics Approximations to Front Propagation

In this note we prove that appropriately scaled threshold dynamics-type algorithms corresponding to the fractional Laplacian of order alpha is an element of (0, 2) converge to moving fronts. When alpha >= 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.