In this talk, I will present several fractional Allen-Cahn problems involving multiple interfaces.
The diffusion operator is given by the fractional Laplacian of order s, where s \in (0,1).
In two dimensions, these equations model the dynamics of multiple dislocation loops in crystalline materials. We show that when s \geq 1/2, each interface evolves independently according to mean curvature flow. In contrast, when s < 1/2, the interfaces evolve by fractional mean curvature flow and are coupled through an interaction term generated by the presence of the other interfaces.

This is a joint works with E. Hasani, T. Sangsawang, and M. Vaughan.