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A Diffusion-Generated Approach to Multiphase Motion

In this article, we present a diffusion-generated approach for evolving multiple junctions. This work generalizes an earlier method by Merriman, Bence, and Osher which alternately diffuses and sharpens characteristic functions for each phase region to produce pure mean curvature flow. Specifically, our new method produces a normal velocity which depends on a positive multiple of the curvature of the interface plus the difference in bulk energy densities for prescribed junction angles. This simple method naturally treats topological mergings and breakings, produces no overlapping regions or vacuums, and can be made very fast. Numerical studies are provided which show that our method agrees with front tracking and a recent variational approach for a variety of examples. Asymptotic expansions are also carried out near junctions to justify our algorithms.