### Convergence of phase-field models and thresholding schemes for multi-phase mean curvature flow

The thresholding scheme is a time discretization for mean curvature flow. It has a natural extension to multi-phase mean curvature flow, which models the slow relaxation of grain boundaries in polycrystals. Recently, Esedoglu and Otto showed that thresholding can be interpreted as minimizing movements for an energy that Gamma-converges to the total interfacial area. In this talk I'll present new convergence results, in particular in the multi-phase case with arbitrary surface tensions. The main result establishes convergence to a weak formulation of (multi-phase) mean curvature flow in the BV-framework of sets of finite perimeter. Furthermore, I will present a similar result for the vector-valued Allen-Cahn equation.

This talk encompasses joint works with Felix Otto, Thilo Simon, and Drew Swartz.