Convergence of the Allen-Cahn Equation to Multiphase Mean Curvature Flow

We present a convergence result for solutions of the vector-valued Allen-Cahn equation. In the spirit of the work of Luckhaus and Sturzenhecker we establish convergence towards a distributional formulation of multiphase mean~curvature flow using sets of finite perimeter. Like their result, ours relies on the assumption that the time-integrated energies of the approximations converge to those of the limit. Furthermore, we apply our proof to two variants of the equation, incorporating external forces and volume constraints.