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.
All Science Journal Classification (ASJC) codes
- Applied Mathematics