Sharp interface limits of scalar and vectorial Allen-Cahn equations

The Allen-Cahn type equations are widely adopted in the description of the evolution of interfaces in continuum mechanics. In this talk we shall  first discuss the relative entropy method for the convergence of the scalar time-dependent Allen-Cahn equation to the mean curvature flow. The superiority of such a method is that it can derive equal-partition of the Allen-Cahn energy without Modica-Ilmanen type maximum principle. Then we shall discuss  applications of such a method to more sophisticated systems.