Convergence of Allen-Cahn equations to De Giorgi's Multiphase Mean Curvature Flow

Multiphase mean curvature is an important evolution equation in various geometrical or physical problems and its connection to the Allen-Cahn equations has a long history. In this talk, we will discuss a conditional convergence result for systems of Allen-Cahn equations to a De Giorgi type solution for multiphase mean curvature flow. Beforehand we will be looking at the gradient flow structure of both the mean curvature flow and the Allen-Cahn equation and discussing De Giorgi's optimal energy dissipation inequality for both equations. Lastly the question will be raised if the conditional convergence can be improved, or to be more specific, what happens if we drop the assumption of energy convergence.