Convergence of Solutions of the Weighted Allen-Cahn Equations to Brakke Flow
Speaker
Prof. Gao-Feng Zheng
School of Mathematics and Statistics, Central China Normal University
Abstract

In this talk, we concern the parabolic Allen-Cahn equation with a potential $K$ with slow diffusion and fast reaction. In particular, the convergence of solutions  to a generalized Brakke's mean curvature flow is established in the limit of a small parameter $\epsilon \rightarrow 0$. More precisely, we show that a sequence of Radon measures, associated to energy density of solutions to the parabolic Allen-Cahn equation, converges to a weight measure of an integral varifold. Moreover, the limiting varifold evolves by a vector which is the difference between the mean curvature vector and the normal part of ${\nabla K}/{2K}$.