Convergence of the Allen-Cahn equation to Brakke's motion by mean curvature 论文

摘要

The equation du /dt = Au -(\/ 2 )f(u ) was introduced by Allen and Cahn to model the evolution of phase boundaries driven by isotropic surface tension. Here / = F r and F is a potential with two equal wells. We prove that the measures d\ = ((/2)\Du \ 2 + {l/)F{u )) dx converge to Brakke's motion of varifolds by mean curvature. In consequence, the limiting interface is a closed set of finite %? n ~x-measure for each t > 0 and of finite ^"-measure in spacetime. In particular the limiting interface is a "thin" subset of the level-set flow (which can fatten up) and satisfies the maximum principle when tested against smooth, disjoint surfaces moving by mean curvature. The main tools are Huisken's monotonicity formula, Evans-Spruck's lower density bound and equipartition of energy. In addition, drawing on Brakke's regularity theory, there is almost-everywhere regularity for generic (i.e., nonfattening) initial condition.