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

Abstract

The equation du ε /dt = Au ε -(\/ε 2 )f(u ε ) 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 dμ\ = ((ε/2)\Du ε \ 2 + {l/ε)F{u ε )) 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.