CONVERGENCE OF THE ALLEN-CAHN EQUATION TO BRAKES MOTION BY MEAN-CURVATURE

The equation partial derivative u(epsilon)/partial derivative t = DELTAu(epsilon) - (1/epsilon2)f(u(epsilon)) was introduced by Allen and Cahn to model the evolution of phase boundaries driven by isotropic surface tension. Here f = F' and F is a potential with two equal wells. We prove that the measures dmu(t)epsilon = ((epsilon/2)\Du(epsilon)\2+ (1/epsilon)F(u(epsilon)))dx converge to Brakke's motion of varifolds by mean curvature. In consequence, the limiting interface is a closed set of finite H(n-1)-measure for each t greater-than-or-equal-to 0 and of finite H(n)-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.