zbMATH — the first resource for mathematics

Convergence of the Allen-Cahn equation to Brakke’s motion by mean curvature. (English) Zbl 0784.53035
The equation \(\partial u^ \varepsilon/\partial t=\Delta u^ \varepsilon-(1/\varepsilon^ 2)\) \(f(u^ \varepsilon)\) 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 \(d\mu_ t^ \varepsilon\equiv\bigl((\varepsilon/2)| Du|^ 2+(1/\varepsilon)F(u)\bigr)dx\) converge to Brakke’s motion of varifolds by mean curvature. In consequence, the limiting interface is a closed set of finite \({\mathcal H}^{n-1}\)-measure for each \(t\geq 0\) and of finite \({\mathcal H}^ n\)-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.

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C40 Global submanifolds
58D25 Equations in function spaces; evolution equations
Full Text: DOI