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.

MSC:

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