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Convergence of the Allen-Cahn equation with constraint to Brakke’s mean curvature flow. (English) Zbl 1479.35531

Summary: In this paper, we consider the Allen-Cahn equation with constraint. In 1994, X. Chen and C. M. Elliott [Proc. R. Soc. Lond., Ser. A 444, No. 1922, 429–445 (1994; Zbl 0814.35044)] studied the asymptotic behavior of the solution of the Allen-Cahn equation with constraint. They proved that the zero level set of the solution converges to the classical solution of the mean curvature flow under the suitable conditions on initial data. In 1993, T. Ilmanen [J. Differ. Geom. 38, No. 2, 417–461 (1993; Zbl 0784.53035)] proved the existence of the mean curvature flow via the Allen-Cahn equation without constraint in the sense of Brakke. We proved the same conclusion for the Allen-Cahn equation with constraint.

MSC:

35K57 Reaction-diffusion equations
53E10 Flows related to mean curvature
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