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Convergence of the Allen-Cahn equation with constraint to Brakke’s Mean curvature flow. (English) Zbl 06780899
Summary: In this paper, we consider the Allen-Cahn equation with constraint. In 1994, Chen and Elliott [9] 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, Ilmanen [20] 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
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
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