Advances in Differential Equations

Convergence of the Allen-Cahn equation with constraint to Brakke's mean curvature flow

Keisuke Takasao

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Abstract

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.

Article information

Source
Adv. Differential Equations Volume 22, Number 9/10 (2017), 765-792.

Dates
First available in Project Euclid: 27 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.ade/1495850459

Subjects
Primary: 35K57: Reaction-diffusion equations 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.)

Citation

Takasao, Keisuke. Convergence of the Allen-Cahn equation with constraint to Brakke's mean curvature flow. Adv. Differential Equations 22 (2017), no. 9/10, 765--792. https://projecteuclid.org/euclid.ade/1495850459


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