Convergence of Perturbed Allen Cahn Equations to Forced Mean Curvature Flow
Luca Mugnai and Matthias Röger
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Submission date: 10. Feb. 2009
published in: Indiana University mathematics journal, 60 (2011) 1, p. 41-75
DOI number (of the published article): 10.1512/iumj.2011.60.3949
MSC-Numbers: 53C44, 35K55, 49Q15
Keywords and phrases: Allen-Cahn equation, sharp interface limits, motion by mean curvature
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We study perturbations of the Allen--Cahn equation and prove the convergence to forced mean curvature flow in the sharp interface limit. We allow for perturbations that are square-integrable with respect to the diffuse surface area measure. We give a suitable generalized formulation for forced mean curvature flow and apply previous results for the Allen--Cahn action functional. Finally we discuss some applications.