Convergence Of Solutions Of The Weighted Allen–Cahn Equations To Brakke Type Flow

Keywords

Integral varifolds; Mean curvature vector; Parabolic Allen–Cahn equations; Weight

Abstract

In this paper, we study the parabolic Allen–Cahn equation, which has slow diffusion and fast reaction, with a potential K. In particular, the convergence of solutions to a generalized Brakke’s mean curvature flow is established in the limit of a small parameter ε→ 0. More precisely, we show that a sequence of Radon measures, associated to energy density of solutions to the parabolic Allen–Cahn equation, converges to a weight measure of an integral varifold. Moreover, the limiting varifold evolves by a vector which is the difference between the mean curvature vector and the normal part of ∇ K/ 2 K.

Publication Date

10-1-2018

Publication Title

Calculus of Variations and Partial Differential Equations

Volume

57

Issue

5

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1007/s00526-018-1409-8

Socpus ID

85051703850 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/85051703850

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