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Convergence of the Allen‐Cahn Equation to Multiphase Mean Curvature Flow
Author(s) -
Laux Tim,
Simon Theresa M.
Publication year - 2018
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21747
Subject(s) - mathematics , allen–cahn equation , mean curvature flow , convergence (economics) , curvature , limit (mathematics) , flow (mathematics) , work (physics) , mean curvature , mathematical analysis , multiphase flow , geometry , mechanics , physics , economics , economic growth , mechanical engineering , engineering
We present a convergence result for solutions of the vector‐valued Allen‐Cahn equation. In the spirit of the work of Luckhaus and Sturzenhecker we establish convergence towards a distributional formulation of multiphase mean~curvature flow using sets of finite perimeter. Like their result, ours relies on the assumption that the time‐integrated energies of the approximations converge to those of the limit. Furthermore, we apply our proof to two variants of the equation, incorporating external forces and volume constraints.© 2018 Wiley Periodicals, Inc.