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Convergence of the Allen-Cahn Equation to multi-phase mean-curvature flow

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arxiv 1606.07318 v2 pith:UT22VABV submitted 2016-06-23 math.AP

classification math.AP
keywords convergenceequationallen-cahnflowmean-curvaturemulti-phaseresultapply
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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 multi-phase 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 a volume constraint.

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