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The limit of the anisotropic double-obstacle Allen–Cahn equation

Published online by Cambridge University Press:  14 November 2011

Charles M. Elliott  and
Reiner Schätzle
Charles M. Elliott
Affiliation:
Centre for Mathematical Analysis and its Applications, School of Mathematical Sciences, University of Sussex, Falmer, Brighton BN1 9QH, U.K. e-mail: C.M.Elliott@sussex.ac.uk
Reiner Schätzle
Affiliation:
Universität Bonn, Institut für Angewandte Mathematik, Wegelerstraße 6, D-53115 Bonn, Germany e-mail: reiner@iam.uni-bonn.de
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Abstract

In this paper, we prove that solutions of the anisotropic Allen–Cahn equation in doubleobstacle form

where A is a strictly convex function, homogeneous of degree two, converge to an anisotropic mean-curvature flow

when this equation admits a smooth solution in ℝn. Here VN and R respectively denote the normal velocity and the second fundamental form of the interface, and More precisely, we show that the Hausdorff-distance between the zero-level set of φ and the interface of the above anisotropic mean-curvature flow is of order O(ε2).

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 126 , Issue 6 , 1996 , pp. 1217 - 1234
Copyright
Copyright © Royal Society of Edinburgh 1996

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