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Mathematics of Computation

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Threshold dynamics for anisotropic surface energies

Authors: Matt Elsey and Selim Esedoḡlu
Journal: Math. Comp. 87 (2018), 1721-1756
MSC (2010): Primary 65M12; Secondary 35K93
Published electronically: October 19, 2017
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Abstract: We study extensions of Merriman, Bence, and Osher's threshold dynamics scheme to weighted mean curvature flow, which arises as gradient descent for anisotropic (normal dependent) surface energies. In particular, we investigate, in both two and three dimensions, those anisotropies for which the convolution kernel in the scheme can be chosen to be positive and/or to possess a positive Fourier transform. We provide a complete, geometric characterization of such anisotropies. This has implications for the unconditional stability and, in the two-phase setting, the monotonicity, of the scheme. We also revisit previous constructions of convolution kernels from a variational perspective, and propose a new one. The variational perspective differentiates between the normal dependent mobility and surface tension factors (both of which contribute to the normal speed) that results from a given convolution kernel. This more granular understanding is particularly useful in the multiphase setting, where junctions are present.

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Additional Information

Matt Elsey
Affiliation: Courant Institute, New York, New York, 10012
Address at time of publication: Unaffiliated

Selim Esedoḡlu
Affiliation: Department of Mathematics, University of Michigan, Ann Arnor, Michigan 48109-2025

Received by editor(s): April 22, 2016
Received by editor(s) in revised form: January 29, 2017
Published electronically: October 19, 2017
Additional Notes: The first author’s work was supported by NSF grant OISE-0967140. The second author’s work was supported by NSF grant DMS-1317730.
Article copyright: © Copyright 2017 American Mathematical Society