Threshold dynamics for anisotropic surface energies
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- by Matt Elsey and Selim Esedoḡlu PDF
- Math. Comp. 87 (2018), 1721-1756 Request permission
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.References
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Additional Information
- Matt Elsey
- Affiliation: Courant Institute, New York, New York, 10012
- Address at time of publication: Unaffiliated
- MR Author ID: 869207
- Email: elsey.matt@gmail.com
- Selim Esedoḡlu
- Affiliation: Department of Mathematics, University of Michigan, Ann Arnor, Michigan 48109-2025
- Email: esedoglu@umich.edu
- 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.
- © Copyright 2017 American Mathematical Society
- Journal: Math. Comp. 87 (2018), 1721-1756
- MSC (2010): Primary 65M12; Secondary 35K93
- DOI: https://doi.org/10.1090/mcom/3268
- MathSciNet review: 3787390