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Analysis of Diffusion Generated Motion for Mean Curvature Flow in Codimension Two: A Gradient-Flow Approach

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Abstract

The Merriman–Bence–Osher (MBO) scheme, also known as diffusion generated motion or thresholding, is an efficient numerical algorithm for computing mean curvature flow (MCF). It is fairly well understood in the case of hypersurfaces. This paper establishes the first convergence proof of the scheme in codimension two. We concentrate on the case of the curvature motion of a filament (curve) in \({{\mathbb{R}}^3}\). Our proof is based on a new generalization of the minimizing movements interpretation for hypersurfaces (Esedoglu–Otto ’15) by means of an energy that approximates the Dirichlet energy of the state function. As long as a smooth MCF exists, we establish uniform energy estimates for the approximations away from the smooth solution and prove convergence towards this MCF. The current result that holds in codimension two relies in a very crucial manner on a new sharp monotonicity formula for the thresholding energy. This is an improvement of an earlier approximate version.

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Acknowledgements

The authors thank Selim Esedoglu, Felix Otto, and Drew Swartz for useful discussion. The support by the Purdue Research Foundation and the hospitality of the Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany, are highly noted.

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Authors and Affiliations

  1. Department of Mathematics, University of California, Berkeley, CA, 94720, USA

    Tim Laux

  2. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

    Nung Kwan Yip

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  1. Tim Laux
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Correspondence to Tim Laux.

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Communicated by A. Garroni

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Laux, T., Yip, N.K. Analysis of Diffusion Generated Motion for Mean Curvature Flow in Codimension Two: A Gradient-Flow Approach. Arch Rational Mech Anal 232, 1113–1163 (2019). https://doi.org/10.1007/s00205-018-01340-x

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