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Discrete-to-continuum
crystalline curvature flows
Antonin Chambolle, Daniele De Gennaro and Massimiliano Morini |
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Vol. 19 (2026), No. 4, 783–822
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Abstract |
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We consider here a fully discrete variant of the implicit variational scheme for mean curvature flow, see Almgren et al. (1993) and Luckhaus and Sturzenhecker (1995), in a setting where the flow is governed by a crystalline surface tension defined by the limit of pairwise interactions energy on the discrete grid. The algorithm is based on a new discrete distance from the evolving sets, which prevents the occurrence of the spatial drift and pinning phenomena identified in Misiats and Yip (2016) and Braides et al. (2010) in a similar discrete framework. We provide the first rigorous convergence result holding in any dimension, for any initial set and for a large class of purely crystalline anisotropies, in which the spatial discretization mesh can be of the same order or coarser than the time step. |
Keywords
crystalline flows, discrete-to-continuum approximation,
minimizing movements, distributional solutions
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Mathematical Subject Classification
Primary: 35D30, 35K93, 49M25, 53E10, 65K10
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Milestones
Received: 31 May 2024
Revised: 1 March 2025
Accepted: 8 June 2025
Published: 17 May 2026
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Authors |
| Antonin Chambolle | |
| CEREMADE (CNRS UMR 7534) Université Paris-Dauphine, PSL University Paris France |
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| Daniele De Gennaro | |
| Bocconi University Milano Italy |
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| Massimiliano Morini | |
| Dipartimento di Matematica Università degli Studi di Parma Parma Italy |
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| © 2026 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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