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Existence and uniqueness for anisotropic and crystalline mean curvature flows


Authors: Antonin Chambolle, Massimiliano Morini, Matteo Novaga and Marcello Ponsiglione
Journal: J. Amer. Math. Soc. 32 (2019), 779-824
MSC (2010): Primary 53C44, 49M25, 35D30, 35K93
DOI: https://doi.org/10.1090/jams/919
Published electronically: April 11, 2019
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Abstract: An existence and uniqueness result, up to fattening, for crystalline mean curvature flows with forcing and arbitrary (convex) mobilities, is proven. This is achieved by introducing a new notion of solution to the corresponding level set formulation. Such solutions satisfy a comparison principle and stability properties with respect to the approximation by suitably regularized problems. The results are valid in any dimension and for arbitrary, possibly unbounded, initial closed sets. The approach accounts for the possible presence of a time-dependent bounded forcing term, with spatial Lipschitz continuity. As a result of our analysis, we deduce the convergence of a minimizing movement scheme proposed by Almgren, Taylor, and Wang (1993) to a unique (up to fattening) ``flat flow'' in the case of general, including crystalline, anisotropies, solving a long-standing open question.


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Additional Information

Antonin Chambolle
Affiliation: Centre de Mathématiques Appliquées, École Polytechnique, 91128 Palaiseau Cedex, France
Email: antonin.chambolle@cmap.polytechnique.fr

Massimiliano Morini
Affiliation: Dipartimento di Scienze, Matematiche, Fisiche e Informatiche, Università di Parma, Parco Area delle Scienze, 7/A, Parma, Italy
Email: massimiliano.morini@unipr.it

Matteo Novaga
Affiliation: Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy
Email: novaga@dm.unipi.it

Marcello Ponsiglione
Affiliation: Dipartimento di Matematica, Università di Roma La Sapienza, Piazzale Aldo Moro 5, 00185 Rome, Italy
Email: ponsigli@mat.uniroma1.it

DOI: https://doi.org/10.1090/jams/919
Keywords: Geometric evolution equations, minimizing movements, crystalline mean curvature motion, level set formulation
Received by editor(s): September 15, 2017
Received by editor(s) in revised form: October 10, 2018, and January 11, 2019
Published electronically: April 11, 2019
Additional Notes: The first author was partially supported by the ANR, programs ANR-12-BS01-0014-01 “GEOMETRYA” and ANR-12-BS01-0008-01 “HJnet”.
The third author was partially supported by an invited professorship of the Ecole Polytechnique, Palaiseau.
The second and fourth authors were partially supported by the GNAMPA grant 2016 “Variational methods for nonlocal geometric flows”.
Article copyright: © Copyright 2019 American Mathematical Society