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 a solution satisfies the comparison principle and a stability property 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 byproduct of the analysis, the problem of the convergence of the Almgren-Taylor-Wang minimizing movements scheme to a unique (up to fattening) ''flat flow'' in the case of general, possibly crystalline, anisotropies is settled.

Existence and uniqueness for anisotropic and crystalline mean curvature flows / Chambolle, A.; Morini, M.; Novaga, M.; Ponsiglione, M.. - In: JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY. - ISSN 0894-0347. - 32:3(2019), pp. 779-824. [10.1090/jams/919]

Existence and uniqueness for anisotropic and crystalline mean curvature flows

Ponsiglione M.
2019

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 a solution satisfies the comparison principle and a stability property 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 byproduct of the analysis, the problem of the convergence of the Almgren-Taylor-Wang minimizing movements scheme to a unique (up to fattening) ''flat flow'' in the case of general, possibly crystalline, anisotropies is settled.
2019
Crystalline mean curvature motion; geometric evolution equations; level set formulation; minimizing movements
01 Pubblicazione su rivista::01a Articolo in rivista
Existence and uniqueness for anisotropic and crystalline mean curvature flows / Chambolle, A.; Morini, M.; Novaga, M.; Ponsiglione, M.. - In: JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY. - ISSN 0894-0347. - 32:3(2019), pp. 779-824. [10.1090/jams/919]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1344536
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