Existence and uniqueness for anisotropic and crystalline mean curvature flows

Antonin Chambolle (1) , Massimiliano Morini (2) , Matteo Novaga (3) , Marcello Ponsiglione (4)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
2 Dipartimento di Matematica e Informatica [Parma]
3 Dipartimento di Matematica [Pisa]
4 Sapienza University of Rome - Dipartimento di Matematica "Guido Castelnuovo" [Roma I]

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 by-product 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.

Mots clés: Geometric evolution equations; level set formulation; Minimizing movements; Crystalline mean curvature motion

Domaines: Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]

Antonin Chambolle, Massimiliano Morini, Matteo Novaga, Marcello Ponsiglione. Existence and uniqueness for anisotropic and crystalline mean curvature flows. Journal of the American Mathematical Society, 2019, pp.1. ⟨10.1090/jams/919⟩. ⟨hal-01463880v2⟩