Existence and Uniqueness for a Crystalline Mean Curvature Flow

Abstract

Abstract. An existence and uniqueness result, up to fattening, for a class of crystalline mean curvature flows with natural mobility is proved. The results are valid in any dimension and for arbitrary, possibly unbounded, initial closed sets. The comparison principle is obtained by means of a suitable weak formulation of the flow, while the existence of a global-in-time solution follows via a minimizing movements approach.

Minimizing Movements for Mean Curvature Flow of Partitions

“…A typical example of such a motion is anisotropic mean curvature flow: given a norm φ on R n (called anisotropy), the equation for the anisotropic mean curvature flow of hypersurfaces parametrized as Γ t reads as β(ν)V = −div Γt [∇φ(ν)] on Γ t , (1.1) where V denotes the normal velocity of Γ t in the direction of the unit outer normal ν of Γ t and β is the mobility, a positive kinetic coefficient [30]. Anisotropic mean curvature flow is called crystalline provided the boundary of the Wulff shape W φ := {φ ≤ 1} lies on finitely many hyperplanes; in this quite interesting case, equation (1.1) must be properly interpreted, due to the nondifferentiability of φ ; see for instance [1,29,52,28,12,13,31,20,32,17,18]. Equation (1.1) (sometimes referred to as the two-phase evolution) can be generalized to the case of networks in the plane, and more generally to the case of partitions of space (sometimes called the multiphase case): here the evolving sets are intrisically nonsmooth, since the presence of triple junctions (in the plane), or multiple lines, quadruple points etc.…”

Approximation of General Facets by Regular Facets with Respect to Anisotropic Total Variation Energies and Its Application to Crystalline Mean Curvature Flow

“…Recently, A. Chambolle, M. Morini, and M. Ponsiglione [19] established a unique global solvability of the flow V D . /Ä for arbitrary convex and not necessarily bounded initial data, in an arbitrary dimension.…”

Generalized crystalline evolutions as limits of flows with smooth anisotropies

“…More precisley, a norm ψ is said to be φ-regular if the associated ψ-Wulff shape W ψ satisfies a uniform inner φ-Wulff shape condition at all points of its boundary. Such a condition implies that the φ-curvature k φ of ∂W ψ is bounded above and it enables us to show that a distributional formulation in the spirit of [17] still holds true. Next, owing to the simple observation that the φ-regular mobilities are dense, we succeed in extending the notion of solution to general mobilities by an approximation procedure.…”

“…Let us now briefly describe the most recent progress on the problem. In [17], the first global-in-time existence and uniqueness result for the level set flow associated to (1.1), valid in all dimensions, for arbitrary (possibly unbounded) initial sets, and for general (including crystalline) anisotropies φ was established, but under the particular choice ψ = φ (and g = 0). The main contribution of that work is the observation that the variant of the ATW scheme proposed in [15,14]) converges to solutions that satisfy a new stronger distributional formulation of the problem in terms of distance functions.…”