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

Abstract

We show that every bounded subset of a euclidean space can be approximated by a set that admits a certain vector field, the so‐called Cahn‐Hoffman vector field, that is subordinate to a given anisotropic metric and has a square‐integrable divergence. More generally, we introduce a concept of facets as a kind of directed sets, and show that they can be approximated in a similar manner. We use this approximation to construct test functions necessary to prove the comparison principle for viscosity solutions of the level set formulation of the crystalline mean curvature flow that were recently introduced by the authors. As a consequence, we obtain the well‐posedness of the viscosity solutions in an arbitrary dimension, which extends the validity of the result in the previous paper.© 2018 Wiley Periodicals, Inc.

Existence and uniqueness for anisotropic and crystalline mean curvature flows

“…(iv) (Generic non-fattening): As in the classical case, for any given uniformly continuous initial datum u 0 all but countably many sublevels do not produce any fattening. (v) (Comparison with other notions of solutions): Our solution-via-approximation u coincides with the classical viscosity solution in the smooth case and with the Giga-Požár viscosity solution [38,39] whenever such a solution is well-defined, that is, when g is constant, φ is purely crystalline and the initial set is bounded. (vi) (Phase-field approximation): When g is constant, a phase-field Allen-Cahn type approximation of u holds.…”

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.…”

Generalized crystalline evolutions as limits of flows with smooth anisotropies

“…Remark 3.9 (Comparison with the Giga-Pozar solution). When φ is purely crystalline and g ≡ c for some c ∈ R the unique level set solution in the sense of Definition 3.6 coincides with the viscosity solution constructed in [33,34].…”

“…However, when the anisotropy φ in (1.1) is non-differentiable or crystalline, the lack of smoothness of the involved differential operators makes it much harder to pursue the aforementioned approaches. In fact, in the crystalline case the problem of finding a suitable weak formulation of (1.1) in dimension N ≥ 3 leading to a unique global-in-time solution for general initial sets has remained open until the very recent works [17,33,16,34].…”

“…By completely different methods, in [33] and more recently in [34], the authors succeed in extending the viscosity approach of [27] to the case N = 3 and to the general case N ≥ 3, respectively. In fact, as in [27] they are able to deal with very general equations of the form…”