Characterization of facet breaking for nonsmooth mean curvature flow in the convex case

Abstract

We investigate the breaking and bending phenomena of a facet of a three-dimensional crystal which evolves under crystalline mean curvature flow. We give necessary and sufficient conditions for a facet to be calibrable, i.e. not to break or bend under the evolution process. We also give a criterion which allows us to predict exactly where a subdivision of a non-calibrable facet takes place in the evolution process.

Anisotropic mean curvature on facets and relations with capillarity

“…This result was already known for convex facets [21], and in that context the two conditions are actually equivalent. For general̃︀ ϕ-convex facets (Definition 5.6), condition (1.14) is not sufficient for ϕ-calibrability (Example 5.7).…”

“…This corresponds to the case when the right hand side of the second line in (1.13) is not constant anymore. We shall call any vector field solution of the above mentioned minimization problem an optimal selection in the (possibly non ϕ-calibrable) facet F. Remarkably, it is possible to prove [21] that a facet is ϕ-calibrable if and only if its "mean velocity" is less than or equal to the mean velocity of any subset of the facet. (7) We say that F is strictly ϕ-calibrable if it is ϕ-calibrable and there is no B ⊂ F, B ≠ ∅, having mean velocity equal to that of F.…”

“…Both in the Euclidean and in the anisotropic case, there is also a necessary and sufficient condition for a smooth enough convex body to be a ψ-Cheeger set. It appeared at first in [51] for m = 2 and ψ Euclidean; in [21] for m = 2, ψ ∈ M(R 2 ); in [4] for m ≥ 2 and ψ the Euclidean norm; finally in [37] in the whole generality. This latter result is recalled in Theorem 3.8 below.…”

“…The notion of calibrability can be given for any convex anisotropy ϕ [20], [21], and in any dimension m ≥ 1 [5], [37]. (6) Noncalibrable facets allow to construct explicit examples of facet breaking-bending phenomena, see again [20], [21]: indeed, it seems reasonable that, at least at time t = 0, the facet breaks in correspondance of the jump set of its curvature and bends if the curvature is continuous and not constant.…”

“…In particular, we are interested in examples of facets F ⊂ ∂E which admit a subunitary vector field allowing to define an anisotropic mean curvature not easily expressible in terms of a scalar function. The study of anisotropic mean curvature of facets is related to crystalline mean curvature flow [68], [70], [71], [2], [48], [49]: for instance, the constancy of the crystalline mean curvature makes a facet translate parallely to itself in normal direction, at least for a short time, thus preventing the facet-breaking and bending phenomena [21].…”